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1. Braids arithmetic
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• May 12th 1868." .
• lbs lbs 63; .
• 384 : 2240 :: 21805 .
• 65415 .
• 763175 .
• He then divides 763175 by 6 to obtain 3; 127195 / 16 / 8.
• When butter sells at 1 / 71/4 (this is, one shilling, seven and a quarter pence) for 22 oz.
• How much money at 389; per cent will yield as much interest as £490 at 4 per cent? .
• A bankrupt's debts amount to £; 5130 and his effects to £ 3729 / 18 / 9.
• Suppose the arms of a deceitful balance to be to each other as 12 to 11½.
• Suppose the arms of a deceitful balance to be to each other as 1to 10 and suppose a weight of 35 lbs.
• of sugar for 3; 13 / 2 / 7.
• A piece of calico 25 inches wide is valued at 2 / 1 per yard.
• Find the value of 1725 stones of hay at 64 /- per 100 stones.
• Find the amount of 1386 gallons at £ 44 / 11 /- for 252 gallons.
• A butcher buys a piece of linen measuring 26 yards at 2 / 789; per yard.
• Find the insurance on £ 340 / 15 / 6 at 3 /- per cent.
• If 3 cwt 13 lbs of soap cost 3; 14 / 2 / 9 what will 7 cwt 2 qrs 25 lbs cost? .
• 66 : 20 :: 11 .
• 4620 : 1260 .
• He then multiplies 1260 by 11 to get 13860 and divides by 4620 to obtain 3d.
• If 12 /- are given for the carriage of 2 cwt 3 qrs for 192 miles, how much should be given at that rate for the carriage of 8 cwt 1 qrs for 128 miles? .
• Find the interest of £ 456 / 12 /- for 43/4 years at 489; per cent.
• Find the interest of £; 512 / 10 /- for 133 days at 5 per cent.
• If the quartern loaf sells at 789;d when wheat sells at 50 /- a quarter, what ought the 6d loaf to weigh when wheat sells at 80 /- a quarter? .
• If 118 men eat 80 qrs of wheat in 108 days, how many qr will 88 men eat in a year and 107 days? .
• If 45 10d loaves can be made from 6 bushels of wheat, how many 8d loaves be got from 9 qrs? .
• If 3; 100 gain £ 5 in a year, what will £ 650 gain in 219 days? .
• If 3 persons are boarded 4 weeks for £ 7, how long ought 14 persons to be boarded for 3; 112? .
• A's £ 200, B's 3; 133 / 6 / 8, C's 63;113 / 7 / 9 and D's £78 / 11 / 2.
• 63; S d 63; Ac.Ro.
• He multiplies 48000 by 2957 and divides by 126067.
• 3512655/126067.
• 2350459/126067.
• 1210996/126067.
• Divide a common consisting of 165 acres among A, B and C according to their rents which are £900, 63;1050 and £800 respectively.
• They receive of net freight for a voyage £; 315 / 14 / 6.
• must be mixed together to form a compound worth 689;d per lb? .
• 5d 1/2 1 @ 5d .
• 6d 1/2 1 @ 6d .
• 7d 1+ 1/2 + 1/2 4 @ 7d .
• He brackets the 3 lines on the left writing 689;d to the left of the bracket, and he brackets the 3 lines on the right writing Ans.
• A bale of flannel containing 336 yards of which the average cost was 1 / 6 per yard consists of 4 kinds, which cost respectively 1 / 1, 1 / 3, 1 / 8 and 1 / 10 per yard.
• He then computes 14 : 4 :: 336 to get 96 @ 1 / 1 .
• Also 14 : 2 :: 336 to get 48 @ 1 / 3 , 14 : 3 :: 336 to get 72 @ 1 / 8 , and 14 : 5 :: 336 to get 120 @ 1 / 10.
• A pennyweight is 1/20 of a Troy ounce and is 1.55 grams.
• Braid obtains 4 @ 218, 1 @ 220, 1 @ 224, and 2 @ 230.
• Braid divides 1728 by 1080 to get 1 with remainder 448.
• 4, 9, 12 and 36.
• 2 5; 2 5; 3 5; 3 = 36 Ans.
• 1211/12.
• 132/12.
• 7/9 of 4/5 of 121/2.
• 1980/3168.
• 589;/22 .
• 3456/6048 = 384/672 = 48/84 = 8/14 = 4/7 Ans.
• 1980/3168 = 220/352 = 55/88 = 5/8 Ans.
• 187500/450000 = 375/900 = 75/180 = 15/36 = 5/12 Ans.
• 3168/4608 = 352/512 = 44/64 = 11/16 Ans.
• 589;/22 = 11/44 = 1/4 Ans.
• 6/7, 2/5, 5/12, 3/4.
• 2/3 5; 8 5; 9 = 144/216 .
• 5/8 5; 3 5; 9 = 135/216 .
• 7/9 5; 3 5; 8 = 168/216 .
• 2/3 5; 8/8 5; 9/9 = 144/216 etc.] .
• 1/2, 7/8, 2/3, 11/12.
• 1/2 7; 12 = 12/24 .
• 7/8 ÷ 3 = 21/24 .
• 2/3 ÷ 8 = 16/24 .
• 11/12 ÷ 2 = 22/24.
• 1/2, 2/3, 5/6, 7/12.
• £ (75;205;125;4)/9.
• Guin (45;215;12)/5.
• A guinea is 3; 1 / 1 /- or 21 shillings.
• far 5 / (6 5; 4 5; 12 5; 20).
• lbs (4 5; 12 5; 20) / 7.
• A farthing is 1/4 pence.
• (9 / 2d) / (63; 1 / 1 /-).
• Reducing to pence gives 110/252 = 55/126 Ans.
• 6 oz 2 dwt 1089; grains / 1 lb.
• 3; 117/95.
• 3/7, 2/9, 1/3, 9/10.
• 473/4, 295/7, and 359/14.
• 985/8, 3511/12, and 265/6.
• 11/15 guinea, 4/5 63;, and 3/5 Shilling.
• 3/7 5; 90 = 270 .
• 2/9 5; 70 = 140 .
• 1/3 5; 210 = 210 .
• 9/10 5; 63 = 567 .
• From 56 take 21 4/15.
• From 35 take 16 5/16.
• From 6/7 63; take 3/5 guinea.
• From 3; 114/9 take £33/8.
• From 11/12 of 3/8 take 2/3 of 1/7.
• 56/1 5; 15 = 840 .
• 319/15 5; 1 = 319 .
• He subtracts 319 from 840 to get 521, then divides by 15 giving 3411/15 Ans.
• Multiply 11/24 by 12/1.
• Multiply 10/13 by 71/1.
• Multiply 2711/12 by 85/6.
• Divide 3/7 by 6/11.
• Divide 21/25 by 14/15.
• Divide 12/13 by 6.
• Divide 11/15 by 8.
• Divide 692/11 by 8/9.
• .00142857 .
• Reduce 655/2149 to a decimal.
• He divides 3 in by 12 to get .25 in, then 1.25 ft by 3 to get .416 yards, then 157.416 by 1760 to get .089441287 mile.
• Value .603125 acres.
• He multiplies by 4 to get 2.412500 roods.
• Then he multiplies .412500 by 40 to get 16.500000 (square) poles, then .500000 by 301/4 to get 15.125000.
• Finally he converts .125000 to 1/8.
• Value .483125 miles.
• Value 5776;.2957795.
• 4/10 = 2/5 Ans.
• 75/100 = 15/20 = 3/4 Ans.
• 625/1000 = 125/200 = 25/40 = 5/8 Ans.
• Reduce .64 3/4, .0061/8, and .4867/8 to vulgar fractions.
• 3895/8000 = 779/1600 Ans.
• .3 = 3/9 = 1/3 Ans.
• .90 = 90/99 = 10/11 Ans.
• Reduce .63, .108, and .148 to vulgar fractions.
• .63 = 63/99 = 21/33 = 7/11 Ans.
• .108 = 108/999 = 36/333 = 12/111 = 4/37 Ans.
• .148 = 148/999 = 4/27 Ans.
• Reduce .14634, .857142, and .615384 to vulgar fractions.
• .14634 = 14634/99999 = 4878/33333 = 1626/11111 = 6/41 Ans.
• .857742 = 857142/999999 = 285714/333333 = 95238/111111 = 6/7.
• .615384 = 615384/999999 = 205128/333333 = 68376/111111 = 22792/37037 = 8/13.
• 375/9000 = 5/12 Ans.
• 1875/9000 = 375/1800 = 5/24 Ans.
• 6/90 = 2/30 = 1/15 Ans.
• .003 = 3/900 = 1/300 Ans.
• .0083 = 75/9000 = 1/120 Ans.
• Reduce .0185, .0046296, and .7621951 to vulgar fractions.
• Add together 89.8125 + 271.05 + .375 + 127.9 + .01875 + 68.28945.
• Add together .01825 + 17.5 + .00375 + 199.25 + 144 + 310.0125.
• Reduce to decimals and then add together 213/4 + 197/16 + 45/8 + 151/2 + 834/5 + 453/80.
• Add 21.75, 19.4375, 4.625, 15.5, 83.8 and 45.0375 to get 190.15 Ans.
• Reduce to decimals and then add together £; 21 / 10 /- + 63; 8 / 17 / 6 + 63; 4 / 18 / 9 + 63; 3 / 3 / 6 + 63; 9 / 18 / 789; + 63; 1 / 16 / 288;.
• Add 21.5, 8.875, 4.9375, 3.175, 9.93125, 1.809375 to get 50.2281125 Ans.
• Add together .83 + 7.416 + .31855 + 6.25 + 4.38 + 29.627.
• 7.41666|6 .
• .31855|0 .
• 48.83521 6 .
• Add together 17.5 +182.75 + .4 + 19.85 + .008125 + 89.655.
• Reduce and add together £ 44 / 7 / 689; + 63; 9 / 15 / 10 + 63; 6 / 8 / 8 + 63; 12 / 19 / 7 + 63; 10 / 0 / 088; + 63;9 / 0 / 289; + 63; 13 / 9 / 590;.
• 9.7916666|6 .
• 12.9791666|6 .
• 10.1041666|6 .
• 9.1041666|6 .
• 3; 10 / 0 / 088; = 10.0010416 .
• £ 9 / 0 / 289; = 9.010416 .
• 3; 13 / 9 / 590; = 13.4739583 .
• Reduce and add together 121/3 + 35/6 + 415/12 + 851/8 + 71/2 +692/9 + 55/24.
• 41.416|6 .
• 85.125|0 .
• Add together 9.45 + 5.3 + 13.83 + 1.76235 + 16.42135 + 157.025641 + 19.142857.
• 16.42135|135135 .
• 157.02564|102564 .
• 19.14285|714285 .
• 222.97341 164106 .
• Repeats of 1, 2, 3 and 6 all work with a repeat of 6.
• Reduce and add together 943/16 + 585/27 + 15212/13 + 142/3 + 456/7 + 97/24 + 769/11.
• 94.1875|000000 .
• 58.1851|851851 .
• 45.8571|428571 .
• 9.2916|666666 .
• 76.8181|818181 .
• 541.9294 201169 .
• Add together .83 + 7.416 + .31855 + 6.25 + 4.38 + 29.627.
• From 5.53125 take 1.25.
• From 213.5 take 1.8125.
• Reduce to decimals and from 1731/5 take 991/16.
• Reduce to decimals and from £; 81 / 12 / 6 take £ 37 / 9 / 11/2.
• £; 81.625 - 63; 37.45625 = 63; 44.16875.
• From 96.3135 take 37.3.
• 69.3135 .
• 31.98016 .
• From 81.7175 take 73.561.
• 81.7175 .
• 73.5615615 .
• 8.1559384 .
• From 34.815 take 5.47325.
• 34.85185185 .
• 29.37860185 .
• 7.726180 .
• 91.138 .
• Reduce to decimals and from £;9 1 / 18 / 889; take £ 55 / 4 / 11¾.
• £; 91.9354166 - 63; 55.2489583 = 63; 36.6864583.
• 5;4.5 .
• 4 125 .
• 37.125 .
• Multiply .24165 by .175.
• Long multiplication gives .00515625 .
• Multiply .3518 by 124.
• Long multiplication gives 11.528125 .
• Multiply 63.416 by 32.5.
• 63.416 .
• 5;32.5 .
• 31 7083 .
• 2061.0416 .
• Multiply .49838 by 12.64.
• Multiply 365.481 by .00325.
• 365.481 .
• 5; .00325 .
• 1827409 .
• 1.18781590 .
• So the correct answer should be 1.18781590 .
• Multiply .27185 by 1.426.
• .27185 .
• 5;1.426 .
• 163111 .
• 27185185 .
• 52 6153846 .
• 701 5384613 .
• 6138 4615384 .
• 6896.1230769 .
• Multiply .92937 by 1500.
• 5;1500 .
• 5; .3 .
• 30.7518 .
• Divide .5136 by 2.715.
• Braid divides using long division and gives up after .189190.
• Braid, after some calculation, ends up dividing 978 by 8548 and gets .1141 (actually he forgets to put the decimal point in).
• He should have divided 977.9 by 8548 and the correct answer is then .11440102948058025269..

2. Egyptian Papyri
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 41 59 .
• 1 59 .
• 2 118 .
• 32 1888 .
• 41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0 .
• to see that 41 = 32 + 8 + 1.
• Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them.
• 41 59 .
• 1 59 .
• 2 118 .
• 32 1888 .
• 2419 .
• 59 41 .
• 1 41 .
• 4 164 .
• 16 656 .
• 32 1312 .
• 2419 .
• 41 = 1.20 + 0.21 + 0.22 + 1.23 + 0.24 + 1.25 .
• 59 = 1.20 + 1.21 + 0.22 + 1.23 + 1.24 + 1.25.
• 1 65 .
• 2 130 .
• 16 1040 .
• Now we look for numbers in the right hand column which add up to 1495.
• We see that 1040 + 260 + 130 + 65 = 1495 and we tick the rows in which these numbers occur: .
• 1 65 .
• 2 130 .
• 16 1040 .
• 16 + 4 + 2 + 1 = 23, .
• so 1495 divided by 65 is 23.
• 1 65 .
• 2 130 .
• 16 1040 .
• Now look for the numbers in the right hand column which add to a number n with 1500-65 < n Ͱ4; 1500.
• 1040 + 260 + 130 + 65 = 1495 .
• 1 65 .
• 2 130 .
• 16 1040 .
• 16 + 4 + 2 + 1 = 23, .
• so 1500 divided by 65 is 23 and 5/65 = 1/13 remaining.
• Hence the answer is 23 1/13.
• Now it might be supposed that doubling the unit fraction 1/5 would be easy and yield the sum of the unit fractions 1/5 + 1/5.
• For example twice 1/5 would be written as 1/3 + 1/15.
• Note that Ahmes did not need to give the double of 1/n for n even since it is just 1/m where n = 2m.
• 1/5 1/3 + 1/15 .
• 1/7 1/4 + 1/28 .
• 1/9 1/6 + 1/18 .
• 1/11 .
• 1/13 .
• 1/15 1/10 + 1/30 .
• 1/17 1/12 + 1/51 + 1/68 .
• This is discussed in [',' R J Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA., 1982).','6] and further ideas, adding and correcting information from [',' R J Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA., 1982).','6], is given in [',' E M Bruins, Egyptian arithmetic, Janus 68 (1-3) (1981), 33-52.
• ','17], [',' E M Bruins, Reducible and trivial decompositions concerning Egyptian arithmetics, Janus 68 (4) (1981), 281-297.
• ','18], [',' R J Gillings, The Egyptian Mathematical Leather Role - line 8 : How did the scribe do it?, Historia Math.
• 8 (4) (1981), 456-457.
• 23 (1978), 181-191; 358.
• Problem 21: Complete 2/3 and 1/15 to 1.
• 2/3 + 1/15 + x = 1.
• In this case multiply each fraction by 15 to obtain .
• 10 + 1 + y = 15.
• [Of course it would not appear in this form but rather "complete 10 and 1 to 15".] .
• Now the answer to the red auxiliary equation is 4 so the original equation had solution twice 5; (twice 5; 1/15).
• From the doubling table we see that double 1/15 is 1/10 + 1/30.
• Doubling this gives 1/5 + 1/15 which is the required solution to Problem 21.
• x + x/4 = 15.
• He takes x = 4 5; 3 = 12.
• Then x/4 = 3, so x + x/4 = 15 as required.
• 2 Tipaza, 1990 (Algiers, 1998), 125-145.
• ','31].
• Let us now see how to multiply, using Egyptian methods, 1 + 1/3 + 1/5 by 30 + 1/3.
• 1 1 + 1/3 + 1/5 .
• 2 2 + 2/3 + 1/3 + 1/15 = 3 + 1/15 .
• 4 6 + 1/10 + 1/30 .
• 8 12 + 1/5 + 1/15 .
• 16 24 + 1/3 + 1/15 + 1/10 + 1/30 .
• 2/3 2/3 + 1/6 + 1/18 + 1/10 + 1/30 .
• 1/3 1/3 + 1/12 + 1/36 + 1/20 + 1/60 .
• Now here the row beginning 2/3 has been computed from 2/3 of 1 is 2/3, 2/3 of 1/3 is double 1/9 which is 1/6+1/18, 2/3 of 1/5 is double 1/15 which is 1/10 + 1/30.
• Next find the numbers in the left hand column which add to 30+1/3.
• 1 1 + 1/3 + 1/5 .
• 2 3 + 1/15 .
• 4 6 + 1/10 + 1/30 .
• 8 12 + 1/5 + 1/15 .
• 16 24 + 1/3 + 1/15 + 1/10 + 1/30 .
• 2/3 2/3 + 1/6 + 1/18 + 1/10 + 1/30 .
• 1/3 1/3 + 1/12 + 1/36 + 1/20 + 1/60 .
• 46 + 1/5 + 1/10 + 1/12 + 1/15 + 1/30 + 1/36.
• 1 9 .
• 1/9 1 .
• 1 8 .
• 2 16 .
• Notice that the solution is equivalent to taking π = 4(8/9)2 = 3.1605.
• 12 (3) (1985), 261-268.
• 12 (3) (1985), 261-268.
• 12 (3) (1985), 261-268.
• The calculation begins by working out the area of the base: 4 5; 4 = 16.
• Then the area of the top is worked out: 2 5; 2 = 4.
• Next the product of the side of the base with the side of the top is computed: 4 5; 2 = 8.
• Finally the product of 1/3 of the height with the previous sum of 28 is taken and the scribe writes:- .

3. Fair book
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• James Walker's Fair Book of 1852 .
• He founded Madras College in 1832 in St Andrews, the school being based on the Madras System which involved schoolboys tutoring their fellow pupils.
• The school grew rapidly after its foundation and by 1838 there were 800 pupils at the College, by 1845 there were around 900 pupils, and by 1860 there were well over 1000 pupils.
• In the archive of the school there is a jotter of about 80 pages of a schoolboy who studied at the College in 1852.
• If the length of the keel of tonnage be 100 feet and the extreme breadth of the ship 35 feet.
• log 35 = 1.544068 .
• log 17.5 = 1.243038 .
• 4.787106 .
• log 94 = 1.973128 .
• 651.5 = 2.813978 .
• 96 = 1.
• Note the common notation 5 / 10 for 5 shillings and 10 pence.
• What is the value of an ox measuring 7 ft 3 in in girth and 5 ft 4 in in length at 5 / 10 a stone, reckoning the offal 1/3 the value of the four quarters.
• He uses a rule squaring the girth in feet, multiplying by the length in feet, then multiplying the answer by 5/21 to obtain the weight in stones.
• 63;25..
• He calculates as for the previous problem multiplying 9.35 by 9.35 by 5.75 by 5/21.
• He then takes his answer of 125.3 and multiplies by 200/121.
• Let AB or AC be 100 links, and BC 136 links, what is the angle BAC.
• Walker takes a half of 136, then uses the sine rule to obtain half the angle at A.
• He should get 4276;50 but (by a copying error) gets 4576;50.
• He then doubles this to get the correct answer of 8576;40 (confirming that the above was only a copying error and that he had it correct in his rough working).
• Let AB or AC be 100 links, and BC 63.5 links, what is the angle BAC.
• 3676;54' .
• 5176;34' .
• 8676;51' .
• Let AB be 100 links, and the perpendicular or tangent BC 71.5.
• Finds AB + BC = 171.5 and AB - BC = 28.5.
• and solves (using six figure logs) to obtain x = 976;26.
• He then adds and subtracts this from 45 to obtain 5476;26 and 3576;34.
• He next uses the sine rule Sine 976;26 : 171.5 : : Sine 45 : x .
• 122.9 : 90 : : 71.5 : A .
• and solves to obtain 3576;34.
• 3576;34'.
• Six figure logs are used to calculate the area as 143000 square links.
• This is then divided by 100,000 to get acres.
• The decimal fraction is then multiplied by 4 to obtain roods (4 roods to 1 acre).
• Then the decimal fraction is multiplied by 301/4 to obtain square yards (301/4 sq yds to a sq pole).
• 4 ac 1 r 28 p 24 yds.
• 3 ac 1 r 21 p 13 yds.
• What is the area of a triangular field, 3 sides, 816, 1048 and 1270 links.
• 4 ac 1 r 1 p 13 yds.
• 3 r 6 p 21 yds .
• This time Walker uses the obvious method of computing the area as 1/2 base times height.
• 4 ac 1 r 3 p 11 yds.
• What is the area of a triangular field ABC the base AB being 823 links, the angle A = 3776;10' and B = 6476;52'.
• He calculates the third angle of the triangle to be 7876;58.
• This is an arithmetic error, it should be 7776;58.
• He seems to take this to be his answer - so he forgets to multiply by sin 6476;52.
• If AB = 1232 links, BC = 283, CD = 1059, DA = 282 and BD = 1170; what is the area.
• If AB = 700 links, BC = 416, CD = 669, DA = 325 and AC = 793 links.
• What is the area and rent of a ridge of grass, the length being 965 links, the breadth 23 links at one end and 21 links at the other, at 10 guineas per acre.
• Calculates half of 1260+984 times 567, then converts to acres, roods, poles, square yards.
• 6 ac 1 r 17 p 22 yds.
• What is the area of a field in the form of a trapezoid, its parallel sides being 1051 and 850 links, and their perpendicular distance 436 links.
• 4 ac 0 r 23 p 1 yd.
• If AB = 834, BC = 673, CD = 635, DA = 539 links and the angle A = 8776;20: Required area of field.
• 4 ac 1 r 18 p 7 yds.
• What is the area of a four sided field its diagonals being 1000 and 850 links and the angle at their intersection 6576;25'.
• Sine 6576;25' 5; 500 5; 850 = area.
• 3 ac 3 r 18 p 10 yds.
• What is the area of a four sided field, its diagonals being 940 and 898 links, and the angle at their intersection 4976;15'.
• 3 ac 0 r 31 p 17 yds.
• If the diagonal AC = 848 links, the angles BAC = 3076;56; CAD = 6976;10; BCA = 6276;15; and ACD = 4176;12; what is the are of the field? .
• Walker computes ADC = 6976;38, then uses the sine rule to find DC = 845.4.
• Suppose that from one corner, a measurer cannot see all the other corners of a field, but takes his observations from a point of rising ground at A, and that its angles are as follows; BAC = 10576;; CAD = 5976;30; DAE = 12976; and EAB = 6676;30'; and the lines AB = 480, AC = 550, AD = 665, AE = 730.
• 6 ac 1 r 0 p 23 yds.
• These contain up to 18 measured lines.
• Find the area of a triangular field, the three sides being 840, 460, 1120 links.
• 2 ac 1 r 2 p 16 yds.
• 4 ac 1 r 32 p.
• 5 ac 3 r 14 p.
• Converts 8 poles to 5000 sq links by multiplying by 625, then divides by 21 to get 2382/21 links.
• Finally reduces 105/690 = 21/138 = 7/46.
• He coverts this to 187500 square links.
• Divide a common of 244 ac 3 r 30 p among A, B, C, and D, whose estates, on which their claims are founded, are respectively £500, 400, 150, 100 a year; the quality of each being 20/18/15/12.
• Walker divides 500, 450, 150, 100 by 20, 18, 15, 12 respectively.
• He then sums the answers 25, 25, 10, 8 1/3 to obtain 68 1/3.
• 68 1/3 : 25 : : 244.3.30 : A .
• by first multiplying 681/3 : 25 by 3 to obtain 205 : 75 then dividing by 5 to obtain 41 : 15.
• He converts 244 ac 3 r 30 p to 39190 poles and finds 15/41 of that using long multiplication and division.
• A = 89 ac 2 r 17 p 33/41.
• Similar calculations for C and D give 35 ac 3 r 15 p 5/41 and 28 ac 3 r 19 p 11/41respectively.
• If an absciss of 9 corresponds to an ordinate of 12, what is the ordinate of which the absciss is 25.
• √9 : W30;16 : : 12 : x .
• 3 : 4 : : 12 : x .
• If an absciss is 16 and its ordinate 12 what is the parameter.
• Computes using long multiplication and division that area is 1 acre.
• If the base of a parabolic segment is 500 links and its absciss 100 makes an angle of 5576; with it; what is the area of the segment.
• Walker computes (using six figure logs) the height as 100 sin 5576; = 81-91.
• If the 2 parallel ends of a zone of a parabola be 10 and 6, and the part of the absciss perpendicular to and connecting the middle of those ends be 4; what will be the area of the zone.
• He writes (10 + 62 16) 4 5; 2/3 = Ans.
• If the two parallel ends of a zone of a parabola are 200 and 180 and the part of the absciss connecting the middle of those ends are 120 and makes an angle of 5076;; required area of zone.
• The length of a base line within a field curvilinear on the other side is 315 links and 11 equidistant ordinates erected thereon measure 70, 86, 96, 104, 109, 110, 108, 105, 99, 90 and 85 links, respectively, what is the area of the space between the base line and the curvilinear side of the field.
• The average ordinate is calculated by adding the 11 ordinates to obtain 1062, then dividing by 11 to get 96.54.
• 30410.
• The calculation is via 210 : 150 : : ͩ0;(168 x 42) : x giving: .
• When the transverse axis is 180, the conjugate 60 and the two abscisses 144 and 36, what is the ordinate.
• The calculation is via 180 : 60 : : ͩ0;(144 x 36) : x giving: .
• Note that the square root is done long hand without the use of logs or without the obvious 12 5; 6.
• 50 : 70 : : 15 : x.
• Note that Walker recognises √225 = 15.
• This gives x = 21.
• He then calculates 35 77; 21 = 14 or 56.
• 30 : 90 : : ͩ0;(152- 122): x so x = 27.
• But 45 77; 27 = 72 or 18.
• 72 or 18.
• ͩ0;(14 x 56) : 20 : : 70 : x .
• Transverse axis = 144 + 36 = 180.
• 144 5; 36 : 242: : 1802: x2.
• Walker computes 22.5 + √(22.52 - 182) = 36.
• 182 : 45 5; 12 : : 36 : x .
• For the circumference he uses the approximate expression π/2 5; √(a2 + b2) where a, b are the lengths of the major and minor axes.
• The area is (exactly) π/4 5; a b .
• Walker computes (1402 + 1202)/2 = 17000.
• He then multiplies by 3.1416 (long-hand) to obtain 409.6.
• 614.1.
• The actual answer should be 614.4 and Walker's error comes from the fact that he calculates the square root of 38250 as 195.5 ending the calculation too soon since the correct answer should be 195.576.
• 1332.855216.
• The correct answer is 1332.865.
• Computes 480 x 600 x .7854 = 226195.2 (note π/4 = .7854).
• 2 ac 1 r 1 p 27 yds.
• Computes 210 x 180 x .7854 =296881.2 (note π/4 = .7854).
• 296881.2 .
• 13194.7.
• 1 ac 1 r 17 p 4 yds.
• √(20 x 5) : 6 : : 15 : x .
• 10 : 6 : : 15 : x .
• The calculation 609 + 116 = 725 is not written down.
• √(725 x 116) : 280 : : 609 : x .
• Computes √(602+ 802)+ 80 = 180.
• 602 : 25 x 160 : : 180 : x .
• The calculation 352 5; 20 5; .3927 is carried out long-hand.
• 9621.150.
• 32572.108.
• Calculation (402+ 322)12x .3927 = 12365.337.
• (202 + 162) 6 5; .3927 = .
• But calculation incomplete and left as 3936 5; .3927.
• Calculation 122 5; 30 5; .7854 5; 8/15 is carried out.
• As previous problem: calculation 82 5; 20 5; .7854 5; 8/15 is carried out.
• 563.165.
• As previous problem: calculation 202 5; 50 5; .7854 5; 8/15 is carried out.
• (202 5; 8 + 152 5; 3 + 20 5; 15 5; 4) 25 = 126875 .
• Then 126875 5; .05236 = 6643.175 (note .05236 = π/60) .
• 6643.175 .
• (312 5; 8 + 242 5; 3 + 31 × 24 5; 4) 46 = 570032 .
• Then 570032 5; .05236 = 29846.875 (note .05236 = π/60) .
• (35.525; 8 + 322 5; 3 + 35.5 5; 32 5; 4)38 = 543020 .
• Then 543020 5; .05236 = 28432.52720 (note .05236 = π/60) .
• Walker's first attempt is to add log 4 to √3 = 1.732050.
• 23 5; 0.11785113 (note 0.11785113 = 1/(3√8)) .
• Walker computes 62 5; 6 = 216 and 63 5; 1 = 216.
• Walker computes 82 5; 3.4641016 = 221.7025024 .
• 221.702.
• Find the surface and solidity of a dodecahedron, of which the side is 12 ft.
• Walker computes 122 5; 20.6457788 = 2972.9921472 and 123 5; 7.66311896 = 13241.86956288.
• 2972.992 and 13241.869.
• Walker computes 202 5; 8.660254 = 3461.01600 and 203 5; 2.18169499 = 17453.55992000.
• 3461.016 and 17453.5.
• Walker computes 302 5; 8.660254 = 7794.228600 and 303 5; 2.18169499 = 58905.76473000.
• 3 ft 6 in times 4 ft 3 in is 14.875 sq ft or 14 sq ft 126 sq in.
• Walker writes this as 14..
• Now if we multiply by 1 ft 9 in we get 26.03125 cubic ft or 26 cu ft 54 cu in.
• 24 / 13/4.
• (The correct answer is 24 / 1.) .
• Walker computes 24 5; 24 5; 14 = 8064 and 24 5; 24 5; 24 = 13824.
• (Of course he should have calculated 24 5; 24 5; 10 which is much easier!).
• He converts to cubic feet by dividing by 1728.
• Find the content of a block of freestone of which the dimensions taken in different places are as follows; the lengths 13 ft 5 in and 12 ft 7 in, breadths 5 ft 10 in, 5 ft 7 in and 5 ft 1 in, and depths 4 ft 9 in, 4 ft 7 in and 4 ft 2 in.
• 321..
• 63;19..
• 13/1.
• How many cubic yards have been dug out of part of a quarry, the areas of 5 perpendicular sections taken at right angles across the line of excavation at the common distance of 9 feet from one another being respectively 75, 210, 379, 712 and 924 square feet.
• Here A = 75 + 924 = 999; 4B = 4 (210 + 712) = 3688; and 2C = 2 5; 379 = 758.
• Therefore (A + 4B + 2C) 5; D/3 = (999 + 3688 + 758) 5; 3 ÷ 27.
• 56 represent a longitudinal section of a piece of ground over which a railway is to be made the line AB being at the bottom of the cutting: let the breadth of the road be 30 ft, the ratio of the slopes 1 1/2 to 1; the depths of the cross sections bg = 22 ft, ch = 14 ft, di = 16 ft, ek = 12 ft and fl = 28 ft, and their distances Ag = 360 ft, gh = 300 ft, hi = 180 ft, ik = 252 ft, hl = 342 ft and lB = 294 ft.
• Find the area of a square cistern, of which the side is 96 inches, or its content at 1 inch deep, in imperial bushels.
• Walker uses six figure logs to compute the volume in cubic inches, then subtracts 3.346 from the answer (so dividing by 2218.192) to get: .
• 4.154 bushels.
• Find the area of a rectangular vessel 144 inches long and 84 inches broad, or its content at 1 in deep in imperial gallons and bushels.
• Divides volume in cubic inches by 277.274 to get 43.62 gall and divides by 2218.192 to get 5.453 bushels.
• Find the area of a couch frame 130 inches long, and 80 in broad, or its content at 1 in deep, in imperial bushels.
• height 52 1/2 in or its content at 1 in deep and this of hot soup.
• Find the area of a triangle, of which the base is 50 feet and the perpendicular 20 in or its content at 1 inch deep, in imperial gallons, and lbs of green starch.
• Find the area of a trapezium, of which the diagonal is 78 in and the perpendiculars falling down from the opposite angles 23 and 151/2 inches, or its content at 1 inch deep, in imperial gallons and lbs tallow.
• 5.415 gall and 47.81 lbs.
• Find the area of a trapezoid, of which the parallel sides are 143 and 121 inches, and their perpendicular distance 10 inches, or its content at 1 inch deep, in imperial gallons and bushels.
• Find the area of a circular tun, of which the diameter is 60 inches or its content at 1 inch deep, in imperial gallons.
• Walker computes 60 5; 60 (using six figure logs) then divides by 353.036.
• 10.19 gallons.
• Using six figure logs, Walker squares the length of the side and multiplies by 3.6339124.
• He carries out this calculation twice, the first time converting to gallons by dividing by 277.274, the second times converting to bushels by dividing by 2218.192.
• Find the area of a regular octagon, of which the side is 100 inches or its content at i inch deep, in imperial gallons and bushels.
• Using six figure logs, Walker squares the length of the side and multiplies by 4.8284271.
• He carries out this calculation twice, the first time converting to gallons by dividing by 277.274, the second times converting to bushels by dividing by 2218.192.
• 174.1 gall and 21.76 bush.
• Find the area in imperial gallons of a curvilinear vessel, of which the traverse diameter is 148 in and 13 perp.
• ordinates of which the common distance is 15 in are as follows: 82.4, 96.5, 112, 119.2, 121.3, 122.4, 123, 122.6, 121.2, 118.9, 112, 96.3, 82.4, with a small segment of 2 in high at each end.
• The imperial gallon is not a measure on area! Walker computes A = 82.4 + 82.4, B = 96.5 + 119.2 + 122.4 + 122.6 + 118.9 + 96.3, C = 112 + 121.3 + 123 + 121.2 + 112.
• Next he calculates 96.5 5; 2 = 193, multiplies again by 2 to obtain 386, divides by 3 to get 128.6 which he adds to 4047.4 to obtain 4176.
• Using six-figure logs he cubes 30 and divides by 277.274 to get 97.4 gall and divides by 2218.192 to get 12.17 bushels.
• How many imperial gallons of wort will a back contain, of which the length is 110 in, the breadth 90 in, and the depth 10 in.
• 357.1 gallons.
• Using six-figure logs, he computes 48 5; 48 5; 60 and divides by 30.609 to obtain lbs.
• Again he computes 48 5; 48 5; 60 and divides by 353.03604 to obtain 391.6 gallons.
• On the first occasion he divides by 2218.192 to obtain bushels, on the second occasion he divides by 26.76 to obtain the lbs of hot soap, and on the third occasion he divides by 40.3 to obtain the lbs of raw starch.
• Using six-figure logs, he twice calculates 40 5; 40 5; 32.
• First time he divides by 106.773 to obtain 479.7 gall.
• Using six-figure logs, he calculates 154 5; 54.96 5; 50 and divides by 353.035 to obtain 1198 gallons.
• Using six-figure logs, he calculates 48 5; 48 5; 60 and divides by 3, and then by 353.035 to obtain 130.2 gallons.
• Using six-figure logs, he calculates 60 5; 60 5; 90 and divides by 3, and then by 1289.288 to obtain 83.76 bushels.
• Using six-figure logs, he twice calculates 90 5; 90 5; 90.
• 168.1 gallons.
• 55.1 imp.
• 221.2 imp.
• Find the content of a vessel in the form of a frustum of a rectangular pyramid, of which the depth is 180 in, sides of one base 100 and 60 in, of the other 80 and 40 in, in imperial gallons.
• He writes the log as 1.693213 instead of 1.653213.
• He computes 45 5; 45 5; 3 = 6002 (which of course is incorrect because of the miscopying of the log).
• He squares 15 and adds to this, multiplies by 15, then multiplies by .5236 to obtain the volume in cubic inches.
• 64 5; 36 = 2304 and 48 5; 27 = 1290 (this should be 1296 - probably a copying error from tables).
• He then multiplies 2304 by 1290 and takes the square root to get 1728.
• This answer is correct since in the calculation he uses the log of 48 5; 27 which was right.
• He calculates 2304 + 1290 + 1728, multiplies by 48, then divides by 3.
• Finally he divides by 277.274 to obtain the answer 307.1 imp gall.
• 10 5 35.75 ..
• 10 15 38.
• 10 25 39.25 ..
• 10 35 40.75 ..
• 41.5 41.125 .
• (39.375 5; 39.375) / 353.036 5; 10 = 43.91 .
• (41.125 5; 41.125) / 353.036 5; 10 = 47.9 .
• (35.875 5; 35.875) / 353.036 5; 10 = 36.45 .
• (37.625 5; 37.625) / 353.036 5; 10 = 40.1 .
• 43.91 + 47.9 + 36.45 + 40.1 = 168.36 .
• 168.36 + 10 = 178.36 (The 10 here is the drip) .
• 13 6.5 97.5 ..
• 98.1 97.8 .
• 10 18 95.8 ..
• 96.4 96.1 .
• 10 28 94.3 ..
• 93.9 94.1 .
• 10 38 93.2 ..
• 93.1 .
• (97.8 5; 97.8) / 353.036 5; 13 = 352.2 .
• (96.1 × 96.1) / 353.036 5; 10 = 261.5 .
• (94.1 × 94.1) / 353.036 5; 10 = 261.5 .
• (93.1 × 93.1) / 353.036 5; 10 = 261.5 .
• 352.2 + 261.5 + 261.5 + 261.5 = 1110 .
• 1110 + 38 = 1148 (The 38 here is the crown) .
• 9 59.1 ..
• 64.1 63.95 .
• 10.6 62.1 ..
• Find the content of a cask, of which the bung diameter is 31 inches, the head diameter 24 inches, and the length 32 1/2 inches and the perpendicular distance mn being 8 inches.
• 24 5; 24 + 31 ×; 31 × 2 = 2498.
• (2498 5; 32.5)/1059.108 = 76.66 .
• 26.8 5; 26.8 + 31.7 5; 31.7 5; 2 = 2727.4 (misprint for 2727.2, the log is correct) .
• (2727.4 5; 32.7)/1059.108 = 84.21 .
• 84.21 gallons .
• 21.2 5; 21.2 + 29 5; 29 5; 2 = 2131.4 .
• (29 5; 29 - 21.2 5; 21.2) 5; 2/5 = 156.6 .
• 2131.4 - 156.6 = 1974.8 .
• (1974.8 5; 47.4)/1059.108 = 88.39 .
• 24 5; 24 + 31 ×; 31 × 2 = 2498 .
• (2498 5; 46)/1059.108 = 108.5 .
• 23 5; 23 + 29 5; 29 5; 2 = 2211 .
• (2211 × 36)/1059.108 = 75.15 .
• 75.15 gallons .
• Find the content of a cask, of which the bung diameter is 31 inches, the head diameter 23 inches, and the length 40 inches and the perpendicular mn 1.4 inches.
• 31 5; 31 + 23 5; 23 + 29.2 5; 29.2 5;4 = 4900 .
• (4900 5; 40)/2118.217 = 92.53 .
• Find the mean diameter, and thence the content in imperial gallons of a cask, of which the bung diameter is 31 inches, the head diameter 23 inches, and the length 50 inches and the perpendicular mn 1.4 inches.
• Walker divides 23 by 31 and obtains .7419.
• He then calculates 8 5; .613 (which he labels Table area) to obtain 4.904.
• (27.904 5; 27.904 5; 50)/353.036 = 110.2 .
• He incorrectly looks up log 31 writing 1.491212 instead of 1.491362] .
• He then calculates 4 5; .682 to obtain 2.728.
• Walker divides 23 by 29 and obtains .7931.
• He then calculates 6 5; .691 to obtain 4.146.
• To 4.146 he adds 23 to obtain 27.146 .
• (27.146 5; 27.146 x 27)/353.036 = 56.35 .
• He then calculates 6.5 5; .677 to obtain 4.4.
• (26.7 x 26.7 5; 47)/353.036 = 94.91 .
• 94.91 gallons .
• Adds to get 1.1387 which he multiplies (using logs) by 30 to obtain 34.16.
• 34.16 gallons.
• 32 = 1.9337 .
• Adds to get 2.5381 which he multiplies (using logs) by 47 to obtain 119.2.
• He then calculates 3 5; .606 to obtain 1.818.
• To 1.818 he adds 19 to obtain 20.818 (which he labels mean diam.) .
• (20.818 x 20.818 x 24)/353.036 = 29.46.
• He then computes 5 5; 12 = 60 from which he subtracts 11 (which he labels 1/2 of 22 = bung diam.).
• 22 5; 4 = 88) to obtain 16.4.
• Walker divides 23 by 29 and obtains .7931.
• He then calculates 6 5; .610 (he writes .79 = .610) to obtain 3.66.
• (26.66 5; 26.66 5; 27)/353.036 = 54.35.
• He then computes 10 5; 5 = 50 from which he subtracts 14.5 (which he labels 1/2 of 29).
• He multiplies the resulting 35.5 by the 54.35 and divides the result by 116 (which he labels 29 5; 4) to obtain 16.63.
• What is the ullage of a standing hogshead, of which the length is 29.1 inches, the bung diameter 28.5 inches, the head diameter 24 inches, and the depth of liquor 18 inches.
• 24 5; 24 + 28.5 5; 28.5 5; 2 = 2200.5.
• (2200.5 5; 29.1)/1059.108 = 60.46 .
• He then computes 18 5; 11 = 198.0 from which he subtracts 14.55 (which is 1/2 5; 29.1).
• The 60.46 comes from the antilog of 1.781474 but when he uses it again in the next calculation he clearly looks it up in the tables and gets it wrong, writing 1.782902.] .
• 29.8 5; 29.8 + 33.8 5; 33.8 5; 2 = 3172.
• (3172 5; 39.5)/1059.108 = 118.3 .
• He then computes 30 5; 11 = 330 from which he subtracts 19.75 (which is 1/2 5; 39.5).
• Walker computes √(8/90) 5; 1600 = 477.
• [Here 8 = 2 5; 4, the weight of powder in lbs] .
• √90 : √8 : : 1600 : x .
• 9.5 : 2.9 : : 1600 .
• Walker computes √(4/48) 5; 1600 = 462.
• A 51/2 inch shell, weighing 16 lbs, is fired with 1 lbs of powder, with what velocity is it discharged.
• Walker computes √(2/16) 5; 1600 = 565.6.
• Walker computes ͩ0;(1/8) 5; 1600 = 565.6.
• Walker computes √2.8 5; 175.5 = 293.6.
• Walker computes √4.04 5; 175.5 = 357.2.
• Walker computes √6.75 5; 175.5 = 455.9.
• Walker computes W30;12.8 5; 147.3 = 527.
• Walker computes √7.9 5; 147.3 = 414.
• Walker computes (2000 5; 2000)/64 = 62500 .
• From what height must a body fall to acquire a velocity of 1600 ft per second.
• Walker computes (1600 5; 1600)/64 = 40000 .
• Walker computes (294 5; 294)/64 = 1350 .
• Walker computes √6.75 5; 175.5 = 455.9.
• (455.9 5; 455.9)/ 64 = 3248.
• 2000/455.9 = 4.386 = 3376;30.
• He then writes 3376;30 = 3.1031 which he multiplies by 3248 to obtain 10086.
• He then writes 3276;45 = 3.2968 which he multiplies by 3246 to obtain 10708.
• (10708 - 10086)/2 = 311.
• 10086 + 311 = 10397 .
• 3376;30 elevation and 10397 feet (he appears to write 10.397 feet) .
• Walker computes W30;12.8 5; 147.3 = 527.
• (527 5; 527)/ 64 = 4340.
• 2000/527 = 3.795 = 3476;49.
• He then writes 3476;49 = 2.7631 which he multiplies by 4340 to obtain 11992.
• 11992 feet and 3476;49.
• What is the greatest range of a 10 inch shell, of which the diameter is 9.84 inches, when discharged with a velocity of 1700 ft per sec and the elevation necessary.
• Walker computes √9.84 5; 147.3 = 462.
• (462 5; 462)/ 64 = 3336.
• 1700/462 = 3.679 = 3576;10.
• He then writes 3576;10 = 2.6749 which he multiplies by 3336 to obtain 892.3.
• 892 feet and 3576;10.
• The range of a shell at an elevation of 4576; was found to be 3500 feet, at what elevation must the piece be set, to strike an object at the distance of 2920 feet, with the same charge of powder.
• [log] Sine 90 = 10.000000 (he works here with log 1010 sin) .
• Adds to get 13.465383 from which he subtracts log 3500 = 3.544068 which gives 9.921313.
• He then writes 9.921313 = 5676;32' (again log Sine 5676;32 = 9.921313) which he divides by 2 to obtain 2876;16'.
• 2876;16'.
• (485 5; 485 5; 196)/5120000 = 9.
• (500 5; 500 5; 90)/5120000 = 4.394.
• At what time will a shell range 4000 feet, when discharged at an elevation of 4576;.
• R : Tang 4576; : : 4000 : x .
• Writing the log of Tang 4576; and Sine 9076; as 10, i.e.
• the log of 1010tan 4576; and 1010sin 9076;, he computes .
• (4000 5; Tang 4576;)/Sine 9076;, takes the square root, then divides by 4 to obtain 15.80.
• A shell discharged at an elevation of 4076; ranges 3000 ft.
• R : Tang 4076; : : 4000 : x .
• Writing the log of Tang 4076; as 9.923813 and of Sine 9076; as 10, i.e.
• the log of 1010tan 4576; and of 1010sin 9076;, he computes .
• (3000 5; Tang 4076;)/Sine 9076;, takes the square root, then divides by 4 to obtain 12.54.
• A shell when discharged at an elevation of 3576; ranges 1000 feet what is its greatest height.
• R : Tang 3576; : : 1000 : x .
• Writing the log of Tang 3576; as 9.845227 (he writes 9.849227 in error but this is a copying error only since adding 3 gives 12.845227) and of Sine 9076; as 10, i.e.
• the log of 1010tan 3576; and of 1010sin 9076;, he computes .
• (1000 5; Tang 3576;)/Sine 9076; = 700.
• With what impetus, velocity, and charge of powder must a 13 inch shell be discharged at an elevation of 3276;12' to strike an object at the distance of 3250 feet.
• Walker doubles 3276;12 and halves 3250 writing .
• Sine 6476;25 : rad : : 1625 : x .
• As before he uses six figure logs with log Sine as the log of 1010sin etc in the calculations.
• (1625 x Sine 9076;)/Sine 6476;25 = 1802 feet.
• (340 5; 340 5; 196)/5120000 = 4.403 (this is not very accurate since Walker writes the log of 340 as 2.530418 when it should be 2.531479.
• How far will a shot range on a plane which ascends 876;15 and on another which descends 876;15; the impetus being 3000 feet, and the elevation of the piece 3276;20' the elevation above the plane in the first case 2476;15, in the second 4076;45'.
• How much powder will throw a 13 inch shell 4244 feet on an inclined plane which ascends 876;15', the elevation of the mortar being 3276;30.
• In what time will a shell strike a plane which was 1076;, when discharging with an impetus of 2304 feet, at an elevation of 4576;.
• At what elevation must a mortar be pointed to range 2662 feet on a plane which ascends 1076;, the impetus being 2000 feet.
• Find the weight of an iron ball, of which the diameter is 6.41 inches.
• 43 : 6.413 : : 9 : x .
• Find the weight of an iron ball, of which the diameter is 7.018 inches.
• 43 : 7.0183 : : 9 : x .
• Find the diameter of an 18 lb iron ball.
• Using the results of the previous problem, d3 should be 18 5; (43/9).
• Now 43/9 = 7.1111111..
• which, interestingly, Walker writes as 7.1111/9.
• Using six figure logs he multiplies 7.1111/9 by 18 and takes the cube root to obtain .
• As in the previous problem, Walker computes 36 5; 7.111 1/9 and takes the cube root to obtain .
• As in the previous problem, Walker computes 42 5; 7.111 1/9 and takes the cube root to obtain .
• [Note that he writes 7.1191/2 instead of 7.1111/9 but since he has the correct log it must be a copying error.] .
• What is the weight of an iron shell, the external and internal diameter being 11.1 and 8 inches.
• (11.13 - 83) x 9 ÷ 64 = Ans .
• Then he computes (3 log 11.1 - 3 log 8 + log 9) which is clearly wrong.
• (9.83 - 73) 5; 9 ÷ 64 = Ans .
• 93/57.3 = 12.72 lbs.
• 6 5; 57.3 then takes the cube root.
• 8.019 inches.
• 20 5; 14 5; 12 ÷ 30 = 112.
• How much powder will fill a cubical box of which the side is 18 inches.
• 18 5; 18 5; 18 ÷ 30 = 194.4.
• 1/6 (40 5; 41 5; 42) .
• 40 5; 41 5; 42 ÷ 6 = 11480.
• 20 5; 21 5; 41 ÷ 6 = 2870.
• 30 5; 31 5; 121 ÷ 6 = 18755.
• 50 5; 51 5; 52 ÷ 6 = 23100.
• 25 5; 26 5; 24 ÷ 6 = 2600.0.
• 23100-2600 = 19500 .
• 30 5; 31 5; 61 ÷ 6 = 9455.
• 9 5; 10 5; 19 ÷ 6 = 285.
• x2 = 3002 + (1000 - x)2 .
• x2 = 9000 + 10000000 - 2000 x + x2 .
• (80 x /2)4 = (5331/3)9 .
• x2=900 + 1600 .
• This gives x2 = 160000 or 90000 so x = 400 or 300.
• If from a right-angled triangle, of which the base is 18 feet, and the perpendicular 24 feet, a triangle of which the area is 54 square feet be cut off by a line parallel to the perpendicular, what will be the sides of the latter.
• Walker computes 18 5; 12 = 216 using six figure logs.
• 216 : 54 : : 18 : x .
• Again he solves 2 : 1 : : 24 : x to obtain x = 12.
• He then uses Pythagoras to compute the hypotenuse ͩ0;(122 + 92) = 15.
• = 12 feet, Hyp.
• = 15 feet.
• If from a triangle, of which the three sides are 13, 14, 15, a triangular area of 24 was cut off by a line parallel to the longest side, what will be the length of the sides of the triangle containing that area.
• √84 : √24 : : 13 : x .
• √84 : √24 : : 14 : x .
• √84 : √24 : : 15 : x .
• giving x = 1497/13.
• giving x = 1206/13.
• He now has the three sides as 130, 1497/13, and 1206/13.
• A roof which measures 30 feet 8 in by 16 feet 6 inches, is to be covered with lead, at 8 lbs per square foot, find the expense of the lead @ 41/6 a cwt.
• 30.66 2/3 5; 16.5 5; 8 7; 112.
• 63;74..
• If the expense of paving a semicircular plot at 3 / 6 a yard amounts to 63;19..
• 1/9 1/2 what was the diameter of the circle.
• 63;19..
• 1/91/2 is converted to 4581.5 (pence).
• 4581.5 5; 2 5; 9 ÷ 42 .
• A triangle of which the three sides are 140, 180, 82 and is inscribed in a circle, what is the diameter of the circle.
• Sine 6076; : Sine 9076; : : 12 : x .
• A straight line 330 links long, drawn from the right angle of a right-angled triangle, divides the hypotenuse into two segments which respectively measure 217 and 480.12 links, what is the area of the triangle.
• 267 : Sine 4576; : : 330 : Sine C .
• After calculating the angle to be 6076;55 he adds 4576;, subtracting the answer from 18076; to get BDC = 7476;5 .
• Sine 4576; : 267 : : Sine 7476;5 : BC .
• giving 363.1.
• Sine 4576; : 480.12 : : 105°55 : AB .
• He computes Sine 7476;5 5; 480.12 ÷ Sine 4576;.
• Multiplies the answer by 363.1, divides by 2, then divides by 100000 to obtain 1.185.
• 2 : Sine 5676;18 : : 2.4 : Sine BEF .
• Calculates 8676;49 .
• Sine 5676;18 : 2 : : Sine 3676;53 : FE .
• Sine 8676;49 : 1.442 : : Sine 676;32 : AE .
• Calculates 19.12.
• C = 3076;30, AE = 10.92 (writes 10:92), BC = 19.12.
• In order to ascertain the height of an inaccessible object CD, I selected two stations, A and B, on a level with the object's base, but in a different direction, I found the horizontal angle DAB 8776;5; the horizontal angle ABD 5376;15; the vertical angle CAD 4776;30 and the distance AB 283 feet; what was the height of the object, my eye being 5 feet from the ground.
• 18076; - (5376;15 + 8776;5) = 3976;40 = ADB .
• 9076; - 4776;30 = ACD .
• Sine 3976;40 : 283 : : 5376;15 : AD .
• Sine 4276;30 : AD : : 4776;30 : AC .
• Lord Melville's statue in St Andrews Square Edinburgh is 16 ft high and stands on the top of a column 136 feet high; at what distance from the base of the column in the same horizontal plane will the statue appear under the greatest possible vertical angle, what will that angle be, and at what distaance may the statue be viewed under an angle of 376;.
• A piece of cable 3 feet in length and 9 inches in girt weighs 22 lbs, what will the cable weigh per fathom of which the girt is 12 inches.
• 9 : 12 .
• 20 : 50 : : 1120 : x .
• 2000 : 2450 : : 1120 : x .
• 4 : 49 : : 112 : x .
• 63;20..
• If 20 grain of gold gild a globe which weighs 512 ounces, how many grains will gild a globe of the same kind of wood that weighs 1331 ounces.
• 3.1416 x2 = 20 .
• x = √20/3.1416 .
• 512 : 1331 : : x3 : y3 .
• 512 : 1331 : : (20/3.1416)(3/2): y3 [This doesn't seem right.
• He is now using x = √(20/3.1416).] .
• From this he calculates y (using logs), squares it and multiplies by 3.1416 to obtain 37.8 .
• D2 5; 3.1416 5; 144 5; 3.5 = Ans .
• 723 : 81 3 : : 250 x.
• calculated with logs as 10.09 .
• calculated with logs as 14.37 .
• (1/40)2 5; .7854 x = 1728 .
• 55 miles 4 f 10.5 yds.
• 934.5 : 16800000000 : : 11 : x .
• Walker calculates x (using logs) then multiplies by 480, divides by 437.5, then divides by 19640.
• 300 5; 200 = content in sq.
• 300 5; 8 = 2400 5; 2 = 4800 content of ditch .
• 200 5; 8 = 1600 5; 2 = 3200 content of ditch .
• He then divides by 300 5; 200 by 8000 to obtain 7.32 .
• 13.88 5; 277.274 = (x2 + 9 x2/25 + 3 x2/5).7854 5; 4 .
• 25 5; 277.274 = Content in Cubic Inches .
• He then computes 102 5; .7854 = 78.54 .
• 2025; .7854 = 314.16.
• He then takes the square root of the product 78.54 5; 314.16 to obtain 157.08 .
• 60 AC = 8.221849 .
• 66 AC = 8.180456 .
• From this he gets the angle 1376;30'23" (log of the cosine) .
• He multiples by 2 to get 2776;0'46" .
• 30 : 2776;0'46 : : 66 : Sine .
• Obtains 8776;44 .
• He adds 2776;0'46 to this, and subtracts from 180 to obtain 6576;15'14" .
• R : 30 : : Sine 8776;44 = 9.999660 .
• 30 = 1.477121 .
• 29.97 = 1.476781 .
• In the oblique angled triangle ABC, let the side BC = 532, AB + AC = 637 and the angle BAC = 4076;30, to find the others.
• AC = 402.1, AB = 234.8, C = 2476;16 .
• Given the base 428, the vertical angle 4976;16, and the sum of the other two sides 918; find the rest.

4. Perfect numbers
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, see for example [',' C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid&#8217;s Elements, Centaurus 20 (4) (1976), 269-275.','17] where detailed justification for this idea is given.
• So for example the aliquot parts of 10 are 1, 2 and 5.
• These occur since 1 = 10/10, 2 = 10/5, and 5 = 10/2.
• Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e.
• 6 = 1 + 2 + 3, .
• 28 = 1 + 2 + 4 + 7 + 14, .
• 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 .
• 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 .
• The result which is if interest to us here is Proposition 36 of Book IX of the Elements which states [',' T L Heath, The thirteen books of Euclid&#8217;s Elements (New York, 1956).','2]:- .
• (the sum) 5; (the last) = 7 5; 4 = 28, .
• As a second example, 1 + 2 + 4 + 8 + 16 = 31 which is prime.
• Then 31 5; 16 = 496 which is a perfect number.
• + 2k-1 = 2k - 1.
• If, for some k > 1, 2k - 1 is prime then 2k-1(2k - 1) is a perfect number.
• Nicomachus divides numbers into three classes: the superabundant numbers which have the property that the sum of their aliquot parts is greater than the number; deficient numbers which have the property that the sum of their aliquot parts is less than the number; and perfect numbers which have the property that the sum of their aliquot parts is equal to the number (see [',' M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.','8], or [',' D&#8217;Ooge (tr.), Nicomachus, Introduction to arithmetic (New York, 1926).','1] for a different translation):- .
• However Nicomachus has more than number theory in mind for he goes on to show that he is thinking in moral terms in a way that might seem extraordinary to mathematicians today (see [',' M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.','8], or [',' D&#8217;Ooge (tr.), Nicomachus, Introduction to arithmetic (New York, 1926).','1] for a different translation):- .
• Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with (see [',' M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.','8], or [',' D&#8217;Ooge (tr.), Nicomachus, Introduction to arithmetic (New York, 1926).','1]):- .
• every perfect number is of the form 2k-1(2k - 1), for some k > 1, where 2k - 1 is prime.
• Let us look in more detail at Nicomachus's description of the algorithm to generate perfect numbers which is assertion (4) above (see [',' M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.','8], or [',' D&#8217;Ooge (tr.), Nicomachus, Introduction to arithmetic (New York, 1926).','1]):- .
• First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect.
• Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm [',' D&#8217;Ooge (tr.), Nicomachus, Introduction to arithmetic (New York, 1926).','1]:- .
• Ibn al-Haytham proved a partial converse to Euclid's proposition in the unpublished work Treatise on analysis and synthesis when he showed that perfect numbers satisfying certain conditions had to be of the form 2k-1(2k - 1) where 2k - 1 is prime.
• Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus.
• Medizin 24 (1) (1987), 21-30.','6] and [',' S Brentjes, Eine Tabelle mit vollkommenen Zahlen in einer arabischen Handschrift aus dem 13.
• (4) 8 (2) (1990), 239-241.','7].
• Some even believed the further unjustified and incorrect result that 2k-1(2k - 1) is a perfect number for every odd k.
• Charles de Bovelles, a theologian and philosopher, published a book on perfect numbers in 1509.
• In it he claimed that Euclid's formula 2k-1(2k - 1) gives a perfect number for all odd integers k, see [',' N Miura, Charles de Bovelles and perfect numbers, Historia Sci.
• 34 (1988), 1-10.','10].
• The fifth perfect number has been discovered again (after the unknown results of the Arabs) and written down in a manuscript dated 1461.
• It is also in a manuscript which was written by Regiomontanus during his stay at the University of Vienna, which he left in 1461, see [',' E Picutti, Pour l&#8217;histoire des sept premiers nombres parfaits, Historia Math.
• 16 (2) (1989), 123-136.','14].
• In 1536, Hudalrichus Regius made the first breakthrough which was to become common knowledge to later mathematicians, when he published Utriusque Arithmetices &#9417; in which he gave the factorisation 211 - 1 = 2047 = 23 .
• With this he had found the first prime p such that 2p-1(2p - 1) is not a perfect number.
• He also showed that 213 - 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number 212(213 - 1) = 33550336.
• J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid's Elements.
• The next step forward came in 1603 when Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes).
• Cataldi was able use his list of primes to show that 217- 1 = 131071 is prime (since 7502 = 562500 > 131071 he could check with a tedious calculation that 131071 had no prime divisors).
• From this Cataldi now knew the sixth perfect number, namely 216(217 - 1) = 8589869056.
• Cataldi also used his list of primes to check that 219 - 1 = 524287 was prime (again since 7502 = 562500 > 524287) and so he had also found the seventh perfect number, namely 218(219 - 1) = 137438691328.
• He writes in Utriusque Arithmetices &#9417; that the exponents p = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37 give perfect numbers 2p-1(2p - 1).
• He is, of course, right for p = 2, 3, 5, 7, 13, 17, 19 for which he had a proof from his table of primes, but only one of his further four claims 23, 29, 31, 37 is correct.
• For example Descartes, in a letter to Mersenne in 1638, wrote [',' M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.','8]:- .
• For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number.
• He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic.
• 1 2 3 4 5 6 7 8 9 10 11 12 13 .
• 1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 .
• Just as 7, the exponent of 127, is prime, I say that 126 is a multiple of 14.
• Shortly after writing this letter to Mersenne, Fermat wrote to Frenicle de Bessy on 18 October 1640.
• In this letter he gave a generalisation of results in the earlier letter stating the result now known as Fermat's Little Theorem which shows that for any prime p and an integer a not divisible by p, ap-1- 1 is divisible by p.
• He showed that 223 - 1 was composite (in fact 223 - 1 = 47 5; 178481) and that 237 - 1 was composite (in fact 237 - 1 = 223 5; 616318177).
• In fact assuming that perfect numbers are of the form 2p-1(2p - 1) where p is prime, the question readily translates into asking whether 237 - 1 is prime.
• Fermat not only states that 237 - 1 is composite in his June 1640 letter, but he tells Mersenne how he factorised it.
• (iii) If n is prime, p a prime divisor of 2n- 1, then p - 1 is a multiple of n.
• Fermat proceeds as follows: If p is a prime divisor of 237 - 1, then 37 divides p - 1.
• As p is odd, it is a prime of the form 2 5; 37m+1, for some m.
• The next case to try is 223 (the case m = 3) which succeeds and 237 - 1 = 223 5; 616318177.
• In 1644 he published Cogitata physica mathematica &#9417; in which he claimed that 2p - 1 is prime (and so 2p-1(2p - 1) is a perfect number) for .
• p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 .
• to tell if a given number of 15 or 20 digits is prime, or not, all time would not suffice for the test.
• A suggestion as to the rule he used in giving his list is made in [',' S Drake, The rule behind &#8217;&#8217;Mersenne&#8217;s numbers&#8217;&#8217;, Physis-Riv.
• 13 (4) (1971), 421-424.','9].
• Primes of the form 2p- 1 are called Mersenne primes.
• In 1732 he proved that the eighth perfect number was 230(231 - 1) = 2305843008139952128.
• Then in 1738 Euler settled the last of Cataldi's claims when he proved that 229 - 1 was not prime (so Cataldi's guesses had not been very good).
• It should be noticed (as it was at the time) that Mersenne had been right on both counts, since p = 31 appears in his list but p = 29 does not.
• In two manuscripts which were unpublished during his life, Euler proved the converse of Euclid's result by showing that every even perfect number had to be of the form 2p-1(2p - 1).
• He was able to prove the assertion made by Descartes in his letter to Mersenne in 1638 from which we quoted above.
• (4n+1)4k+1 b2 .
• where 4n+1 is prime.
• He claimed that 2p-1(2p - 1) was perfect for p = 41 and p = 47 but Euler does have the distinction of finding his own error, which he corrected in 1753.
• The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in Cogitata physica mathematica &#9417;.
• In fact Euler's perfect number 230(231 - 1) remained the largest known for over 150 years.
• Mathematicians such as Peter Barlow wrote in his book Theory of Numbers published in 1811, that the perfect number 230(231 - 1):- .
• The first error in Mersenne's list was discovered in 1876 by Lucas.
• He was able to show that 267 - 1 is not a prime although his methods did not allow him to find any factors of it.
• Lucas was also able to verify that one of the numbers in Mersenne's list was correct when he showed that 2127 - 1 is a Mersenne prime and so 2126(2127- 1) is indeed a perfect number.
• Lucas made another important advance which, as modified by Lehmer in 1930, is the basis of computer searches used today to find Mersenne primes, and so to find perfect numbers.
• p = 3, 7, 127, 170141183460469231731687303715884105727, ..
• However checking whether the fourth term of this sequence, namely 2p - 1 for p = 170141183460469231731687303715884105727, is prime is well beyond what is possible.
• In 1883 Pervusin showed that 260(261- 1) is a perfect number.
• In 1903 Cole managed to factorise 267 - 1, the number shown to be composite by Lucas, but for which no factors were known up to that time.
• 267 - 1 = 147573952589676412927.
• Then he wrote 761838257287 and underneath it 193707721.
• Without speaking a work he multiplied the two numbers together to get 147573952589676412927 and sat down to applause from the audience.
• [It is worth remarking that the computer into which I (EFR) am typing this article gave this factorisation of 267 - 1 in about a second - times have changed!] .
• In 1911 Powers showed that 288(289 - 1) was a perfect number, then a few years later he showed that 2107- 1 is a prime and so 2106(2107- 1) is a perfect number.
• In 1922 Kraitchik showed that Mersenne was wrong in his claims for his largest prime of 257 when he showed that 2257- 1 is not prime.
• Intelligencer 7 (2) (1985), 66-68.','20]):- .
• In fact Sylvester proved in 1888 that any odd perfect number must have at least 4 distinct prime factors.
• Today (2018) 50 perfect numbers are known, 288(289- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer.
• At the moment the largest known Mersenne prime is 277n232n917 - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 277n232n916 (277n232n917 - 1).

5. Alcuin's book
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• We have kept to the standard numbering in our list of the puzzles so, for example, there are puzzles 11, 11(a) and 11(b).
• 1500 5; 5 5; 12 = 90000.
• 2x + x/2 + x/4 + 1 = 100.
• He merely checks that 36 satisfies the given conditions by checking that 72 + 18 + 9 + 1 = 100.
• 2x + x + 3x/6 + 2 = 100.
• 2x + 2x/4 = 100.
• x + y + 2z =100, and 10x + 5y + z =100.
• I have material which is 100 feet long and 80 feet wide.
• As in the previous puzzle, 10 5;10 which is 100 tunics.
• After the 30th village there will be 230 = 1,073,741,824 men in the army.
• M1, M2, M3, S1, S2, S3 =----- .
• M2, M3, S2, S3 -----= M1, S1 .
• M2, M3, S2, S3, M1 =----- S1 .
• M2, M3, M1 -----= S2, S3, S1 .
• M2, M3, M1, S1 =----- S2, S3 .
• M1, S1 -----= M2, M3, S2, S3 .
• M1, S1, M2, S2 =----- M3, S3 .
• S1, S2 -----= M1, M2, M3, S3 .
• S1, S2, S3 =----- M1, M2, M3 .
• S2 -----= S1, S3, M1, M2, M3 .
• S2, M2 =----- S1, S3, M1, M3 .
• -----= S2, M2, S1, S3, M1, M3 .
• M, W -----= C1, C2 .
• M -----= C1, C2, W .
• -----= C1, C2, M, W .
• M, W -----= C1, C2 .
• M -----= C1, C2, W .
• This divides the field into 40 5; 25 = 1000 rectangles of the required size.
• There is an irregular field which is 100 yards on each side, 50 metres on one front, 60 yards in the middle, and 50 yards on the other front.
• Alcuin uses an approximate method, assuming that the area is the same as that of a rectangle 1/3(50+50+60) yards by 100 yards.
• Then the area is πr2 = 40000/π = 12732 square yards (approximately).
• There is a field which is 150 feet long.
• There is a four-sided city which has one side of 1100 feet, another side of 1000 feet, a front of 600 feet, and a final side of 600 feet.
• The area of a house is 1200 square feet so there are 523 house areas in the field.
• There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 feet.
• The basilica is 2880 inches long and 2880/23 = 125.2.
• A wine cellar is 100 feet long and 64 feet wide.
• We divide each side into a strip 14 feet wide and one 16 feet wide and put the casks in one way in the 14 foot strip and the other way in the 16 foot strip, so we can fit in 2 5; 25 + 4n5; 14 = 106 on each side of the passageway: a total of 212 .
• Fill up the 28 foot side with 4n× 25 = 100 casks.
• On the 32 foot side put in 8n5; 12 = 96 casks to leave a gap ofn16n× 32 feet at one end.
• Put in 4n× 4 = 16 casks to leave a gap of 4n5; 16 feet.
• Then x = 1 and z = 14 so there is 1 man, 5 women and 14 children who are servants in the household.
• If y = 15 then x = 11 and z = 74.
• the mother receives 1/2(1/4 + 5/12) = 1/3 of 960 pounds, which is 320 pounds, .
• the daughter receives 1/2(7/12 + 0) = 7/24 of 960 pounds, which is 280 pounds.
• And if God will grant you one more year than that, and you shall live to be 100." How old was the boy at the time the old man greeted him? .
• This gives 11x = 50, so x = 4 pounds and 6/11 of a pound.
• This is what each master builder receives, while the apprentice receives 2 pounds and 3/11 of a pound.
• x + y + z = 100 and 3x + y + z/24 = 100.
• x + y + z = 100 and 5x + y + z/20 = 100.
• Subtract the first from the second to get 4x = (19/20)z or 80x = 19z.
• Thus x is divisible by 19 and, by the second equation, cannot be larger than 19.
• (x + x) + x/2 + x/4 + 1 = 100.
• After each has had a piglet in the second corner there will be 83 = 512 pigs.
• Again each produces a litter of 7 in the centre of the sty so at this final stage there will be 86 = 262144 pigs in the sty.
• Continue until we get to the 49th step and the 51st step which total to 100 pigeons.
• If I gave you one of my years to add to this, then you will live to be 100 years old." How old was the boy at the time? .
• Then 3(2x) + 1 = 100.
• Then x + x + x + 1 = 100.
• There must have been 10650/50 = 215 people in the crowd.
• x + y + z = 12 and 2x + y/2 + z/4 = 12.
• If you add me to one of the quarters, there will be 100." How many students are in the school? .
• 3(2x)/4 + 1 =100.
• How many pints do 100 measures of wine contain, and how many cups do 100 measures contain? .
• There are 4800 pints and 28800 cups in 100 measures.
• The total amount of wine is 40 + 30 + 20 + 10 = 100 measures.
• A certain abbot of a monastery was in charge of 12 monks.
• Each monk will receive 204/12 = 17 eggs.

6. Babylonian Pythagoras
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 4 times 4 is 16.
• 25 (4) (1998), 366-378.
• ','18].
• W K Loftus identified this as an important archaeological site as early as 1850 but excavations were not carried out until much later.
• Finally the Tell Dhibayi tablet was one of about 500 tablets found near Baghdad by archaeologists in 1962.
• It has on it a diagram of a square with 30 on one side, the diagonals are drawn in and near the centre is written 1,24,51,10 and 42,25,35.
• Assuming that the first number is 1; 24,51,10 then converting this to a decimal gives 1.414212963 while √2 = 1.414213562.
• Calculating 30 5; [ 1;24,51,10 ] gives 42;25,35 which is the second number.
• J., 1999).','2] and [',' G G Joseph, The crest of the peacock (London, 1991).','4], conjecture that the Babylonians used a method equivalent to Heron's method.
• In fact as Joseph points out in [',' G G Joseph, The crest of the peacock (London, 1991).','4], one needs only two steps of the algorithm if one starts with x = 1 to obtain the approximation 1;24,51,10.
• Now this certainly takes many more steps to reach the sexagesimal approximation 1;24,51,10.
• 1 1.500000000 1;29,59,59 .
• 2 1.250000000 1;14,59,59 .
• 3 1.375000000 1;22,29,59 .
• 4 1.437500000 1;26,14,59 .
• 5 1.406250000 1;24,22,29 .
• 6 1.421875000 1;25,18,44 .
• 7 1.414062500 1;24,50,37 .
• 8 1.417968750 1;25, 4,41 .
• 9 1.416015625 1;24,57,39 .
• 10 1.415039063 1;24,54, 8 .
• 11 1.414550781 1;24,52,22 .
• 12 1.414306641 1;24,51;30 .
• 13 1.414184570 1;24,51; 3 .
• 14 1.414245605 1;24,51;17 .
• 15 1.414215088 1;24,51;10 .
• 16 1.414199829 1;24,51; 7 .
• 17 1.414207458 1;24,51; 8 .
• 18 1.414211273 1;24,51; 9 .
• 19 1.414213181 1;24,51;10 .
• , 15.
• For example the Pythagorean triple 3, 4 , 5 does not appear neither does 5, 12, 13 and in fact the smallest Pythagorean triple which does appear is 45, 60, 75 (15 times 3, 4 , 5).
• However, as Joseph comments [',' G G Joseph, The crest of the peacock (London, 1991).','4]:- .
• He has pointed out that if the Babylonians used the formulas h = 2mn, b = m2-n2, c = m2+n2 to generate Pythagorean triples then there are exactly 16 triples satisfying n ≤ 60, 3076; ≤ t ≤ 4576;, and tan2t = h2/b2 having a finite sexagesimal expansion (which is equivalent to m, n, b having 2, 3, and 5 as their only prime divisors).
• 37 (1995), 29-47.
• ','17], claims that the tablet is connected with the solution of quadratic equations and has nothing to do with Pythagorean triples:- .
• Other authors, although accepting that Plimpton 322 is a collection of Pythagorean triples, have argued that they had, as Viola writes in [',' T Viola, On the list of Pythagorean triples (&#8217;&#8217;Plimpton 322&#8217;&#8217;) and on a possible use of it in old Babylonian mathematics (Italian), Boll.
• 1 (2) (1981), 103-132.','31], a practical use in giving a:- .
• and so 80x = 2500 or, in sexagesimal, x = 31;15.
• It asks for the sides of a rectangle whose area is 0;45 and whose diagonal is 1;15.
• If the sides are x, y we have xy = 0.75 and x2 + y2 = (1.25)2.
• Subtract from x2 + y2 = 1;33,45 to get x2 + y2 - 2xy = 0;3,45.
• Take the square root to obtain x - y = 0;15.
• Divide x2 + y2 - 2xy = 0;3,45 by 4 to get x2/4 + y2/4 - xy/2 = 0;0,56,15.
• Add xy = 0;45 to get x2/4 + y2/4 + xy/2 = 0;45,56,15.

7. Pi history
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and W30;10 = 3.162 have been traced to much earlier dates: though in defence of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary.
• In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 5; (8/9)2 = 3.16 as a value for π.
• The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC).
Go directly to this paragraph
• 223/71 < π < 22/7.
• If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.
• Consider a circle of radius 1, in which we inscribe a regular polygon of 3 5; 2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 5; 2n-1 sides, with semiperimeter an.
• where K = 3 5; 2n-1.
• an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K), .
• an+1bn = (bn+1)2nnnnnnn.
Go directly to this paragraph
• Zu Chongzhi (430-501 AD) 355/113 .
Go directly to this paragraph
• 800 ) 3.1416 .
Go directly to this paragraph
• 1430) 14 places .
Go directly to this paragraph
• Viete (1540-1603) 9 places .
Go directly to this paragraph
• Roomen (1561-1615) 17 places .
Go directly to this paragraph
• Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.
• One of the earliest was that of Wallis (1616-1703) .
Go directly to this paragraph
• 2/π = (1.3.3.5.5.7.
• π/4 = 1 - 1/3 + 1/5 - 1/7 + ..
• This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).
Go directly to this paragraph
• In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series.
• (-1 ≤ x Ͱ4; 1)nnn.
• π/6 = (1/√3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ..
• The 10th term is 1/(19 5; 39√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.
• In 1706 Machin found such a formula: .
Go directly to this paragraph
• One of them, an Englishman named Shanks, used Machin's formula to calculate π to 707 places, publishing the results of many years of labour in 1873.
Go directly to this paragraph
• 1701: Machin used an improvement to get 100 digits and the following used his methods: .
• 1789: Vega got 126 places and in 1794 got 136 .
Go directly to this paragraph
• 1841: Rutherford calculated 152 digits and in 1853 got 440 .
• Shanks knew that π was irrational since this had been proved in 1761 by Lambert.
Go directly to this paragraph
• He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528th place, after which all his digits were wrong.
Go directly to this paragraph
• In 1949 a computer was used to calculate π to 2000 places.
Go directly to this paragraph
• Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference.
Go directly to this paragraph
• The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states "3.14159 andc.
Go directly to this paragraph
• Euler adopted the symbol in 1737 and it quickly became a standard notation.
Go directly to this paragraph
• The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got .
Go directly to this paragraph
• π = 355/113 = 3.1415929 .
• 2 5; 0.7857 / π = 1/2 .
• from which he got the highly creditable value of π = 3.1428.
• It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934.
Go directly to this paragraph
• In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.
• 246, 1897) .
• Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π? .
• Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
• However, if π is normal then the first million digits 314159265358979..
• Even if π is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.
• 3.14159265358979323846264..
• JOC/EFR August 2001 .

8. Prime numbers
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
• Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number.
Go directly to this paragraph
• The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form.
Go directly to this paragraph
• He devised a new method of factorising large numbers which he demonstrated by factorising the number 2027651281 = 44021 × 46061.
• The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31 5; 11 is composite.
• Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime.
Go directly to this paragraph
• For example 211 - 1 = 2047 = 23 5; 89 is composite, though this was first noted as late as 1536.
• The number M19 was proved to be prime by Cataldi in 1588 and this was the largest known prime for about 200 years until Euler proved that M31 is prime.
Go directly to this paragraph
• In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun.
• The largest is M77n232n917 which has 23n249n425 decimal digits.
• As mentioned above he factorised the 5th Fermat Number 232 + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.
• He was able to show that not only is the so-called Harmonic series &#8721; (1/n) divergent, but the series .
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ..
• Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers).
Go directly to this paragraph
• π(n) = n/(log(n) - 1.08366) .
• π(n) = ∫; (1/log(t) dt where the range of integration is 2 to n.
• The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem.
Go directly to this paragraph
• The result was eventually proved (using powerful methods in Complex analysis) by Hadamard and de la Vallee Poussin in 1896.
Go directly to this paragraph
• Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.
Go directly to this paragraph
• Are there infinitely many primes of the form n2 + 1 ? .
• 251, 257, 263, 269 has length 4.
• n2 - n + 41 is prime for 0 ≤ n ≤ 40.
• Are there infinitely many primes of this form? The same question applies to n2 - 79 n + 1601 which is prime for 0 ≤ n ≤ 79.
• Are there infinitely many primes of the form n# + 1? (where n# is the product of all primes ≤ n.) .
• Are there infinitely many primes of the form n# - 1? .
• Are there infinitely many primes of the form n! + 1? .
• Are there infinitely many primes of the form n! - 1? .
• The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in January 2018) is M77n232n917 which has 23n249n425 decimal digits.
• The largest known twin prime pair is 2n996n863n034n895 5; 21n290n000 77; 1, with 388 342 decimal digits.
• The largest known factorial prime (prime of the form n! 77; 1) is 208n003! - 1.
• It is a number of 1n015n843 digits and was announced in 2016.
• The largest known primorial prime (prime of the form n# 77; 1 where n# is the product of all primes ≤ n) is 1n098n133# + 1.
• It is a number of 476n311 digits and was announced in 2012.

9. Pell's equation
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• nx2 + 1 = y2 .
• y2 - nx2 = 1 .
• b2 - na2 = 1 and d2 - nc2 = 1 .
• (bd + nac)2 - n(bc + ad)2 = 1 .
• (bd - nac)2 - n(bc - ad)2 = 1.
• as a solution of Pell's equation nx2 + 1 = y2.
• y = (b2 + na2)/2 = (2b2 - 2)/2 = b2 - 1 .
• So Brahmagupta was able to show that if he could find (a, b) which "nearly" satisfied Pell's equation in the sense that na2 + k = b2 where k = 1, -1, 2, -2, 4, or -4 then he could find one, and therefore many, integer solutions to Pell's equation.
• For example, if we attempt to solve 23x2 + 1 = y2 we see that a = 1, b = 5 satisfies 23a2 + 2 = b2 so, by the above argument, x = 5, y = 24 satisfies Pell's equation.
• x = 25;55;24 = 240,n y = 242 + 235;52 = 1151 .
• x = 11515,n y = 55224 .
• x = 552480,n y = 2649601 .
• 83x2 + 1 = y2 .
• 835;12 - 2 = 92 .
• (1476, 13447), .
• (39695544, 361643617), .
• (6509827161, 59307347962), .
• (1067571958860, 9726043422151), .
• (175075291425879, 1595011813884802) .
• The next step forward was taken by Bhaskara II in 1150.
• He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell's equation nx2 + 1 = y2 starting off from any "close" pair (a, b) with na2 + k = b2.
• If (m2 - n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta's method to find a solution to Pell's equation nx2 + 1 = y2.
• This happens when an equation of the form nx2 + t = y2 is reached where t is one of 1, -1, 2, -2, 4, -4.
• 61x2 + 1 = y2.
• Using the above method he chooses m so the (m + 8)/3 is an integer, making sure that m2 - 61 is as small as possible.
• x = 226153980, ny = 1766319049 .
• as the smallest solution to 61x2 + 1 = y2.
• Secondly the algorithm always reaches a solution of Pell's equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is reached and then applying Brahmagupta's method.
• If experience of the algorithm is only via examples then, knowing how to proceed when k = -1, 2, -2, 4, or -4 is reached, it is natural to switch to Brahmagupta's method at that point.
• 103x2 + 1 = y2.
• 1035;12 - 3 = 102.
• Choose m so that m + 10 is divisible by -3 with m2 - 103 as small as possible leads to m = 11 and we obtain .
• 1035;72 - 6 = 712.
• Next we must choose m so that 7m + 71 is divisible by -6 and m2 - 103 as small as possible.
• 1035;202 + 9 = 2032.
• Continuing, choose m so that 20m+203 is divisible by 9 and m2 - 103 as small as possible.
• Take m = 11 to get the equation .
• 1035;472 + 2 = 4772.
• x = 22419,n y = 227528.
• 97x2 + 1 = y2 .
• 975;12 + 3 = 102 .
• 975;72 + 8 = 692 .
• 975;202 + 9 = 1972 .
• 975;532 + 11 = 5222 .
• 975;862 - 3 = 8472 .
• 975;5692 - 1 = 56042 .
• The European interest began in 1657 when Fermat issued a challenge to the mathematicians of Europe and England.
• 61x2 + 1 = y2.
• There followed an exchange of letters between these mathematicians during 1657-58 which Wallis published in Commercium epistolicum in 1658.
• Frenicle de Bessy tabulated the solutions of Pell's equation for all n up to 150, although this was never published and his efforts have been lost.
• 313x2 + 1 = y2.
• x = 1819380158564160, ny = 32188120829134849 .
• In 1658 Rahn published an algebra book which contained an example of Pell's equation.
• Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell's equation based on the exchange of letters he had published in Commercium epistolicum in 1658.
• We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n.
• He gave the basis for the continued fractions approach to solving Pell's equation which was put into a polished form by Lagrange in 1766.
• Lagrange published his Additions to Euler's Elements of algebra in 1771 and this contains his rigorous version of Euler's continued fraction approach to Pell's equation.
• For example W30;19 has the continued fraction expansion .
• The convergent immediately before the point from which it repeats is 170/39 and Lagrange's theory says that .
• 19x2 + 1 = y2.
• To find the infinite series of solutions take the powers of 170 + 39W30;19.
• (170 + 39W30;19)2 = 57799 + 13260W30;19 .
• (170 + 39W30;19)3 = 19651490 + 4508361W30;19 .
• x = 4508361, ny = 19651490 .
• Here are the first few powers of (170 + 39W30;19), starting with its square, which gives the first few solutions to the equation 19x2 + 1 = y2 .
• 57799 + 13260W30;19 .
• 19651490 + 4508361W30;19 .
• 6681448801 + 1532829480W30;19 .
• 2271672940850 + 521157514839W30;19 .
• 772362118440199 + 177192022215780W30;19 .
• 262600848596726810 + 60244766395850361W30;19 .
• 89283516160768675201 + 20483043382566906960W30;19 .

10. Babylonian mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• (2) 1 (3) (1992), 173-180.
• They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian ..
• As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation.
• Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC.
• The table gives 82 = 1,4 which stands for .
• 82 = 1, 4 = 1 × 60 + 4 = 64 .
• and so on up to 592 = 58, 1 (= 58 5; 60 +1 = 3481).
• a/b = a 5; (1/b) .
• 4 0; 15 .
• 5 0; 12 .
• 6 0; 10 .
• 10 0; 6 .
• 12 0; 5 .
• 15 0; 4 .
• 16 0; 3, 45 .
• 18 0; 3, 20 .
• 27 0; 2, 13, 20 .
• Now the table had gaps in it since 1/7, 1/11, 1/13, etc.
• 1/13 = 7/91 = 7 5; (1/91) = (approx) 7 5; (1/90) .
• A scribe would give a number close to 1/7 and then write statements such as (see for example [',' G G Joseph, The crest of the peacock (London, 1991).','5]):- .
• Multiply 1;48 by 1,40 to get the answer 3,0.
• 2/35; 2/3 x + 100 = x .
• This is why the scribe computed 2/3 5; 2/3 subtracted the answer from 1 to get (1 - 4/9), then looked up 1/(1 - 4/9) and so x was found from 1/(1 - 4/9) multiplied by 100 giving 180 (which is 1; 48 times 1, 40 to get 3, 0 in sexagesimal).
• A problem on a tablet from Old Babylonian times states that the area of a rectangle is 1, 0 and its length exceeds its breadth by 7.
• Compute half of 7, namely 3; 30, square it to get 12; 15.
• To this the scribe adds 1, 0 to get 1; 12, 15.
• 40 (1956), 185-192.','10] Berriman gives 13 typical examples of problems leading to quadratic equations taken from Old Babylonian tablets.
• Retranslation and analysis, Amphora (Basel, 1992), 315-358.','26].

11. Weather forecasting
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 2.1 First Attempts .
• Observing the skies and drawing the correct conclusions from these observations was crucial to people's survival [','W Wiedlich, Eine kleine Geschichte der Wettervorhersage (General-Anzeiger, 27.11.2007)','26].
• Nowadays, we are more independent of weather conditions due to central heating, air conditioners, greenhouses and so forth, but weather forecasts are more accurate than they ever were [','P Bethge, J Blech, M Dworschak, O Stampf, Das gefuhlte Wetter (Spiegel, 23/2000) ','21].
• television and the fact that weather reports often have a higher audience rate than the preceding news broadcasts, illustrate that forecasting has become a highly competitive business [','P Bethge, J Blech, M Dworschak, O Stampf, Das gefuhlte Wetter (Spiegel, 23/2000) ','21].
• In 340 BC, Aristotle wrote his book Meteorologica &#9417;, where he tried to explain the formation of rain, clouds, wind and storms.
• For example, he believed that heat could cause water to evaporate [',' http://yale.edu/ynthi/curriculum/units/1994/5/94.05.01.x.html, L Alter','47].
• But he also jumped to quite a few wrong conclusions, such as that winds form "as the Earth exhales" [','P Bethge, J Blech, M Dworschak, O Stampf, Das gefuhlte Wetter (Spiegel, 23/2000) ','21], which were rectified from the Renaissance onwards.
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• Many of these proverbs are based on very good observations and are accurate, as contemporary meteorologists have discovered [','W Wiedlich, Eine kleine Geschichte der Wettervorhersage (General-Anzeiger, 27.11.2007)','26].
• In 1592, Galileo Galilei (1564-1642) developed the world's first thermometer.
• His student Evangelista Torricelli (1608-1647) invented the barometer in 1643, which allowed people to measure atmospheric pressure.
• Five years later, Blaise Pascal (1623-1662) proved that pressure decreases with altitude.
• This discovery was verified by Edmond Halley (1656-1742) in 1686; Halley also was the first one to map trade winds.
• One of the first to study storms was the scientist and politician Benjamin Franklin (1706-1790).
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• One of the first weather observation networks, which operated from 1654 to 1670, was established by the Tuscan nobleman Ferdinand II [','R Wengenmayr, Wettervorhersage.
• For fifteen years (1780-1795), 39 weather stations, spread across Northern America, Europe and Russia, collected meteorological data three times per day, using standardised equipment [','R Wengenmayr, Wettervorhersage.
• The invention of the electrical telegraph in 1837 by Samuel Morse facilitated the production of weather forecasts, as data and any other weather observations now could easily and swiftly be transmitted to another country and even to another continent.
• Observation wards began to appear all over Europe and Northern America, but it was not until the Crimean War (1853-1856) that people realised the benefits of weather forecasts.
• The French Emperor Napoleon III later ordered that the weather for that day should be analysed; he learned that the storm could have been predicted and that warnings could have been transmitted by telegraph [','M Birke, Wettervorhersage & Wetterdienst (University of Regensburg, 2009)','12].
• The British Meteorological Department issued regular gale warnings from 1861 onwards; the first US storm-warning system began to operate ten years later, dwarfing the European services with its size and funds [',' http://eh.net/encyclopedia/article/craft.weather.forecasting.history, E D Craft','33].
• The first weather satellite, TIROS 1 (Television and Infrared Observation Satellite) was launched in 1960.
• The first mathematician who thought of applying mathematics to weather forecasting was the Norwegian Vilhelm Bjerknes (1862-1951).
• In 1898 he formulated his circulation theorem: in a nutshell, it explains the evolution and the subsequent decay of circulations in fluids.
• Possibly even more importantly, the theorem also marks "the move of Vilhelm Bjerknes into meteorology" [','A J Thorpe, H Volkert, M ZiemiaDski, The Bjerknes&#8217; Circulation Theorem: A Historical Perspective (Bulletin of the American Meteorological Society, Volume 84, Issue 4, April 2003), 471-480','11, p.
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• During a visit to the United States in 1905, he presented his theories.
• Most of Bjerknes's important results were published in On the Dynamics of the Circular Vortex with Applications to the Atmosphere and to Atmospheric Vortex Wave Motion in 1921.
• [',' http://www.bjerknes.uib.no/pages.asp?id=114&kat=3&lang=1, S Gronas','27] .
• The first attempt to use mathematics in order to predict the weather was made by the British mathematician Lewis Fry Richardson (1881-1953), who simplified Bjerknes' equations so that solving them became more feasible.
• In 1919, he resumed his former job there, but resigned only a year later when the Meteorological Office became part of the Air Ministry.
• Richardson also included a research department in charge of refining the models [','R Stewart, Weather Forecasting by Computer (2009)','10; 1, pp.
• 157-158].
• This forecast factory is "remarkably similar to descriptions of modern multiple-processor supercomputers used in weather forecasting today" [','R Stewart, Weather Forecasting by Computer (2009)','10].
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• 161] .
• In 1916, Richardson decided to join the Friends Ambulance Unit (that had been formed for conscientious objectors) and serve in the First World War, out of curiosity.
• Nonetheless, he decided to include his calculations in his seminal book Weather Prediction by Numerical Process, which was published in 1922.
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• 161].
• Meanwhile, the groundbreaking research of meteorologists such as Jacob Bjerknes (1897-1975) and Carl-Gustaf Rossby (1898-1957) furthered scientists' knowledge of the atmosphere and helped pave the way for the eventual triumph of numerical weather forecasting.
• In 1939, when he held a professorship at the Massachusetts Institute of Technology (MIT), he discovered the so-called Rossby waves, which are meanders of large-scale airflows in the atmosphere.
• fC = 2 5; ω 5; sin φ .
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, p.
• The only day with conditions good enough for the invasion was 6th June [','R Stewart, Weather Forecasting by Computer (2009)','10; 1, pp.
• 189-195].
• The Turbulent History of Weather Prediction from Franklin&#8217;s Kite to El Nino (Hoboken NJ, 2002)','1, pp.
• 195-196].
• D-Day not only changed world history, it also highlighted the importance of weather forecasts [','R Stewart, Weather Forecasting by Computer (2009)','10].
• The mathematician John von Neumann (1903-1957), one of the fathers of computer science, was the first one to think of using computers to predict the weather.
• In 1946, he presented his ideas to a group of leading meteorologists, including Carl-Gustaf Rossby and the young, gifted mathematician Jule Charney (1917-1981).
• In 1950, the ENIAC successfully produced a 24-hour forecast.
• This took about 24 hours, but the computer developed by von Neumann's team was much faster (this computer produced its first forecast in 1952), and from 1955 onwards, numerical forecasts generated by computers were issued on a regular basis.
• In 1948, Charney developed the quasi-geostrophic approximation, which reduces several equations of atmospheric motions to only two equations in two unknown variables [','R S Harwood, Atmospheric Dynamics (Chapter 1: Basics, Chapter 5: Balance of Forces in Synoptic Scale Flow, Chapter 13: Quasi-Geostrophic Equations) (University of Edinburgh, 2005) ','14, chapter 13].
• In 1963, a six-layer model based on the primitive equations was used for producing a forecast.
• Today, the world's leading meteorological centres use the most powerful computers on the planet; the new computer at the British Met Office for example is capable of 125 trillion calculations per second [','http://www.metoffice.gov.uk','42].
• The discovery that the atmosphere is chaotic, by MIT researcher Edward Lorenz (1917-2008) in 1961, came as a serious blow to all these high-flying ideas.
• For this he ran a shortened forecasting model on his computer, and to his great surprise, inputting data that differed from previously entered values only in the fourth decimal place, significantly changed the weather the computer predicted [',' http://www-history.mcs.st-andrews.ac.uk/Biographies/Lorenz_Edward.html, J J O&#8217;Connor, E F Robertson','38].
• Chaotic behaviour is commonly known as the "butterfly effect", a term coined due to the title of a talk Lorenz gave in 1972: Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Meteorologists had presumed that small weather changes in some specified places would affect the weather in other places, but after Lorenz's discovery, they had to accept that it did not matter if a butterfly flapped its wings in Brazil, or Bulgaria, or Bangladesh, the result might still be a tornado in Texas (or somewhere else for that matter).
• Kapitel 6: Dynamik der Atmosphare (University of Bern, 2009), 118-123','15, p.
• with &#961; being the density and p being the pressure.
• The symbol &#8711; is called Nabla and is a vector differential operator.
• in the "air belt" of 1-2 km altitude above the Earth's surface, is exposed to drag against the ground.
• Here, u is the zonal wind, parallel to the circles of latitude; v is the meridional wind, parallel to the circles of longitude; and w is the vertical wind component [','U Langematz, Vorlesung 6: Dynamik I (FU Berlin, 2009) www.geo.fu-berlin.de/met/ag/strat/lehre/sose09/Vorlesung_Mittlere_Atmosphaere/','16].
• where &#961;0(z) is the base-state density, an exponentially decreasing function of height.
• A six-hour forecast for North America for 11 April 2010, with fine resolution, produced on 11 April 2010 using the North American Mesoscale Model (NAM) of the U.
• Assuming that hydrostatic balance applies limits the smallest possible grid spacing to about 5 - 10 km [','http://www.dwd.de','29], which is not fine enough a resolution for detailed, accurate forecasts of small-scale weather events like thunderstorms.
• In order to make a forecast for the future (at time step t + [6;t), you do not start at the present time step t, but at the previous step t - [6;t, and the forecast leaps over the time step t (with [6;t denoting the size of the time step, that is the difference between two points in time).
• Generally, in a terrain-following coordinate system, the grid-spacing in the λ-direction is given by [6;λ; similarly, [6;φ and [6;ζ represent the spacing in the φ-direction and the ζ-direction, respectively.
• The value of a variable ψ at xl is given by ψl; and the finite difference for ψl is given using the values of ψ;l+1 and ψ;l-1, i.e.
• However, the lowest power of the difference of x, [6;x, in E gives the order of the approximation.
• In general, centred finite difference approximations are better than forward or backward approximations, which can also be derived from the Taylor expansions for ψ;l+1 and ψ;l-1.
• Not only space, but also time has to be discretized, and time derivatives can also be represented as finite difference approximations, that is in terms of values at discrete time levels [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• A time step is denoted by [6;t, and a discrete time level is given by tn = t0 +nΔt with t0 being the initial time for integration.
• The explicit scheme is much easier to solve than the implicit one, as it is possible to compute the new value of ψl at time n+1 for every grid point, provided the values of ψl are known for every grid point at the current time step n [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• The implicit scheme, on the other hand, is absolutely stable, but it results in a system of simultaneous equations, so is more difficult to solve [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• The terms fψ are integrated over big time steps [6;t.
• These time steps are then subdivided into several small time steps [6;τ, over which the terms sψ are integrated [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
• In the case that sψ = 0, we get, using a 2[6;t leapfrog interval, .
• The superscript m is the time step counter for the integration over the small time steps [6;τ within the leapfrog interval used above.
• The term f nψ is constant throughout the small time step integrations, but the value of ψ;n+1 is not known before the last one of these integrations has been completed.
• The term is the result of a process called averaging; you assume that the mean value of ψ;n+1 does not vary as fast with respect to both space and time than deviations from the mean would.
• In 1976, the Australian and Canadian weather services were the first ones to adopt this method, which is now used by a range of weather services across the globe; the European Forecasting Centre ECMWF in Reading, for example, adopted it in 1983 [','F Baer, The Spectral Method: Its Impact on NWP (2004) ','20].
• This corresponds to a grid length of roughly 25 km [','F Grazzini, A Persson, User Guide to ECMWF forecast products, version 4.0 (Meteorological Bulletin M3.2, 14.03.2007)','7] (the DWD's and the Met Office's global model has a resolution of 40 km).
• R(x: a0, a1, ..
• where the um are the complex expansion coefficients and M is the maximum wave number [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• There are several methods which convert differential equations to discrete problems, for example the least-square method or the Galerkin method, and which can be used in order to choose the time derivative such that the residual function is as close to zero as possible [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• Most commonly, Fast Fourier Transforms are used, but in principle all transform methods make it possible to switch between a spectral representation and a grid-point representation [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• Furthermore, products with more than two components suffer from aliasing, meaning that waves that are too short to be resolved for a certain grid resolution falsely appear as longer waves [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• a set of grid points) [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
• Admittedly, increasing the grid resolution involves the risk that errors in the initial data are multiplied when more grid points are used [','http://met.no','41], so mathematical models will have to take this into account.
• The 24-hour forecast for North America for 12 April 2010 with medium resolution, produced on 11 April 2010 using the NOAA's Global Forecast System (GFS)Click on the picture to see a larger version .
• A 162-hour forecast for North America for 17 April 2007 with medium resolution, produced on 11 April 2007 using the NOAA's Global Forecast System.Click on the picture to see a larger version .
• A six-day forecast nowadays is now as accurate as a one-day forecast in 1968 [','http://www.dwd.de','29].
• Current one-day forecasts are accurate in 9 out of 10 cases, and three-day forecasts still have a hit rate of 70%.
• Still, meteorologists at the ECMWF estimate that the global economic loss due to inaccurate weather forecasts amounts to up to a billion Euros per year [','W Wiedlich, Eine kleine Geschichte der Wettervorhersage (General-Anzeiger, 27.11.2007)','26], so meteorologists hope that improved forecast quality will reduce this number.
• JOC/EFR May 2011 .

12. Bakhshali manuscript
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The paper [',' R C Gupta, Centenary of Bakhshali manuscript&#8217;s discovery, Ganita Bharati 3 (3-4) (1981), 103-105.','8] describes this discovery along with the early history of the manuscript.
• The Bakhshali Manuscript is the name given to the mathematical work written on birch bark and found in the summer of 1881 near the village Bakhshali (or Bakhshalai) of the Yusufzai subdivision of the Peshawar district (now in Pakistan).
• Dr Hoernle presented a description of the BM before the Asiatic Society of Bengal in 1882, and this was published in the Indian Antiquary in 1883.
• He gave a fuller account at the Seventh Oriental Conference held at Vienna in 1886 and this was published in its Proceedings.
• A revised version of this paper appeared in the Indian Antiquary of 1888.
• In 1902, he presented the Bakhshali Manuscript to the Bodleian Library, Oxford, where it is still (Shelf mark: MS.
• In 1927-1933 the Bakhshali manuscript was edited by G R Kaye and published with a comprehensive introduction, an English translation, and a transliteration together with facsimiles of the text.
• 11 (2) (1976), 112-124.','6] gives the range 200 - 400 AD as the most probable date.
• In [',' M N Channabasappa, Mathematical terminology peculiar to the Bakhshali manuscript, Ganita Bharati 6 (1-4) (1984), 13-18.','5] the same author identifies five specific mathematical terms which do not occur in the works of Aryabhata and he argues that this strongly supports a date for the Bakhshali manuscript earlier than the 5th century.
• Joseph in [',' G G Joseph, The crest of the peacock (London, 1991).','3] suggests that the evidence all points to the:- .
• L V Gurjar in [',' L V Gurjar, Ancient Indian Mathematics and Vedha (Poona, 1947).','1] claims that the manuscript is no later than 300 AD.
• What does the manuscript contain? Joseph writes in [',' G G Joseph, The crest of the peacock (London, 1991).','3]:- .
• As an example, here is how 3/4 - 1/2 would be written.
• 21 (1) (1986), 51-61.','9] and some of these lead to indeterminate equations.
• 5 x1 + x2 + x3 = x1 + 7 x2+ x3 = x1 + x2 + 8 x3 = k.
• Then 4 x1 = 6 x2 = 7 x3 = k - (x1 + x2 + x3).
• For integer solutions k - (x1 + x2 + x3) must be a multiple of the lcm of 4, 6 and 7.
• The Bakhshali manuscript takes k - (x1 + x2 + x3) = 168 (this is 4 5; 6 5; 7) giving x1 = 42, x2 = 28, x3 = 24.
• This is not the minimum integer solution which would be k = 131.
• x1 = 21k/131, x2 = 14k/131, x3 = 12k/131 .
• so we obtain integer solutions by taking k = 131 which is the smallest solution.
• This solution is not given in the Bakhshali manuscript but the author of the manuscript would have obtained this had he taken k - (x1 + x2 + x3) = lcm(4, 6, 7) = 84.
• So 30 days are required when each has 13 5; 30/6 - 10 = 55 dinaras.
• The rule of three is the familiar way of solving problems of the type: if a man earns 50 dinaras in 8 days how much will he earn in 12 days.
• 8 50 12 .
• phala 5; iccha/pramana .
• or in the case of the example 50 5; 12/8 = 75 dinaras.
• 13 5; n/6 = 3 5; n/2 +20 .
• so n = 30 and each has 13 5; 30/6 - 10 = 55 dinaras.
• Taking Q = 41, then A = 6, b = 5 and we obtain 6.403138528 as the approximation to ͩ0;41 = 6.403124237.
• The Bakhshali manuscript also uses the formula to compute W30;105 giving 10.24695122 as the approximation to W30;105 = 10.24695077.
• Correct answer is 22.068076490713 .
• Bakhshali formula gives 29.816105242176 .
• Correct answer is 29.8161030317511 .
• If we took 889 = 302 - 11 instead of 292 + 48 we would get .
• Bakhshali formula gives 29.816103037078 .
• Correct answer is 29.8161030317511 .
• 11 (2) (1976), 112-124.','6] derives from the Bakhshali square root formula an iterative scheme for approximating square roots.
• He finds in [',' M N Channabasappa, The Bakhshali square-root formula and high speed computation, Ganita Bharati 1 (3-4) (1979), 25-27.','7] that it is 38% faster than Newton's method in giving ͩ0;41 to ten places of decimals.

13. Babylonian numerals
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 1 5; 104 + 2 5; 103 + 3 5; 102 + 4 5; 10 + 5.
• If one thinks about it this is perhaps illogical for we read from left to right so when we read the first digit we do not know its value until we have read the complete number to find out how many powers of 10 are associated with this first place.
• The Babylonian sexagesimal positional system places numbers with the same convention, so the right most position is for the units up to 59, the position one to the left is for 60 5; n where 1 ≤ n ≤ 59, etc.
• 1 5; 603 + 57 5; 602 + 46 5; 60 + 40 .
• Here is 1,57,46,40 in Babylonian numerals .
• Here is an example from a cuneiform tablet (actually AO 17264 in the Louvre collection in Paris) in which the calculation to square 147 is carried out.
• For example if we write 0.125 then this is 1/10 + 2/100 + 5/1000 = 1/8.
• So 1/3 has no finite decimal fraction.
• Similarly the Babylonian sexagesimal fraction 0;7,30 represented 7/60 + 30/3600 which again written in our notation is 1/8.
• To illustrate 10,12,5;1,52,30 represents the number .
• 10 5; 602 + 12 5; 60 + 5 + 1/60 + 52/602 + 30/603 .
• which in our notation is 36725 1/32.
• If I write 10,12,5,1,52,30 without having a notation for the "sexagesimal point" then it could mean any of: .
• 0;10,12, 5, 1,52,30 .
• 10;12, 5, 1,52,30 .
• 10,12; 5, 1,52,30 .
• 10,12, 5; 1,52,30 .
• 10,12, 5, 1;52,30 .
• 10,12, 5, 1,52;30 .
• 10,12, 5, 1,52,30 .
• in addition, of course, to 10, 12, 5, 1, 52, 30, 0 or 0 ; 0, 10, 12, 5, 1, 52, 30 etc.
• Theon's answer was that 60 was the smallest number divisible by 1, 2, 3, 4, and 5 so the number of divisors was maximised.
• A base of 12 would seem a more likely candidate if this were the reason, yet no major civilisation seems to have come up with that base.
• Now an angle of an equilateral triangle is 6076; so if this were divided into 10, an angle of 676; would become the basic angular unit.
• The reason has to involve the way that counting arose in the Sumerian civilisation, just as 10 became a base in other civilisations who began counting on their fingers, and twenty became a base for those who counted on both their fingers and toes.
• This gives a way of finger counting up to 60 rather than to 10.
• Although 5 is nothing like as common as 10 as a number base among ancient peoples, it is not uncommon and is clearly used by people who counted on the fingers of one hand and then started again.

14. Indian Sulbasutras
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• For example (5, 12, 13), (12, 16, 20), (8, 15, 17), (15, 20, 25), (12, 35, 37), (15, 36, 39), (5/2 , 6, 13/2), and (15/2 , 10, 25/2) all occur.
• It is an approximate method based on constructing a square of side 13/15 times the diameter of the given circle as in the diagram on the right.
• This corresponds to taking π = 4 5; (13/15)2 = 676/225 = 3.00444 so it is not a very good approximation and certainly not as good as was known earlier to the Babylonians.
• For example in the Baudhayana Sulbasutra, as well as the value of 676/225, there appears 900/289 and 1156/361.
• In different Sulbasutras the values 2.99, 3.00, 3.004, 3.029, 3.047, 3.088, 3.1141, 3.16049 and 3.2022 can all be found; see [',' R P Kulkarni, The value of π known to Sulbasutrakaras, Indian J.
• 13 (1) (1978), 32-41.','6].
• In [',' R C Gupta, New Indian values of π from the Manava sulba sutra, Centaurus 31 (2) (1988), 114-125.','3] the value π = 25/8 = 3.125 is found in the Manava Sulbasutras.
• In [',' A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 &#8217;&#8217;Nauka&#8217;&#8217; (Moscow, 1974), 220-222; 302.','9] in addition to examining the problem of squaring the circle as given by Apastamba, the authors examine the problem of dividing a segment into seven equal parts which occurs in the same Sulbasutra.
• √2 = 1 + 1/3 + 1/(3 5; 4) - 1/(3 5; 4 5; 34) = 577/408 .
• which is, to nine places, 1.414215686.
• Compare the correct value √2 = 1.414213562 to see that the Apastamba Sulbasutra has the answer correct to five decimal places.
• Datta, in 1932, made a beautiful suggestion as to how this approximation may have been reached.
• In [',' B Datta, The science of the Sulba (Calcutta, 1932).','1] Datta considers a diagram similar to the one on the right.
• 1 + 1/3 + 1/(3 5; 4) .
• Now Datta argues in [',' B Datta, The science of the Sulba (Calcutta, 1932).','1] that to improve the "not quite a square" the Sulbasutra authors could have calculated how broad a strip one needs to cut off the left hand side and bottom to fill in the missing part which has area (1/12)2.
• 2 5; x 5; (1 + 1/3 + 1/12) = (1/12)2.
• This has the solution x = 1/(3 5; 4 5; 34) which is approximately 0.002450980392.
• 1 + 1/3 + 1/(3 5; 4) - 1/(3 5; 4 5; 34) .
• 2 5; x 5; (1 + 1/3 + 1/12) - x2 = (1/12)2 .
• for x which leads to x = 17/12 - √2 which is approximately equal to 0.002453105.
• In [',' R C Gupta, Baudhayana&#8217;s value of √2, Math.
• Education 6 (1972), B77-B79.','4] Gupta gives a simpler way of obtaining the approximation for √2 than that given by Datta in [',' B Datta, The science of the Sulba (Calcutta, 1932).','1].

15. Mental arithmetic
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• [Wallis] occupied himself in finding (mentally) the integral part of the square root of 3 5; 1040; and several hours afterwards wrote down the result from memory.
• This is A C Aitken, see [',' I M L Hunter, An exceptional talent for calculative thinking, British Journal of Psychology 53 (3) (1962), 243-258.','9] , [',' A C Aitken, The art of mental calculation: with demonstrations, Trans.
• Royal Society for Engineers, London 44 (1954), 295-309.','1] and [',' P C Fenton, To catch the spirit: The memoir of A C Aitken (Otago, 1995).','7], and we shall consider his methods later in this article.
• Zerah Colburn's powers in calculating are described in [',' W W Rouse Ball, Mathematical Recreations and Essays (London, 1940).','2], [',' F D Mitchell, Mathematical prodigies, American Journal of Psychology 18 (1907), 61-143.','12] and [',' E W Scripture, Mathematical prodigies, American Journal of Psychology 4 (1891), 1-59.','13].
• in 1804 and he visited Europe in 1812, when only eight years old to demonstrate his skills:- .
• Among questions asked him at this time were to raise 8 to the 16th power; in a few seconds he gave the answer 281,474,976,710,656 which is correct.
• Asked for the factors of 247,483 he replied 941 and 263; asked for the factors of 171,395 he gave 5, 7, 59 and 83, asked for the factors of 36,083 he said there were none.
• Asked for the square of 4,395 he hesitated but on the question being repeated he gave the correct answer, namely 19,316,025.
• Questioned as to the cause of his hesitation, he said he did not like to multiply four figures by four figures, but he said 'I found out another way; I multiplied 293 by 293 and then multiplied this product twice by 15'.
• On another occasion when asked for the product of 21,734 by 543 he immediately replied 11,801,562; and being questioned explained that he had arrived at this by multiplying 65,202 by 181.
• George Parker Bidder was born in 1806 at Moreton Hampstead in Devonshire, England.
• He was not one to lose his skills when educated and wrote an interesting account of his powers in [',' G P Bidder, On mental calculation, Minutes of Proceedings, Institution of Civil Engineers 15 (1856), 251-280.','3].
• One of Bidder's sons was able to multiply two numbers of 15 digits but he was slow, and less accurate, compared with his father.
• Bidder wrote, see [',' G P Bidder, On mental calculation, Minutes of Proceedings, Institution of Civil Engineers 15 (1856), 251-280.','3]:- .
• As examples of Dase's calculating ability here is an example: 79532853 5; 93758479 = 7456879327810587, time taken 54 seconds.
• Had he performed on stage he might well have brought the crowds in according to the description given by H W Adams, see [',' F D Mitchell, Mathematical prodigies, American Journal of Psychology 18 (1907), 61-143.','12], [',' F D Mitchell, Mathematical prodigies, American Journal of Psychology 18 (1907), 61-143.','12]:- .
• He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until, in not more than one minute, said he 133,491,850,208,566,925,016,658,299,941,583,225! .
• Only at the age of 15 did I feel I might develop a real power and for some years about that time, without telling anyone, I practised mental calculation from memory like a Brahmin Yogi, a little extra here, a little extra there, until gradually what had been difficult at first became easier and easier..
• To give just one example, see [',' J Fauvel and P Gerdes, African slave and calculating prodigy : bicentenary of the death of Thomas Fuller, Historia Mathematica 17 (2) (1990), 141-151.','6]:- .
• Royal Society for Engineers, London 44 (1954), 295-309.','1] and by an Edinburgh psychologist in [',' I M L Hunter, An exceptional talent for calculative thinking, British Journal of Psychology 53 (3) (1962), 243-258.','9].
• In [',' I M L Hunter, An exceptional talent for calculative thinking, British Journal of Psychology 53 (3) (1962), 243-258.','9] Hunter describes Aitken reciting the first 1000 digits of π to him:- .
• The total time taken is 150 sec.
• In 1873, Shanks carried this to 707 decimals; but it was not until 1948 that it was discovered that the last 180 of these were wrong.
• Now, in 1927 I had memorised those 707 digits for an informal demonstration to a students society, and naturally I was rather chagrined, in 1948, to find that I had memorised something erroneous.
• When π was calculated to 1000 and indeed more decimals, I re-memorised it.
• 1961 = 37 5; 53 = 442 + 52 = 402 + 192 .
• For example if asked for the decimal expansion of 1/851 he would think of 851 as 23 5; 37, if asked for the square root of 851 then he thought of it as 292 + 10, if asked for the decimal expansion of 17/851 then he would think of it as almost 0.02.
• If I go for a walk and if a motor car passes and it has the registration number 731, I cannot but observe it is 17 times 43.
• [Given a number] is it a prime of the form 4n+1, and so expressible as the sum of two squares in one way only? Is it the numerator of a Bernoullian number, or one occurring in some continued fraction? And so on.
• On asked to compute the decimal expansion of 1/697 he explained his method.
• He saw immediately that 697 = 17 5; 41.
• I mentally worked out a 41st and divided it by 17 at the same time.
• You've got to alternate back and forward I've got to be aware that a 41st is point, 0, 2, 4, 3, 9, and be dividing by 17 along.
• He said, see [',' I M L Hunter, An exceptional talent for calculative thinking, British Journal of Psychology 53 (3) (1962), 243-258.','9]:- .
• One can divide by a number like 59, or 79, or 109, or 599, and so on, by short division.
• 6 ) 1.0 1 6 9 4 9 1 5 2..
• 0.0 1 6 9 4 9 1 5 2 5..
• Write it as 15/69.
• 7 ) 15.2 1 7 3 9 1 3 0..
• 0.2 1 7 3 9 1 3 0 4..
• In fact 5/23 = 0.2173913043478260869565, a recurring decimal with a period of 22 digits.
• There are other possibilities: For example, the mental calculator is, or should be, very familiar with the factorisation of numbers; he should know not merely that 23 time 13 is 299, but that 23 times 87 is 2001.
• For example 5/23 is equal to 435/2001; and if we note that 435 is the same as 434.999999999..
• 217 391 304 347 ..
• Even at the moment of registering 56088, I have checked it by dividing it by 8, so 7011, and this by 9 gives 779.
• I recognise 779 as 41 by 19.
• And 41 by 3 is 123, while 19 by 24 is 456.
• When Hunter interviewed Aitken in 1961 he had before him a record of the 1930's test and he asked Aitken if he remembered being asked to recite a random sequence of words.

16. Fair book insert
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In the archive of the Madras College, St Andrews, there is a jotter of a schoolboy who studied at the College in 1852.
• What is the area of a rhomboid, of which the base is 640 links, the slope end 436 links, and the inclined angle 4776;20'.
• The antilog of 0.313136 is required.
• The nearest value from the tables is 0.311966 with antilog 2.051.
• The difference 0.313136-0.311966 gives 170.
• Adding a 0 gives 1700 which is then divided by 212 to obtain 80 to give the corrected antilog of 0.313136 as 2.05180.
• This is then divided by 100,000 to get acres.
• The decimal fraction is then multiplied by 4 to obtain roods (4 roods to 1 acre).
• Then the decimal fraction is multiplied by 301/4 to obtain square yards (301/4 sq yds to a sq pole).
• What is the area of a triangle, of which the base is 1254 links, and the perpendicular height 847 links.
• Divide by 100000 to get 5 acres.
• Finally multiply .70880 by 30 1/4 to obtain 21.44120 to get 21.44 yds [again square yards and yards are not distinguished.] .
• Answer is 5 a 1 r 9 p 21.44 yds.
• Two of the sides of a triangle are 621 and 842 links respectively, and the included angle 2776;19'; what is its area.
• If two of the sides of a triangle are 1580 and 1228 links, and the included angle 2776;19'; what is its area.
• Answer is worked out as 1 a 1 r 28 p 26.6 yds.
• What is the area of a triangle, of which the three sides are 428, 614, and 517 feet.
• When the four sides of a trapezium inscribed in a circle are 800, 610, 508, and 420 feet; what is the area of the trapezium.
• What is the area of the trapezium ABCD the diagonal BD being 896 and the perpendiculars AM, CN 437, 351 links.
• The area is calculated as (437 + 351)/2 times 896 using six figure logs and converted to acres, roods, sq poles and sq yds.
• Let M denote Nelson's Monument on the Calton Hill of Edinburgh, L the tower at the end of the quay at Leith, I the island of Inchkeith, P the pier at Pettycur, and B the school at Burntisland: also let PB = 10110 feet, the angles PBL = 2976;56', IBL = 4076;27', LBM = 876;24', MPB = 8676;52', LBM = 476;35' and IPL = 3776;52'.
• A modern value, for comparison, would be 41851509 ft (at least that is the equatorial value).
• Method is as the previous question, but the answer left in feet as 15832.
• The answer is given as 11.30 miles.
• The velocity of sound is taken as 114.2 ft per sec.
• (Today the value is 108.7 ft per sec.) The answer is converted to miles (by subtracting the log of 5280) to get 5.4072 miles.
• 5280 is multiplied by 8 before logs are used to divide by 114.2.
• The Area of a Survey is 48 Acres 3 Roods 10 Poles and the area of a similar Plan constructed to a Scale of 3 Chains, on 12 Poles to the inch is:_ 12 Acres 0 Roods 32 1/2 Poles; required the scale to which the Plan was constructed.
• First both areas are converted to Poles, that of the Survey being 7810 Poles and that of the similar Plan is 1952 1/2 Poles.
• 1952 1/2 : 7810 : : 9 : x.
• Hence the Plan was laid down to a Scale of 6 Chains, on 24 Poles to 1 Inch." .
• It is also interesting that Walker is inconsistent in handling powers of 10.
• Here he uses log .07957747 = 8.900791 but in the next two problems uses log .07957747 = (bar 2).900791.
• The answer is given in square miles 49164612.
• He obtains the answer 22855.1.
• The first is used (with six figure logs) to give A = 3276;58 (this answer is in fact wrong since log 252 is used instead of log 256 in the calculation).
• 5776;2 is obtained.
• Sine 3276;5 : 252 : : Sine 5776;2 : AB .
• Walker uses the same method as the previous question to obtain A = 4076;49, C = 4976;11 and AB = 393.5.
• Solution given using methods as above is correct hypotenuse is 693.1 and the angles are 4376;50 and 4676;10.
• The hypotenuse of a right angled triangle is 325, and the angle adjacent to the base 4276;36 find the other acute angle and sides.
• First calculates the other angle as 4776;24 then uses the same type of proportional equations as before to calculate the sides: .
• Sine 90 : 325 : : Sine 4276;36 : AB .
• Sine 90 : 325 : : Sine 4776;24 : CB.
• As before powers of 10 are inserted in a way that is understood, so log (1/61) = 8.214670.
• 61 .
• 2|159 .
• log 79.5 = 1.900367 .
• log 34.5 = 1.537819 .
• log(1/61)= 8.214670 .
• 2|19.928580 .
• 2276;56 9.964290 .
• 4576;52 .
• What is the area of the field ABCDEF, AB being = 640, Aa = 400, FA = 260, Ab = 1020, bC = 200, AD = 1220, Dc =250, cC = 190 and DB = 1130 links.
• What is the surface of a cone its axis being 16 ft and the diameter of its base 2 ft 10 in.
• He then divides by 144 to obtain 6.305 sq ft.
• Next he computes 34 times π and divides by 24 (12 to convert to feet, and the 2 from the formula).
• He then multiplies the answer by 16 to obtain 71.20.
• He first computes the area of the base by squaring the length of the side of the regular pentagon and multiplying by 1.7204774.
• He makes an error when adding the two parts of the sum 28.27 + 15.70 obtaining 53.97 instead of 43.97.
• He converts 16 ft 9 in to 16.75 ft and multiples 53.97 by this, ten divides by 2.

17. The number e
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This was in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers.
• A few years later, in 1624, again e almost made it into the mathematical literature, but not quite.
• In 1647 Saint-Vincent computed the area under a rectangular hyperbola.
• Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm.
• Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1.
• Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places.
• In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x).
• In 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of (1 + 1/n)n as n tends to infinity.
• In 1684 he certainly recognised the connection between logarithms and exponents, but he may not have been the first.
• As far as we know the first time the number e appears in its own right is in 1690.
• It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium &#9417;.
• Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731.
• He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum &#9417; that he gave a full treatment of the ideas surrounding e.
• e = 1 + 1/1! + 1/2! + 1/3! + ..
• and that e is the limit of (1 + 1/n)n as n tends to infinity.
• Euler gave an approximation for e to 18 decimal places, .
• e = 2.718281828459045235 .
• In fact taking about 20 terms of 1 + 1/1! + 1/2! + 1/3! + ..
• For, if the continued fraction for (e - 1)/2 were to follow the pattern shown in the first few terms, 6, 10, 14, 18, 22, 26, ..
• There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in 1854.
• In fact one needs about 120 terms of 1 + 1/1! + 1/2! + 1/3! + ..
• In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula i-i = √(eπ).
• Certainly it was Hermite who proved that e is not an algebraic number in 1873.
• In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different.
• In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.
• Anyone wishing to see e to 10,000 places - look here.
• JOC/EFR September 2001 .

18. Real numbers 3
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Were the real numbers consistent? Would an inconsistency appear one day and much of the mathematical building come tumbling down? Some of the intuitive difficulties that began to be felt revolved around the fact that the real numbers were not countable, that is, they could not be put in 1-1 correspondence with the natural numbers.
• Cantor proved that the real numbers were not countable in 1874.
• He produced his famous "diagonal argument" in 1890 which gave a second, more striking, proof that the real numbers were not countable.
• L = {n1 , n2 , n3 , n4 , ..
• , 51 becomes Z, then code all the punctuation marks, and then make all the remaining numbers up to 99 translate to an empty space.
• c = 0.01020304050607080910111213141516171819202122232425..
• Then continue with the 10000 pairs of 2-blocks 0000, 0001, 0002, ..
• ., 0099, 0100, 0101, ..
• Let us use c to describe a paradox which was discovered in 1905.
• This will enable us to explain Richard's paradox, discovered by Jules Richard in 1905.
• If the n-th block of c does not describe a real number (most of course will not even be meaningful in English) then set the n-th digit of r(n) to be 1.
• Emile Borel introduced the concept of a normal real number in 1909.
• Then if it is a "random" number the digit 1 should occur about 1/10 of the time so, if we denote by N(1,n) the number of times 1 occurs in the first n decimal digits, then N(1,n)/n should tend to 1/10 as n tends to infinity.
• Similarly for the all digits i in the set {0, 1, 2, ..
• ., 9) we should have N(i,n)/n tending to 1/10 as n tends to infinity.
• There were still an uncountable number of non-normal numbers, however, which was easily seen by taking the subset of all real numbers with no digit equal to 1.
• This was achieved first by Sierpinski in 1917.
• Three major paradoxes were due to Burali-Forti in 1897, Russell in 1902, and Richard in 1905.
• Poincare (1908) and Weyl (1918) complained that analysis had to be based on a concept of the real numbers which eliminated the non-constructive features.
• Godel proved some striking theorem in 1930.
• In 1936 Gentzen proved arithmetic consistent, but only by using transfinite methods which were less accepted than arithmetic itself.
• In 1933 David Champernowne, who was an undergraduate at Cambridge University and a friend of Alan Turing, devised Champernowne's number.
• ., 9, 10, 11, ..
• 0.12345678910111213141516171819202122232425262728293031323334353637383940 ..
• In The Construction of Decimals Normal in the Scale of Ten published in the Journal of the London Mathematical Society in 1933, Champernowne proved that his number was normal in base 10.
• In 1937 Mahler proved that Champernowne's number was transcendental.
• 0.p(1)p(2)p(3)..
• In 1946 Copeland and Erdős proved that the number .
• 0.2357111317192329313741434753596167717379838997101103107109113127131137139 ..
• In fact although no irrational algebraic number has yet been proved to be absolutely normal nevertheless it was conjectured in 2001 that this is the case.
• In 1927 Borel came up with his "know-it-all" number.
• If the n-th block of c translates into a true/false question then set the n-th digit of k to be 1 if the answer to the question is true, and 2 if the answer is false.
• Borel devotes a whole book [',' E Borel, Les nombre inaccessible (Gauthier-Villiars, Paris, 1952).','1], which he published in 1952, to discuss another idea, namely that of an "inaccessible number".
• In 1936 Turing published a paper called On computable numbers.
• If the n-th block is not a valid programme to output a real number, then make the n-th digit of t equal to 1.

19. Real numbers 1
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Pythagoras seems to have thought that "All is number"; so what was a number to Pythagoras? It seems clear that Pythagoras would have thought of 1, 2, 3, 4, ..
• , W30;17 were not commensurable with 1:- .
• Theodorus was writing out for us something about roots, such as the sides of squares whose area was 3 or 5 units, showing that the sides are incommensurable with the unit: he took the examples up to 17, but there for some reason he stopped.
• Heimonen, in [',' I Grattan-Guinness, Numbers, ratios, and proportions in Euclid&#8217;s Elements : How did he handle them?, Historia Mathematica 23 (1996), 355-375.','10], looks at the views of different historians concerning the discovery of the irrational numbers:- .
• 27">Knorr set out a new theory, trying especially to explain better why Theodoros stopped just at the square root of 17.
• We should note that Euclid never identified the ratio 2 : 1 with the number 2.
• , W30;17 are incommensurable with a segment of unit length.
• Stifel, in his Arithmetica Integra &#9417; (1544) argues that irrationals must be considered valid:- .
• A major advance was made by Stevin in 1585 in La Theinde &#9417; when he introduced decimal fractions.
• He argued strongly in L'Arithmetique &#9417; (1585) that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature.
• One further comment by Stevin in L'Arithmetique &#9417; is worth recording.

20. Planetary motion
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This situation is currently approximated in textbooks by the two-body problem, which itself became amenable to analysis only after the work of Newton (1642-1727) [','Isaac Newton: The Mathematical Principles of Natural Philosophy, London 1687.','1].
• (The theory was developed first in terms of circles based on the heliocentric configuration invented by Copernicus (1473-1543) [','Nicolaus Copernicus: On the Revolutions of the Heavenly Spheres, Nurnberg 1543.','2].) In this case, the motion of each individual planet occurs in isolation, entirely unaffected by any other member of the system.
• Moreover the topic is of great historical significance - since the discovery of the two laws stated above actually took place during the period 1600-1630 [','The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609.
• They were validated in his later work: Epitome of Copernican Astronomy, Book V, Frankfurt 1621.','3], under the kinematical circumstances described above: see Kepler's Planetary Laws.
• We start from what was almost certainly the earliest definition of an ellipse (because it can be derived from the plane section of a cone in three easy steps, as set out in [','A E L Davis: &#8217;Some plane geometry from a cone ..
• &#8217;Mathematical Gazette, forthcoming, July 2007.','4]).
• Now from [6;QHB, .
• Then two geometrical equivalences can be derived from [6;APH, again shown in the figure: .
• Applying Pythagoras' theorem to [6;APH, we derive: .
• r2 =a2(1 - e2)sin2 β + a2(cos2 β + 2e cos β + e2) .
• = a2(1 + 2e cos β + e2cos2β).
• r = AP = a(1 + e cos β).
• = a2(1 + 2e cos β + e2).
• AQ = a(1 + 2e cos β + e2)89; .
• AQ = a(1 + e cos β + 1/2 e2sin2 β + ..
• The characteristic property of orbital motion in its most general form is generally stated dynamically, but it was in fact first proved as a kinematical relation in Book I, Prop.1 of Newton's work, already cited [','Isaac Newton: The Mathematical Principles of Natural Philosophy, London 1687.','1] (at that early stage, the concept of mass had not yet been introduced).
• r dθ/dt = 2π ab/T 5; 1/r.nnnnnnnnnnnn(10) .
• dβ/dt = 2π a/T 5; 1/r.nnnnnnnnnnnn(12) .
• dt/dβ = T/2π 5; r/a = T/2π; (1 + e cos β)nnnnfrom (5).
• r = a(1 + e cos β).
• Radial acceleration = (2π)2 a3/T2 5; 1/r2 towards the Sun.
• a3/T2 = 1/(2π)2.h2/l.
• Bernard Cohen: The Birth of a New Physics, Norton 1985 (up-dated), p.166.','5] (who presumably chose the notation to commemorate the discoverer of this relationship) to write the result: .
• (It was proved, in a dynamical context, in Book I, Prop.15 of Newton's work, already cited.) However, it is possible to formulate a rational basis for the above deduction, founded on geometry - and so to produce a theoretical proof of Law III which would have been not so far beyond the conceptual understanding of a pre-Newtonian mathematician [','A E L Davis: &#8217;Kepler&#8217;s potential proof of his Third Law&#8217; in Miscellanea Kepleriana, ed.

21. Kepler's Laws
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The greatest achievement of Kepler (1571-1630) was his discovery of the laws of planetary motion.
• These are found in Astronomia Nova &#9417; 1609, underpinned by important work in Epitome &#9417; Book V (1621).
• Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova &#9417; Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
• Kepler originally investigated the orbit of Mars because that was the task allocated to him by Tycho Brahe (1546-1601), when Kepler joined him in Prague around 1600.
• Tycho had amassed a vast store of observations extending over 30 years; these are probably the most accurate that would ever be made with the naked eye, since Galileo (1564-1642) had introduced the telescope into astronomy soon afterwards (in 1610).
• Geocentricity was obviously in accordance with the evidence of the senses - as well as being the only arrangement acceptable to the Church - by contrast with the heliocentric configuration, which was proposed by Copernicus (1473-1543).
• Kepler was introduced to Copernicanism as a student at the University of Tubingen by his teacher, Michael Maestlin (1550-1631).
• In the earlier chapters of Astronomia Nova &#9417; Kepler embarked on a programme of 'reducing the observations' (this term means removing, as far as possible, all effects due to the observer's position in time and space).
• He found the heavy calculations, and the necessary checking, 'mechanical and tedious', as he remarked later: he did not have the benefit of logarithms, which were not invented until 1614, by Napier (1550-1617).
• (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome &#9417; Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
• 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
• Sometime in the years 1594-1604, Kepler studied the Conics of Apollonius, and expressed great admiration for it, citing it throughout his optical and stereometrical work - yet he never referred to any of its propositions in connection with his astronomy.
• The stages are classified according to the chapters of Astronomia Nova &#9417; in whch each construction appears: .
• Even when nothing more is known about these curves one can imagine applying a mathematical 'grading-machine' to sort them by width, to produce BF', BF'', BF, respectively, as shown in Figure (2) (constructed as special cases by the same procedure, taking β = 9076;).
• There is good reason to believe that this was the earliest plane definition of an ellipse, (because it can be derived directly from a section of a cone in three easy steps [','A E L Davis, &#8217;Some plane geometry from a cone: the focal distance of an ellipse at a glance&#8217;, Mathematical Gazette, forthcoming July 2007.','1]), as well as the definition most commonly, if not exclusively, used by Kepler's contemporaries: it is just the ratio-property of the ordinates.
• Kepler had already invented the term 'focus' in Astronomiae Pars Optica &#9417; (1604) in connection with his work on vision, though he did not realize its connection with his astronomy at that juncture - in Astronomia Nova &#9417; he simply referred to the point A as punctum eccentricum, or eccentric point.
• In Astronomia Nova &#9417; Ch.
• Because of its importance the proof has been reproduced more than once [','A E L Davis, &#8217;Kepler&#8217;s Road to Damascus&#8217;, Centaurus 35 (1992) pp.156-157;','2]; though modernized in style, and reordered - to increase its impact - clearly it has not been altered in substance, since it relies on nothing more 'advanced' than Pythagoras' Theorem, and properties of similar triangles.
• Because of its importance the proof has been reproduced several times [','A E L Davis, &#8217;Kepler&#8217;s Road to Damascus&#8217;, Centaurus 35 (1992) pp.156-157;','2]; though modernized in style, and reordered - to increase its impact - clearly it has not been altered in substance, since it relies on nothing more 'advanced' than Pythagoras' Theorem, and properties of similar triangles.
• Kepler's practical problem in Astronomia Nova &#9417;, however, was to discover a way of measuring the time taken to reach the typical position (P) of the planet at an intermediate point of the orbit.
• Eccentric sector QAC = circle sector QBC + [6;QBA = segment QHC + [6;QHA.
• [6;PHA / [6;QHA = b/a.
• Ellipse sector PAC = ellipse segment PHC + [6;PHA = b/a (segment QHC + [6;QHA).
• So, at the end of Astronomia Nova &#9417;, Kepler had discovered that, when the planet's path was an ellipse, time in orbit appeared to be precisely proportional to the area swept out.
• Fortunately, the apparent irreconcilability of the two representations motivated him eventually to come to a more precise understanding of the mathematics of an elliptic orbit: see [','A E L Davis, Kepler&#8217;s unintentional ellipse - a celestial detective story, Mathematical Gazette 82, no.493 (1998), p.42;','3].) .
• Area of circle sector QAC = 1/2 a2.β + 1/2 a2e sinβ, .
• Area of ellipse sector PAC = 1/2 ab.β + 1/2 abe sinβ.
• He went on to tackle the problem of the motions of planets, and their causes, in Epitome Astronomiae Copernicanae &#9417;.
• This consists of seven books, though only Book V (1621), supported in places by Book IV (1620), contains the really innovative work.
• And further confirmation of the motions is given in Harmonice Mundi &#9417;, 1618 Book V, Chapter 3, where there are some quantified references to a single planet, in addition to the main discussion which involves the planetary system.
• In De Caelo &#9417; I, 3, Aristotle had declared that there were only two simple motions, circular and linear.
• Accordingly, on this authority, Kepler was able to match each one of the pair of results (the curve, and the independently-determined representation of time) that he had discovered in Astronomia Nova &#9417;, to one of these mutually perpendicular components of motion.
• He suggested this a few years before the rotation was actually established, c.1610, from observations of sunspots by Galileo, Scheiner (a minor contemporary astronomer), and independently by Harriot (c.1560-1621).
• He selected magnetism, having come across a recently-published book, De Magnete &#9417;, 1600 by Gilbert (1544-1603), which stated that the Earth should be regarded as a giant magnet.
• (Unfortunately Kepler's successors have often failed to distinguish these causes clearly, because they have only considered his early opinions in Astronomia Nova &#9417;, when he had not yet sorted things out.) Of course Kepler's final views on causes were entirely wrong in modern eyes, but they were eminently sensible: one traditional, the other making use of the most recent knowledge of the day.
• It was not until 1687 that Newton (1642-1727) gave a quantified definition of mass [','A E L Davis: &#8217;The Mathematics of the Area Law: Kepler&#8217;s successful proof in Epitome Astronomiae Copernicanae (1621)&#8217;, Archive for History of Exact Sciences 57, 5 (2003), especially Section 6.','4].
• t ∝ β + e sinβCoordinates(in terms of β)r = a(1 + e cosβ) .
• δt/δβ ∝ r or δβ/δt ͩ3; 1/rComponentsof motionδr/δβ = -ae sinβ .
• 'Impulsion' of Sun's rays ͩ3; 1/r Kepleriancauses[Strength of planet's fibres ∝ e] .

22. Word problems
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The relevant paper was published in 1882 with a second paper in the following year.
• The 1882 paper Gruppentheoret5ische Studien &#9417; contains the first appearance of a group presentation as we know it.
• He knew intuitively how to tell whether a word in the free group is trivial - make all cancellations of aa-1 and a-1a for every generator a appearing in the word.
• Heinrich Tietze published the paper Uber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten &#9417; in 1908.
• The topological part of this paper was based on the notion of the fundamental group introduced by Poincare in 1895.
• In 1910 Max Dehn published Uber die Topologie des dreidimensionalen Raumes &#9417;.
• He made the group theory problems explicit in his 1911 paper Uber unendliche diskontinuierliche Gruppen &#9417;.
• Dehn also proved in this 1911 paper that a finitely presented group can have a subgroup which is not finitely presented.
• In 1912 Dehn studied the word problem and the conjugacy problem for the fundamental groups of orientable closed 2-dimensional manifolds.
• He showed in Transformationen der Kurven auf zweiseitigen Flachen &#9417; (1912) that in special cases one could solve the word problem using a direct approach - called Dehn's algorithm today.
• In this case one could construct a finite list of words in the group generators, u1 , v1 , u2 , v2 , ..
• Dehn solved the word problem for the trefoil knot group in 1914.
• We note, as an aside, that Waldhausen solved the word problem for all knot groups in 1968.
• In 1921 Jakob Nielsen published Om Regning med ikke kommutative Faktoren og dens Anvendelse in Gruppeteorien &#9417;.
• In this paper he showed how, given a set of words { u1 , u2 , ..
• He also gave a method for solving the membership problem in free groups, that is an algorithm to determine whether a given word w in a free group was contained in the subgroup generated by { u1 , u2 , ..
• Membership Problem: Let G = < X | R > be a finite presentation, and U a finite set of words { u1 , u2 , ..
• The Membership Problem is also called the generalised word problem when { u1 , u2 , ..
• He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
• Nielsen returned to the problem of a free group of countable rank in 1955.
• If K is determined by a given infinite set S = { a1 , a2 , ..
• } of elements generating K, then there exists a function L(n) such that all elements of K of length not exceeding n are contained in the subgroup KL generated by the subset { a1 , a2 , ..
• In 1926 Emil Artin solved the word problem for braid groups.
• Let us now look at 1-relator groups.
• Magnus solved the problem that Dehn had given him in 1930.
• In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word problem for this class of groups.
• In 1933 Church produced what today is called "Church's thesis" which states that l-definability is the precise notion of "computable function".
• In 1936 Alan Turing produced the notion of a Turing machine and proposed that a "computable function" was one which could be computed by a Turing machine.
• In 1938 Church proposed that the word problem for groups should be proved insoluble using the new rigorous definitions of computable.
• In 1946 and 1947 the first breakthrough was made by Emil Post.
• Post Correspondence Problem: Given an alphabet A = { a1 , a2 , ..
• He succeeded, and published a proof that the word problem for semigroups was insoluble in 1947.
• In 1954 Bill Boone published a paper in which he defined the quasi-Magnus problem and proved it insoluble.
• dissertation of 1952.
• By 1956 Boone had proved that the word problem for groups was insoluble.
• However, Petr Sergeevich Novikov had been working on the problem and announced in 1952 that he had a proof of the insolubility of the word problem for groups.
• He died in 1952 shortly before Novikov announced his results.
• Graham Higman, Bernhard Neumann and Hanna Neumann introduced what is now called an HNN extension in 1949.
• Higman proved the Higman embedding theorem in 1961: .
• Take H(S) = < a, b, c, d | a-ibai = c-idci, i &#8712; S > where S is a recursively enumerable set which is not recursive.
• Then H(S) is the free product of the free group < a, b > with the free group < c, d > with the subgroups < a-ibai | i in S > and < c-idci | i &#8712; S > amalgamated.
• Sergei Ivanovich Adian, a student of Petr Sergeevich Novikov, proved a whole range of decision problems in 1957 and 1958 based on the idea of a Markov property.
• Adian proved the following theorem in 1957: .
• In 1959 Gilbert Baumslag, Bill Boone and Bernhard Neumann proved the following theorem: .
• Finally let us note that Mihailova introduced the membership problem in 1958 and proved: .
• Miller, in 1971, took the study of G = F 5; F further proving that one cannot decide whether L = G.

23. Brachistochrone problem
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The brachistochrone problem was posed by Johann Bernoulli in Acta Eruditorum &#9417; in June 1696.
• Galileo in 1638 had studied the problem in his famous work Discourse on two new sciences.
• Returning to Johann Bernoulli he stated the problem in Acta Eruditorum &#9417; and, although knowing how to solve it himself, he challenged others to solve it.
• He was President of the Royal Society during the years 1695 to 1698 so it was natural that Newton send him his solution to the brachistochrone problem.
• The May 1697 publication of Acta Eruditorum &#9417; contained Leibniz's solution to the brachistochrone problem on page 205, Johann Bernoulli's solution on pages 206 to 211, Jacob Bernoulli's solution on pages 211 to 214, and a Latin translation of Newton's solution on page 223.
• Band 2 : Der Briefwechsel mit Pierre Varignon : Erster Teil : 1692-1702, Die Gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli (Basel, 1988).','1].
• Use y' = dy/dx = cot α and sin2α = 1/(1 + cot2α) = 1/(1 + y'2) to get .
• for a constant h (= 1/2 k2)).
• The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this equation.
• y(1+y'2) = h(1 - cos t)(1+sin2t/(1-cos t)2) .
• = h(1 - cos t + sin2t/(1-cos t)) .
• = h((1 - cos t)2+ sin2t)/(1 - cos t) .
• Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem, namely that of finding the curve for which the time taken by a particle sliding down the curve under uniform gravity to its lowest point is independent of its starting point.
• Despite the friendly words with which Johann Bernoulli described his brother Jacob Bernoulli's solution to the brachistochrone problem (see above), a serious argument erupted between the brothers after the May 1697 publication of Acta Eruditorum &#9417;.
• Band 2 : Der Briefwechsel mit Pierre Varignon : Erster Teil : 1692-1702, Die Gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli (Basel, 1988).','1].
• 27 (1997), 257-276.','10] where as well as the mathematical details the author studies the psychological side.
• The methods which the brothers developed to solve the challenge problems they were tossing at each other were put in a general setting by Euler in Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti &#9417; published in 1744.
• Lagrange, in 1760, published Essay on a new method of determining the maxima and minima of indefinite integral formulas.

24. Measurement
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• There were 28 digits in a cubit, 4 digits in a palm, 5 digits in a hand, 3 palms (so 12 digits) in a small span, 14 digits (or a half cubit) in a large span, 24 digits in a small cubit, and several other similar measurements.
• Also ten is an unfortunate number into which to divide a unit of measurement since it only divides naturally into 1/2 , 1/5 , 1/10 .
• Basing subdivisions on 12, mean that 1/2 , 1/3 , 1/4 , 1/6 , 1/12 are natural subdivisions, giving much more range for trading quantities.
• The main series has ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
• One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
• In France, on the other hand, there was no standardisation and as late as 1788 Arthur Young wrote in "Travels during the years 1787, 1788, 1789" published in 1793:- .
• Gabriel Mouton, in 1670, had suggested that the world should adopt a uniform scale of measurement based on the mille, which he defined as the length of one minute of the Earth's arc.
• The French proposed 4576; which conveniently fell in France, the British proposed London, and the United States proposed the 38th parallel which was conveniently close to Thomas Jefferson's estate.
• Diplomatic wording allowed an international agreement to be reached, but in March 1791 Borda, as chairman of the Commission of Weights and Measures, proposed using instead of the length of a pendulum, the length of 1/10,000,000 of the distance from the pole to the equator of the Earth.
• However between these dates the French Revolution progressed to the stage where the Academie des Sciences was abolished in August 1793 but before that Borda, Lagrange and Laplace had computed a provisional value for the metre based on the survey carried out by Cassini de Thury in 1740.
• It did not last long for, on 12 February 1812, Napoleon returned the country to its former units.
• The decimal metric system was required to be used by law in the Low Countries in 1820.
• In 1830 Belgium became independent of Holland and made the metric system, together with its former Greek and Latin prefixes, the only legal measurement system.
• In 1840 the French government reintroduced the metric system but it took many years before use of the old measures died out.
• It became legal in Britain in 1864 but a law which was passed by the House of Commons to require its use throughout the British Empire never made it through its final stages on to the statute books.
• Similarly in the United States it became legal in 1866, although its use was not made compulsory.
• The German states passed legislation in 1868 which meant that on the unification of these states to form Germany, use of the metric system was made compulsory.
• It is interesting that many leading British scientists were opposed to the introduction of the metric system in Britain in 1864, which is one reason that it only became legal but not compulsory.
• In 1870 an International Conference was convened by the French in Paris.
• Wishing that any decision be a truly international one, the conference was postponed and met again in 1872.
• The outcome was the setting up of the International Bureau of Weights and Measures, to be situated in Paris, and the Convention of the Metre of 1875 which was signed by seventeen nations.
• In 1889 the International Bureau of Weights and Measures replaced the original metre bar in Paris by a new one and at the same time had copies of the bar sent to every country which had signed up to the Convention of the Metre.
• The metre was redefined again in 1983, this time as the distance which light travels in a vacuum in 1/299,792,458 seconds.
• Note that in all these redefinitions, the length of the metre was always taken as close as possible to the value fixed in 1799 by data from the Delambre-Mechain survey.
• Now Borda had argued against using the length of a pendulum which beats at the rate of one second to define the metre in 1791 on the reasonable grounds that the second was not a fixed unit but could change with time.
• Indeed the second, then defined as 1/86,400 of the mean solar day, does change but a fixed definition was introduced in 1956 by the International Bureau of Weights and Measures, as 1/31,556,925.9747 of the length of the tropical year 1900.
• It was changed in 1964 to 9,192,631,770 cycles of radiation associated with a particular change of state of the caesium-133 atom.
• By 1983 when the metre was defined in terms of the second, Borda's objection was no longer valid as the definition of the second by then did not have the astronomical definition which was indeed variable.

25. References for Bourbaki 1
• 35 (1988), 43-49.
• Context 10 (2) (1997), 297-342.
• Intelligencer 15 (1993), 27-35.
• L Baulieu, Bourbaki's art of memory : Commemorative practices in science: historical perspectives on the politics of collective memory, Osiris (2) 14 (1999), 219-251.
• Contemp., Amsterdam, 1998), 75-123.
• L Baulieu, Dispelling a myth : questions and answers about Bourbaki's early work, 1934-1944, in The intersection of history and mathematics, (Birkhauser, Basel, 1994), 241-252.
• Intelligencer 8 (4) (1986), 84-85.
• A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.
• A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt.
• (1) (1998), 8-15.
• Intelligencer 2 (4) (1979-80), 175-180.
• J Delsarte, Compte rendu de la reunion Bourbaki du 14 janvier 1935, Gaz.
• 84 (2000), 16-18.
• Monthly 77 (1970), 134-145.
• 20 (2) (1975), 66-76.
• 14(47) (1971), 50-61.
• Nauk 28 (3)(171) 1973), 205-216.
• A 77 (1972), 447-460.
• 18 (2) (1969), 13-25.
• H Freudenthal, The truth about Bourbaki (Dutch), Euclides (Groningen) 61 (10) (1985/86), 330.
• Wissensch., XII (Steiner, Wiesbaden, 1985), 607-611.
• Intelligencer 7 (2) (1985), 18-22.
• 49 (1) (2002), 1-10.
• C Houzel, The influence of Bourbaki (Italian), in Italian mathematics between the two world wars (Italian), Milan/Gargnano, 1986 (Pitagora, Bologna, 1987), 241-246.
• K Krickeberg, Comment: "Twenty-five years with Nicolas Bourbaki 1949-1973" by A Borel, Mitt.
• (2) (1998), 16.
• 2 (1) (1980), 16-29.
• Intelligencer 8 (2) (1986), 5.
• 24 (2) (1975/76), 169-187.
• Intelligencer 21 (3) (1999), 16-17.
• O Pekonen, Nicolas Bourbaki in memoriam (Finnish), in In the forest of symbols (Finnish) (Art House, Helsinki, 1992), 55-71.
• 4 (1959), 673-678.
• M Senechal, The continuing silence of Bourbaki - an interview with Pierre Cartier, June 18, 1997, Math.
• Intelligencer 20 (1) (1998), 22-28.
• 6(41) (2001), 100-110; 388.
• (2) 3 (1957), 289-297.
• 3 (1961), 23-35.

26. References for Real numbers 1
• 4 (3) (1977), 73-85.
• 5 (1) (1978), 1-14.
• 4 (1) (1984), 25-63.
• I Grattan-Guinness, Numbers, ratios, and proportions in Euclid's Elements : How did he handle them?, Historia Mathematica 23 (1996), 355-375.
• A Heimonen, How were the irrational numbers discovered? (Finnish), Arkhimedes (6) (1997), 10-16.
• 25/26 (3-4) (1978), 120; 208.
• 25/26 (2) (1978), 57-69; 111.
• 1951 (1951), 19-38.
• 9 (1-2) 2001/03), 95-113.
• 46 (1945), 242-264.

27. Calculus history
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ..
• A(1 + 1/4 + 1/42 + 1/43 + ..
• Luca Valerio (1552-1618) published De quadratura parabolae &#9417; in Rome (1606) which continued the Greek methods of attacking these type of area problems.
Go directly to this paragraph
• He showed, using these methods, that the integral of xn from 0 to a was an+1/(n + 1) by showing the result for a number of values of n and inferring the general result.
Go directly to this paragraph
• He applied this to the integral of xm from 0 to 1 which he showed had approximate value .
• + (n-1)m)/nm+1.
• Roberval then asserted that this tended to 1/(m + 1) as n tends to infinity, so calculating the area.
• Descartes produced an important method of determining normals in La Geometrie &#9417; in 1637 based on double intersection.
Go directly to this paragraph
• Barrow was in some way to blame for this since the publisher of Barrow's work had gone bankrupt and publishers were, after this, wary of publishing mathematical works! Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript.
• It was not published until 1711.
• Similarly his Method of fluxions and infinite series was written in 1671 and published in English translation in 1736.
• Newton's next mathematical work was Tractatus de Quadratura Curvarum &#9417; which he wrote in 1693 but it was not published until 1704 when he published it as an Appendix to his Optiks.
• Leibniz learnt much on a European tour which led him to meet Huygens in Paris in 1672.
Go directly to this paragraph
• He also met Hooke and Boyle in London in 1673 where he bought several mathematics books, including Barrow's works.
Go directly to this paragraph
• By 1675 Leibniz had settled on the notation .
• His results on the integral calculus were published in 1684 and 1686 under the name 'calculus summatorius', the name integral calculus was suggested by Jacob Bernoulli in 1690.
Go directly to this paragraph
• However when Berkeley published his Analyst in 1734 attacking the lack of rigour in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning.
Go directly to this paragraph

28. Set theory
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• For example Albert of Saxony, in Questiones subtilissime in libros de celo et mundi &#9417;, proves that a beam of infinite length has the same volume as 3-space.
Go directly to this paragraph
• In 1847 he considered sets with the following definition .
Go directly to this paragraph
• Bolzano gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.
Go directly to this paragraph
• Cantor's early work was in number theory and he published a number of articles on this topic between 1867 and 1871.
Go directly to this paragraph
• An event of major importance occurred in 1872 when Cantor made a trip to Switzerland.
Go directly to this paragraph
• Numerous letters between the two in the years 1873-1879 are preserved and although these discuss relatively little mathematics it is clear that Dedekind's deep abstract logical way of thinking was a major influence on Cantor as his ideas developed.
Go directly to this paragraph
• In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory.
Go directly to this paragraph
• A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set theory was becoming the centre of controversy.
Go directly to this paragraph
• an xn + an-1 xn-1 + an-2 xn-2 + .
• + a1 x + a0 = 0, .
• |an| + |an-1| + |an-2| + ..
• + |a1| + |a0| + n.
• These give roots 0, 1, -1.
• Putting them in 1-1 correspondence with the natural numbers is now clear but ordering them in order of index and increasing magnitude within each index.
• In the same paper Cantor shows that the real numbers cannot be put into one-one correspondence with the natural numbers using an argument with nested intervals which is more complex than that used today (which is in fact due to Cantor in a later paper of 1891).
Go directly to this paragraph
• In his next paper, the one that Cantor had problems publishing in Crelle's Journal, Cantor introduces the idea of equivalence of sets and says two sets are equivalent or have the same power if they can be put in 1-1 correspondence.
Go directly to this paragraph
• At this stage Cantor does not use the word countable, but he was to introduce the word in a paper of 1883.
Go directly to this paragraph
• Cantor published a six part treatise on set theory from the years 1879 to 1884.
Go directly to this paragraph
• When Lindemann proved that π is transcendental in 1882 Kronecker said .
Go directly to this paragraph
• His fifth work in the six part treatise was published in 1883 and discusses well-ordered sets.
Go directly to this paragraph
• In 1884 Cantor wrote 52 letters to Mittag-Leffler each one of which attacked Kronecker.
Go directly to this paragraph
• However, despite a wealth of important work in the years after 1884, there is some indication that he never quite reached the heights of genius that his remarkable papers showed over the 10 year period from 1874 to 1884.
Go directly to this paragraph
• Although not of major importance in the development of set theory it is worth noting that Peano introduced the symbol for 'is an element of' in 1889.
Go directly to this paragraph
• In 1885 Cantor continued to extend his theory of cardinal numbers and of order types.
Go directly to this paragraph
• In 1895 and 1897 Cantor published his final double treatise on sets theory.
Go directly to this paragraph
Go directly to this paragraph
• It is believed that Cantor discovered this paradox himself in 1885 and wrote to Hilbert about it in 1886.
Go directly to this paragraph
• In 1899 Cantor discovered another paradox which arises from the set of all sets.
Go directly to this paragraph
• The 'ultimate' paradox was found by Russell in 1902 (and found independently by Zermelo).
Go directly to this paragraph
• Lebesgue defined 'measure' in 1901 and in 1902 defined the Lebesgue integral using set theoretic concepts.
Go directly to this paragraph
• The first person to explicitly note that he was using such an axiom seems to have been Peano in 1890 in dealing with an existence proof for solutions to a system of differential equations.
Go directly to this paragraph
• Again in 1902 it was mentioned by Beppo Levi but the first to formally introduce the axiom was Zermelo when he proved, in 1904, that every set can be well-ordered.
Go directly to this paragraph
• Godel showed, in 1940, that the Axiom of Choice cannot be disproved using the other axioms of set theory.
Go directly to this paragraph
• Zermelo in 1908 was the first to attempt an axiomatisation of set theory.
Go directly to this paragraph

29. Burnside problem
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A Group G is said to be periodic if for all g &#8712; G there exists n &#8712; N with gn = 1.
• A Group G is said to be periodic of bounded exponent if there exists n &#8712; N with gn = 1 for all g &#8712; G.
• 33 (1902), 230-238.','1] introduced what he termed "a still undetermined point" in the theory of groups: .
• For a fixed n let Fmn denote the subgroup of Fm generated by gn for g &#8712; G.
• B(m, 3) is finite of order ≤ 32m-1 .
• B(2, 4) is finite of order ≤; 212.
• (2) 3 (1905), 435-440.','2]) .
• Theorem (Schur, 1911 [','I Schur, Uber Gruppen periodischer substitutionen, Sitzungsber.
• (1911), 619-627.','3]) .
• 182 (1940), 158-177.','6], appeared specifically addressing the RBP, and not until 1950 that the term "Restricted Burnside Problem" was coined by Magnus [','W Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann of Math.
• 52 (1950), 111-126; Errata, Ann.
• 57 (1953), 606.','7].
• 9 (1933), 154-156 / Math.
• Zeit 32 (1930), 315-318.','4] (independently) showed that B(m, 3) has order 3c, c = m + mC2 + mC3 and is a metabelian group of nilpotency class 3.
• 1940Sanov [','I N Sanov, Solution of Burnside&#8217;s problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math.
• 10 (1940),166-170 (Russian).','5] proved that B(m, 4) is finite.
• 3 (1955), 233-244 .','9] established that B0(2, 5) exists.
• 52 (1956), 381-390.','10] proved that B0(m, 5) exists.
• P Hall and G Higman [','P Hall, P and G Higman, On the p-length of p-soluble groups and reduction theorems for Burnside&#8217;s Problem, Proc.
• (3) 6 (1956), 1-42','11] showed that B0(m, 6) exists and has order 2a3b where a = 1 + (m - 1)3c , b = 1 + (m - 1)2m , c = m + mC2 + mC3 and is hence soluble of derived length 3.
• 2 (1958), 764-786.','12] proved that B(m, 6) is finite, a contribution which was described as a "heroic piece of calculation" by one reviewer.
• 23, (1958) 3-34 (Russian).
• Translations (2) 36 (1964), 63-99.','13] showed that B0(m, p) exists for all p prime.
• Theorem (Hall-Higman, 1956 [','P Hall, P and G Higman, On the p-length of p-soluble groups and reduction theorems for Burnside&#8217;s Problem, Proc.
• (3) 6 (1956), 1-42','11]) .
• Suppose that n = p1k1.
• Even earlier it was known for n odd by Feit-Thompson (the "odd-order paper" of 1962), and at the time of publication must have been a reasonable conjecture.
• In 1989 Zelmanov announced his proof of a positive solution of the Restricted Burnside Problem and was awarded a Fields medal for this in 1994.
• 27 (1959), 749-752.','14], but no definitive proof was forthcoming.
• 1975, (1964), 273-276.','15] provided a counter-example to the General Burnside Problem -- an infinite, finitely generated, periodic group.
• 32 (1968), 212-244; 251-524; 709-731.','16] proved that B(m, n) is infinite for n odd, n ≥ 4381 with an epic combinatorial proof based upon Novikov's earlier efforts.
• This saddened Britton since he was close to publishing himself, but he continued and finished in 1970.
• His paper was published in 1973, but Adian discovered that it was wrong.
• [Translated from the Russian by J Lennox and J Wiegold (Berlin, 1979).]','17] proved that B(m, n) is infinite if n odd, n ≥ 665, improving the Adian-Novikov result of 1968.

30. Real numbers 2
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• By the time Stevin proposed the use of decimal fractions in 1585, the concept of a number had developed little from that of Euclid's Elements.
• If we move forward almost exactly 100 years to the publication of A treatise of Algebra by Wallis in 1684 we find that he accepts, without any great enthusiasm, the use of Stevin's decimals.
• Euler, in Complete introduction to algebra (1771) wrote in the introduction:- .
• Grabiner writes [',' J V Grabiner, The origins of Cauchy&#8217;s rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).','2]:- .
• Cauchy, in Cours d'analyse &#9417; (1821), did not worry too much about the definition of the real numbers.
• Bolzano, on the other hand, showed that bounded Cauchy sequence of real numbers had a least upper bound in 1817.
• 81 (1956), 391-395.','22] where Rychlik describes it as "not quite correct".
• In [',' B van Rootselaar, Bolzano&#8217;s theory of real numbers, Arch.
• 2 (1964/1965), 168-180.','28] van Rootselaar disagrees saying that "Bolzano's elaboration is quite incorrect".
• However in J Berg's edition of Bolzano's Reine Zahlenlehre &#9417; which was published in 1976, Berg points out that Bolzano had discovered the difficulties himself and Berg found notes by Bolzano which proposed amendments to his theory which make it completely correct.
• However his contributions led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions.
• In 1851 he published results on transcendental numbers removing the dependence on continued fractions.
• 0.1100010000000000000000010000..
• Dedekind worked out his theory of Dedekind cuts in 1858 but it remained unpublished until 1872.
• Weierstrass gave his own theory of real numbers in his Berlin lectures beginning in 1865 but this work was not published.
• The first published contribution regarding this new approach came in 1867 from Hankel who was a student of Weierstrass.
• Two years after the publication of Hankel's monograph, Meray published Remarques sur la nature des quantites &#9417; in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit".
• Three years later Heine published a similar notion in his book Elemente der Functionenlehre &#9417; although it was done independently of Meray.
• to be equivalent if the sequence of rational numbers a1 - b1, a2 - b2 , a3 - b3 , a4 - b4 , ..
• Cantor also published his version of the real numbers in 1872 which followed a similar method to that of Heine.
• As we mentioned above, Dedekind had worked out his idea of Dedekind cuts in 1858.
• Another definition, similar in style to that of Heine and Cantor, appeared in a book by Thomae in 1880.
• Thomae added further explanation to his idea of "formal arithmetic" in the second edition of his text which appeared in 1898:- .
• he asked, that the constructions led to systems which would not produced contradictions? He wrote in 1903:- .
• Hilbert had taken a totally different approach to defining the real numbers in 1900.

31. References for Forgery 1
• [English translation of [',' H Bordier and E Mabille, Une Fabrique de Faux Autographes, Ou Recit de L&#8217;Affaire Vrain Lucas (Paris 1870).','1]].

32. Forgery 1
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• He became a professor at the Sorbonne in 1846 when he was 53 years old and he continued to produce important, highly original mathematics texts until the final years of his long life.
• He was elected a full member of the Academie des Sciences in 1851 and the work of the Academy became a major interest in his life.
• By 1854 a passion which he had for collecting manuscripts of historical importance seems to have moved into a new phase when he began to forge documents.
• Lewis writes in the introduction to [',' H Bordier and E Mabille, Une Fabrique de Faux Autographes, Ou Recit de L&#8217;Affaire Vrain Lucas (Paris 1870).','1]:- .
• By 1854, the one-time respectable law clerk was a budding master at his new trade.
• When Vrain-Lucas approached Chasles in 1861 offering to sell him letters between famous people from history, Chasles bought them eagerly and asked Vrain-Lucas if he could seek out more.
• Remarkably all these people wrote in French [',' H Bordier and E Mabille, Une Fabrique de Faux Autographes, Ou Recit de L&#8217;Affaire Vrain Lucas (Paris 1870).','1]:- .
• It was not easy money, for he spend most of every day working on his forgeries [',' H Bordier and E Mabille, Une Fabrique de Faux Autographes, Ou Recit de L&#8217;Affaire Vrain Lucas (Paris 1870).','1]:- .
• In 1867 Chasles approached the Academie des Sciences with his "proof" that Pascal had discovered the law of universal gravitation before Newton.
• In 1869 Vrain-Lucas was arrested and tried for forgery.
• From 1861 to 1869 Chasles paid Lucas between 140 000 and 150 000 francs for the false documents and for books to which Lucas had given spurious provenances.
• The witness replied [',' H Bordier and E Mabille, Une Fabrique de Faux Autographes, Ou Recit de L&#8217;Affaire Vrain Lucas (Paris 1870).','1]:- .
• They certainly were unsophisticated, for it does not take an expert to know that Plato, Socrates, Cleopatra, and Lazarus would not have written in 18th century French.
• One might wonder whether being age 68 when he was first approached by Vrain-Lucas in 1861, he might have been losing his faculties through old age.
• If this is so then he had not lost the ability to produce mathematics of the highest quality since his solution of the problem to determine the number of conics tangent to five given conics which he found in 1864 was remarkable, particularly as it corrected a previous incorrect solution by the outstanding mathematician Steiner.
• Again Chasles' Traite des sections coniques &#9417; (1865) is a text of major importance, and he was undertaking this work throughout the period he was purchasing documents from Vrain-Lucas.

33. function concept
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• E T Bell wrote in 1945:- .
• As O Petersen wrote in 1974 in [',' O Petersen, Logistics and the theory of functions, Arch.
• d&#8217;Hist.
• Sciences 24 (94) (1974), 29-50.','22]:- .
• 16 (1) (1976/77), 37-85.','32], that Oresme was getting closer in 1350 when he described the laws of nature as laws giving a dependence of one quantity on another.
• 16 (1) (1976/77), 37-85.','32]:- .
• In 1638 he studied the problem of two concentric circles with centre O, the larger circle A with diameter twice that of the smaller one B.
• At almost the same time that Galileo was coming up with these ideas, Descartes was introducing algebra into geometry in La Geometrie &#9417;.
• In a paper in 1698 on isoperimetric problems Johann Bernoulli writes of "functions of ordinates" (see [',' A P Youschkevitch, The concept of function up to the middle of the 19th century, Arch.
• 16 (1) (1976/77), 37-85.','32]).
• Tech., Berlin, 2000), 128-181.
• One can say that in 1748 the concept of a function leapt to prominence in mathematics.
• This was due to Euler who published Introductio in analysin infinitorum &#9417; in that year in which he makes the function concept central to his presentation of analysis.
• However Introductio in analysin infinitorum &#9417; was to change the way that mathematicians thought about familiar concepts.
• The function concept had led Euler to make many important discoveries before he wrote Introductio in analysin infinitorum &#9417;.
• 1/12 + 1/22 + 1/32 + 1/42 + ..
• He showed that the sum was π2/6, publishing the result in 1740.
• Let us return to the contents of Introductio in analysin infinitorum &#9417;.
• In 1746 d'Alembert published a solution to the problem of a vibrating stretched string.
• Euler published a paper in 1749 which objected to this restriction imposed by d'Alembert, claiming that for physical reasons more general expressions for the initial form of the string had to be allowed.
• 16 (1) (1976/77), 37-85.','32]:- .
• In 1755 Euler published another highly influential book, namely Institutiones calculi differentialis &#9417;.
• The first problems with Euler's definition of types of functions was pointed out in 1780 when it was shown that a mixed function, given by different formulas, could sometimes be given by a single formula.
• The clearest example of such a function was given by Cauchy in 1844 when he noted that the function .
• However, a more serious objection came through the work of Fourier who stated in 1805 that Euler was wrong.
• Monthly 105 (1) (1998), 59-67.','17] and [',' N Luzin, Function II, Amer.
• Monthly 105 (3) (1998), 263-270.','18] that confusion regarding functions had been due to a lack of understanding of the distinction between a "function" and its "representation", for example as a series of sines and cosines.
• Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
• Condorcet seems to have been the first to take up Euler's general definition of 1755, see [',' A P Youschkevitch, The concept of function in the works of Condorcet (Russian), in Studies in the history of mathematics, No.
• &#8217;&#8217;Nauka&#8217;&#8217;, Moscow, 1974), 158-166, 301.','31] for details.
• In 1778 the first two parts of Condorcet intended five part work Traite du calcul integral &#9417; was sent to the Paris Academy.
• Lacroix, who had read Condorcet's unfinished work, wrote in 1797:- .
• Cauchy, in 1821, came up with a definition making the dependence between variables central to the function concept.
• He wrote in Cours d'analyse &#9417;:- .
• Fourier, in Theorie analytique de la Chaleur &#9417; in 1822, gave the following definition:- .
• Dirichlet, in 1837, accepted Fourier's definition of a function and immediately after giving this definition he defined a continuous function (using continuous in the modern sense).
• Dirichlet also gave an example of a function defined on the interval [ 0, 1] which is discontinuous at every point, namely f(x) which is defined to be 0 if x is rational and 1 if x is irrational.
• In 1838 Lobachevsky gave a definition of a general function which still required it to be continuous:- .
• Hankel, in 1870, deplored the confusion which still reigned in the function concept:- .
• In 1876 Paul du Bois-Reymond made the distinction between a function and its representation even clearer when he constructed a continuous function whose Fourier series diverges at a point.
• This line was taken further in 1885 when Weierstrass showed that any continuous function is the limit of a uniformly convergent sequence of polynomials.
• Earlier, in 1872, Weierstrass had sent a paper to the Berlin Academy of Science giving an example of a continuous function which is nowhere differentiable.
• it created a sensation and, according to Hankel, disbelief when du Bois-Reymond published it in 1875.
• He wrote in 1899:- .
• Where have more modern definitions taken the concept? Goursat, in 1923, gave the definition which will appear in most textbooks today:- .
• Just in case this is not precise enough and involves undefined concepts such as 'value' and 'corresponds', look at the definition given by Patrick Suppes in 1960:- .
• A is a relation (x)(x &#8712; A (y)(z)(x = (y, z)).
• We write y A z if (y, z) &#8712; A.

34. Bourbaki 1
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• They are Henri Cartan, who is 29 years old and has been teaching at Strasbourg since 1931, and Andre Weil who was appointed in 1933 and is 27.
• The year is 1934 and for weeks Cartan has been asking Weil how he would teach different aspects of the differential and integral calculus.
• Weil, like Cartan, is unhappy with the recommended text, Goursat's Traite d'Analyse &#9417;, and has been suggesting to him better ways to introduce various concepts in the calculus.
• They talk about writing a book of 1000 pages to be published within six months.
• They decide that they will meet regularly at Cafe Capoulade and set themselves the goal that they must have agreed the syllabus of the book by the summer of 1935.
• By the summer of 1935 the group had decided that they would write under the name Nicolas Bourbaki.
• Certainly the name came from General Charles Soter Bourbaki was a French general who had fought in the Franco-Prussian war of 1870-71.
• Intelligencer 8 (4) (1986), 84-85.','10] for Boas's own account of these events.
• Monthly 77 (1970), 134-145.','15]:- .
• 3 (1961), 23-35.','39] that the anarchic character of the congresses, which led to the shouting, was really by design:- .
• Intelligencer 2 (4) (1979-80), 175-180.','13] explains the consequences of this:- .
• Intelligencer 2 (4) (1979-80), 175-180.','13]:- .
• Intelligencer 2 (4) (1979-80), 175-180.','13]:- .
• Already in 1935 Bourbaki had taken the decision to produce a series of books which were linearly ordered in the sense that no reference could be made except to books earlier in the linear progression.
• It took much longer than the members of Bourbaki had imagined in 1935 for the first material to be published, which did not happen until 1939.
• Armand Borel explains the subtle title that was chosen for the whole work [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.','11]:- .
• The title "Elements de Mathematique" was chosen in 1938.
• Eilenberg reviewed this Fascicule de resultats &#9417; of Chapter 1 of Book I and wrote:- .
• The second Bourbaki publication came in 1940 when Chapters 1 and 2 of Book III appeared:- .
• Two more publications appeared in 1942, namely Chapters 3 and 4 of Book III, and Chapter 1 of Book II.

35. Black holes
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The publication of Isaac Newton's Philosophiae Naturalis Principia Mathematica in 1687 set out his ideas on the notion of gravity, outlining his law of universal gravitation.
• F = (G m1 m2)/r2 .
• The distance between the centre of gravity of each of the masses is denoted r, and G is the universal gravitational constant, with value around 6.674 10-11 m3 kg-1 s-2 (m = metres, kg = kilograms, s = seconds).
• With his understanding of the fundamentals of gravity, John Michell (1724-1793), having previously studied twin stars, in 1784 went on to propose the idea that there could exist a body sufficiently massive that even light could not escape.
• Michell referred to these objects as 'dark stars' (see [',' A Sundermier, Black Holes, Symmetry Magazine (1 December 2016).','31]).
• Another mathematical 'proof' was however offered independently of Michell by Pierre-Simon Laplace in 1799 in favour of what Michell had proposed, but with different conclusions on the ratios of density and size.
• Moreover, his proof was only provided after the insistence of German astronomer Franz Xaver von Zach (1754-1832), who demanded more than the brief quantitative reasoning that was given in Laplace's original 1796 paper Exposition du Systeme du Monde.
• In fact it had been noted by Hermann Minkowski in 1907 that Einstein's Special Theory of Relativity published in 1905 meant that time was just another dimension.
• Then from their frame of reference, the ball appears to move at 100 km/h.
• With the train analogy, the difference in measured velocities of the ball from the two frames of reference is 100 km/h, however, with light, there can be no difference in measurement, even if it were measured from a frame of reference travelling at almost the speed of light.
• Einstein wrote back in 1916 [',' J Eisenstaedt, The Early Interpretation of the Schwarzschild Solution, in D Howard and J Stachel (eds), Einstein and the History of General Relativity: Einstein Studies 1 (Birkhauser, Boston, 1989), 213-233.','20]:- .
• He reported that this problem of the formation of a black hole [',' M Bartusiak, Black Hole: How an idea abandoned by Newtonians, hated by Einstein, and gambled on by Hawking became loved (Yale University Press, New Haven-London, 2015).','1]:- .
• In 1930, on a voyage from India to England, Chandrasekhar had calculated that a white dwarf much heavier than the Sun could not exist, and that it would undergo a collapse into a singularity with infinite density.
• On 11 January 1935, with Eddington's apparent approval, Chandrasekhar was to deliver his results to a meeting of the Royal Astronomical Society in London.
• At a meeting in Paris in 1939, Eddington maintained his disapproval of Chandrasekhar's ideas, despite the quiet, growing support for Chandrasekhar from the likes of Niels Bohr, Wolfgang Pauli and Paul Dirac.
• Previously known as 'darks stars', 'collapsars' and 'gravitationally completely collapsed objects' to name a few terms, the expression 'black hole' is popularly attributed to physicist John Wheeler, who admitted the term was offered to him by an audience member of one of his lectures in 1967.
• The use of the words 'black hole' had previously been used in 1963 at an astrophysics conference in Dallas, as claimed by science writer Marcia Bartusiak in [',' M Bartusiak, Black Hole: How an idea abandoned by Newtonians, hated by Einstein, and gambled on by Hawking became loved (Yale University Press, New Haven-London, 2015).','1].
• Within two years of the discovery of the neutron by James Chadwick in 1932, German and Swiss astronomers, respectively Walter Baade and Fritz Zwicky, proposed the existence of neutron stars at a meeting of the American Physical Society.
• Similar conclusions on neutron stars had also been made by Russian scientist Lev Davidovich Landau in 1931 before neutrons were discovered.
• Though in this publication Oppenheimer and Snyder had offered the first modern description of black holes, the paper came out on 1 September 1939, the day of Germany's invasion of Poland triggering the start of the Second World War, and thus received little attention.
• Even Einstein attempted to prove the impossibility of their existence in his paper published in 1939, only a month after Oppenheimer and Snyder's.
• In 1967, she, along with radio astronomer Antony Hewish, picked up radio pulses from an unknown source.
• In 1970, Hawking, using quantum theory and general relativity, was able to show that black holes can actually emit radiation, giving a theoretical argument for their existence in 1974.
• What is now widely accepted as the first discovered black hole, Cygnus X-1 was first 'seen' in 1964, and was generally recognised as a black hole by the 1990s.
• It was the subject of a bet in 1974 between Hawking and American theoretical physicist Kip Thorne, which Hawking conceded in 1990, admitting that Cygnus X-1 was indeed in all likelihood a black hole, based on the sufficient observational data they had.
• Launched in 1990, the HST has allowed astronomers to conclude that black holes are probably common to the centres of all galaxies.
• Observed on 14 September 2015, the Collaboration announced on 11 February 2016 that they had made the first observational discovery of gravitational waves (disturbances in the curvature of space-time generated by accelerated masses).

36. Fractal Geometry
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• ','3] However, on July 18, 1872, Karl Weierstrass presented a paper at the Royal Prussian Academy of Sciences showing that for a a positive integer and 0 < b < 1 .
• A E Gerald (Addison -Wesley, 1993).','14] Thus, Weierstrass's proof stands as the first rigorously proven example of a function that is analytic, but not differentiable.
• In 1883 Georg Cantor, who attended lectures by Weierstrass during his time as a student at the University of Berlin [','J J O&#8217;Connor, J.J., and E F Robertson.
• `8;(1) - ψ(0) = 1 however ψ' (x) dx = 0 [','G A Edgar, ed.
• In fact, self-similarity is a feature of fractals, and the Cantor set is an early example of a fractal, though self-similarity was not defined until 1905 (by Cesaro, who was analysing the paper by Helge von Koch discussed below) and fractals were not defined until Mandelbrot in 1975, [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] thus Cantor would not have thought of it in those terms.
• In a paper published in 1904, Swedish mathematician Helge von Koch constructed using geometrical means the now-famous von Koch curve and hence the Koch snowflake, which is three von Koch curves joined together.
• An absolutely key concept in the study of fractals, aside from the aforementioned self-similarity and non-differentiability, is that of Hausdorff dimension, a concept introduced by Felix Hausdorff in March of 1918.
• He was forced to give up his post as a professor at the University of Bonn in 1935, and even though he continued to work on set theory and topology, his work could only be published outside of Germany.
• Namely, "the Julia set of f is the boundary of the set of points z &#8712; C that escape to infinity under repeated iteration by f (z)." [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] .
• Introducing Fractal Geometry (Cambridge, 2000).','7] Julia published a 199-page paper in 1918 called Memoire sur l'iteration des fonctions rationelles &#9417;, which discussed much of his work on iterative functions and describing the Julia set.
• ','11] .
• ','10] .
• (x0 , x1 , x2 , ..
• ) where for each i &#8712; N we have xi = f (xi-1).
• In 1938, the year after Besicovitch's last paper on Hausdorff dimension, Paul Levy produced a comprehensive treatment on the property of self-similarity.
• Benoit Mandelbrot was born in 1924 in Warsaw, Poland and, like Hausdorff, he was also Jewish, though his family managed to escape life under the Third Reich in 1936 by leaving Poland for France, where family and friends helped them set up their new lives.
• In fact, in 1945, Mandelbrojt showed his nephew the works of Fatou and Julia, though the young Mandelbrot initially did not take much of an interest.
• ','13] .
• Mandelbrot's education was very uneven, and completely interrupted in 1940, when Mandelbrot and his family were forced to flee the Nazis again.
• ','13] Unfortunately, this would bring him into direct conflict with the teaching style of "Bourbaki", a group of mathematicians whose belief in solving problems analytically (as opposed to visually) dominated the teaching of mathematics in France at the time.
• ','13] who was a professor at there from 1920 until his retirement in 1959 [','---.
• ','12].
• In 1967, while still there, Mandelbrot wrote his landmark essay, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension [','B Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.
• Science, New Series 156 3775 (May 5, 1967): 636-638.','8], in which he linked the idea of previous mathematicians to the real world -- namely coastlines, which he claimed were "statistically self-similar".
• Science, New Series 156 3775 (May 5, 1967): 636-638.','8] .
• With the ability to see, for the first time, what these sets looked like in their limits, Mandelbrot came up with the idea of mapping the values of c &#8712; C for which the Julia set for the function fc (z) = z2 + c is connected.
• M = {c &#8712; C | fc(n)(z) is finite as n → ∞} .

37. Chinese problems
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A good runner can go 100 paces while a poor runner covers 60 paces.
• The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit.
• Open the first canal and the cistern fills in 1/3 day; with the second, it fills in 1 day; with the third, in 21/2 days; with the fourth, in 3 days, and with the fifth in 5 days.
• B leaves the east gate and walks straight ahead a distance of 16 pu, when he just sees A.
• The land area is 13 mou and 71/2 tenths of a mou.
• Now a pile of rice is against the wall with a base circumference 60 chi and an altitude of 12 chi.
• What is the volume? Another pile is at an inner corner, with a base circumference of 30 chi and an altitude of 12 chi.
• What is the volume? Another pile is at an outer corner, with base circumference of 90 chi and an altitude of 12 chi.
• In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken.
• The sum of the base and height of the triangle is 17 bu.
• If 100 fowls are bought for 100 qian, how many cockerels, hens and chickens are there? .

38. Sundials
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The ecliptic plane meets the equatorial plane at approximately 23.576;.
• (Regardless of the exact location of the zodiac constellations, the ecliptic was divided into 12 equal arcs of 3076; each, leaving most of the constellations off-centred and often not entirely in their designated 3076; region.) The sun's motion along the ecliptic circle takes a (solar) year.
• 247p, 3fold plates : ill, facsims ; 26cm.','1]:- .
• Before the Greeks developed the sundial into the forms Vitruvius lists, the more ancient civilizations of Egypt and Mesopotamia had shadow measuring devices as early as 1500 B.C.
• Journal for the History of Astronomy, 2001.
• 41(1): p.
• 157-167.','4] .
• 41(1): p.
• 157-167.','4] .
• 276(1257): p.
• 276(1257): p.
• 41(1): p.
• 157-167.','4] .
• "The shadow would sweep around such a dial more rapidly in the early morning and late afternoon than around midday, but the Egyptians simply divided the dial into 12 15° sectors or 'hours'.
• 41(1): p.
• 157-167.','4] Further Egyptian development in timekeeping seems to have waned until the Assyrian invasion in the 7th century B.C.[',' J Fermor, Timing the sun in Egypt and Mesopotamia.
• 41(1): p.
• 157-167.','4] .
• 45(1): p.
• 8-12.','7] This is as opposed to modern designs that have their gnomon slanted parallel to the earth's axis.
• 45(1): p.
• 8-12.','7]:- .
• 45(1): p.
• 8-12.','7] .
• 45(1): p.
• 8-12.','7] .
• 45(1): p.
• 8-12.','7] .
• The Journal of Hellenic Studies, 1981.
• 101: p.
• 101-112.','9] In such a dial, the noon line would run from the base of a perpendicular line between the gnomon's tip and the dial surface.) The hyperbolae were centred on this noon line.
• The Journal of Hellenic Studies, 1981.
• 101: p.
• 101-112.','9]:- .
• The Journal of Hellenic Studies, 1981.
• 101: p.
• 101-112.','9] .
• The Journal of Hellenic Studies, 1981.
• 101: p.
• 101-112.','9] The basic principle of the spherical sundial was that it mirrored the celestial sphere in which the sun travels.
• In spite of their noncircular nature, for latitudes below 4576; [which includes the whole of the Mediterranean Sea] the seasonal hour lines between meridian and horizon are very closely approximated by the great circles which do pass through corresponding seasonal hour points on solstitial and equinoctial curves.

39. EMS History
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Edinburgh Mathematical Society 1883-1933 .
• After being an assistant to James Clerk Maxwell at the Cavendish Laboratory he was Regius Professor of Mathematics at St Andrews 1877-1879, then Professor at Edinburgh.
• Chrystal would write perhaps the finest such text and publish it three years later - Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges published in 1886.
• It is suggested that the Society be formed, in the first instance, of all those who shall give in their names on or before February 2, 1883, and who are (1) present or former students in either of the Advanced Mathematical Classes of Edinburgh University, (2) Honours Graduates in any of the British Universities, or (3) recognised Teachers of Mathematics; and that, after the above mentioned date, members be nominated and elected by ballot in the usual manner.
• If there are any of your friends who might take an interest in the Society, kindly inform them of its objects, and invite them to attend the Preliminary Meeting, to be held in the MATHEMATICAL CLASS ROOM here, on Friday, February 2, 1883, at Eight p.m., at which meeting your presence is respectively requested.
• Later in 1883 Thomas Muir was elected to be the second president and he gave his presidential address on The Promotion of Research with Special Reference to the Present State of the Scottish Universities and Secondary Schools on 8 February 1884.
• The Scottish degree in 1883 .
• Up to 1860 an undergraduate in a Scottish university (St Andrews (founded 1411), Glasgow (1450), Aberdeen (1494), and Edinburgh (1582)) studied for an M.A., essentially a set course consisting of English, Latin, Greek, Mental Philosophy, Mathematics and Natural Philosophy.
• The Society develops in 1883-86 .
• From 1883 to 1886 the Society held 31 meetings at which talks were given (the first meeting only dealt with the setting up of the Society).
• He entered the University of St Andrews in 1859 and he graduated with an M.A., completing the course in 1863.
• he earned in 1863 two years later.
• Mackay was appointed as a Mathematics Master at Perth Academy in 1863, spending three years in this post while he attended the Theological Hall with the intention of entering the United Presbyterian Church.
• However, he decided that he would make teaching his career and in 1866 he was appointed as a Mathematics Master at the Edinburgh Academy; he held this post until he retired in 1904.
• in 1884.
• Edmund Whittaker's influence - colloquia of 1913 and 1914 .
• He was appointed Professor of Mathematics at Edinburgh in 1912 following the death of George Chrystal in November of the previous year.
• After six years as Royal Astronomer of Ireland (1906-12), he was appointed to the chair at Edinburgh.
• His influence on the Edinburgh Mathematical Society was rapid, for in 1913 they ran the first mathematical colloquium to he held in the UK.
• Scottish education in 1928 .
• They had started up again in 1926 due to Turnbull's enthusiasm and were held in St Andrews.
• Thomas MacRobert was, like Whittaker, educated at Cambridge (1907-10).
• He spent his whole career in Glasgow, being appointed an assistant to George Gibson, the professor, in 1910 without having undertaken any research.
• He became professor when Gibson retired in 1927.
• Whittaker was right wing, an Anglican who converted to become a Roman Catholic in 1927, and had a remarkable ability to keep pace with the latest mathematical developments to which he made substantial contributions.
• MacRobert and Whittaker joined the Society at around the same time (1911, 1912 respectively).
• In 1927 MacRobert proposed expanding this into the Journal of the Edinburgh Mathematical Society which would publish articles of pedagogic interest, historical articles, etc.
• An EMS Committee meeting took place on 6 March 1931 in Glasgow.
• The next EMS Committee meeting took place on 1 May 1931 in Edinburgh.
• An indication of the strength of feelings is shown by the fact that MacRobert prevented the Society holding meetings in Glasgow from 1931 until he retired in 1957.
• From 1931 onwards the number of schoolteachers in the Society became less, and it moved towards its position today of being essentially a research Society for university staff.

40. Ten classics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The History of the T'ang records (see [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','1]):- .
• Thirty students were recruited from the lower ranks of society and divided into two classes each of 15 students.
• Multiply the number of units left over when counting in threes by 70, add to the product of the number of units left over when counting in fives by 21, and then add the product of the number of units left over when counting in sevens by 15.
• If the answer is 106 or more then subtract multiples of 105.
• It is known that Zu Chongzhi found the very good approximation to π, namely 355/113 , and it is thought that this book used clever methods to find areas and volumes using limiting processes.

41. Mathematical games
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Similar problems appear in Fibonacci's Liber Abaci &#9417; written in 1202 and the familiar St Ives Riddle of the 18th Century based on the same idea (and on the number 7).
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• Fibonacci, already mentioned above, is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13, ..
• One of the earliest mentions of Chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth etc.
• One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 5; 3 board (using normal knight's moves).
• ., n2 to fill the squares of an n 5; n board so that each row, each column and both main diagonals sum to the same number.
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• Durer's famous engraving of Melancholia made in 1514 includes a picture of a magic square.
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• Veblen in 1908 used matrix methods to study magic squares.
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• It appears in the 1550 edition of his book De Subtililate &#9417;.
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• To take all the rings off requires (2n+1 - 1)/3 moves if n is odd and (2n+1 - 2)/3 moves if n is even.
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• He is best known for his translation of 1621 of Diophantus's Arithmetica.
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• Bachet, however, is also famed as a collector of mathematical puzzles which he published in 1612 Problemes plaisans et delectables qui font par les nombres &#9417;.
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• Euler, in 1759 following a suggestion of L Bertrand of Geneva, was the first to make a serious mathematical analysis of it, introducing concepts which were to become important in graph theory.
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• The Thirty Six Officers Problem, posed by Euler in 1779, asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 5; 6 square so that no rank or regiment will be repeated in any row or column.
• The generalised problem, in how many ways can n queens be placed on an n 5; n board so that no two attack each other, was posed by Franz Nauck in 1850.
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• In 1874 Gunther and Glaisher described methods for solving this problem based on determinants.
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• There is a unique solution (up to symmetry) to the 6 5; 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for one penny.
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• In 1857 Hamilton described his Icosian game at a meeting of the British Association in Dublin.
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• Jacques and Sons, makers of high quality chess sets, for £25 and patented in London in 1859.
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• The problem, posed in 1850, asks how 15 school girls can walk in 5 rows of 3 each for 7 days so that no girl walks with any other girl in the same triplet more than once.
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• Solutions for n = 9, 15, 27 were given in 1850 and much work was done on the problem thereafter.
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• Edouard Lucas invented the Towers of Hanoi in 1883.
• The problem of tiling an 8 5; 8 square with a square hole in the centre was solved in 1935.
• This problem was shown by computer to have exactly 65 solutions in 1958.
• In 1953 more general polyominoes were introduced.
• A 3 5; 4 5; 5 rectangular prism can be made from the 3-dimensional pentominoes.
• The aim of this game is to assemble a 3 5; 3 5; 3 cube.
• A slightly older game (1921) but still a cube game is due to P A MacMahon and called 30 Coloured Cubes Puzzle.
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• (Can you prove there are exactly 30 such cubes?) Choose a cube at random and then choose 8 other cubes to make a 2 5; 2 5; 2 cube with the same arrangement of colours for it's faces as the first chosen cube.
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• Each face of the 2 5; 2 5; 2 cube has to be a single colour and the interior faces have to match in colour.
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• Here is the first one composed by Smullyan in 1925 when he was 16 years old at THIS LINK.
• He published some of Smullyan's retrograde analysis chees problems in 1973.
• He also reported on a computing game in 1973.
• Invented in 1974, patented in 1975 it was put on the market in Hungary in 1977.
• However it did not really begin as a craze until 1981.
• By 1982 10 million cubes had been sold in Hungary, more than the population of the country.
• The cube consists of 3 5; 3 5; 3 smaller cubes which, in the initial configuration, are coloured so that the 6 faces of the large cube are coloured in 6 distinct colours.
• The 9 cubes forming one face can be rotated through 4576;.

42. Egyptian mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858.
• For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6.
• 1 (1) (1974), 93-94.
• Joseph [',' G G Joseph, The crest of the peacock (London, 1991).','8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind.
• The angle between the base and one of the faces is 5176; 50' 35".
• The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034.
• On the other hand the cotangent of the slope angle of 5176; 50' 35" is very close to π/4.
• In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.
• 12 (2) (1985), 107-122.','38] Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid.
• He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 51/2 to 7 and the fact that (11/14) 5; 4 = 3.1428 and is close to π is nothing more than a coincidence.
• Later a more accurate value of 3651/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account.

43. Matrices and determinants
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• There are two fields whose total area is 1800 square yards.
• One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard.
• If the total yield is 1100 bushels, what is the size of each field.
• 1 2 3 .
• 3 1 1 .
• 8 1 1 .
• 36 1 1 .
• Cardan, in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [',' E Knobloch, Determinants, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 766-774.','7] calls mother of rules ! This rule gives what essentially is Cramer's rule for solving a 2 5; 2 system although Cardan does not make the final step.
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• In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the way the Chinese methods described above were constructed.
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• Using his 'determinants' Seki was able to find determinants of 2 5; 2, 3 5; 3, 4 5; 4 and 5 5; 5 matrices and applied them to solving equations but not systems of linear equations.
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• In 1693 Leibniz wrote to de l'Hopital.
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• 10 + 11x + 12y = 0 .
• 10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30 .
• His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678.
• It contains the first published results on determinants proving Cramer's rule for 2 5; 2 and 3 5; 3 systems and indicating how the 4 5; 4 case would work.
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• Cramer gave the general rule for n 5; n systems in a paper Introduction to the analysis of algebraic curves (1750).
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• In 1764 Bezout gave methods of calculating determinants as did Vandermonde in 1771.
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• In 1772 Laplace claimed that the methods introduced by Cramer and Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of systems of linear equations without actually calculating it, by using determinants.
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• Lagrange, in a paper of 1773, studied identities for 3 5; 3 functional determinants.
• 1/6 [z(x'y" - y'x") + z'(yx" - xy") + z"(xy' - yx')].
• The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms.
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• It was Cauchy in 1812 who used 'determinant' in its modern sense.
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• In 1826 Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients.
• Jacobi published three treatises on determinants in 1841.
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• Cayley, also writing in 1841, published the first English contribution to the theory of determinants.
• Eisenstein in 1844 denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers except for the lack of commutativity.
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• The first to use the term 'matrix' was Sylvester in 1850.
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• After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics.
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• Cayley quickly saw the significance of the matrix concept and by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix.
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• Cayley in 1858 published Memoir on the theory of matrices which is remarkable for containing the first abstract definition of a matrix.
• Cayley also proved that, in the case of 2 5; 2 matrices, that a matrix satisfies its own characteristic equation.
• He stated that he had checked the result for 3 5; 3 matrices, indicating its proof, but says:- .
• In fact he also proved a special case of the theorem, the 4 5; 4 case, in the course of his investigations into quaternions.
• In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan.
• Frobenius, in 1878, wrote an important work on matrices On linear substitutions and bilinear forms although he seemed unaware of Cayley's work.
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• He cites Kronecker and Weierstrass as having considered special cases of his results in 1874 and 1868 respectively.
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• The nullity of a square matrix was defined by Sylvester in 1884.
• In 1896 Frobenius became aware of Cayley's 1858 Memoir on the theory of matrices and after this started to use the term matrix.
• Despite the fact that Cayley only proved the Cayley-Hamilton theorem for 2 5; 2 and 3 5; 3 matrices, Frobenius generously attributed the result to Cayley despite the fact that Frobenius had been the first to prove the general theorem.
• An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory.
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• An important early text which brought matrices into their proper place within mathematics was Introduction to higher algebra by Bocher in 1907.
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• Turnbull and Aitken wrote influential texts in the 1930's and Mirsky's An introduction to linear algebra in 1955 saw matrix theory reach its present major role in as one of the most important undergraduate mathematics topic.
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44. References for Babylonian mathematics
• G G Joseph, The crest of the peacock (London, 1991).
• 33 (1) (1998), 1-23.
• 40 (1956), 185-192.
• 17 (1) (1986), 22-31.
• L Brack-Bernsen and O Schmidt, Bisectable trapezia in Babylonian mathematics, Centaurus 33 (1) (1990), 1-38.
• 15 (1953), 412-422.
• E M Bruins, Fermat problems in Babylonian mathematics, Janus 53 (1966), 194-211.
• 17 (1955), 16-23.
• 21 (1976), 61-70; 353.
• E M Bruins, Computation in the old Babylonian period, Janus 58 (1971), 222-267.
• M Caveing, La tablette babylonienne AO 17264 du Musee du Louvre et le probleme des six freres, Historia Math.
• 12 (1) (1985), 6-24.
• 10 (Basel, 1992), 103-114; 404; 410.
• 33 (1) (1981), 57-64.
• Indeterminate analysis in Babylonian mathematics, Osiris 8 (1948), 12-40.
• Conflicting interpretations of Babylonian mathematics, Isis 31 (1940), 405-425.
• Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
• 27 (1964), 139-141.
• Retranslation and analysis, Amphora (Basel, 1992), 315-358.
• J Hoyrup, Babylonian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 21-29.
• 26 (1) (1996), 155-162.
• ACM 15 (7) (1972), 671-677.
• ACM 19 (2) (1976), 108.
• 67 (1947), 305-320.
• J S Liu, A general survey of Babylonian mathematics (Chinese), Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 16 (1) (1993), 80-87.
• K Muroi, A circular field problem in the late Babylonian metrological-mathematical text W 23291-x, Ganita-Bharati 19 (1-4) (1997), 86-90.
• (2) 6 (3) (1997), 229-230.
• K Muroi, Babylonian mathematics - ancient mathematics written in cuneiform writing (Japanese), in Studies on the history of mathematics (Kyoto, 1998), 160-171.
• (2) 7 (3) (1998), 199-203.
• K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
• 34 (1988), 11-19.
• K Muroi, Interest calculation of Babylonian mathematics: new interpretations of VAT 8521 and VAT 8528, Historia Sci.
• 39 (1990), 29-34.
• (2) 1 (3) (1992), 173-180.
• (2) 1 (1) (1991), 59-62.
• (2) 5 (3) (1996), 249-254.
• 3 (1976), 417-439.
• 8 (1) (1953), 31-63.
• Cuneiform Studies 1 (1947), 219-240.
• 96 (1944), 29-39.
• Near Eastern Studies 5 (1946), 203-214.
• G Sarton, Remarks on the study of Babylonian mathematics, Isis 31 (1940), 398-404.
• Nauk 7 (1940), 102-153.
• Nauk 8 (1941), 293-335.

45. Arabic mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.
• The works [',' E S Kennedy et al., Studies in the Islamic Exact Sciences (1983).','6] and [',' J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.','17] are on "Islamic mathematics", similar to [',' A A al&#8217;Daffa, The Muslim contribution to mathematics (London, 1978).','1] which uses the title the "Muslim contribution to mathematics".
• Other authors try the description "Arabic mathematics", see for example [',' R Rashed, Entre arithmetique et algebre: Recherches sur l&#8217;histoire des mathematiques arabes (Paris, 1984).','10] and [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','11].
• The most important Greek mathematical texts which were translated are listed in [',' J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.','17]:- .
• The more minor Greek mathematical texts which were translated are also given in [',' J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.','17]:- .
• As Rashed writes in [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','11] (see also [',' R Rashed, Entre arithmetique et algebre: Recherches sur l&#8217;histoire des mathematiques arabes (Paris, 1984).','10]):- .
• and 1/x, 1/x2, 1/x3, ..
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• Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work [',' R Rashed, L&#8217;extraction de la racine n-ieme et l&#8217;invention des fractions decimales (XIe - XIIe siecles), Arch.
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• 18 (3) (1977/78), 191-243.','18]:- .
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• He wrote a treatise on cubic equations, which [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','11]:- .
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• Al-Baghdadi (born 980) looked at a slight variant of Thabit ibn Qurra's theorem, while al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k-1(2k - 1) where 2k - 1 is prime.
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• Al-Haytham, is also the first person that we know to state Wilson's theorem, namely that if p is prime then 1+(p-1)! is divisible by p.
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• It is called Wilson's theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result.
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• Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Arabic mathematics.
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• He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself.
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• Nasir al-Din al-Tusi (born 1201), like many other Arabic mathematicians, based his theoretical astronomy on Ptolemy's work but al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
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• Sharaf al-Din al-Tusi (born 1201) invented the linear astrolabe.
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• The importance of the Arabic mathematicians in the development of the astrolabe is described in [',' J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.','17]:- .
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46. Jaina mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• (Siwan) 18 (3) (1984), 98-107.','16] Jha looks at the contributions of Jainas from the 5th century BC up to the 18th century AD.
• The topics are listed in [',' G G Joseph, The crest of the peacock (London, 1991).','2] as:- .
• This led to the work described in [',' R C Gupta, Chords and areas of Jambudvipa regions in Jaina cosmography, Ganita Bharati 9 (1-4) (1987), no.
• 1-4, 51-53.','3] on a mathematical topic in the Jaina work, Tiloyapannatti by Yativrsabha.
• 2588 = 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056.
• The paper [',' R C Gupta, The first unenumerable number in Jaina mathematics, Ganita Bharati 14 (1-4) (1992), 11-24.','6] describes the way that the first unenumerable number was constructed using effectively a recursive construction.
• Next, r1 = g(rq) is defined by a complicated recursive subprocedure, and then as before a new larger number n1 = f(r1) is defined.
• Jaina mathematics recognised five different types of infinity [',' G G Joseph, The crest of the peacock (London, 1991).','2]:- .
• In the Bhagabati Sutra rules are given for the number of permutations of 1 selected from n, 2 from n, and 3 from n.
• Similarly rules are given for the number of combinations of 1 from n, 2 from n, and 3 from n.
• ., 10, etc.
• The value of π in Jaina mathematics has been a topic of a number of research papers, see for example [',' R C Gupta, Madhavacandra&#8217;s and other octagonal derivations of the Jaina value π = W30;10, Indian J.
• 21 (2) (1986), 131-139.','4], [',' R C Gupta, On some rules from Jaina mathematics, Ganita Bharati 11 (1-4) (1989), 18-26.','5], [',' R C Gupta, Circumference of the Jambudvipa in Jaina cosmography, Indian J.
• 10 (1) (1975), 38-46.','7], and [',' S K Jha and M Jha, A study of the value of π known to ancient Hindu & Jaina mathematicians, J.
• 13 (1990), 38-44.','17].
• The approximation π = W30;10 seems one which was frequently used by the Jainas.
• 12 (2) (1977), 127-136.','23] points out that, according to the Jaina school, the greatest possible number of eclipses in a year is four.
• The data in the Surya Prajnapti implies a synodic lunar month equal to 29 plus 16/31 days; the correct value being nearly 29.5305888.
• However, in [',' S D Sharma, and S S Lishk, Length of the day in Jaina astronomy, Centaurus 22 (3) (1978/79), 165-176.','22] Sharma and Lishk present an alternative hypothesis which would allow the data to be of Indian origin.

47. Nine chapters
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• For example in [',' K Chemla, Relations between procedure and demonstration : Measuring the circle in the &#8217;Nine chapters on mathematical procedures&#8217; and their commentary by Liu Hui (3rd century), in History of mathematics and education: ideas and experiences (Essen, 1992) (1996), 69-112.','8] Chemla shows that Chinese mathematicians certainly understood how to give convincing arguments that their methodology for solving particular problems was correct.
• ., 12: .
• Suppose a field has width 1+ 1/2 + 1/3 + ..
• + 1/n.
• What must its length be if its area is 1? .
• Problems 12 to 18 involve the extraction of square roots, and the remaining problems involve the extraction of cube roots.
• A good runner can go 100 paces while a poor runner covers 60 paces.
• The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit.
• Open the first canal and the cistern fills in 1/3 day; with the second, it fills in 1 day; with the third, in 21/2 days; with the fourth, in 3 days, and with the fifth in 5 days.
• 20/(x/2) = (20 + x + 14)/1775.
• Then x2 + x(20 + 14) = 2 (205;1775), or .
• x2 + 34x = 71000.
• Consider the fact that Britain changed to a decimal currency in 1970.

48. Golden ratio
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 1/x = x/(1 - x) gives x2 + x - 1 = 0 so x = (√;5-1)/2.
• Then the golden ratio is 1/x = (√5 + 1)/2 = 1.6180339887498948482..
• However, when Fibonacci produced Liber Abaci &#9417; he used many Arabic sources and one of them was the problems of Abu Kamil.
• In Liber Abaci &#9417; he gives the lengths of the segments of a line of length 10 divided in the golden ratio as W30;125 -5 and 15 - W30;125.
• Pacioli wrote Divina proportione &#9417; which is his name for the golden ratio.
• He also states the result given in Liber Abaci &#9417; on the lengths of the segments of a line of length 10 divided in the golden ratio.
• See [',' L Curchin and R Herz-Fischler, De quand date le premier rapprochement entre la suite de Fibonacci et la division en extreme et moyenne raison?, Centaurus 28 (2) (1985), 129-138.','6] for further details.
• The first known calculation of the golden ratio as a decimal was given in a letter written in 1597 by Michael Maestlin, at the University of Tubingen, to his former student Kepler.
• He gives "about 0.6180340" for the length of the longer segment of a line of length 1 divided in the golden ratio.
• The correct value is 0.61803398874989484821..
• He, like the annotator of Pacioli's Euclid, knows that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio and he states this explicitly in a letter he wrote in 1609.
• The result that the quotients of adjacent terms of the Fibonacci sequence tend to the golden ratio is usually attributed to Simson who gave the result in 1753.
• It appears in a publication of 1634 which appeared two years after Albert Girard's death.
• The first known use of the term appears in a footnote in Die reine Elementar-Matematik &#9417; by Martin Ohm (the brother of Georg Simon Ohm):- .
• The first edition of Martin Ohm's book appeared in 1826.
• 20 (2) (1982), 146-158.','9], examines the evidence and reaches the conclusion that 1835 marks the first appearance of the term.
• The article [',' J Mawhin, Au carrefour des mathematiques, de la nature, de l&#8217;art et de l&#8217;esoterisme: le nombre d&#8217;or, Rev.
• 169 (2-3) (1998), 145-178.','11] discusses whether the golden section is a universal natural phenomenon, to what extent it has been used by architects and painters, and whether there is a relationship with aesthetics.
• JOC/EFR July 2001 .

49. Debating topics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Each year in the Chinese calendar is named after one of 12 animals.
• Base 10 must result from counting on 10 fingers.
• Is 1 a number? .
• Building all numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is very clever indeed.
• We have looked at how numbers are built from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• The equation x2 + 1 = 0 has no real number solution.

50. Fund theorem of algebra
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The formula when applied to the equation x3 = 15x + 4 gave an answer involving ͩ0;-121 yet Cardan knew that the equation had x = 4 as a solution.
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• Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'.
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• Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.
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• Viete gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algebre &#9417;, However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C.
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• Now Harriot knew that a polynomial which vanishes at t has a root x - t but this did not become well known until stated by Descartes in 1637 in La geometrie &#9417;, so Albert Girard did not have much of the background to understand the problem properly.
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• A 'proof' that the FTA was false was given by Leibniz in 1702 when he asserted that x4 + t4 could never be written as a product of two real quadratic factors.
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• D'Alembert in 1746 made the first serious attempt at a proof of the FTA.
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• |z1| < |c|, |w1| < |c|.
• Firstly, he uses a lemma without proof which was proved in 1851 by Puiseau, but whose proof uses the FTA! Secondly, he did not have the necessary knowledge to use a compactness argument to give the final convergence.
• In 1749 he attempted a proof of the general case, so he tried to proof the FTA for Real Polynomials: .
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• His proof in Recherches sur les racines imaginaires des equations &#9417; is based on decomposing a monic polynomial of degree 2n into the product of two monic polynomials of degree m = 2n-1.
• Now Euler knew a fact which went back to Cardan in Ars Magna &#9417;, or earlier, that a transformation could be applied to remove the second largest degree term of a polynomial.
• In 1772 Lagrange raised objections to Euler's proof.
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• Laplace, in 1795, tried to prove the FTA using a completely different approach using the discriminant of a polynomial.
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• In his doctoral thesis of 1799 he presented his first proof and also his objections to the other proofs.
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• Gauss's proof of 1799 is topological in nature and has some rather serious gaps.
• In 1814 the Swiss accountant Jean Robert Argand published a proof of the FTA which may be the simplest of all the proofs.
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• Argand had already sketched the idea in a paper published two years earlier Essai sur une maniere de representer les quantities imaginaires dans les constructions geometriques &#9417;.
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• In this paper he interpreted i as a rotation of the plane through 9076; so giving rise to the Argand plane or Argand diagram as a geometrical representation of complex numbers.
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• Now in the later paper Reflexions sur la nouvelle theorie d'analyse &#9417; Argand simplifies d'Alembert's idea using a general theorem on the existence of a minimum of a continuous function.
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• In 1820 Cauchy was to devote a whole chapter of Cours d'analyse &#9417; to Argand's proof (although it will come as no surprise to anyone who has studied Cauchy's work to learn that he fails to mention Argand !) This proof only fails to be rigorous because the general concept of a lower bound had not been developed at that time.
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• The Argand proof was to attain fame when it was given by Chrystal in his Algebra textbook in 1886.
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• Two years after Argand's proof appeared Gauss published in 1816 a second proof of the FTA.
• A third proof by Gauss also in 1816 is, like the first, topological in nature.
• Gauss introduced in 1831 the term 'complex number'.
• The term 'conjugate' had been introduced by Cauchy in 1821.
• In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots.
• The first proof that the only commutative algebraic field containing R was given by Weierstrass in his lectures of 1863.
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• It was published in Hankel's book Theorie der complexen Zahlensysteme &#9417;.
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• Weierstrass noted in 1859 made a start towards a constructive proof but it was not until 1940 that a constructive variant of the Argand proof was given by Hellmuth Kneser.
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• This proof was further simplified in 1981 by Martin Kneser, Hellmuth Kneser's son.
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51. Ptolemy Manuscript
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Unfortunately, [',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1] .
• The oldest of these manuscripts was created no earlier than the late thirteenth century[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• Isis 22(2) (1935) 533-539.','3] .
• The American Journal of Philology 62(2) (1941) 244-246.','4] The forebear of these two branches was not the Ptolemaic original, but a younger manuscript later than Ptolemy.
• Such a manuscript does not survive today, but must be assumed [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• The nature of such errors point to a manuscript that utilized capital letters, which were used in manuscripts prior to the ninth century[',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1].
• It is possible that this missing link could have been written as far back as late antiquity.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• As time went on, however, distinguishing between original errors, copying errors, and improvements became more and more difficult.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• This was a major turning point in the history of the Geography, marking the beginning of a new proliferation of manuscripts.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• The probable explanation of this renewed interest in Ptolemy is the work of Byzantine scholar Maximos Planudes (c.1255-1305).
• discovered through many toils the geographia of Ptolemy, which had disappeared for many years.(Kugeas in [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• While a certain amount of errors in transmission are to be expected (noted as early as the seventh century[',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1]), emendations have also been made which are not errors, as can be seen by comparing this family to others.
• These alterations seem to have been made due to discrepancies with the text and the maps drawn to the text's specifications.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• Classical Philology 35(3) (1940) 333-336.','5] Did Ptolemy have a map of the world in front of him, from which he wrote the text? Did Ptolemy create world and/or regional maps based on his text? Or did the maps come later, produced by a student or reader of Ptolemy's text? .
• Adopting the scheme of Berggren and Jones, I will denote two classes of manuscripts, A and B, based on the number of regional maps associated with the manuscripts.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• In both versions, the regional maps follow the projection described for them by Ptolemy, namely a cylindrical projection in which distances along the central latitude and longitude are in the proper ratio.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• The world maps of the A version, nonetheless, have the same, rough northern coastline for this unknown territory.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• To fit all of the maps into the manuscript, smaller maps were needed, and so the regional maps were divided into more manageable sizes.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• Did Ptolemy draw a map or maps to accompany his text? Berggren and Jones reason for the affirmative [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• As Ptolemy insists in 1.17, the way to detect and eliminate inconsistencies such as those he detects in [his predecessor] Marinos' writing is to draw a map.
• Furthermore, if the point of writing the text was to instruct how to draw a world map, surely he must have tried it himself.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• The American Journal of Philology 62(2) (1941) 244-246.','4] .
• Neither manuscript version A nor B could have contained the maps, which must always have occupied large sheets.[',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1] For the world map to accommodate all of the cities listed in the catalogue, at best it would need to be one by two metres.
• Most maps of any size were often displayed publicly, painted or affixed to walls, not in manuscripts.[',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• Further evidence that the maps do not originate with Ptolemy has been put forward by Diller: [',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1] .
• The American Journal of Philology 62(2) (1941) 244-246.','4] Berggren and Jones point to Maximos Planudes as the mapmaker.
• The poem most likely means that Planudes [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• Because of the lack of practically of fitting the maps into the relatively small manuscripts before 1300, [',' Ptolemy, Ptolemy&#8217;s geography : an annotated translation of the theoretical chapters / J.
• The American Journal of Philology 62(2) (1941) 244-246.','4] .

52. Euclid's definitions
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 1.1.
• In [',' W R Knorr, &#8217;Arithmetike stoicheiosis&#8217; : on Diophantus and Hero of Alexandria, Historia Math.
• 20 (2) (1993), 180-192.','2] Knorr argues convincingly that this work is in fact due to Diophantus.

53. Indian mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh.
• An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
• One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
• The way that the contributions of these mathematicians were prompted by a study of methods in spherical astronomy is described in [',' K Shankar Shukla, Early Hindu methods in spherical astronomy, Ganita 19 (2) (1968), 49-72.','25]:- .
• Student 53 (1-4) (1985), 204-208','26]:- .
• Student 53 (1-4) (1985), 204-208','26].
• Madhava also gave other formulae for π, one of which leads to the approximation 3.14159265359.
• The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835.
• See for example [',' K M Marar and C T Rajagopal, Gregory&#8217;s series in the mathematical literature of Kerala, Math.
• Student 13 (1945), 92-98.','15].
• 25 (1) (1998), 1-21.','12] for more details.

54. Pi chronology
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 1Rhind papyrus2000 BC13.16045 (= 4(8/9)2) .
• 2Archimedes250 BC33.1418 (average of the bounds) .
• 3Vitruvius20 BC13.125 (= 25/8) .
• 4Chang Hong13013.1622 (= W30;10) .
• 5Ptolemy15033.14166 .
• 6Wang Fan25013.155555 (= 142/45) .
• 7Liu Hui26353.14159 .
• 8,Zu Chongzhi48073.141592920 (= 355/113) .
• 9Aryabhata49943.1416 (= 62832/20000) .
• 10Brahmagupta64013.1622 (= W30;10) .
• 11Al-Khwarizmi80043.1416 .
• 12Fibonacci122033.141818 .
• 14Al-Kashi1430143.14159265358979 .
• 15Otho157363.1415929 .
• 16Viete159393.1415926536 .
• 17Romanus1593153.141592653589793 .
• 18Van Ceulen1596203.14159265358979323846 .
• 19Van Ceulen1596353.1415926535897932384626433832795029 .
• 20Newton1665163.1415926535897932 .
• 21Sharp169971 .
• 22Seki Kowa170010 .
• 24Machin1706100 .
• 25De Lagny1719127Only 112 correct .
• 26Takebe172341 .
• 28von Vega1794140Only 136 correct .
• 32Lehmann1853261 .
• FergusonJan 1947710Desk calculator .
• Smith, Wrench19491120Desk calculator .
• Reitwiesner et al.19492037ENIAC .
• GenuysJan 195810000IBM 704 .
• FeltonMay 195810021PEGASUS .
• Guilloud195916167IBM 704 .
• Shanks, Wrench1961100265IBM 7090 .
• Guilloud, Bouyer19731001250CDC 7600 .
• Tamura19822097144MELCOM 900II .
• Kanada, Yoshino, Tamura198216777206HITACHI M-280H .
• Ushiro, KanadaOct 198310013395HITACHI S-810/20 .
• GosperOct 198517526200SYMBOLICS 3670 .
• BaileyJan 198629360111CRAY-2 .
• Kanada, TamuraSept 198633554414HITACHI S-810/20 .
• Kanada, TamuraOct 198667108839HITACHI S-810/20 .
• Kanada, Tamura, KuboJan 1987134217700NEC SX-2 .
• Kanada, TamuraJan 1988201326551HITACHI S-820/80 .
• ChudnovskysAug 19891011196691 .
• ChudnovskysAug 19912260000000 .
• Kanada, TakahashiAug 199751539600000HITACHI SR2201 .
• Kanada, TakahashiSept 1999206158430000HITACHI SR8000 .
• There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the nth hexadecimal digit of π to be computed without the preceeding n - 1 digits.
• Plouffe discovered a new algorithm to compute the nth digit of π in any base in 1997.

55. Indian numerals
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• First we will examine the way that the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 evolved into the form which we recognise today.
• However we must not forget that many countries use symbols today which are quite different from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and unless one learns these symbols they are totally unrecognisable as for example the Greek alphabet is to someone unfamiliar with it.
• There were separate Brahmi symbols for 4, 5, 6, 7, 8, 9 but there were also symbols for 10, 100, 1000, ..
• In [',' G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).','1] Ifrah lists a number of the hypotheses which have been put forward.
• Ifrah proposes a theory of his own in [',' G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).','1], namely that:- .
• 18 (1) (1983), 23-38.','7].
• 37 (4) (1987), 365-392.','8].
• A third hypothesis is put forward by Joseph in [',' G G Joseph, The crest of the peacock (London, 1991).','2].
• He is asked to name all the numerical ranks beyond a koti which is 107.
• He lists the powers of 10 up to 1053.
• Taking this as a first level he then carries on to a second level and gets eventually to 10421.
• He writes in [',' G G Joseph, The crest of the peacock (London, 1991).','2]:- .

56. Greek numbers
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Here are the symbols for the numbers 5, 10, 100, 1000, 10000.
• Acrophonic 5, 10, 100, 1000, 10000.
• For 5, 10, 100, 1000, 10000 there will be only one puzzle for the reader and that is the symbol for 5 which should by P if it was the first letter of Pente.
• Here is 1-10 in Greek acrophonic numbers.
• What is slightly more surprising is that the system had intermediate symbols for 50, 500, 5000, and 50000 but they were not new characters, rather they were composite symbols made from 5 and the symbols for 10, 100, 1000, 10000 respectively.
• Most of these forms are older than the main form of the numerals we have considered being more typical of the period 1500 BC to 1000 BC.
• For example 11, 12, ..
• , 19 were written: .
• alphabetical 11-19.
• First form of 1000, ..
• Second form of 1000, ..
• The symbol M with small numerals for a number up to 9999 written above it meant that the number in small numerals was multiplied by 10000.
• The number to be multiplied by 10000, 10000000, etc is written after the M symbol and is written between the parts of the number, a word which is best interpreted as 'plus'.
• As an example here is the way that Apollonius would have written 587571750269.
• Apollonius's representation of 587571750269.
• Archimedes designed a similar system but rather than use 10000 = 104 as the basic number which was raised to various powers he used 100000000 = 108 raised to powers.
• The first octet for Archimedes consisted of numbers up to 108 while the second octet was the numbers from 108 up to 1016.
• Using this system Archimedes calculated that the number of grains of sand which could be fitted into the universe was of the order of the eighth octet, that is of the order of 1064.
• JOC/EFR January 2001 .

57. Trigonometric functions
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Menelaus proved a property of plane triangles and the corresponding spherical triangle property known the regula sex quantitatum &#9417;.
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• Ptolemy was the next author of a book of chords, showing the same Babylonian influence as Hipparchus, dividing the circle into 36076; and the diameter into 120 parts.
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• This allowed him to calculate the chord subtended by angles of 3676;, 7276;, 6076;, 9076; and 12076;.
• Using these methods Ptolemy found that sin 30' (30' = half of 1°) which is the chord of 1° was, as a number to base 60, 0 31' 25".
• This same table was reproduced in the work of Brahmagupta (in 628) and detailed method for constructing a table of sines for any angle were give by Bhaskara in 1150.
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• Chapters of Copernicus's book giving all the trigonometry relevant to astronomy was published in 1542 by Rheticus.
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• In 1533 Regiomontanus's work De triangulis omnimodis &#9417; was published.
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• Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing.
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• The first use of sin in a book was in 1634 by the French mathematician Herigone while Cavalieri used Si and Oughtred S.
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• It is just the sine turned (versed) through 9076;.
• The name tangent was first used by Thomas Fincke in 1583.
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• The term cotangens was first used by Edmund Gunter in 1620.
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• The common abbreviation used today is tan by we write tan whereas the first occurrence of this abbreviation was used by Albert Girard in 1626, but tan was written over the angle .
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• cot was first used by Jonas Moore in 1674.
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• The term 'trigonometry' first appears as the title of a book Trigonometria by B Pitiscus, published in 1595.
• Johann Bernoulli found the relation between sin-1z and log z in 1702 while Cotes, in a work published in 1722 after his death, showed that .
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• in 1722 while Euler, in 1748, gave the formula (equivalent to that of Cotes) .
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58. Squaring the circle
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• One of the oldest surviving mathematical writings is the Rhind papyrus, named after the Scottish Egyptologist A Henry Rhind who purchased it in Luxor in 1858.
• This is quite a good approximation, corresponding to a value of 3.1605, rather than 3.14159, for π.
• Themistius states [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','1]:- .
• Now Archimedes is famed for his introduction of the spiral curve, but why did he introduced this curve? The authors of [',' J Delattre and R Bkouche, Why ruler and compass?, in History of Mathematics : History of Problems (Paris, 1997), 89-113.','7] suggest three reasons:- .
• Mathematicians in India were interested in the problem (see for example [',' T Hayashi, A new Indian rule for the squaring of a circle: Manava&#8217;sulbasutra 3.2.9-10, Gadnita Bharati 12 (3-4) (1990), 75-82.','11]) while in China mathematicians such as Liu Hsiao of the Han Dynasty showed himself to be one of the prominent of those attempting to square the circle in around 25 AD.
• In [',' T Albertini, La quadrature du cercle d&#8217;ibn al-Haytham : solution philosophique ou mathematique?, J.
• 9 (1-2) (1991), 5-21, 132.','6] the work of al-Haytham on squaring the circle is discussed.
• Not long after the work of al-Haytham, Franco of Liege in 1050 wrote a treatise De quadratura circuli &#9417; on squaring the circle.
• 26 (98) (1976), 59-105.','8] and [',' M Folkerts and A J E M Smeur, A treatise on the squaring of the circle by Franco of Liege, of about 1050.
• 26 (99) (1976), 225-253.','9] and in it Franco examines three earlier methods based on the assumption that π is 25/8 , 49/16 or 4.
• One such false proof, given by Saint-Vincent in a book published in 1647, was based on an early type of integration.
• For example Johann Bernoulli gave a method of squaring the circle through the formation of evolvents and this method is described in detail in [',' J E Hofmann, Johann Bernoullis Kreisrektifikation durch Evolventenbildung, Centaurus 29 (2) (1986), 89-99.','12].
• The historian of mathematics, Montucla, made squaring the circle the topic of his first historical work published in 1754.
• A major step forward in proving that the circle could not be squared using ruler and compasses occurred in 1761 when Lambert proved that π was irrational.
• It only led to a greater flood of amateur solutions to the problem of squaring the circle and in 1775 the Paris Academie des Sciences passed a resolution which meant that no further attempted solutions submitted to them would be examined.
• The popularity of the problem continued and there are many amusing stories told by De Morgan on this topic in his book Budget of Paradoxes which was edited and published by his wife in 1872, the year after his death.
• The final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is it is not the root of any polynomial equation with rational coefficients.
• As an example of the former type of claim, the New York Tribune published a letter in 1892 in which the author claimed to have rediscovered a secret going back to Nicomedes which proved that π = 3.2.
• Among the correct approximate constructions to square the circle was one by Hobson in 1913.
• This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..
• for π instead of 3.14159265..
• In the Journal of the Indian Mathematical Society in 1913 in a paper named Squaring the circle Ramanujan gave a construction which was equivalent to giving the approximate value of 355/113 for π, which differs from correct value only in the seventh decimal place.
• 97">Note.- If the area of the circle be 140,000 square miles, then [the side of the square] is greater than the true length by about an inch.
• Among other constructions given by Ramanujan in 1914 (Approximate geometrical constructions for π, Quarterly Journal of Mathematics XLV (1914), 350-374) was a ruler and compass construction which was equivalent to taking the strange yet remarkable approximate value for π to be (92+ 192/22)1/4.
• Now this is 3.1415926525826461253..
• which differs from π only in the ninth decimal place (π = 3.1415926535897932385..

59. Elliptic functions
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse.
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• In 1679 Jacob Bernoulli attempted to find the arc length of a spiral and encountered an example of an elliptic integral.
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• Jacob Bernoulli, in 1694, made an important step in the theory of elliptic integrals.
• ds/dt = 1/ͩ0;(1 - t4) .
• dt/ͩ0;(1 - t4) .
• dt/ͩ0;(1 - t2) .
• ∫ t2 dt/ͩ0;(1 - t4) .

60. General relativity
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Kepler's laws of planetary motion and Galileo's understanding of the motion and falling bodies set the scene for Newton's theory of gravity which was presented in the Principia in 1687.
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• Laplace looked at the stability of the solar system in Traite du Mecanique Celeste &#9417; in 1799.
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• Some profound remarks about gravitation were made by Maxwell in 1864.
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• In 1900 Lorentz conjectured that gravitation could be attributed to actions which propagate with the velocity of light.
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• In 1907, two years after proposing the special theory of relativity, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity.
• After the major step of the equivalence principle in 1907, Einstein published nothing further on gravitation until 1911.
• Then he realised that the bending of light in a gravitational field, which he knew in 1907 was a consequence of the equivalence principle, could be checked with astronomical observations.
• He had only thought in 1907 in terms of terrestrial observations where there seemed little chance of experimental verification.
• Einstein published further papers on gravitation in 1912.
• In 1913 Einstein and Grossmann published a joint paper where the tensor calculus of Ricci and Levi-Civita is employed to make further advances.
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• When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said .
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• It was the second half of 1915 that saw Einstein finally put the theory in place.
• Le Verrier, in 1859, had noted that the perihelion (the point where the planet is closest to the sun) advanced by 38" per century more than could be accounted for from other causes.
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• This last possibility would replace the 1/d2 by 1/dp, where p = 2+ε for some very small number ε.
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• By 1882 the advance was more accurately known, 43'' per century.
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• From 1911 Einstein had realised the importance of astronomical observations to his theories and he had worked with Freundlich to make measurements of Mercury's orbit required to confirm the general theory of relativity.
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• Freundlich confirmed 43" per century in a paper of 1913.
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• Also in the 18 November paper Einstein discovered that the bending of light was out by a factor of 2 in his 1911 work, giving 1.74".
• In fact after many failed attempts (due to cloud, war, incompetence etc.) to measure the deflection, two British expeditions in 1919 were to confirm Einstein's prediction by obtaining 1.98" 77; 0.30" and 1.61" ± 0.30".
• Hilbert applied the variational principle to gravitation and attributed one of the main theorem's concerning identities that arise to Emmy Noether who was in Gottingen in 1915.
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• In fact Emmy Noether's theorem was published with a proof in 1918 in a paper which she wrote under her own name.
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• A special case of Emmy Noether's theorem was written down by Weyl in 1917 when he derived from it identities which, it was later realised, had been independently discovered by Ricci in 1889 and by Bianchi (a pupil of Klein) in 1902.
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• Immediately after Einstein's 1915 paper giving the correct field equations, Karl Schwarzschild found in 1916 a mathematical solution to the equations which corresponds to the gravitational field of a massive compact object.
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• In December 1915 Ehrenfest wrote to Lorentz referring to the theory of November 25, 1915.
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61. Chinese numerals
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1899 a major discovery was made at the archaeological site at the village of Xiao dun in the An-yang district of Henan province.
• What was ||| ? It could be 3, or 21, or 12, or even 111.
• For example Sun Zi, in the first chapter of the Sunzi suanjing &#9417;, gives instructions on using counting rods to multiply, divide, and compute square roots.
• Xiahou Yang's Xiahou Yang suanjing &#9417; written in the 5th century AD notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved to the left by 1, 2, 3, or 4 squares.
• Similarly to divide by 10, 100, 1000, or 10000 the rods are moved to the right by 1, 2, 3, or 4 squares.
• What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.

62. References for Jaina mathematics
• G G Joseph, The crest of the peacock (London, 1991).
• R C Gupta, Chords and areas of Jambudvipa regions in Jaina cosmography, Ganita Bharati 9 (1-4) (1987), no.
• 1-4, 51-53.
• R C Gupta, Madhavacandra's and other octagonal derivations of the Jaina value π = W30;10, Indian J.
• 21 (2) (1986), 131-139.
• R C Gupta, On some rules from Jaina mathematics, Ganita Bharati 11 (1-4) (1989), 18-26.
• R C Gupta, The first unenumerable number in Jaina mathematics, Ganita Bharati 14 (1-4) (1992), 11-24.
• 10 (1) (1975), 38-46.
• R C Gupta, Errata: "Chords and areas of Jambudvipa regions in Jaina cosmography", Ganita Bharati 10 (1-4) (1988), 124.
• A Jain, Some unknown Jaina mathematical works (Hindi), Ganita Bharati 4 (1-2) (1982), 61-71.
• 11 (2) (1976), 85-111.
• 14 (1) (1979), 31-65.
• 24 (3) (1989), 163-180.
• 30 (2-4) (1995), 103-131.
• 28 (4) (1993), 303-308.
• 22 (4) (1987), 359-371.
• (Siwan) 18 (3) (1984), 98-107.
• 13 (1990), 38-44.
• 12 (2) (1977), 106-113.
• 12 (1) (1977), 33-44.
• 14 (1) (1979), 1-15.
• 12 (2) (1977), 173-180.
• S D Sharma, and S S Lishk, Length of the day in Jaina astronomy, Centaurus 22 (3) (1978/79), 165-176.
• 12 (2) (1977), 127-136.

63. Voting
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This voting system is the one in which each voter casts a single vote for the candidate they wish to elect and this was proved in 1952 by Kenneth May.
• His suggestion was written down in a manuscript which was available to Nicholas of Cusa in the 15th century but a fuller description of Llull's proposal for a voting system was made in another of his manuscripts which was only discovered and published in 2001.
• In 1433 Cusa, having studied Llull's idea and realising that it had deficiencies, proposed a different system which would always result in a winner.
• Llull's idea of a fair election was proposed again 500 years later by Condorcet in his Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix &#9417; published in 1785.
• A second edition of Condorcet's work, including much new material, was published in 1805 with a different title, namely Elements du calcul des probabilites et son application aux jeux de hasard, a la loterie et aux jugements des hommes &#9417;.
• 7 (2) (1990), 99-108.','6]:- .
• One of them displays a matrix for pairwise comparisons; this is a work written in 1299, nearly 600 years before the matrix notation was believed to have been invented by C L Dodgson.
• In fact it was shown in 1951 by Kenneth Arrow, who won the Nobel Prize for Economics in 1972, that no voting system can be fair in the sense that it will produce a winner who will be preferred to every other candidate and still guarantee that a decisive result will be the outcome of the election.
• In 1944 von Neumann and Morgenstern published Theory of Games and Economic Behaviour which allowed concepts such as tactical voting to be put into a precise mathematical form.
• The quota Q = k/(m+1)+1 is calculated.
• The author of [',' F Abeles, C L Dodgson and apportionment for proportional representation, Gadnita Bharati 3 (3-4) (1981), 71-82.','4] argues that:- .

64. References for Pi history
• A Ahmad, On the π of Aryabhata I, Ganita Bharati 3 (3-4) (1981), 83-85.
• 67 (2) (1994), 83-91.
• P Beckmann, A history of π (Boulder, Colo., 1971).
• E M Bruins, With roots towards Aryabhata's π-value, Ganita Bharati 5 (1-4) (1983), 1-7.
• G L Cohen and A G Shannon, John Ward's method for the calculation of pi, Historia Mathematica 8 (2) (1981), 133-144.
• 25 (1960), 183-195.
• R C Gupta, The value of π in the 'Mahabharata', Ganita Bharati 12 (1-2) (1990), 45-47.
• R C Gupta, On the values of π from the Bible, Ganita Bharati 10 (1-4) (1988), 51-58.
• R C Gupta, New Indian values of π from the 'Manava'sulba sutra', Centaurus 31 (2) (1988), 114-125.
• R C Gupta, Lindemann's discovery of the transcendence of π : a centenary tribute, Ganita Bharati 4 (3-4) (1982), 102-108.
• Education 9 (1975), B1-B5.
• Education 9 (3) (1975), B45-B48.
• Education 7 (1973), B17-B20.
• 37 (1989), 1-16.
• C Jami, Une histoire chinoise du 'nombre π', Archive for History of Exact Sciences 38 (1) (1988), 39-50.
• 13 (1990), 38-44.
• (Siwan) 16 (3) (1982), 54-59.
• 13 (1) (1978), 32-41.
• (2) 3 (3) (1994), 185-199.
• 18 (1952), 25-30.
• 63 (5) (1990), 291-306.
• (2) 7 (1) (1986), 1-8.
• Intelligencer 7 (2) (1985), 69-72.
• 69 (449) (1985), 218-219.
• 25 (3-4) (1989), 74-77.
• I Tweddle, John Machin and Robert Simson on inverse-tangent series for π, Archive for History of Exact Sciences 42 (1) (1991), 1-14.
• (2) 4 (2) (1994), 139-157.

65. Physical world
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In Apologia &#9417; written in 1600, but unpublished, Kepler argues that accuracy in "saving the phenomena" cannot distinguish which mathematical theory might correspond to reality.
• 4, 4+3, 4+6, 4+12, 4+24, 4+48, 4+96, 4+192, ..
• divided by 10 to get .
• 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6, ..
• Now the distances of the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn from the Sun (taking the distance of the Earth as 1) are .
• 0.39, 0.72, 1.0, 1.52, -, 5.2, 9.5 .

66. Classical light
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• He wrote Optica &#9417; in about 300 BC in which he studied the properties of light which he postulated travelled in straight lines.
• About the same time at Roger Bacon was working on optics in England, Witelo was studying mirrors and refraction of light and wrote up his findings in Perspectiva &#9417; which was a standard text on optics for several centuries.
• Following this there was improved understanding of using a lens, and by 1590 Zacharius Jensen even used compound lenses in a microscope.
• In fact an important discovery had been made earlier by Thomas Harriot when he discovered the sine law of refraction of light in 1601, but he did not publish the result.
• Galileo turned his telescope on Jupiter in 1610 and observed its four major moons.
• In 1611 Kepler published Dioptrice &#9417; which was another important work on optics.
• Willebrord Snell discovered the sine law of refraction of light in 1621 but, like Harriot, he did not publish the result.
• The first to publish the law was Descartes in 1637.
• In was contained in La Dioptrique &#9417; published as a supplement to Discours de la method pour bien conduire sa raison et chercher la verite dans les sciences &#9417;.
• 36 (3) (1993), 253-294.','47] for details) and Fermat initially assumed that they had reached a different law since they had started from different assumptions.
• See [',' L Rozenfel&#8217;d, Gravitational effects of light (Russian), in Einstein collection, 1980-1981 &#8217;&#8217;Nauka&#8217;&#8217; (Moscow, 1985), 255-266; 335.','38] for details of why Descartes was so strongly convinced.
• In 1647 Cavalieri published an important contribution to optics when he gave the relationship between the curvature of a thin lens and its focal length.
• Inspired by Kepler's discoveries on light, James Gregory had begun to work on lenses and in Optica Promota&#9417; (1663) he described the first practical reflecting telescope now called the Gregorian telescope.
• In 1672 Newton published his theory of colour in the Philosophical Transactions of the Royal Society and in it he gave experimental evidence that light is composed of minute particles.
• In [',' N Kipnis, History of the principle of interference of light (Basel, 1991).','3] Nakajima discusses the Newton-Hooke controversy of 1672:- .
• This seems to have caused some confusion in the interpretation of the optical controversy between Newton and Hooke in 1672.
• [We] present a new interpretation of the optical controversy of 1672.
• The effect of the argument was to prevent Newton publishing his complete theory of light until after the death of Hooke in 1703.
• We should point out, however, that Newton's views did undergo changes between 1672 and the publication of Opticks in 1704.
• These are examined carefully in [',' A E Shapiro, Light, pressure, and rectilinear propagation : Descartes&#8217; celestial optics and Newton&#8217;s hydrostatics, Studies in Hist.
• 5 (1974), 239-296.','41].
• Huygens was developing his wave theory of light at this time and by 1678 he had it worked out in all its mathematical details although he did not publish his Treatise on light until 1690.
• After Hooke's death, Newton published Opticks in 1704.
• In 1676 Romer used data from the eclipses of Jupiter's moons to get the first reasonable value for the speed of light.
• Not everyone in the 18th century agreed, however, and when Euler published his work on optics Nova theoria lucis et colorum &#9417; in 1746 it argued strongly for a wave theory of light.
• Euler's theory was in fact the second version of his wave theory of light and details of both theories are considered in [',' J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
• 37 (1) (1980), 59-79.
• Little progress had been made between Newton's Opticks of 1704 and Euler's optical work.
• Perhaps the most significant was James Bradley's calculation of the velocity of light in 1727.
• In [',' J Eisenstaedt, Dark bodies and black holes, magic circles and Montgolfiers : light and gravitation from Newton to Einstein, in Einstein in context (Cambridge, 1993), 83-106.','22] Hakfoort studies the work of Nicolas Beguelin of 1772:- .
• He published his results in 1801, describing the pattern of dark and light bands seen on the screen behind the holes.
• His explanation of interference, from his own words of 1807, is as follows [',' R Baierlein, Newton to Einstein (Cambridge, 1992).','1]:- .
• Malus's discovery of the polarisation of light by reflection was published in 1809 and his theory of double refraction of light in crystals was published in the following year.
• (New Haven, Conn., 1970).','10] for details.
• Brewster's publication in 1811 gave what is now known as Brewster's law, namely that the maximum polarisation of a beam of light occurs when it strikes the surface of a transparent medium so that the refracted ray makes an angle of 9076; with the reflected ray.
• Dark lines in the spectrum of light had first been observed in 1802 by William Wollaston but the correct explanation of them had to wait a few years until a more thorough investigation by Joseph von Fraunhofer who measured the exact positions of over 500 such lines.
• In 1817 the French Academie des Sciences proposed as their prize topic for the 1819 Grand Prix a mathematical theory to explain diffraction.
• Fresnel wrote a paper giving the mathematical basis for his wave theory of light and in 1819 the committee, with Arago as chairman, and including Poisson, Biot and Laplace met to consider his work.
• He wrote [',' R Baierlein, Newton to Einstein (Cambridge, 1992).','1]:- .
• Arago stated in his report on Fresnel's entry for the prize to the Academie des Sciences [',' R Baierlein, Newton to Einstein (Cambridge, 1992).','1]:- .
• In the 1820s and 1830s diffraction was studied by a number of scientists; Fraunhofer published his theory in 1823 while twelve years later Airy mathematically calculated the diffraction pattern produced by a circular aperture.
• Fizeau, in 1849, was the first person to calculate the speed of light without using an astronomical method.
• The light which passed through the wheel was sent on a journey of 17.3 kilometres before being reflected back to interfere with light which had passed through the partially reflecting mirror.
• He found that it took 0.00056 seconds to make the 17.3 km journey and he calculated a speed of 300,000 kilometres per second with an error of 1000 km per sec.
• In 1860 Bunsen and Kirchhoff observed dark lines in the spectrum of a light source passed though burning substances.
• In 1845 Faraday studied the effect of a magnetic field on plane-polarised light.
• In 1846 Faraday gave a lecture at the Royal Institution in which he put forward his view that there is a unity in the forces of nature.
• One of Maxwell's first contributions to light was the creation of the first colour photograph in 1861.
• In 1862 Maxwell realised that electromagnetic phenomena are related to light when he discovered that they travelled at the same speed.
• In 1864 Maxwell wrote a paper in which he stated (see [',' R Baierlein, Newton to Einstein (Cambridge, 1992).','1]):- .
• Planck, who made one of the next major breakthoughts described in Light through the ages: Relativity and quantum era, said on the occasion of the centenary of Maxwell's birth in 1931, that this theory:- .

67. References for Indian mathematics
• 1.
• G G Joseph, The crest of the peacock (London, 1991).
• R Mukherjee, Discovery of zero and its impact on Indian mathematics (Calcutta, 1991).
• R C Gupta, A bibliography of selected Sanskrit and allied works on Indian mathematics and mathematical astronomy, Ganita Bharati 3 (3-4) (1981), 86-102.
• Education 9 (4) (1975), B65-B75.
• Education 11 (4) (1977), B80-B81.
• R C Gupta, Indian mathematics abroad up to the tenth century A.D., Ganita Bharati 4 (1-2) (1982), 10-16.
• R C Gupta, Indian mathematics and astronomy in the eleventh century Spain, Ganita Bharati 2 (3-4) (1980), 53-57.
• 25 (1) (1998), 1-21.
• P Jha, Indian mathematics in the Dark Age, Ganita-Bharati 17 (1-4) (1995), 75-79.
• R Lal and R Prasad, Integral solutions of the equation Nx2+1= y2in ancient Indian mathematics (cakravala or the cyclic method), Ganita Bharati 15 (1-4) (1993), 41-54.
• Student 13 (1945), 92-98.
• V Mishra and S L Singh, Height and distance problems in ancient Indian mathematics, Ganita-Bharati 18 (1-4) (1996), 25-30.
• 74 (1977), 31-43.
• S Parameswaran, Putumana Somayaji, Ganita Bharati 14 (1-4) (1992), 37-44.
• K R Rajagopalan, State of Indian mathematics in the southern region up to 10th century A.D., Ganita Bharati 4 (1-2) (1982), 78-82.
• 35 (2) (1986), 91-99.
• R Rashed, Indian mathematics in Arabic, in The intersection of history and mathematics (Basel, 1994), 143-148.
• 26 (2) (1991), 185-207.
• Education 7 (1973), A44-A45.
• K Shankar Shukla, Early Hindu methods in spherical astronomy, Ganita 19 (2) (1968), 49-72.
• Student 53 (1-4) (1985), 204-208 .

68. References for Real numbers 2
• J V Grabiner, The origins of Cauchy's rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).
• 3 (2) (1976), 161-166.
• 50 (1-2) (1997), 131-158.
• R P Burn, Irrational numbers in English language textbooks, 1890-1915 : constructions and postulates for the completeness of the real numbers, Historia Math.
• 19 (2) (1992), 158-176.
• 5 (1990), 67-73.
• 4 (3) (1977), 73-85.
• 5 (1) (1978), 1-14.
• 1 (1) (1977), 9-20.
• "Nauka", Moscow, 1973), 176-180; 338.
• 4 (1) (1984), 25-63.
• 23 (1978), 71-76; 357.
• 23 (1978), 56-70; 357.
• (1) (1981), 106-107.
• 25/26 (3-4) (1978), 120; 208.
• 25/26 (2) (1978), 57-69; 111.
• 81 (1956), 391-395.
• Logic 8 (1) (1987), 25-44.
• 11 (1-2) (1975), 24-27.
• 9 (1-2) 2001/03), 95-113.
• 2 (1964/1965), 168-180.

69. Topology history
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1736 Euler published a paper on the solution of the Konigsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis &#9417;.
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• In 1750 he wrote a letter to Christian Goldbach which, as well as commenting on a dispute Goldbach was having with a bookseller, gives Euler's famous formula for a polyhedron .
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• Euler published details of his formula in 1752 in two papers, the first admits that Euler cannot prove the result but the second gives a proof based dissecting solids into tetrahedral slices.
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• The route started by Euler with his polyhedral formula was followed by a little known mathematician Antoine-Jean Lhuilier (1750 -1840) who worked for most of his life on problems relating to Euler's formula.
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• In 1813 Lhuilier published an important work.
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• Mobius published a description of a Mobius band in 1865.
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• Johann Benedict Listing (1802-1882) was the first to use the word topology.
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• Listing wrote a paper in 1847 called Vorstudien zur Topologie &#9417; although he had already used the word for ten years in correspondence.
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• In 1861 Listing published a much more important paper in which he described the Mobius band (4 years before Mobius) and studied components of surfaces and connectivity.
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• Riemann had studied the concept in 1851 and again in 1857 when he introduced the Riemann surfaces.
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• c = m1a1 + m2a2 + ..
• m1a1 + m2a2 + ..
• The idea of connectivity was eventually put on a completely rigorous basis by Poincare in a series of papers Analysis situs &#9417; in 1895.
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• Euler's convex polyhedra formula had been generalised to not necessarily convex polyhedra by Jonquieres in 1890 and now Poincare put it into a completely general setting of a p-dimensional variety V.
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• This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
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• Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set.
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• Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano-Weierstrass theorem which states .
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• Hilbert used the concept of a neighbourhood in 1902 when he answered in the affirmative one of his own questions, namely .
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• In 1906 Frechet called a space compact if any infinite bounded subset contains a point of accumulation.
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• A few years later in 1914 Hausdorff defined neighbourhoods by four axioms so again there were no metric considerations.
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• Hadamard introduced the word 'functional' in 1903 when he studied linear functionals F of the form .
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• Frechet continued the development of functional by defining the derivative of a functional in 1904.
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• Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series.
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• Schmidt's work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Frechet.
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• A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces.
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• The collection of methods developed by Poincare was built into a complete topological theory by Brouwer in 1912.
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70. Orbits
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The first to propose a system of planetary paths which would set the scene for major advances was Copernicus who in De revolutionibus orbium coelestium &#9417; (1543), argued that the planets and the Earth moved round the Sun.
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• You can see a diagram from De revolutionibus orbium coelestium &#9417; showing Copernicus's solar system at THIS LINK.
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• In 1600 Kepler became assistant to Tycho Brahe who was making accurate observations of the planets.
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• After Brahe died in 1601 Kepler continued the work, calculating planetary paths to unprecedented accuracy.
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• Both these laws were first formulated for the planet Mars, and published in Astronomia Nova &#9417; (1609).
• You can see a diagram from Astronomia Nova &#9417; showing Kepler's elliptical path for Mars at THIS LINK.
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• Kepler's third law, that the squares of the periods of planets are proportional to the cubes of the mean radii of their paths, appeared in Harmonice mundi &#9417; (1619) and, perhaps surprisingly in view of the above comments, was widely accepted right from the time of its publication.
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• In 1679 Hooke wrote a letter to Newton.
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• All this was in the two plague years of 1665-1666..
• In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance from the Sun.
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• Despite the claims by Newton in the above quote, he had in fact proved this result in 1680 as a direct result of the letters from Hooke.
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• Newton indeed reworked his proof and sent a nine page paper De motu corporum in gyrum &#9417; to Halley.
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• He received Newton's complete manuscript by April of 1687 but there were many problems not the least being that Newton tried to prevent the publication of the Book III when Hooke claimed priority with the inverse square law of force.
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• A bright comet had appeared on 14 November 1680.
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• You can see a diagram of the orbit of the comet of 1680 from the Principia at THIS LINK.
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• When he computed the orbits for three comets which had appeared in 1537, 1607 and one Halley observed himself in 1682, he found that the characteristics of the orbits were almost identical.
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• Halley deduced they were the same comet and later was able to identify it with one which had appeared in 1456 and 1378.
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• Taking the perturbations into account Halley predicted the comet would return and reach perihelion (the point nearest the Sun) on 13 April 1759.
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• The comet was actually first seen again in December 1758 reaching perihelion on 12 March 1759.
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• In 1713 a second edition of the Principia, edited by Roger Cotes, appeared.
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• Euler developed methods of integrating linear differential equations in 1739 and made known Cotes' work on trigonometric functions.
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• He drew up lunar tables in 1744, clearly already studying gravitational attraction in the Earth, Moon, Sun system.
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• Clairaut and d'Alembert were also studying perturbations of the Moon and, in 1747, Clairaut proposed adding a 1/r4 term to the gravitational law to explain the observed motion of the perihelion, the point in the orbit of the Moon where it is closest to the Earth.
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• However by the end of 1748 Clairaut had discovered that a more accurate application of the inverse square law came close to explaining the orbit.
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• He published his version in 1752 and, two years later, d'Alembert published his calculations going to more terms in his approximation than Clairaut.
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• Cassini made a measurement of an arc of longitude in 1712 but obtained a result which wrongly suggested that the Earth was elongated at the poles.
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• In 1736 Maupertuis obtained the correct result verifying Newton's predictions.
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• This superimposed effect has a period of 18.6 years and was first observed by Bradley in 1730 but not announced until 18 years later when he had observed the full cycle.
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• The Paris Academie des Sciences offered Prizes for work on this topic in 1748, 1750 and 1752.
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• In 1748 Euler's studies of the perturbation of Saturn's orbit won him the Prize.
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• Lagrange won the Academie des Sciences Prize in 1764 for a work on the libration of the Moon.
• He also won the Academie des Sciences of 1766 for work on the orbits of the moons of Jupiter where he gave a mathematical analysis to explain an observed inequality in the sequence of eclipses of the moons.
• The first comet to have an elliptical orbit calculated which was far from a parabola was observed by Messier in 1769.
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• The Academie des Sciences Prize of 1772 for work on the orbit of the Moon was jointly won by Lagrange and Euler.
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• Lagrange submitted Essai sur le probleme des trois corps &#9417; in which he showed that Euler's restricted three body solution held for the general three body problem.
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• For Jupiter these bodies are called Trojan planets, the first to be discovered being Achilles in 1908.
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• The Trojan planets move 6076; in front and 6076; behind Jupiter at what are now called the Lagrangian points.
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• Lagrange introduced the method of variation of the arbitrary constants in a paper in 1776 stating that the method was of interest in celestial mechanics and, in special cases, had been already been used by Euler, Laplace and himself.
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• Lagrange published further major papers in 1783 and 1784 on the theory of perturbations of orbits using methods of variations of the arbitrary constants and, in 1785, applied his theory to the orbits of Jupiter and Saturn.
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• An important development occurred on 13 March 1781 when the astronomer William Herschel (father of John Herschel) observing in his private observatory in Bath, England found .
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• This work was to culminate in the publication of Mecanique celeste &#9417; (1799) in which, among many other important results, he claimed to prove the stability of the solar system.
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• Laplace's work of 1787, that of Adams of 1854 and later Delaunay's work described below eventually provided solutions.
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• Observations of Uranus in the early years of the 19th Century showed there were problems with its orbit and by 1830 Uranus had departed by 15" from the best fitting ellipse.
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• The next body to be discovered in the solar system was the minor planet Ceres, discovered in 1801.
• In 1766 J D Titus and in 1772 J E Bode had noted that .
• (1+4)/10, (3+4)/10, (6+4)/10, (12+4)/10, (24+4)/10, (48+4)/10, (96+4)/10 .
• gave the distances of the 6 known planets from the Sun (taking the Earth's distance to be 1) except there was no planet at distance 2.8.
• A search was made for a planet at distance 2.8 and on 1 January 1801 G Piazzi discovered such a body.
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• On 11 February Piazzi fell ill and ended his observations.
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• Johann Encke, a student of Gauss, computed (using Gauss's method) an elliptical orbit for the comet of 1818.
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• Papers published by Hamilton in 1834 and 1835 made major contributions to the mechanics of orbiting bodies.
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• as did the significant paper published by Jacobi in 1843 where he reduced the problem of two actual planets orbiting a sun to the motion of two theoretical point masses.
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• Bertrand extended Jacobi's work in 1852.
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• In 1836 Liouville studied planetary theory, the three body problem and the motion of the minor planets Ceres and Vesta.
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• Delaunay, famed for his work on the orbit of the Moon, investigated the perturbations in a paper of 1842.
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• Arago urged Le Verrier to work on the problem and on 1 June 1846 Le Verrier showed that the irregularities could be explained by an unknown planet and he determined the coordinates at which the planet would be found.
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• Le Verrier's personal triumph however was somewhat diminished when, on 15 October, a letter was published from the English astronomer Challis claiming that John Couch Adams of Cambridge University had made similar calculations to those of Le Verrier which he had completed in September 1845.
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• The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.
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• In 1865 Delaunay suggested that the discrepancies arose from a slowing of the Earth's rotation due to tidal friction, an explanation which is today believed to be correct.
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• Le Verrier had published an account of his theory of Mercury in 1859.
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• Le Verrier's search proved in vain and by 1896 Tisserand had concluded that no such perturbing body existed.
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• G W Hill published an account of his lunar theory in 1878.
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• Bruns proved in 1887 that apart from the 10 classical integrals, 6 for the centre of gravity, 3 for angular momentum and one for energy, no others could exist.
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• In 1889 Poincare proved that for the restricted three body problem no integrals exist apart from the Jacobian.
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• In 1890 Poincare proved his famous recurrence theorem, namely that in any small region of phase space trajectories exist which pass through the region infinitely often.
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• Poincare published 3 volumes of Les methods nouvelle de la mecanique celeste &#9417; between 1892 and 1899.
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• Poincare introduced further topological methods in 1912 for the theory of stability of orbits in the three body problem.
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• The comet of 1680 .

71. Greek sources II
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• First let us see what facts Heath knew when he wrote his famous book [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1], A history of Greek mathematics, which he began in 1913.
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• Hence, Heath can deduce that [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1]:- .
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• Eutocius's commentary on Archimedes On the Sphere and Cylinder II includes a quotation from Diocles solving the following problem of Archimedes (see for example [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1]):- .
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• Heath deduces from the quotes in Eutocius that Diocles [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1]:- .
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• Heath also writes [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1]:- .
• It presents an Arabic translation of Diocles On burning mirrors, from a manuscript copied in 1462, together with an English translation and commentary.
• The publication of this work in 1976 was an event of major importance in the history of mathematics adding a previously lost piece to the jigsaw.
• As we mentioned above he was the ruler of Pergamum from 241 to 197 BC.
• One papyrus states [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1921).','1]:- .

72. Ring Theory
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Familiar examples of rings such as the real numbers, the complex numbers, the rational numbers, the integers, the even integers, 2 5; 2 real matrices, the integers modulo m for a fixed integer m, will almost certainly be given in the Abstract Algebra book as will many beautiful theorems on rings but what will be missing are the reasons systems satisfying these particular axioms have been singled out for such intensive study.
• For example Legendre and Gauss investigated integer congruences in 1801.
• This theorem, proved as recently as 1995, states: .
• In 1847 Lame announced that he had a solution of Fermat's Last Theorem and sketched out a proof.
• 4 = 25;2 and 4 = (1 + √-3)5;(1 -√-3).
• Gauss had proved around 1801 that numbers of the form a + bͩ0;-1, where a, b are integers, could be written uniquely as a product of prime numbers of the form a + bͩ0;-1 in an analogous manner to the unique decomposition of an integer as a product of prime integers.
• In fact, numbers of the form a + bω +cω2 where a, b, c are integers and ω is a complex cube root of 1, also have unique factorisation, and this can be used to prove the n = 3 case of Fermat's last Theorem.
• The argument following Lame's announcement was settled by Kummer who pointed out that he had published an example in 1844 to show that the uniqueness of such decompositions failed and in 1846 he had restored the uniqueness by introducing "ideal complex numbers".
• The popular story that Kummer invented "ideal complex numbers" in an attempt to correct an error in this proof of Fermat's Last Theorem is almost certainly false; see Edwards [','Edwards, H.M., Fermat&#8217;s Last Theorem, (Berlin 1977).
• ','1].
• In 1847, just after Lame's announcement, Kummer used his "ideal complex numbers" to prove Fermat's Last Theorem for all n < 100 except n = 37, 59, 67 and 74.
• Dedekind also introduced the word "module" (early spelling: "modul") in 1871 although its initial definition was considerably more restricted than the present definition, being first introduced as a subgroup of the additive group of a ring; the term is now used for a "vector space with coefficients from a ring".
• Prime numbers were generalised to prime ideals by Dedekind in 1871.
• In 1882 an important paper by Dedekind and Heinrich Weber accomplished two things; it related geometric ideas with rings of polynomials and extended the use of modules.
• Hilbert, motivated by studying invariant theory, studied ideals in polynomial rings proving his famous "Basis Theorem" in 1893.
• Primary ideals were introduced in 1905 by Lasker in the context of polynomial rings.
• (Lasker was World Chess Champion from 1894 to 1921.) Lasker proved the existence of a decomposition of an ideal into primary ideals but the uniqueness properties of the decomposition were not proved until 1915 by Macaulay.
• ','2] that algebra texts such as that of Heinrich Weber [','Weber, H., Lehrbuch der Algebra, (Braunschweig, 1895-1896).
• ','4] in 1895 contained axioms for groups similar to many present-day texts.
• In about 1921 she made the important step, which we commented on earlier, of bringing the two theories of rings of polynomials and rings of numbers under a single theory of abstract commutative rings.
• Discrimination made it difficult for her to publish her work and it was not until Van der Waerden's important work on Modern Algebra [','Van der Waerden, B.L., Moderne Algebra (2 Vols) (Berlin 1930, 1931).
• ','3] was published in 1930 that Noether's results become widely known.
• In 1843 inspiration struck Hamilton - the generalisation was not to three dimensions but to four dimensions and the commutative property of multiplication no longer held.
• Matrices with their laws of addition and multiplication were introduced by Cayley in 1850 while, in 1870, Pierce noted that the now familiar ring axioms held for square matrices - another early example of the axiomatic approach to rings.
• In 1905 he proved that every finite division ring (a ring in which every non-zero element has a multiplicative inverse) is commutative and so is a field.
• In 1908 Wedderburn had the important idea of splitting the study of a ring into two parts, one part he called the radical, the part which was left being called semi-simple.
• The Wedderburn theory was extended to non-commutative rings satisfying both ascending and descending finiteness conditions (called chain conditions) by Artin in 1927.
• The breakthrough here was made in 1945 by Jacobson who was a student of Wedderburn's using ideas of Perlis in 1942.
• It is interesting to note that this fundamental work by Jacobson hinges on the idea of the "Jacobson radical" of a ring which is an analogue of a group theory idea due to Frattini as early as 1885.

73. Mathematics and Architecture
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Salingaros writes in [',' Z Sagdic, Ottoman architecture: relationships between architectural design and mathematics in architect Sinan&#8217;s works, in Nexus III : architecture and mathematics, Ferrara, June 4-7, 2000 (Pisa, 2000), 123-132.','27]:- .
• The golden number is (1 + √5)/2 = 1.618033989 and an angle based on this will have size arcsec(1.618033989) = 51° 50'.
• Now the sides of the Great Pyramid rise at an angle of 5176; 52'.
• Is this a coincidence? F Rober, in 1855, was the first to argue that the golden number had been used in the construction of the pyramids.
• The authors of [',' L Pepe, Architecture and mathematics in Ferrara from the thirteenth to the eighteenth centuries, Nexus III : architecture and mathematics, Ferrara, 2000 (Pisa, 2000), 87-104.','23], however, suggest reasons for the occurrence of many of the nice numbers, in particular numbers close to powers of the golden number, as arising from the building techniques used rather than being deliberate decisions of the architects.
• Berger, in [',' E Berger, Bauwerk und Plastik des Parthenon, in Antike Kunst (Basel, 1980).','11], makes a study of the way that the Pythagorean ideas of ratios of small numbers were used in the construction of the Temple of Athena Parthenos.
• The length of the Temple is 69.5 m, its width is 30.88 m and the height at the cornice is 13.72 m.
• height : width : length = 16 : 36 : 81 .
• In the same work of 1855 he also argued that the golden number was used in the construction of the Temple of Athena on the Parthenon.
• We are fortunate to know quite a bit about the mathematical methods of ancient architecture through the work De architectura &#9417; by Vitruvius.
• One of the remarkable parts of De architectura &#9417; is Book 5 where Vitruvius discusses acoustics.
• Before we leave Vitruvius's De architectura &#9417; it is worth noting that, although today we see Vitruvius more as a practical man rather than as a scholar, nevertheless Cardan included him in his list of the twelve leading thinkers of all time.
• Argan writes [',' G C Argan, Brunelleschi (Milan, 1955).','1]:- .
• He followed Alberti (1435), Durer (1525) and Burgi (1604) when in 1630 he constructed a mechanical device that enabled one to draw accurate geometric perspective.
• In 1687 he was appointed to the chair of architecture at the Academie Royale.
• An example of a person who excelled in architecture and mathematics was Aronhold who taught at the Royal Academy of Architecture at Berlin from 1851.
• Aronhold was appointed professor at the Royal Academy of Architecture in 1863.
• From 1852 to 1861 Brioschi was professor of applied mathematics at the University of Pavia.
• Wiener studied engineering and architecture at the University of Giessen from 1843 to 1847.
• in architecture in 1926.

74. Time 1
• JOC/EFR December 2018 .

75. Greek sources 1
• JOC/EFR December 2018 .

76. Decimal time
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The new calendar was accepted and came into force at the beginning of year II, in the autumn of 1793.
• On 1 November 1795 (11 brumaire by the new calendar) a law was passed which required the creation of clocks with ten hours in the day, 100 minutes in an hour, and 100 seconds in a minute.
• Now the earth would rotate 40 degrees in an hour and, since the metre had been designed so that one quarter meridian was 10 million metres, each degree of latitude would be 100 kilometres long.
• His great five volume work Traite de Mecanique Celeste &#9417;, the first two volumes of which appeared in 1799, was written using the new units of time and angle.
• In 1884 an international conference was held in Washington in the United States with the aim of choosing an internationally agreed line for the zero of longitude.
• Of course there is a strong connection between time and angle since the earth rotates through the circumference of a circle in a day, in fact in 10 grads per hour with these units.
• His table, put before the commission on 7 April 1897, ignored powers of 10 in the factors (since no effort was required to multiply or divide by 10).

77. Cartography
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The wall painting was discovered in 1963 near the modern Ankara in Turkey.
• This line is, to a quite high degree of accuracy, 3676; north and Eratosthenes chose it since it divided the world as he knew it into two fairly equal parts and defined the longest east-west extent known.
• Right at the beginning Ptolemy identifies two distinct types of cartography, the first being [',' J L Berggren and A Jones, Ptolemy&#8217;s Geography : An annotated translation of the theoretical chapters (Princeton, 2000).','1]:- .
• The second type is [',' J L Berggren and A Jones, Ptolemy&#8217;s Geography : An annotated translation of the theoretical chapters (Princeton, 2000).','1]:- .
• He followed previous cartographers in dividing the circle of the equator into 36076; and took the equator as the basis for the north-south coordinate system.
• Thus the line of latitude through Rhodes and the Pillars of Hercules (present day Gibraltar) was 3676; and this line divided the world as Ptolemy knew it fairly equally into two.
• Therefore instead of the Mediterranean covering 4276; as it should, it covers 6276; in Ptolemy's coordinates.
• In [',' F J Swetz, The Sea Island mathematical manual : surveying and mathematics in ancient China (Pennsylvania, PA, 1992).','12] Liu Hui's 3rd century work the Sea Island mathematical manual is studied.
• Kartenband (Frankfurt am Main, 2000).','10], has done much to demonstrate that the medieval Islamic geographers had an important influence on the development of geography in Europe up to 1800.
• A detailed description of this projection is given in [',' A Ahmedov and B A Rozenfel&#8217;d, &#8217;&#8217;Cartography&#8217;&#8217; - one of Biruni&#8217;s first essays to have reached us (Russian), in Mathematics in the East in the Middle Ages (Russian) (Tashkent, 1978), 127-153.
• ','17].
• The Catalan World Map produced in 1375 was the work of Abraham Cresques from Palma in Majorca.
• The first steps involved the translation of Ptolemy's Geography into Latin which was begun by Emmanuel Chrysoloras and completed in 1410 by Jacobus Angelus.
• In 1457 he was commissioned by the King of Portugal to produce a new world map containing details of the new lands discovered by the Portuguese explorers, and charts drawn by these explorers were sent to him.
• The first printed version of Ptolemy's Geography appeared in 1475 being the Latin translation referred to above.
• The date of the first edition to contain maps is still disputed but may be the one printed in Rome in 1478 which contained 27 maps.
• New maps were added to various editions to include more accurate and detailed information about Europe, the first being in the Florence edition of 1480 which contained new maps of France, Italy, Spain and Palestine based on recent knowledge.
• The first to show the New World was a new edition of the 1475 Rome edition, which appeared in 1508 with 34 maps.
• The edition which many consider to be the first modern atlas (although the term 'atlas' was not used until Gerardus Mercator coined it around 1578) was published in Strasburg in 1513 with 27 maps of the ancient world and 20 new maps based on recent knowledge produced by Martin Waldseemuller.
• Waldseemuller's map of the world was the first to cover 36076; of longitude and to show the complete coast of Africa.
• Another first for Waldseemuller occurred in an earlier work in 1507 in which he proposed the naming of America (see [',' J N Wilford, The mapmakers : the story of the great pioneers in cartography from antiquity to the space age (New York, 1981).','16] where the following quotation is given):- .
• He set up a new press in Nuremburg in 1472 and announced his intention to publish maps and books including Ptolemy's Geography.
• Werner's most famous work on geography is In Hoc Opere Haec Cotinentur Moua Translatio Primi Libri Geographicae Cl'Ptolomaei &#9417; written in 1514.
• He employed his ideas of perspective on maps, and in particular he collaborated with Johann Stabius in the construction of globes in 1515.
• Apianus, a noted mathematician, began his publishing career with producing a world map Typus orbis universalis &#9417; which he based on an earlier 1520 world map by Martin Waldseemuller.
• His 1524 publication Cosmographia seu descriptio totis orbis &#9417; was a work based largely on Ptolemy which provided an introduction to astronomy, geography, cartography, surveying, navigation, weather and climate, the shape of the earth, map projections, and mathematical instruments.
• In 1530 he published On the Principles of Astronomy and Cosmography, with Instruction for the Use of Globes, and Information on the World and on Islands and Other Places Recently Discovered which made major contributions to cartography.
• In 1533 Gemma Frisius published Libellus de locurum &#9417; which described the theory of trigonometric surveying and in particular contains the first proposal to use triangulation as a method of accurately locating places.
• A new globe which Gerardus Mercator produced in 1541 was the first to have rhumb lines shown on it.
• This work was an important stage in his developing the idea of the Mercator projection which he first used in 1569 for a wall map of the world on 18 separate sheets.
• Edward Wright published mathematical tables to be used in calculating Mercator's projection in 1599, see [',' A V Dorofeeva, From the history of the discovery of the Mercator projection (Russian), Mat.
• v Shkole (3) (1988), i; 81.','20] for details.
• He published the Theatrum orbis terrarum &#9417; in 1570 which consisted of 70 maps presented in a uniform style using the most up-to-date data.
• USA in 1884 had delegates from 26 countries.
• In 1891 there was an International Geographical Congress in Bern which established the International Map of the World.
• The scale was to be 1:1000000 and several nations agreed to cooperate to produce a world map to this standard.

78. Zero
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended).
• For example Mukherjee in [',' R Mukherjee, Discovery of zero and its impact on Indian mathematics (Calcutta, 1991).','6] claims:- .
• If this were true then 0 5; ∞ must be equal to every number n, so all numbers are equal.
• It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
• In 1247 the Chinese mathematician Qin Jiushao wrote Mathematical treatise in nine sections which uses the symbol O for zero.
• A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
• 29 (5) (1998),729--744.','12] write:- .
• In Liber Abaci &#9417; he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that.
• Recently many people throughout the world celebrated the new millennium on 1 January 2000.
• Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001.

79. Classical time
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Later a more accurate value of 3651/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account.
• By the time the Roman architect Vitruvius wrote De architectura &#9417; shortly before 27 BC, he was able to describe 13 different designs of sundial in Book 9 of his work.
• Of course the sun could not be used to tell the time at night and clepsydras or water clocks were in use in Egypt by 1500 BC.
• (5) 73 (11) (1987), 465-476.','15].
• Progress in timekeeping in Europe was non-existent from around 500 AD to 1300 AD, but in other countries progress did continue with mechanical clocks being introduced in China.
• We will return to the Strasbourg astronomical clock in a moment, but first let us consider the 14th century work De proportionibus proportionum &#9417; by Oresme.
• A decision to replace the clock was made and work began in the cathedral on a new clock in 1547.
• However the cathedral returned to be a Roman Catholic one soon after construction of the new clock began, the project was put on hold and only restarted in 1571 when the cathedral was again a Protestant church.
• Gemma Frisius wrote in 1530:- .
• When we have completed a journey of 15 or 20 miles, it may please us to learn the difference of longitude between where we have reached and our place of departure.
• Several large prizes were offered for a solution to the problem of determining longitude and Galileo tried the persuade the Spanish Court in 1616 that he could determine absolute time using Jupiter's moons and, after failing to convince them, tried to persuade Holland of his method when they offered a large prize in 1636.
• Long before his discovery of Jupiter's moons he discovered the fundamental property of the pendulum in 1583.
• The first to succeed in making a pendulum clock was Huygens in 1656, see [',' A Elzinga, Christian Huygens and the elimination of time, Acad.
• (5) 73 (10) (1987), 394-404.','11].
• The ultimate version of the mechanical universe appeared in Newton's Principia in 1687.
• This was a major new idea regarding time, see [',' R T W Arthur, Newton&#8217;s fluxions and equably flowing time, Stud.
• 26 (2) (1995), 323-351.','6], [',' E S de Oliveira Barra, Newton on motion, space and time (Portuguese), Cad.
• (3) 3 (1-2) (1993), 85-115.','9], [',' R Rynasiewicz, By their properties, causes and effects : Newton&#8217;s scholium on time, space, place and motion.
• 26 (1) (1995), 133-153.','16] and [',' R Rynasiewicz, By their properties, causes and effects : Newton&#8217;s scholium on time, space, place and motion.
• 26 (2) (1995), 295-321.','17].
• It was only in the middle of the 19th century that the second law of thermodynamics was proposed by Clausiusand this was the first law to lack symmetry in the direction of time, see [',' G Bierhalter, Zyklische Zeitvorstellung, Zeitrichtung und die fruhen Versuche einer Deduktion des Zweiten Hauptsatzes der Thermodynamik, Centaurus 33 (4) (1990), 345-367.','7].
• Clausius read a paper to the Berlin Academy on 18 February 1850 which contained this second law of thermodynamics.

80. Forgery 2
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1739 Maupertuis became friendly with both du Chatelet and Voltaire, spending some time living at their home at Cirey.
• He had become king of Prussia on the death of his father in 1740.
• He began to invite top people to participate in his Academy and he approached both Voltaire and Maupertuis in 1740.
• It was Voltaire who recommended Maupertuis to Frederick for the position of President of the Berlin Academy in 1740.
• On 12 May 1746 Maupertuis was officially appointed as president of the Berlin Academy and four years later Frederick persuaded Voltaire to come to Berlin too.
• This was not a serious quarrel, so when Konig visited Berlin again in 1751 and passed a manuscript of an article to Maupertuis for publication, the head of the Academy simply accepted it without reading it.
• Beeson writes [','D Beeson, Maupertuis : an intellectual biography (Oxford, 1992).','1]:- .
• 15">Konig relates that he had been tempted to write against Maupertuis's exposition of least-action theory from the moment he first read it, in 1749.
• In March 1751 Konig's paper, approved for publication by Maupertuis, appeared in print.
• On 13 April 1752 the Berlin Academy met to essentially try Konig on a charge of forgery.
• Many members chose not to attend the meeting out of embarrassment and one of those who did said [','D Beeson, Maupertuis : an intellectual biography (Oxford, 1992).','1]:- .
• On 18 September an anonymous pamphlet A reply from an Academician of Berlin to an Academician of Paris defending Konig appeared.
• Frederick now entered the fray publishing on 15 October an anonymous pamphlet Letter to the public defending Maupertuis which, despite claims of anonymity, clearly indicated its author by having the Prussian eagle, the crown and the sceptre on its title page.
• Maupertuis had published Letter on the progress of science in 1752 and Voltaire used this as a way to attack him.
• Beeson writes in [','D Beeson, Maupertuis : an intellectual biography (Oxford, 1992).','1]:- .
• Gerhardt undertook the research in 1898 which provided a convincing argument that the quotation was genuine.
• This suggestion is also impossible, as was shown by Kabitz in 1913, and although the evidence all suggests that the quotation is genuine, it has never been established who the likely recipient was.
• Versions of all four letters quoted by Konig in his pamphlet Appeal to the public were found by Kabitz in a collection of copies of Leibniz letters owned by the Bernoulli's and he published his findings in 1913.

81. Real numbers 1
• JOC/EFR December 2018 .

82. Light 1
• JOC/EFR December 2018 .

83. Bourbaki 1
• JOC/EFR December 2018 .

84. References for Set theory
• 26 (1) (1984), 9-18.
• I H Anellis, Russell's earliest interpretation of Cantorian set theory, 1896-1900, Philos.
• (2) 2 (1) (1987), 1-31.
• I H Anellis, Russell's earliest reactions to Cantorian set theory, 1896-1900, in Axiomatic set theory (Providence, R.I., 1984), 1-11.
• I Angelelli, "Class as one" and "class as many" before modern set theory, Historia Mathematica 6 (3) (1979), 305-309.
• I Copi, The Burali-Forti paradox, Philosophy of Science 25 (1958), 281-286.
• A R Garciadiego, Bertrand Russell and the origin of paradoxes in set theory (Spanish), Mathesis 4 (1) (1988), 113-130.
• H Gispert, La theorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue ..
• 1 (1) (1995), 39-81.
• I Grattan-Guinness (ed.), Selected essays on the history of set theory and logics (1906-1918) by Philip E B Jourdain (Bologna, 1991).
• G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
• Symbolic Logic 2 (1) (1996), 1-71.
• A Kertesz, Georg Cantor (1845-1918) : Schopfer der Mengenlehre (Halle, 1983).
• Symbolic Logic 53 (1) (1988), 2-6.
• DDR 4 (1985), 9-22.
• G H Moore, The Origins of Zermelo's axiomatisation of set theory, Journal of Philosophical Logic 7 (1978), 307-329.
• G H Moore, Ernst Zermelo, A E Harward, and the axiomatization of set theory, Historia Mathematica 3 (2) (1976), 206-209.
• W Purkert, Cantor and the Burali-Forti paradox, The Monist 67 (1984), 92-106.
• A 38 (1986), 313-327.
• G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).

85. Forgery 1
• JOC/EFR December 2018 .

86. Fermat's last theorem
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Pierre de Fermat died in 1665.
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• In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14.
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• The n = 7 case was finally solved by Lame in 1839.
Go directly to this paragraph
• On 1 March of that year Lame announced to the Paris Academie that he had proved Fermat's Last Theorem.
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• Wantzel claimed to have proved it on 15 March but his argument .
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• The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846.
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• In 1915 Jensen proved that the number of irregular primes is infinite.
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• Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.
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• In 1983 a major contribution was made by Gerd Faltings who proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn.
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• The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem.
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• In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrucken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.
Go directly to this paragraph
• The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA.
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• If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.
• In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof.

87. References for Babylonian Pythagoras
• G G Joseph, The crest of the peacock (London, 1991).
• A Ahmad, On Babylonian and Vedic square root of 2, Ganita Bharati 16 (1-4) 1994), 1-4.
• C Anagnostakis and B R Goldstein, On an error in the Babylonian table of Pythagorean triples, Centaurus 18 (1973/74), 64-66.
• 17 (1) (1986), 22-31.
• E M Bruins, Fermat problems in Babylonian mathematics, Janus 53 (1966), 194-211.
• 52 (1949), 629-632.
• 41 (1957), 25-28.
• The errors on Plimpton 322, Sumer 11 (1955), 117-121.
• 51 (1948), 332-341.
• M Caveing, La tablette babylonienne AO 17264 du Musee du Louvre et le probleme des six freres, Historia Math.
• 12 (1) (1985), 6-24.
• 37 (1995), 29-47.
• 25 (4) (1998), 366-378.
• 33 (1) (1981), 57-64.
• 8 (3) (1981), 277-318.
• 16 (1953), 54-56.
• 27 (1964), 139-141.
• J Hoyrup, The Babylonian cellar text BM 85200+ VAT 6599: Retranslation and analysis, in Amphora (Basel, 1992), 315-358.
• 26 (1) (1996), 155-162.
• 24 (3-4) (1988), 37-41.
• K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
• (2) 9 (2) (1999), 127-133.
• (2) 1 (1) (1991), 59-62.
• D J de Solla Price, The Babylonian "Pythagorean triangle" tablet, Centaurus 10 (1964/1965), 1-13.
• O Schmidt, On Plimpton 322: Pythagorean numbers in Babylonian mathematics, Centaurus 24 (1980), 4-13.
• 1 (2) (1981), 103-132.

88. Neptune and Pluto
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The first planet to be discovered was Uranus by William and Caroline Herschel on 13 March 1781.
Go directly to this paragraph
• See [','S Drake and C T Kowal, Galileo&#8217;s sighting of Neptune, Scientific American 243 (6) (1980) 52-59.','1].
Go directly to this paragraph
• Lalande (1732-1807), a French astronomer whose tables of the planetary positions were the most accurate until the 19th Century, recorded Neptune on the 8th and 10th of May 1795 without recognising that it was not a star.
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• John Herschel, who we shall see in a moment was to be involved with the discovery of Neptune, recorded Neptune on 14 July 1830 believing it to be a star.
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• Von Lamont (1805-1879), a Scottish born astronomer who lived most of his life in Munich, is famed for his determination of the orbits of moons of Saturn and Uranus, and also for discovering the periodic fluctuation of the Earth's magnetic field.
• Delambre computed tables of planetary positions Tables du Soleil, de Jupiter, de Saturne, d'Uranus et des satellites de Jupiter &#9417; published in 1792.
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• Bouvard (1767-1843), a French astronomer who was director of the Paris Observatory, had already published accurate tables of the orbits of Jupiter and Saturn in 1808 and he now undertook to produce a corrected version of Delambre's tables for Uranus.
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• He published his new tables of Uranus in 1821 but wrote .
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• On 3 July 1841 Adams, while still an undergraduate at Cambridge, wrote .
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• As well as the orbit he had calculated the mass of the planet and its position on 1 October 1845.
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• On 10 November Le Verrier published his first paper on his investigations.
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• On 1 June 1846 Le Verrier published a second paper in which he showed that a variety of other possible causes could not explain the orbit of Uranus, and deduced that the only possible cause could be a planet further from the Sun than Uranus.
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• He gave some details of a possible orbit of the "new planet" with a predicted position for the beginning of 1847.
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• He observed on the nights of 29, 30 July, 4, 12 August and recorded the results.
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• He checked out his methods by comparing the first 39 stars recorded on 12 August and checking that they appeared on his 30 July records.
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• If he had continued his comparison he would have discovered the "new planet" which he had recorded on 12 August but which had not been in the search area on 30 July.
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• On 10 September John Herschel addressed a meeting of the British Association in Southampton.
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• Le Verrier wrote to the German astronomer Galle on 18 September asking him to search for the "new planet" at his predicted location.
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• Lassell began observing on 2 October and on 10 October he discovered Neptune's moon Triton.
• Both had predicted positions which were very close to the actual position but both had predicted orbits which meant that Neptune would only be close to its predicted position around 1840-1850 while at other times (it takes about 165 years to complete one orbit and has not yet completed one since its discovery) it would be far from the positions predicted by both Adams and Le Verrier.
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• Percival Lowell (1855-1916), an American astronomer, was interested in Mars.
• In 1905 Lowell completed his analysis of the data and predicted the existence of a planet beyond Neptune which was responsible for the perturbations.
• By 1905, of course, astronomical observations had greatly improved due to photography.
• A search was begun at the Flagstaff Observatory in 1915 and for two years they photographed the area of the sky in which "Planet X", as Lowell called it, was predicted.
• Lowell presented has paper Memoir on a Trans-Neptunian Planet to the American Academy on 13 January 1915.
• Another American astronomer, William Henry Pickering (1858-1938), actually constructed Lowell's Flagstaff Observatory in 1894.
• He moved to the Harvard College Observatory and, in 1919, he also predicted a position of a trans-Neptunian planet using the discrepancies in both the orbits of Uranus and Neptune as data.
• on the afternoon of February 18, 1930, I suddenly came upon the images of Pluto! The experience was an intense thrill, because the nature of the object was apparent at first sight.
• The planet was photographed every night from then on to confirm the observation and on 13 March 1930, the 75th anniversary of Lowell's birth and the 149th anniversary of Uranus's discovery (it is a remarkable coincidence that these should be the same day), an announcement was made from Flagstaff.

89. ETH history
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Konjunkturen der ETH Zurich 1855 - 2005 &#9417; by D Gugerli, P Kupper and D Speich, Chronos-Verlag, Zurich, 2005.
• The origins of the Federal Polytechnic in Zurich are closely linked to the foundation of modern Switzerland as a federal state in 1848.
• The Swiss Confederation was founded in 1291, although it has to be said that the three founding cantons Uri, Schwyz, and Unterwalden merely renewed an already existing confederacy.
• As a result of Napoleon's Helvetic campaign the Helvetic Republic replaced the old Confederacy in 1798.
• 115-116].
• As an aside, Switzerland chose neutrality at the same time, in 1815.
• Whilst the Confederacy became neutral in 1515, it was more out of necessity than choice as the conflicts between the different cantons did not allow for any external military engagement.
• Returning to the Swiss Confederation, the so-called Restoration (1814-1830) was followed by a Regeneration period.
• 121-122].
• Tensions between the cantons resulted in two so-called "Freischarenzuge" in 1844 and 1845, respectively: Volunteer troops of radical Liberals attempted to overthrow the catholic government of canton Luzern and demanded that Jesuits should be banished from the country.
• After numerous discussions the Swiss Federal Constitution was adopted on 12 September 1848, and thus Switzerland became a federal state.
• A number of polytechnics were founded in Germany from 1825-1836, and then again from 1860 onwards [',' D Gugerli, P Kupper and D Speich, Die Zukunftsmaschine.
• In the autumn of 1855 the Federal Polytechnic opened its doors for students.
• From 1866-1899, the period that is of most interest for this thesis, there were eight Schools: .
• School of Forestry; from 1871 School of Agriculture & Forestry, comprising subdivisions for forestry, agriculture, and cultural engineering .
• Preparatory Course in Mathematics, until 1881 .
• In 1899 the Department of Military Sciences was established, before then courses in that area were taught as elective subjects.
• Nevertheless, in 1912 about 10% of Polytechnic professors were mathematicians.
• This high percentage is partly explained by the fact that they also taught in other Schools [','G Frei and U Stammbach, Hermann Weyl und die Mathematik an der ETH Zurich 1913-1930, (Birkhauser, Basel, 1992)','3, p.
• Guggenbuhl notes that Kappeler attached great value to  a suitable personality and teaching abilities' [',' G Guggenbuhl, Geschichte der Eidgenossischen Technischen Hochschule in Zurich, in: Eidgenossische Technische Hochschule 1855-1955.
• (Ecole Polytechnique Federale, Buchverlag der Neuen Zurcher Zeitung, Zurich, 1955, 1-257)','6, p.
• During the 1880s and 1890s Chemistry and Physics moved into new buildings equipped with state-of-the-art laboratories, and a mechanical engineering laboratory was established in 1900.
• Moreover, a laboratory for material testing was founded in 1880 as part of the Polytechnic; it has since developed into the Swiss Federal Laboratories for Materials Science and Technology (German acronym: EMPA).
• In order to increase the number of entrants, entry requirements were lowered to a certain degree in 1859; students were now examined in 6-7 subjects, had to write an essay in their native language and submit evidence of proficiency in the languages of instruction.
• The Polytechnic developed two strategies: negotiations with secondary schools and, as an interim measure, establishing a Preparatory Course in 1859.
• For several decades it also shared resources - staff and rooms - with the University of Zurich, which had been established in 1833.
• When it moved into its own building in 1864 the University was given its own area.
• 106-109].
• In 1881, partly due to a petition by its alumni association GEP, the Polytechnic revised some of its regulations.
• The German Emperor awarded the Institute of Technology in Charlottenburg, Berlin (today University of Technology Berlin) the right to confer doctorates in 1899, and many German polytechnics followed suit.
• As the American William K Tate wrote in 1913:  The Polytechnic School at Zurich ranks among the world's greatest technical universities' [','M Burri, Hochschul-Rankings: Instrumente der Internationalisierung:','1; quoted in the original].

90. Special relativity
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The classical laws of physics were formulated by Newton in the Principia in 1687.
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• Prompted by Maxwell's ideas, Michelson began his own terrestrial experiments and in 1881 he reported .
• Lorentz wrote a paper in 1886 where he criticised Michelson's experiment and really was not worried by the experimental result which he dismissed being doubtful of its accuracy.
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• Michelson was persuaded by Thomson and others to repeat the experiment and he did so with Morley, again reporting that no effect had been found in 1887.
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• [Michelson and Morley were to refine their experiment and repeat it many times up to 1929.] .
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• Also in 1887 Voigt first wrote down the transformations .
• Voigt corresponded with Lorentz about the Michelson-Morley experiment in 1887 and 1888 but Lorentz does not seem to have learnt of the transformations at that stage.
• Lorentz however was now greatly worried by the new Michelson-Morley experiment of 1887.
• In 1889 a short paper was published by the Irish physicist George FitzGerald in Science.
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• Lorentz was unaware of FitzGerald's paper and in 1892 he proposed an almost identical contraction in a paper which now took the Michelson-Morley experiment very seriously.
• When it was pointed out to Lorentz in 1894 that FitzGerald had published a similar theory he wrote to FitzGerald who replied that he had sent an article to Science but I do not know if they ever published it .
• Larmor wrote an article in 1898 Ether and matter in which he wrote down the Lorentz transformations (still not written down by Lorentz) and showed that the FitzGerald-Lorentz contraction was a consequence.
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• Lorentz wrote down the transformations, now named after him, in a paper of 1899, being the third person to write them down.
• The most amazing article relating to special relativity to be published before 1900 was a paper of Poincare La mesure du temps &#9417; which appeared in 1898.
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• By 1900 the concept of the ether as a material substance was being questioned.
• Poincare, in his opening address to the Paris Congress in 1900, asked Does the ether really exist? In 1904 Poincare came very close to the theory of special relativity in an address to the International Congress of Arts and Science in St Louis.
• June of 1905 was a good month for papers on relativity, on the 5th June Poincare communicated an important work Sur la dynamique de l'electron &#9417; while Einstein's first paper on relativity was received on 30th June.
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• The first paper on special relativity, other than by Einstein, was written in 1908 by Planck.
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• Also in 1908 Minkowski published an important paper on relativity, presenting the Maxwell-Lorentz equations in tensor form.
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• When Poincare lectured in Gottingen in 1909 on relativity he did not mention Einstein at all.
• He gave a lecture in 1913 when he remarked how rapidly relativity had been accepted.
• In 1912 Lorentz and Einstein were jointly proposed for a Nobel prize for their work on special relativity.
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• The recommendation is by Wien, the 1911 winner, and states .
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91. Gregory's observatory
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• On 13 May 1673 he writes again:- .
• It is likely that these 'affairs' related to the Observatory he was hoping to build for, on 10 June 1673, the University of St Andrews commissioned Gregory to go to London to purchase instruments for the Observatory.
• The full text of Gregory's letter to Flamsteed of 19 July 1673 (sent to Flamsteed father's house in Derby) can be read at THIS LINK .
• What was this building? If a large house, then would the whole 3rd storey have been a hall as Gregory describes? That it is aligned with the library and the rest of the College is not totally surprising (the 9 or 10 degrees to the meridian that Gregory describes).
• Drawing of a building located at the foot of Westburn Lane as it appeared in 1693 .
• It were tedious to write down particularlie all the instruments I have brought home, yea a larger letter wold not contein all ther names and sizes, for I have all sort: our largest quadrant is of oak, covered with brasse, 4 foot in radius and actually divided in minutes, of which we can judge 1/3 or 1/4: we have two semisextans, all of brasse, 6 foot in radius, diagonally divided, in which wee can judge 1/6 or 1/7 of a minut.
• Gregory left St Andrews in the summer of 1674 to take up his new appointment at the University of Edinburgh.
• Gregory, writing from Edinburgh, replied to James Frazer on 13 July 1675 explaining his reasons for not replying to his earlier letter and for leaving St Andrews: .
• James Frazer replied to Gregory on 10 August 1675: .
• John Loveday, in Diary of a Tour in 1732, records that: .
• Again it could not have been undertaken since the Senate Minutes of 10 June 1729 record: .
• In November 1736 the University sold off lead and timber from the Observatory and, in 1761, it took down the remaining unsafe walls.
• Something must have survived, however, since "Observatory in ruins" still appears on maps of the town in 1820.

92. 20th century time
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1870 Carl Neumann questioned Newton's law of inertia.
• P G Tait answered Carl Neumann's problem of the inertial clock in 1883 and in doing so he essentially showed that Newton's absolute space was an unnecessary concept, for he could create an absolute space framework.
• Mach published a history of mechanics in 1883.
• In 1898 Poincare wrote a paper in which he asked two highly significant questions about time.
• In 1902 Poincare wrote another paper relevant to our topic.
• Einstein decided that time was the whole key to understanding the universe, see [',' J Ehlers, Concepts of time in classical physics, in Time, temporality, now, Tegernsee, 1996 (Berlin, 1997), 191-200.','14].
• We have already talked about experiments capable of detecting time differences of 1/1000000000 of a second in an hour.
• R J Rudd introduced a genuine free pendulum clock in 1898, then W H Shortt introduced a clock with two pendulums in 1921.
• In 1928 a totally new type of clock was built by W A Marrison at Bell Laboratories, namely the quartz crystal clock.
• In 1949 the National Bureau of Standards in the United States built the first atomic clock, using ammonia.
• The accuracy was such that by 1967 the second was changed from its original astronomical definition as a fraction of a day, to a definition where the second was given as 9,192,631,770 oscillations of the cesium atom's resonant frequency.
• By 1993 the National Institute of Standards and Technology in the United States had built an atomic clock accurate to five parts in 1015.
• Let us now look at another revolution in time which took place in the 20th century with the discovery of quantum mechanics, see [',' G Bruzzaniti, The structure of physical time : the problem of the discretization of spacetime through the evolution of the concept of &#8217;&#8217;chronon&#8217;&#8217; (Italian), Nuncius Ann.
• 3 (2) (1988), 101-147.','10].
• Heisenberg discovered the Uncertainty Principle in 1927.
• The idea was first put forward by Einstein, together with Nathan Rosen and Boris Podolsky in 1935 and it is known by the initials of its proposers as the EPR experiment.
• An interpretation of quantum theory put forward by Hugh Everett in 1957 is the many worlds interpretation.
• Penrose, in [',' R Penrose, The emperor&#8217;s new mind : Concerning computers, minds, and the laws of physics (New York, 1989).','6], takes a different approach but reaches similar conclusions about our perception of time:- .

93. References for Classical light
• N Kipnis, History of the principle of interference of light (Basel, 1991).
• A I Sabra, Theories of light : From Descartes to Newton (Cambridge-New York, 1981).
• E J Atzema, All phenomena of light that depend on mathematics : a sketch of the development of nineteenth-century geometrical optics, Tractrix 5 (1993), 45-80.
• (5) 9 (1985), 255-261.
• 23 (1) (1992), 39-74.
• 3 (2) (1985), 119-151.
• 55 (4) (1998), 401-420.
• 50 (3-4) (1997), 359-393.
• S D'Agostino, Maxwell's dimensional approach to the velocity of light, Centaurus 29 (3) (1986), 178-204.
• (N.S.) 33 (1-3) (1996), 5-51.
• The measurements for absolute electromagnetic units and the velocity of light, Scientia (Milano) 113 (5-8) (1978), 469-480.
• (5) 9 (1985), 147-167.
• 42 (4) (1991), 315-386.
• J Eisenstaedt, Dark bodies and black holes, magic circles and Montgolfiers : light and gravitation from Newton to Einstein, in Einstein in context (Cambridge, 1993), 83-106.
• 39 (3) (1982), 297-310.
• J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
• 37 (1) (1980), 59-79.
• 45 (5) (1988), 521-533.
• C Huygens, Treatise on light, in Great Books of the Western World 34, Encyclopaedia Britannica (Chicago- London- Toronto, 1952), 545-619.
• 63 (2) (1985), 265-274.
• 26 (1) (1987), 63-70..
• 8 (1963), 5-42.
• 11 (1-4) (1969), 390-407.
• H Nakajima, Two kinds of modification theory of light : some new observations on the Newton-Hooke controversy of 1672 concerning the nature of light, Ann.
• 41 (3) (1984), 261-278.
• 13 (1977), 339-376.
• 61 (2) (1993), 108-112.
• XXII: Physics (Russian) (Moscow, 1979), 128-140.
• J Renn, T Sauer and J Stachel, The origin of gravitational lensing : A postscript to Einstein's 1936 Science paper: "Lens-like action of a star by the deviation of light in the gravitational field", Science 84 (1936), 506-507, by A Einstein, Science 275 (5297) (1997), 184-186.
• 51 (2-3) (1998), 347-354.
• L Rosenfeld, The velocity of light and the evolution of electrodynamics, Nuovo Cimento (10) 5 (1956), Supp.
• 1630-1669.
• L Rozenfel'd, Gravitational effects of light (Russian), in Einstein collection, 1980-1981 "Nauka" (Moscow, 1985), 255-266; 335.
• 26 (1) (1982), 1-12.
• 11 (1973/74), 134-266.
• 5 (1974), 239-296.
• A E Shapiro, The evolving structure of Newton's theory of white light and color, Isis 71 (257) (1980), 211-235.
• A E Shapiro, The gradual acceptance of Newton's theory of light and color, 1672-1727, Perspect.
• 4 (1) (1996), 59-140.
• J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein : the formative years, 1879-1909 (Boston, MA, 2000), 231-251.
• M Suffczy'nski, Velocity of light, in Isaac Newton's Philosophiae naturalis principia mathematica, Lublin, 1987 (Singapore, 1988), 69-71.
• 36 (3) (1993), 253-294.
• (1998), Suppl.
• 40, 1-171.
• 37 (2) (1980), 179-187.

94. Mayan mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Hernan Cortes, excited by stories of the lands which Columbus had recently discovered, sailed from Spain in 1505 landing in Hispaniola which is now Santo Domingo.
• After farming there for some years he sailed with Velazquez to conquer Cuba in 1511.
• He was twice elected major of Santiago then, on 18 February 1519, he sailed for the coast of Yucatan with a force of 11 ships, 508 soldiers, 100 sailors, and 16 horses.
• He captured Tenochtitlan before the end of 1519 (the city was rebuilt as Mexico City in 1521) and the Aztec empire fell to Cortes before the end of 1521.
• He joined the Franciscan Order in 1541 when about 17 years old and requested that he be sent to the New World as a missionary.
• Certainly what he then did was to write a book Relacion de las cosas de Yucatan &#9417; (1566) which describes the hieroglyphics, customs, temples, religious practices and history of the Mayans which his own actions had done so much to eradicate.
• The book was lost for many years but rediscovered in Madrid three hundred years later in 1869.
• In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20's up to 19, the next the number of 400's up to 19, etc.
• However although the Maya number system starts this way with the units up to 19 and the 20's up to 19, it changes in the third place and this denotes the number of 360's up to 19 instead of the number of 400's.
• After this the system reverts to multiples of 20 so the fourth place is the number of 18 5; 202, the next the number of 18 5; 203 and so on.
• For example [ 8;14;3;1;12 ] represents .
• 12 + 1 × 20 + 3 5; 18 5; 20 + 14 5; 18 5; 202 + 8 5; 18 5; 203 = 1253912.
• As a second example [ 9;8;9;13;0 ] represents .
• 0 + 13 5; 20 + 9 5; 18 5; 20 + 8 5; 18 5; 202 + 9 5; 18 5; 203 =1357100.
• It contained 13 "months" of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19.
• This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short "month" of only 5 days that was called the Wayeb.
• At any rate having two calendars, one with 260 days and the other with 365 days, meant that the two would calendars would return to the same cycle after lcm(260, 365) = 18980 days.
• What date was the Mayan creation date? The date most often taken is 12 August 3113 BC but we should say straightaway that not all historians agree that this was the zero of this so-called "Long Count".
• [ 8;14;3;1;12 ] .
• 12 + 1 × 20 + 3 5; 18 5; 20 + 14 5; 18 5; 202 + 8 5; 18 5; 203 .
• which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.
• [ 9;8;9;13;0 ] .
• 0 + 13 5; 20 + 9 5; 18 5; 20 + 8 5; 18 5; 202 + 9 5; 18 5; 203 .
• which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.
• [ 9;8;9;13;0 ] = 0 + 13 5; 20 + 9 5; 18 5; 20 + 8 5; 18 5; 202 + 9 5; 18 5; 203 = 1357100 .
• [ 9;8;9;13 ] = 13 + 9 5; 20 + 8 5; 18 5; 20 + 9 5; 18 5; 202 = 67873.
• Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20 5; 67873 = 1357460 which is not equal to 1357100.
• For when we multiple [ 9;8;9;13 ] by 20 we get 9 5; 400 where in [ 9;8;9;13;0 ] we have 9 5; 360.
• 68 (3) (1980), 249-255.','15] demonstrate.
• 10 (1985), 443-453.','19]:- .
• With such crude instruments the Maya were able to calculate the length of the year to be 365.242 days (the modern value is 365.242198 days).
• II 203 (1994), 101-116.','14] describes seven types of frieze ornaments occurring on Mayan buildings from the period 600 AD to 900 AD in the Puuc region of the Yucatan.

95. Arabic numerals
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Now in [',' G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).','1] (where a longer quote is given) Ifrah tries to determine which Indian work is referred to.
• The Arabic text is lost but a twelfth century Latin translation, Algoritmi de numero Indorum &#9417; gave rise to the word algorithm deriving from his name in the title.
• The Latin text certainly describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
• Around the middle of the tenth century al-Uqlidisi wrote Kitab al-fusul fi al-hisab al-Hindi &#9417; which is the earliest surviving book that presents the Indian system.
• A number, say 4376; 21' 14", would have been written as "mj ka yd" in this base 60 version of the "abjad" letters for calculating.
• The numerals from al-Biruni's treatise copied in 1082 .
• In fact a closer look will show that between 969 and 1082 the biggest change in the numerals was the fact that the 2 and the 3 have been rotated through 9076;.
• Perhaps because scribes did not have much experience at writing Indian numerals, they wrote 2 and 3 the correct way round instead of writing them rotated by 9076; so that they would appear correctly when the scroll was rotated to be read.
• Fibonacci writes in his famous book Liber abaci &#9417; published in Pisa in 1202:- .
• JOC/EFR January 2001 .

96. Maxwell's House
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The house where James Clerk Maxwell was born is at 14 India Street, Edinburgh about a fifteen minute walk from the railway station which is in the centre of Edinburgh.
• James Clerk Maxwell was born on 13th June 1831 in Edinburgh at 14 India Street, a house built for his father in that part of Edinburgh's elegant Georgian New Town which was built after the Napoleonic Wars.
• He arranged to have a house built at 14 India Street so that they could be nearer to Isabella.
• The substantial terrace house was built in 1820 and documents relating to the purchase of the house are in the Display Cabinet.
• Their son James Clerk Maxwell was born in the house at 14 India Street and he would eventually inherit the house on the death of his father, retaining the house throughout his life.
• You can see a picture of Glenlair as it was when Maxwell finally left it in 1884.
• A portrait of Jane and Frances as young girls, painted by their mother Elizabeth Cay, now hangs in 14 India Street.
• Frances's father was Robert Hodsham Cay LLD (1758-1810), a judge to the High Court of Admiralty of Scotland.
• She was eight years older than James Clerk Maxwell and she painted pictures of the family almost every day, some of which are now displayed in 14 India Street.
• It was decided that James should attend the Edinburgh Academy and in November 1841 the family travelled from Glenlair, stopping a few days at his uncles house in Penicuik, and at other relations at Newton, before reaching Isabella Wedderburn's house at 31 Heriot Row on 18 November.
• Edinburgh 10 (1880), 331-339.','5] relates James Clerk Maxwell's early days at the Edinburgh Academy:- .
• At 14 India Street, in the Display Cabinet, we [JOC and EFR] studied newspaper cuttings with the headline "They called him Dafty".
• The way they reacted on his first day at school was clear from the state in which he arrived back at 31 Heriot Row [',' L Campbell and W Garnett, The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).','1]:- .
• He spent his summers back at Glenlair and, by Jemima's pictures drawn in the summer of 1843, he seems to have slotted back into his old pastimes.
• Back in Edinburgh James was taken by his father to a meeting of the Royal Society of Edinburgh on 18 December 1843.
• The chair in which James sat to study while at 31 Heriot Row is now in the former dining room at 14 India Street, recovered with a material with a pattern depicting the digital nature of light waves, to honour one of Maxwell's great pieces of work.
• He wrote on 19 June 1844:- .
• Edinburgh 10 (1880), 331-339.','5]:- .
• One by his father, as he states in his diary entry and this is now in the possession of the Royal Society of Edinburgh, the other by James himself and this copy is on view in the Display Cabinet at 14 India Street.
• Edinburgh 10 (1880), 331-339.','5]:- .
• I still possess some of the manuscripts we exchanged in 1846 and early 1847.
• On our [JOC and EFR] visit to 14 India Street, we were fascinated when we were given the chance to examine the notebook containing the manuscripts to which Tait referred.
• For example [',' L Campbell and W Garnett, The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).','1] he wrote The Song of the Edinburgh Academy in 1848:- .
• While in his final year of study for the Mathematical Tripos at Cambridge he wrote a poem A Problem in Dynamics [',' L Campbell and W Garnett, The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).','1] which begins:- .
• One final comment must be made about our visit to 14 India Street.
Go directly to this paragraph
• Another quote, this time by Sir J J Thomson, concerns one of Maxwell's discoveries [',' J J Thomson, James Clerk Maxwell, in James Clerk Maxwell : A Commemorative Volume 1831-1931 (Cambridge, 1931), 1-44.','6]:- .
• Actually this quote by Sir J J Thomson, written as long ago as 1931, is remarkable in almost predicting the Internet.
• Sir James Jeans wrote [',' J Jeans, James Clerk Maxwell&#8217;s method, in James Clerk Maxwell : A Commemorative Volume 1831-1931 (Cambridge, 1931), 91-108.','3], also in 1931 on the centenary of Maxwell's birth:- .
Go directly to this paragraph

97. Harriot's manuscripts
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Thomas Harriot died in 1621.
Go directly to this paragraph
• The story of how knowledge of Thomas Harriot's mathematical genius has come down to us is related in several of the references, see for example [',' J Fauvel, R Flodd and R Wilson (eds.), Oxford figures : 800 years of the mathematical sciences (Oxford, 2000).','1], [',' J W Shirley, Thomas Harriot : a biography (Oxford, 1983).','2], or [',' J W Shirley (ed.), Thomas Harriot : renaissance scientist (Oxford, 1974).','4].
• Early discussions in the newly founded Royal Society of London centred around the search for Harriot's lost papers, but inquiries made from 1662 to 1669 proved fruitless, and it was finally assumed that they had been destroyed.
• The rediscovery of Harriot's papers occurred in 1784.
• Zach had been appointed as tutor to the son of Count de Bruhl so, when de Bruhl was sent to England in 1783 as Saxon Minister, Zach came to London with him.
• The Count de Bruhl visited Petworth in the summer of 1784 and there he found Harriot's manuscripts hidden among the stable accounts.
• The manuscripts had been untouched since Henry Percy's death in 1632, eleven years after Harriot's own death.
• In 1786 Zach proposed to the Oxford University Press that he publish a major biography of Harriot together with an edited edition of the most important of Harriot's manuscripts.
• Thomas Hornsby, the Savilian Professor of Astronomy at Oxford, proposed Zach for an honorary degree which was awarded in 1786.
• Also in 1786 Zach was granted the title Baron von Zach which, like his honorary doctorate, Zach used to promote himself.
• Without doing any editorial work whatsoever, and without writing any biographical material, von Zach sent some of Harriot's papers to the Principal of Brasenose College in 1794 and asked that he forward them to Oxford University Press for publication.
• Three years after receiving Harriot's papers to referee for publication, Robertson was appointed Savilian Professor of Geometry and, in 1810, Savilian Professor of Astronomy.
• Hutton wrote in his Mathematical Dictionary published in 1797 (see, for example [',' J Fauvel, R Flodd and R Wilson (eds.), Oxford figures : 800 years of the mathematical sciences (Oxford, 2000).','1]):- .
Go directly to this paragraph
• However, when the referee of the astronomical papers failed to reply, these too were sent to Robertson as a referee in 1798.
• They had sent the manuscripts back to Petworth House in 1799 having made the decision that publication in their present form was impossible.
Go directly to this paragraph
• In 1822 Robertson, angry that Playfair and others were attacking Oxford University Press, made his reports on the papers public.
Go directly to this paragraph
• It is written in [',' J Fauvel, R Flodd and R Wilson (eds.), Oxford figures : 800 years of the mathematical sciences (Oxford, 2000).','1] that he:- .
• He did, however, begin a serious study of Harriot's manuscripts but he died in 1839 leaving copious notes on the manuscripts but another opportunity for publishing a proper edition was lost.
• The will was located by Henry Stevens of Vermont (1819-1886), see [',' R C H Tanner, Henry Stevens and the associates of Thomas Harriot, in Thomas Harriot: Renaissance scientist (Oxford, 1974), 91-106.','5] and [',' R C H Tanner, The study of Thomas Harriot&#8217;s manuscripts.
• Harriot&#8217;s will, History of science 6 (Cambridge, 1967), 1-16.','6] for full details of this interesting episode.
• The book, Thomas Harriot and his associates was privately printed in London in 1900.

98. Chandrasekhar Eddington
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1930, the Indian physics student Subrahmanyan Chandrasekhar obtained a scholarship to continue his studies in the University of Cambridge after graduating from Presidency College in Madras, India.
• He had also shown that they had a stellar configuration of polytropic index 3 (a polytrope is stellar material in equilibrium under its own gravity, following an equation of state of the form P = K&#961;i1+1/n, where n is the polytropic index).
• His original result for this limit was of 0.91 times the mass of the Sun.
• It was later revised to 2.0, so Chandrasekhar changed his result to 1.44 times the mass of the Sun.
• Inspired by James Jeans' suggestion that the material in the core of a star might not follow Boyle's law, Stoner examined the effects of Fermi-Dirac degeneracy (a jamming of electrons in phase space) in the density of stellar material and derived a value for a limiting mass "above which the gravitational kinetic equilibrium considered will not occur." He published his results (his estimate for the limit was of 1.7 times the mass of the Sun) in: The limiting density in white dwarf stars, (Philosophical Magazine 7 (1929), 63-70); and The equilibrium of dense stars, (Philosophical Magazine 9 (1930), 944-963).
• Anderson also refrained from even mentioning it in his 1929 paper Uber die Grenzdichte der Materie und der Energie, published in Zeitschrift fur Astrophysik 56 (1929), 851-856.
• Both of them were sceptical, because the existence of the limit brought to the forefront a question to which they did not have an answer: what happened to a star whose mass was over the limit? Nevertheless, Milne eventually got Chandrasekhar's paper The highly collapsed configurations of a stellar mass published in the March 1931 issue of the Monthly Notices of the Royal Astronomical Society.
• A shorter version of the paper The maximum mass of ideal white dwarfs, which Chandrasekhar had submitted in 1930, was published in volume 74 of the American Astrophysical Journal later in 1931.
• Chandrasekhar did not abandon the problem, however, and, in 1932, while at the University of Copenhagen, he published a new article Some remarks on the state of matter in the interior of stars in volume 5 of the German Zeitschrift fur Astrophysik.
• Once he finished his PhD, examined by Fowler and Eddington in an almost comic oral during which the two examiners spent much of the time arguing, and was elected a Fellow of Trinity College in 1933, Chandrasekhar visited Russia in 1934, where the enthusiasm of astronomers like Victor Ambartsumian and Lev Landau convinced him to resume his research on the topic.
• Eddington had dismissed Milne's modifications in a meeting of the Royal Astronomical Society in 1929:- .
• In 1934, he finished two papers on his theory of white dwarfs in which he had improved the results of his 1931 paper by obtaining an exact solution to the equation of state, which accounted for inhomogeneous polytropes, through extensive numerical analysis.
• Fowler had introduced him to the Society in 1930, and he had become a fellow in 1933.
• When interviewed in 1977, Chandrasekhar remembered vividly how he had been made fun of by Eddington [',' S Weart, Interview of Subrahmanyan Chandrasekhar on 17 May 1977, Niels Bohr Library, American Institute of Physics.','30]:- .
• William McCrea, who was present in the meeting, later said to K C Wali (in 1979), lamenting not having objected to Eddington's arguments at the time:- .
• Chandrasekhar managed to respond to Eddington's arguments in 1935.
• Peierls tackled the problem again in On Lorentz Invariance in the Quantum Theory (published 1942, submitted 1941), together with Paul Dirac and Maurice Pryce.
• Looking for a reason behind this blunder, Werner Israel posits that it was due to an unorthodox definition of particle that Eddington had proposed in 1923's The Mathematical Theory of Relativity.
• This became clear in 1939 but first let us give one further example of Eddington attacking Chandrasekhar.
• Eddington had been invited in 1936 to lecture at a conference to celebrate the tercentenary of Harvard University.
• He had used the occasion to attack Chandrasekhar who, he said in his lecture [',' A S Eddington, Constitution of the stars, The Scientific Monthly (November 1936), 385-395.','14]:- .
• In 1937 Chandrasekhar took up a position in the United States.
• In 1939, Chandrasekhar published his finalised theory of white dwarfs in An Introduction to the Study of Stellar Structure.
• It was well received: for example Stromgren called it [',' B Stromgren, Review: An Introduction to the Study of Stellar Structure, by S Chandrasekhar, Popular Astronomy 47 (1939), 287-289.','27]:- .
• Ledoux, in [',' P Ledoux, &#8217;An introduction to the study of stellar structure&#8217; de S Chandrasekhar, Ciel et Terre 55 (1939), 412-415.','20], said it deserved to be compared to the works of Eddington, Jeans and Rosseland.
• For example Gamow and Schoenberg write in Neutrino Theory of Star Collapse (1941) [',' G Gamow and M Schoenberg, Neutrino Theory of Stellar Collapse, Physical Review 59 (1941), 539-547.','16]:- .
• F Hoyle, W A Fowler, G R Burbridge and E M Burbridge write in On relativistic astrophysics (1963) [',' 17.
• F Hoyle, W A Fowler, G R Burbridge and E M Burbridge, On relativistic astrophysics, Astrophysics Journal 139 (1963), 909-928.','17]:- .

99. Ledermann interview
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Then came hyperinflation in 1922 when money was completely wiped out.
• In the end, after one year, the new currency was introduced at the rate of 1 to 1012.
• Yes, at that school when I was then about 10 or 11, it was 1920-21.
• When my father went to the war in 1915/16, and I was already starting school, he wrote me letters in Latin.
• Throughout my first semester at University I had still not reached my 18th birthday.
• The only time I was away from Berlin was to do my chemistry lab in Marburg (1931, I think).
• The first of these applied to Alfred Brauer who had been a soldier (EFR: He was wounded), and yes, he was badly wounded, and the second applied to Schur because in 1916 he was an extraordinary professor at Bonn, so had effectively become a Prussian civil servant.
• Fortunately I was already a lecturer here at St Andrews, in 1938, and my brother was a doctor in a medical practice in London.
• There were a lot of German Jews in the Hampstead area, there still are, and in some parts German was spoken more often than English, so he had a few people who consulted him, until his health broke down, and he died in 1949.
• EFR: So, that was you in St Andrews in 1934.
• When I arrived in St Andrews in 1934, I was completely bewildered.
• When that finished, Godfrey Thomson came along, introduced by Aitken, and then came this meeting in 1938, where one of Turnbull's staff left and he was in a fix.
• When Walter returned to St Andrews in 1938 he was appointed during the summer by Turnbull who now had a vacancy since a member of the St Andrews department had left.
• He attended the Edinburgh Mathematical Society Colloquium of 1938 in St Andrews (the last for a number of years due to the war).
• The previous Edinburgh Mathematical Society Colloquium had been held in St Andrews in the summer of 1934.
• JOC/EFR March 2001 .

100. Christianity and Mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The mathematician Freeman Dyson writes in [',' F Dyson, Review of Feynman and of Polkinghorne &#8217;&#8217;Belief in God in an Age of Science&#8217;&#8217;, New York Review of Books (28 May, 1998).','16]:- .
• Russell notes not only the impact of Pythagoras on science but also on Christian thinking [',' B Russell, History of Western Philosophy (London, 1961).','11]:- .
• However, real knowledge could not be gained through the senses [',' B Russell, History of Western Philosophy (London, 1961).','11]:- .
• Russell [',' B Russell, History of Western Philosophy (London, 1961).','11] gives Aristotle's ideas of God:- .
• He wrote a major work De Genesi ad Litteram &#9417;, a literal commentary on Genesis, in 401.
• He writes (see for example [',' E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo (Cambridge, 1998), 271-347.','21]):- .
• Augustine argues against a literal interpretation in many cases using the argument (see for example [',' E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo (Cambridge, 1998), 271-347.','21]):- .
• His respect for the ancient Greek system is reflected in his own words (see for example [',' D Alexander, Rebuilding the matrix (Oxford, 2001).','1]):- .
• Zimmermann, in [',' G Zimmermann, Die Gottesvorstellung des Nicolaus Copernicus, Studia Leibnitiana 20 (1) (1988), 63-79.','30] suggests the following:- .
• Tolosani wrote that Copernicus (see for example [',' E Rosen, Was Copernicus&#8217; Revolutions approved by the Pope, Journal of the History of Ideas 36 (1975), 531-542.','26]):- .
• The arguments put forward by Tolosani against Copernicus were that, in addition to contradicting Holy Scripture, his ideas contradicted Aristotle's physics (see for example [',' E Rosen, Was Copernicus&#8217; Revolutions approved by the Pope, Journal of the History of Ideas 36 (1975), 531-542.','26]):- .
• This retained the mathematical advantages of Copernicus without the problems with contradicting Aristotle or Holy Scripture (see for example [',' E Rosen, Galileo&#8217;s misstatements about Copernicus, Isis 32 (1958), 319-330.','27]):- .
• In another work Brahe writes (see for example [',' E Rosen, Galileo&#8217;s misstatements about Copernicus, Isis 32 (1958), 319-330.','27]):- .
• He began his book with a discussion of its relevance to the Holy Scripture (see for example [',' E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 (1975), 225-231.','25]):- .
• He tackled the question head-on in the Introduction to Astronomia nova &#9417; (1609) (see for example [',' E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 (1975), 225-231.','25]):- .
• He wrote in Harmonice Mundi &#9417; (1619):- .
• The Catholic Church, however, in a move against the ideas of Luther, had declared itself the only authority to interpret the Holy Scripture at the Council of Trent in 1546:- .
• Certainly in Cena de le Ceneri &#9417; (1584) Bruno declared his support for the reality of the heliocentric theory and also claimed that the universe is infinite.
• He explained to him in 1597 that he did not wish to enter the argument because of fear of ridicule.
• By 1613 Galileo believed that his telescopic observations of the moons of Jupiter proved that the Earth and planets revolved round the Sun.
• He was drawn into the controversy, however, through Castelli who had been appointed to the chair of mathematics in Pisa in 1613.
• Galileo, less convinced that Castelli had won the argument, wrote Letter to Castelli to him examining (see for example [',' E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo (Cambridge, 1998), 271-347.','21]):- .
• Galileo used arguments similar to those that we have quoted from Kepler's Astronomia nova &#9417; (1609).
• He points out that theologians cannot tell a mathematician what mathematics he must believe to be true (see for example [',' E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo (Cambridge, 1998), 271-347.','21]):- .
• One of the major points of disagreement was whether an individual could form their own interpretation of the Holy Scripture (the Protestant view) or whether, as the Catholic Church argued and had stated clearly after the Council of Trent in 1546, everyone must accept the interpretation of the Holy Scripture made by the Catholic Church.
• One might reasonably ask why Galileo published Dialogue Concerning the Two Chief Systems of the World - Ptolemaic and Copernican after the judgement of 1616.
• When he did so he was confronted with the alternative version of the ruling of 1616, which was an unsigned document.
• Galileo still had in his possession the certificate Bellarmine had signed and given him in 1616, although Bellarmine had died in 1621 so could not clarify the difference between the two versions.
• This was logical, of course, since the judgement of 1616 had declared it totally false.
• He wrote near the end of his life (see for example [',' D Alexander, Rebuilding the matrix (Oxford, 2001).','1]):- .

101. Inca mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Inca empire which existed in 1532, before the Spanish conquest, was vast.
• For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.
• Garcilaso de la Vega, whose mother was an Inca and whose father was Spanish, wrote (see for example [',' G G Joseph, The crest of the peacock (London, 1991).','5]):- .
• We quote Garcilaso de la Vega again [',' G G Joseph, The crest of the peacock (London, 1991).','5]:- .
• 11 (1981), 1-15.','9] and [',' D Pareja, Pre-Hispanic tools of computation : the quipu and the yupana (Spanish), Rev.
• 4 (1) (1986), 37-56.
• ','11].
• For example [',' M Ascher and R Ascher, Code of the quipu : A study in media, mathematics, and culture (Ann Arbor, Mich., 1981).','2] in which the authors write:- .
• A tantalising glimpse may be contained in the writings of the Spanish priest Jose de Acosta who lived among the Incas from 1571 to 1586.
• He writes in his book Historia Natural Moral de las Indias &#9417; which was published in Madrid in 1596:- .
• JOC/EFR January 2001 .

102. Abstract groups
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This article is based on a lecture given by Peter Neumann (a son of Bernhard Neumann and Hanna Neumann) at a conference at the University of Sussex on 19 March 2001 to celebrate the 90th birthday of Walter Ledermann.
• A group G is a set with a binary operation G 5; G → G which assigns to every ordered pair of elements x, y of G a unique third element of G (usually called the product of x and y) denoted by xy such that the following four properties are satisfied: .
• Let us first note that there were two meanings of the term "abstract group" during the first half of the 20th century - say from 1905 to 1955.
• Galois defined a group in 1832 although it did not appear in print until Liouville published Galois' papers in 1846.
• The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Academie des Sciences in 1829.
• Rene Taton has found evidence in the archives of the Academie which suggest that Cauchy spoke with Galois and persuaded him to withdraw the paper and submit a new version of it for the Grand Prix of 1830.
• Galois was invited by Poisson to submit a third version of his memoir on equations to the Academie and he did so on 17 January 1831.
• Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition.
• This was reinforced when Jordan published his major group theory text Traite des substitutions et des equations algebraique &#9417; in 1870.
• He wrote a paper on groups in 1854 which he published again in two separate journals in 1878.
• In 1878 Cayley wrote:- .
• Burnside, in his book The Theory of Groups of Finite Order published in 1897 gave the following definition:- .
• If his elements are operations then why does he need to postulate the associative law? Rather strangely Burnside does not assume the existence of an identity, although one can infer it from the fact that A and A -1 are in the set and we have closure.
• Burnside repeats exactly the same definition in the second edition of his book which appeared in 1911.
• In 1870 Kronecker gave a definition of a group in a completely different context, namely the context of a class group in algebraic number theory.
• However, Heinrich Weber in 1882 gave a very similar definition to that of Kronecker yet he did tie it in with previous work on groups.
• That this fails for infinite systems was noted by Heinrich Weber in his famous text Lehrbuch der Algebra &#9417; published in 1895.
• JOC/EFR March 2001 .

103. Abstract linear spaces
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1804 he published a work on the foundations of elementary geometry Betrachtungen uber einige Gegenstande der Elementargoemetrie &#9417;.
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• In 1827 Mobius published Der barycentrische Calcul &#9417; a geometrical book which studies transformations of lines and conics.
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• In 1837 Mobius published a book on statics in which he clearly states the idea of resolving a vector quantity along two specified axes.
• Between these two works of Mobius, a geometrical work by Bellavitis was published in 1832 which also contains vector type quantities.
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• In 1814 Argand had represented the complex numbers as points on the plane, that is as ordered pairs of real numbers.
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• He presented these results in a paper to the Irish Academy in 1833.
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• Hamilton's quaternions, published in 1843, was an important example of a 4-dimensional vector space but, particularly with Tait's work on quaternions published in 1873, there was to be some competition between vector and quaternion methods.
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• In 1857 Cayley introduced matrix algebras, helping the move towards more general abstract systems by adding to the different types of structural laws being studied.
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• In 1858 Cayley noticed that the quaternions could be represented by matrices.
Go directly to this paragraph
• In 1867 Laguerre wrote a letter to Hermite Sur le calcul des systemes lineaires &#9417;.
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• Laguerre's work on linear systems was followed up by a paper by Carvallo in 1891.
• Grassmann's contribution Die Ausdehnungslehre &#9417; appeared in several different versions.
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• The first was in 1844 but it was a very difficult work to read, and clearly did not find favour with mathematicians, so Grassmann tried to produce a more readable version which appeared in 1862.
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• Grassmann's 1862 version of Die Ausdehnungslehre &#9417; has a long introduction in which Grassmann gives a summary of his theory.
• Saint-Venant's claim is a fair one since he published a work in 1845 in which he multiples line segments in an analogous way to Grassmann.
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• However, rather typically of Cauchy, in 1853 he published Sur les clefs algebrique &#9417; in Comptes Rendus which describes a formal symbolic method which coincides with that of Grassmann's method (but makes no reference to Grassmann).
• Grassmann complained to the Academie des Sciences that his work had priority over Cauchy's and, in 1854, a committee was set up to investigate who had priority.
• In 1867 he wrote a paper Theorie der complexen Zahlensysteme &#9417; concerning formal systems where combination of the symbols are abstractly defined.
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• He credits Grassmann's Die Ausdehnungslehre &#9417; as giving the foundation for his work.
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• The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888.
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• Grassmann preceduto dalle operazioni della logica deduttiva &#9417; is remarkable.
• It gives the basic calculus of set operation introducing the modern notation ∩, ∪, &#8712; for intersection, union and is an element of.
• It is hard to believe that Peano writes the following in 1888.
• Although never attaining the level of abstraction which Peano had achieved, Hilbert and his student Schmidt looked at infinite dimensional spaces of functions in 1904.
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• Schmidt introduced a move towards abstraction in 1908 introducing geometrical language into Hilbert space theory.
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104. Infinity
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This is particularly true in ancient Greek times, as Knorr writes in [',' W R Knorr, Infinity and continuity in ancient and medieval thought (Ithaca, N.Y., 1982), 112-145.','26]:- .
• Aristotle discussed this in Chapters 4-8 of Book III of Physics (see [',' D D Spalt, Die Unendlichkeiten bei Bernard Bolzano, in Konzepte des mathematisch Unendlichen im 19.
• Jahrhundert (Gottingen, 1990), 189-218.','36]) where he claimed that denying that the actual infinite exists and allowing only the potential infinite would be no hardship to mathematicians:- .
• We will come to Cantor's ideas towards the end of this article but for the moment let us consider the effect Aristotle had on later Greek mathematicians by only allowing the potentially infinite, particularly on Euclid; see for example [',' D D Spalt, Die Unendlichkeiten bei Bernard Bolzano, in Konzepte des mathematisch Unendlichen im 19.
• Jahrhundert (Gottingen, 1990), 189-218.','36].
• I, SCIAMVS 2 (2001), 9-29.','32] have noticed a remarkable way that Archimedes investigates infinite numbers of objects in The Method in the Archimedes palimpsest:- .
• In the first century BC Lucretius wrote his poem De Rerum Natura &#9417; in which he argued against a universe bounded in space.
• In Summa theologia &#9417;, written in the 13th Century, Thomas Aquinas wrote:- .
• The article [',' A A Davenport, The Catholics, the Cathars, and the concept of infinity in the thirteenth century, Isis 88 (2) (1997), 263-295.','15] examines:- .
• Basically what al-Karaji did was to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n =1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one could continue the process indefinitely.
• He refused to change his views and he was burned at the stake in 1600.
• He tackled the topic of infinity in Discorsi e dimostrazioni matematiche intorno a due nuove scienze &#9417; (1638) where he studied the problem of two concentric circles with centre O, the larger circle A with diameter twice that of the smaller one B.
• 54 (2) (1999), 87-99.','25] Knobloch takes a new look at this work by Galileo.
• Cavalieri wrote Geometria indivisibilibus continuorum &#9417; (1635) in which he thought of lines as comprising of infinitely many points and areas to be composed of infinitely many lines.
• The Roman College rejected indivisibles and banned their teaching in Jesuit Colleges in 1649.
• The symboln ∞ n which we use for infinity today, was first used by John Wallis who used it in De sectionibus conicis &#9417; in 1655 and again in Arithmetica infinitorum &#9417; in 1656.
• Immanuel Kant argued in The critique of pure reason (1781) that the actual infinite cannot exist because it cannot be perceived:- .
• Gauss, in a letter to Schumacher in 1831, argued against the actual infinite:- .
• Perhaps one of the most significant events in the development of the concept of infinity was Bernard Bolzano's Paradoxes of the infinite which was published in 1840.
• 69 (Providence, RI, 1988), 79-92.','17], looks at infinity and nonstandard analysis.

105. Trisecting an angle
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Then DAE is an equilateral triangle and so the angle DAE is 6076; and DAC is 3076;.
• Perhaps more surprisingly, angles such as one of 2776; can be trisected - can you solve this? The problem is therefore to trisect an arbitrary angle and the aim is to make the construction using ruler and compass (which is impossible) but failing that to devise some method to trisect an arbitrary angle.
• Now angle EAB is 1/3 of angle CAB.
• But angle CGA = angle GEC + angle ECG = 2 5; CEG = 2 5;EAB as required.
• Just mark off a length of 2 5;AC at the right hand end of the ruler and then slide the ruler with one mark on CD, the other on FC extended until the ruler defines a line passing through A.
• Angle XAC = angle ACF = angle CFA = angle FEA + angle FAE = 2 5; angle FEA = 2 5; angle XAB.
• Heath writes in [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','1]:- .
• The first shows that if AB is a fixed line then locus of a point P such that 2 5; angle PAB = angle PBA is a hyperbola.
• To see this we need to note that, from the property of the hyperbola described above, 2 5; angle PAB = angle PBA.
• But 2 5; angle PAB = angle POB, and 2 5; angle PBA = angle POA (angle at centre of a circle is twice the angle at the circumference standing on the same arc).
• Hence 2 5; angle POB = angle POA as required.
• He writes [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','1]:- .
• In 1837 Wantzel published proofs in Liouville's Journal of:- .

106. References for Quantum mechanics history
• South Africa 47 (1) (1989), 81-101.
• 26 (4) (1996), 545-557.
• L M Brown, Quantum mechanics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 1252-1260.
• 44 (2) (1991), 137-179.
• A B Datsev, The role of Erwin Schrodinger (1887-1961) in the creation and interpretation of quantum mechanics (Bulgarian), Fiz.-Mat.
• 30 (63) (3) (1988), 191-195.
• Acta 56 (5) (1983), 993-1001.
• P A Hanle, Erwin Schrodinger's reaction to Louis de Broglie's thesis on the quantum theory, Isis 68 (244) (1977), 606-609.
• J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
• 37 (1) (1980), 59-79.
• G S Im, Experimental constraints on formal quantum mechanics : the emergence of Born's quantum theory of collision processes in Gottingen, 1924-1927, Archive for History of Exact Sciences 50 (1) (1996), 73-101.
• C W Kilmister, Quantum mechanics 1899-1925 : a survey of concept formation, Bull.
• 13 (9-10) (1977), 239-243.
• P T Matthews, Dirac and the foundation of quantum mechanics, Reminiscences about a great physicist : Paul Adrien Maurice Dirac (Cambridge, 1987), 199-224.
• J Mehra and H Rechenberg, The historical development of quantum theory (5 volumes) (New York-Berlin, 1982-1987).
• Student 55 (2-4) (1987), 71-82.
• L Navarro, On Einstein's statistical-mechanical approach to the early quantum theory (1904-1916), Historia Sci.
• (2) 1 (1) (1991), 39-58.
• 52 (1) (1985), 44-63.
• 51 (4) (1979), 863-914.
• A Pais, Max Born's statistical interpretation of quantum mechanics, Science 218 (4578) (1982), 1193-1198.
• 25 (1) (1995), 183-204.
• 12 (10) (1982), 971-976.
• H Reichenbach, The space problem in the new quantum mechanics, Erkenntnis 35 (1-3) (1991), 29-47.
• 17 (12) (1987), 1205-1220.
• 44 (3) (1997), 323-328.
• 32 (4) (1987), 177-190.

107. Doubling the cube
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Theon of Smyrna quotes a work by Eratosthenes (see Heath [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2]):- .
• This purports to be a letter written by Eratosthenes to King Ptolemy and, although the letter is a forgery, the writer does quote some genuine writings of Eratosthenes [',' M R Cohen and I E Drabkin (trs.), A source book in Greek science (Harvard, 1948).','1]:- .
• 22 (2) (1995), 119-137.','8] this type of argument was not available to Hippocrates so one has to consider not only how he proved the equivalence but also how Hippocrates thought of the result in the first place.
• Heath also suggests in [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2] that Hippocrates may have come to the idea from number theory for he quotes Euclid's Elements Book VIII:- .
• 22 (2) (1995), 119-137.','8] to deduce that compound ratios , while well known to Archimedes, belonged to more modern mathematics than that available to Hippocrates.
• Heath writes [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2]:- .
• We shall use some coordinate geometry in a moment to see that Archytas is correct, but first let us give the construction in the words of Eutocius, unchanged except for the names of the point which I have changed to fit the notation of our diagram and that described above (see for example [',' J Delattre and R Bkouche, Why ruler and compass?, in History of Mathematics : History of Problems (Paris, 1997), 89-113.','7]):- .
• However, Heath [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2] suggests that Eudoxus was:- .
• Plutarch wrote (see for example [',' J Delattre and R Bkouche, Why ruler and compass?, in History of Mathematics : History of Problems (Paris, 1997), 89-113.','7]):- .
• He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2]:- .
• Details of the construction is given in [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2].
• In 1837 Wantzel published proofs in Liouville's Journal of:- .

108. Poincaré - Inspector of mines
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• As part of his studies he undertook field work in Normandy in 1876 and then travelled abroad on trips organised by the Ecole, going to Austria-Hungary in 1877 and to the several Scandinavian countries in the following year.
• However on 1 September 1879 there was an explosion at the Magny pit, which had only been operational from 8 July 1879, and Poincare was quickly on the scene to assess the cause.
• There were twenty-two men in a shift and the explosion occurred at 3.45 in the morning killing sixteen of the twenty-two men who had descended the shaft in the cage at 18.00 the previous evening to begin work.
• Poincare descended the shaft into the mine to start his investigation of the cause of the explosion on 1 September while the attempted rescue operation was in progress.
• Roy and Dugas note in [',' M Roy and R Dugas, Henri Poincare, Ingenieur des Mines, Annales des Mines 193 (1954), 8-23.','2] that Poincare's report:- .
• His report on this visit was signed on 30 November and then on 1 December he was given a lectureship in mathematics at the Faculty of Sciences of Caen.
• He was promoted to chief engineer on 22 July 1893, then to inspector general on 16 June 1910.
• He wrote an article in 1912, the year of his death, on mining.

109. Gravitation
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• This theory survived through the 16th century and the following illustration from a book published in 1547 shows how Aristotle's theory was applied to cannonballs.
• He wrote in 1644 in Principles of Philosophy:- .
• All of these ideas were put forward by Descartes in 1644.
• Huygens pointed out in 1669 that if the ether flowed in a circular motion round the Earth's axis then gravitational attraction should be towards the axis rather than towards the centre of the Earth.
• In 1675 Malebranche published Recherche de la verite which revised Descartes' theory of forces, showing that no force could be associated with the rest state of a body, so everything depended on Descartes' motus force.
• Leibniz came up with other objections, proposing new ideas in Acta eruditorum &#9417; in 1686.
• One year after Leibniz published his article in Acta erditorum &#9417;, Newton published the Philosophiae naturalis principia mathematica &#9417; or Principia as it is always called.
• Newton himself wrote in a letter to Dr Bentley dated 25 February 1693 (see for example [',' I B Cohen, Isaac Newton&#8217;s papers and letters on natural philosophy and related documents (Cambridge, 1958).','1], page 302):- .

110. Knots and physics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1847 Listing published Vorstudien zur Topologie &#9417; and then Riemann published important papers on complex analysis in 1851 and 1857 which investigated connectivity and Riemann surfaces.
• The Scottish mathematical physicists did not know of these papers until much later but Helmholtz, whose paper of 1858 directly influenced them, built much on Riemann's ideas.
• In 1858 Helmholtz published his important paper in Crelle's Journal on the motion of a perfect fluid.
• Helmholtz's paper Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen &#9417; began by decomposing the motion of a perfect fluid into translation, rotation and deformation.
• It is remarkable that Thomson was able to develop his ideas quickly enough that he could publish On vortex atoms in the Proceedings of the Royal Society of Edinburgh still in 1867.
• Tait began to think about knots and Thomson's second paper on vortex atoms, which appeared in 1869, included diagram of knots and links drawn by Tait.
• He was interested in knots because of electromagnetic considerations and in a letter to Tait written on the 4 December 1867 he rediscovered an integral formula counting the linking number of two closed curves which Gauss had discovered, but had not published, in 1833.
• ., nm and showed that the region possessed N = n+n1+n+ ..
• .+ nm cycles and was (N+1)-ly connected.
• However, more than 100 years after they were written these manuscripts were published in [',' P M Harman (ed.), James Clerk Maxwell, The scientific letters and papers of James Clerk Maxwell 1862-1873 II (Cambridge, 1995).','2].
• There are three manuscripts on knots and some time between the second, which Maxwell wrote in October 1868, and the third, which he wrote on 29 December 1868, he had read Listing's 1847 paper Vorstudien zur Topologie &#9417; for in the third manuscript he lists Listing's main results.
• In 1869 Thomson tried to clarify the topological ideas that he was using.
• By 1876 Thomson had made little progress with his ideas of vortex atoms.
• Tait decided to embark on a classification of plane closed curves in 1876, writing in a report to the British Association for the Advancement of Science:- .
• By 1877 Tait had classified all knots with seven crossings but he stopped there.
• He returned to the topic of knots in his address to the Edinburgh Mathematical Society in 1883:- .
• It is greatly desired that someone, with the requisite leisure, should try to extend this list, if possible up to 11 ..
• By then he had received from Kirkman 1581 knot projections with 11 crossings and this time Tait felt that he did not have the time to solve the equivalence problem for these.
• JOC/EFR January 2001 .

111. References for Bolzano's manuscripts
• Biography in Dictionary of Scientific Biography (New York 1970-1990).
• V Jarnik, Bolzano and the foundations of mathematical analysis (Prague, 1981).
• 1 (1952), 65-98.
• 24 (4) (1987), 373-377.
• 24 (4) (1987), 442-446.
• 24 (4) (1987), 406-413.
• 24 (4) (1987), 353-372.
• J Berg, A requirement for the logical basis of scientific theories implied by Bolzano's logic of variation, in Impact of Bolzano's epoch on the development of science (Prague, 1982), 415-425.
• 31 (3) (1998), 121-130.
• P Bussotti, The problem of the foundations of mathematics at the beginning of the nineteenth century : Two lines of thought: Bolzano and Gauss (Italian), Teoria (N.S.) 20 (1) (2000), 83-95.
• 7 (1) (2004), 101-121.
• 52 (3-4) (1999), 343-361.
• 74 (1982), 679-689.
• A Coffa, Bolzano and the birth of semantics, in The Semantic Tradition from Kant to Carnap (Cambridge University Press, Cambridge, 1991), 22-40.
• 24 (4) (1987), 423-441.
• 8 (Moscow, 2001), 210-216, .
• 24 (4) (1987), 452-468.
• 83 (10) (1986), 558-564.
• Logic 12 (3) (1983), 299-318.
• 52 (3-4) (1999), 385-398.
• M Hyksova, Bolzano's inheritance research in Bohemia, Mathematics throughout the ages, Holbaek, 1999/Brno, 2000 (Prometheus, Prague, 2001), 67-91 .
• D M Johnson, Prelude to dimension theory : The geometrical investigations of Bernhard Bolzano, Archive for History of Exact Science 17 (1976), 275-296.
• 6 (3) (1975), 229-269.
• 32 (1982), 667-670.
• 2 (1964/1965), 398-409.
• (4) 42 (1964), 209-214.
• 52 (3-4) (1999), 429-455.
• 28 (4) (2001), 177-183.
• 15 (1974/75), 176-190.
• 24 (4) (1987), 447-451.
• M Nemcova, Frantisek Josef Studnicka and Bernard Bolzano (Czech), Mathematics in the 19th century (Czech), Vyskov, 1994 (Prometheus, Prague, 1996), 115-119.
• 15 (1) (1995), 49-59.
• 52 (3-4) (1999), 363-383.
• 52 (3-4) (1999), 399-427.
• S Russ, Bolzano's analytic programme, The Mathematical intelligencer 14 (3) (1992), 45-53.
• 38 (5) (1993), 249-259.
• 7 (2) (1980), 156-185.
• G Schubring, Bernard Bolzano - Not as Unknown to His Contemporaries as is Commonly Believed?, Historia Mathematica 20 (1993), 45-53.
• J Sebestik, The mathematical system of Bernard Bolzano (Spanish), Mathesis 6 (4) (1990), 393-417.
• 17 (1964), 129-164.
• 52 (3-4) (1999), 479-506.
• 24 (4) (1987), 378-405.
• 26 (2) (1973), 97-112.
• 52 (3-4) (1999), 457-477.
• H Sinaceur, Bolzano et les mathematiques, in Les philosophes et les mathematiques (Ellipses, Paris, 1996), 150-173.
• D D Spalt, Bolzanos Lehre von den messbaren Zahlen 1830-1989, Arch.
• 42 (1) (1991), 15-70.
• 41 (4) (1991), 311-362.
• W Stelzner, Compatibility and relevance: Bolzano and Orlov, in The Third German-Polish Workshop on Logic & Logical Philosophy, Dresden, 2001, Logic Log.
• 10, (2002), 137-171.
• Logic 2 (1981), 11-20.
• J van Benthem, The variety of consequence, according to Bolzano, Studia Logica 44 (4) (1985), 389-403.
• B van Rootselaar, Bolzano's corrections to his Functionenlehre, Janus 56 (1969), 1-21.
• 2 (1964/1965), 168-180.
• 28 (4) (2001), 177-183.
• DDR (2-4) (1981), 128-152.

112. Science in the 17th century
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Although some historians favour the figure of Nicolaus Copernicus (1473-1543) and the heliocentric theory to mark the beginning of the Scientific Revolution, others situate the origin in Francis Bacon (1561-1626) and his description of the scientific method.
• Some other key figures of this period were Tycho Brahe (1546-1601), Rene Descartes (1596-1650), Johannes Kepler (1571-1630), Galileo Galilei (1564-1642) and Isaac Newton (1642-1727).
• By the 16th and 17th centuries, the paradigm started to shift as some natural philosophers were rejecting unproven theories and using precise tools to obtain exact measurements to base their discoveries on observation and experimentation [Hakim 2005, 19].
• The use of new tools to obtain exact observations reached its zenith with the invention of the telescope, improved by Galileo who, turning it to the skies, observed and described the Moon, the Sun, Jupiter and the Milky Way (Sidereus nuncius &#9417;, 1610) and whose optical mechanism was described by Kepler (Dioptrice &#9417;, 1611).
• As Johannes Kepler affirmed in his book Tertius Interveniens &#9417;, written in 1610: .
• The new system of science and philosophy questioned the authority of these thinkers of the past, debated their ideas, and refused some of their claims [',' S Shapin, The scientific revolution (University of Chicago Press, Chicago, IL, 1996).','14].
• Regional and national differences shaped the way that society responded to the transformations imposed by the scientific method [',' R Porter and Mikulas Teich, The Scientific revolution in national context (Cambridge University Press, Cambridge, 1992).','11].
• These scenarios were the perfect breeding ground for the events, theories and ideas that are known as the Scientific Revolution [',' J E Carney, Renaissance and Reformation, 1500-1620: a biographical dictionary (Greenwood Press, Westport, Conn, 2001).','2].
• But it was many years before, around the year 1600, that Francis Bacon, the Lord Chancellor to the King James I and known nowadays as the 'Father of modern science', had set the foundation of a pragmatic view of the world and its knowledge, had opened the door to man's ability to control nature, and stated that knowing the laws of nature will bring 'the empire of man over things' (Novum Organum &#9417;, 1623).
• Other influential English figures from the beginning of the century were William Gilbert (1544-1603), who introduced the concept of Earth magnetism (De magnete &#9417;, 1600), and William Harvey, who discovered and described the circulation of the blood (De motu cordis et sanguinis &#9417;, 1628).
• After 1650, and before the return of the king Charles II, the new science was practiced for its economic benefits, to find new resources for commerce and technology, and for its social status as, for the society of the time, turning to intellectual amusements had become fashionable [',' L Jardine, Britain and the rise of science, BBC History (17 February 2011).','6].
• The societies enjoyed state patronage, promoted collaboration between their members and had mottos that reflected the commitment of their members to experimentation -the Royal Society motto was 'Nullius in verba' (take nobody's word for it) and the one of the Accademia del Cimento, in Florence, was 'Provando e riprovando' (Try and Try again) - [',' S Shapin, The scientific revolution (University of Chicago Press, Chicago, IL, 1996).','14].
• This is the case for Robert Boyle (1627-1691), Robert Hooke (1635-1703), Christopher Wren (1632-1723), Edmond Halley (1656-1742) and Isaac Newton.
• Newton was president of the Royal Society in 1703.
• Historians see the publication of the Principia as the culmination of the Scientific Revolution [',' J Hakim, The story of science: Newton at the center (Smithsonian Books, Washington, 2005).','4], [',' S Shapin, The scientific revolution (University of Chicago Press, Chicago, IL, 1996).','14].
• The answer may be in the universities, institutions that provided a supportive environment for study and research [',' J R McNeill and William Hardy McNeill, The human web: a bird&#8217;s-eye view of world history (W W Norton, New York, 2003).','8].
• [',' W Ruegg, and H De Ridder-Symoens, A history of the university in Europe 1, 1 (Cambridge University Press, Cambridge, 1992).','13].
• In 1661, a young Isaac Newton wrote in his notebook at Cambridge a motto that he probably adapted from Roger Bacon's writings: "Amicus Plato amicus Aristoteles magis amica veritas" (Plato is my friend, Aristotle is my friend, but a greater friend is truth).
• ." [',' J J O&#8217;Connor and E F Robertson, History topic: Mathematics in St Andrews to 1700, MacTutor History of Mathematics (1996).','10].
• During its first two centuries, students in the University received instruction in Arts, mainly in the study of Aristotle [',' J J O&#8217;Connor and E F Robertson, History topic: Mathematics in St Andrews to 1700, MacTutor History of Mathematics (1996).','10].
• After the reforms of 1579, the University appointed for the first time professors that were specialists on specifics subjects, one of them was a professor of Mathematics, attached to St Salvator's College.
• [',' H C Rawson, Treasures of the University: an examination of the identification, presentation and responses to artefacts of significance at the University of St Andrews, from 1410 to the mid-19th century; with an additional consideration of the development of the portrait collection to the early 21st century (Art History Theses, University of St Andrews, 2010).','12] .
• Mathematical teaching and research at the University began in 1668 with the appointment of James Gregory as the first holder of the Regius Chair of Mathematics in St Andrews.
• The chair was not attached to any college, and James Gregory, who had just become a member of the Royal Society, worked in the Upper Hall of the university library [',' R G Cant, The University of St Andrews: A Short History (Oliver and Boyd, Edinburgh, 1946).','1].
• The construction of the observatory continued despite his absence but neither the building nor the instruments intended to be housed there were ever fully brought to use and the structure, erected at the south end of St Mary's College, became derelict in the eighteenth century and was demolished by the nineteenth [',' H C Rawson, Treasures of the University: an examination of the identification, presentation and responses to artefacts of significance at the University of St Andrews, from 1410 to the mid-19th century; with an additional consideration of the development of the portrait collection to the early 21st century (Art History Theses, University of St Andrews, 2010).','12].
• 1610.
• Kepler, Johannes, 1611.
• Kepler, Johannes, 1610.
• Excerpts translated by Ken Negus in 1897, republished in Kenneth G.
• 2001.

113. Egyptian numerals
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Notice that when the number contained too many symbols for the "part" sign to be placed over the whole number, as in 1/249 , then the "part" symbol was just placed over the "first part" of the number.
• Middle Kingdom - around 2100 BC to 1700 BC .
• New Kingdom - around 1600 BC to 1000 BC .

114. Tartaglia versus Cardan
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It was held in the Church of Santa Maria del Giardino, Milan, on 10 August 1548.

115. Size of the Universe
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Halley, in 1679, pointed out that viewing such a transit from two points on the Earth a known distance apart could be used to determine the size of the solar system.
• The transits of June 1761 and 1769 and those of December in 1874 and 1882 were used to obtain an accurate value for the astronomical unit, which is the distance from the Earth to the Sun.
• He estimated that the distance from the Earth to the Moon is 59 times the radius of the Earth and the distance to the Sun is 1,200 times the radius of the Earth, a serious underestimate.
• He did improve the value for the distance to the Sun to 1,500 times the radius of the Earth but this is still such a serious underestimate that it is little improvement.
• Kepler's Third Law, published in 1619, made the relative distances within the solar system known, but until one distance was known accurately the size was still unknown.
• In 1671 the French planned an expedition, one aim of which was to find an accurate distance to Mars.
• Halley, in 1718, noted that three stars, Sirius, Procyon and Arcturus, had moved relative to the ecliptic (the apparent line of the Sun through the stars) since Hipparchus had measured their positions.
• Bradley, in 1728, did discover the aberration of light while trying to determine a stellar parallax.
• There was considerable interest in using the transits of June 1761 and June 1769 for this purpose and many astronomers set out to observe them from a variety of places such as St Helena, the Cape of Good Hope, and India.
• As a result estimates of the distance to the Sun varied by up to 10 million miles.
• He determined the distance to 61 Cygni, announcing his result in 1838.
• Bessel, using a Fraunhofer heliometer to make the measurements, announced his value of 0.314" which, given the diameter of the Earth's orbit, gave a distance of about 10 light years.
• Thomas Henderson measured the parallax of Alpha Centuari in 1839, showing it had a parallax around three times greater that 61 Cygni, so was much closer.
• A major advance in calculating distances came about in 1908 when Henrietta Swan Leavitt observed variable stars in the Large and Small Magellanic Clouds.
• By 1912 this had been refined so that it was a reliable measure of the distance of a Cepheid variable star, for one only needed to determine the period, which is the time between two occurrences of maximum brightness, to obtain a value for the absolute brightness of the star.
• By 1919 he had worked out that it was a disk which had a huge bulge at the centre.
• In 1920 there was a debate between Shapley and Heber Curtis on the distance scale of the universe and on the spiral nebulas.
• Vesto Slipher, who had worked at the Lowell Observatory with the 24-inch telescope there since 1901, had been asked to investigate the spiral nebulas by Percival Lowell.
• By 1912 Slipher had made a major breakthrough when he had obtained a spectrograph of the Andromeda Nebula and found its light shifted towards the blue.
• Slipher continued his work and by 1914 had obtained spectrographs of 15 spiral nebulas.
• In a paper of 1917 he wrote: .
• In the winter of 1920-21 Shapley had asked him to take photographs of the Andromeda Nebula with the 100-inch telescope at Mount Wilson which had been in operation for about 2 years.
• The first observations of the Andromeda Nebula which led to its distance being calculated were made in 1923 by Edwin Hubble, also using the 100-inch telescope at Mount Wilson.
• In 1919 Hubble announced that there was a linear relation between the distance to a spiral galaxy and its speed of recession.
• In a paper written jointly by Hubble and Humason in 1931 they gave data for more spiral galaxies and estimated the constant in the linear relation to be 558.
• Eddington wrote in 1933:- .
• Walter Baade made observations with the 100-inch telescope in 1944 which led to the discovery that there were two type of stars, Population I and Population II stars.
• In 1948 the 200-inch telescope at Mount Palomar became operational and more data could be obtained.
• By 1952 Baade could announce that the distance scale was wrong by a factor of 2, the Andromeda galaxy was twice as far away as Hubble had estimated, and the Hubble constant was about 250.
• It was Allan Sandage, who was just completing his doctorate in 1952, who carried on refining the distance scale using the 200-inch telescope.
• In 1956, in a joint paper with Humason and Nick Mayall, he gave 180 as his estimate of the Hubble constant.

116. Bourbaki 2
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Armand Borel first became acquainted with the Bourbaki team in 1949.
• He described [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.','11] his first Bourbaki Congress:- .
• Monthly 77 (1970), 134-145.','15], gives a similar description of the Congresses and talks about the qualities that members of the team must have:- .
• The plans at this time were to carry out the aims that Bourbaki had set in 1939, namely to publish six books, each consisting of up to 10 chapters: .
• After World War II, publication restarted in 1947 after a break of five years.
• 1951 Book IV Functions of One Real Variable: Chapter IV.
• By 1958 publications of the six books was complete so in the mid 1950s only the finishing touches remained to be done.
• In 1958 Henri Cartan explained why Bourbaki was not growing old [',' H Cartan, Nicolas Bourbaki and contemporary mathematics, Math.
• Intelligencer 2 (4) (1979-80), 175-180.','13]:- .
• Armand Borel explains in [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.','11]:- .
• 78">In the fifties [the six books] were essentially finished, and it was understood the main energies of Bourbaki would henceforth concentrate on the sequel; it had been in the mind of Bourbaki very early on (after all, there was still no "Traite d'Analyse" &#9417;).
• Grothendieck presented to the next Congress a detailed draft of two chapters, the first containing preliminaries to the book on manifolds and categories of manifolds, while the second contained differentiable manifolds, and the differential formalism [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.','11]:- .
• But by 1958 when the original six books were completed, the first few of these books were already almost 20 years out of date.
• By 1980 they had produced the two planned summary chapters on differential and analytic manifolds, seven chapters on commutative algebra, eight chapters on Lie groups and Lie algebras, and two chapters on spectral theories.
• Cartier gave what he saw as the reasons in [',' M Senechal, The continuing silence of Bourbaki - an interview with Pierre Cartier, June 18, 1997, Math.
• Intelligencer 20 (1) (1998), 22-28.','36]:- .
• Cartier said [',' M Senechal, The continuing silence of Bourbaki - an interview with Pierre Cartier, June 18, 1997, Math.
• Intelligencer 20 (1) (1998), 22-28.','36]:- .

117. Modern light
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• He wrote an article for Encyclopaedia Britannica in 1878 in which he described how light is propagated as a transverse wave, and that its consists of electromagnetic radiation with specific wavelengths.
• Albert Michelson, however, who was spending study leave in Helmholtz's laboratory in Berlin in 1881, tried to carry out Maxwell's experiment.
• Lorentz showed in 1885, however, that the aether drag theory was not possible.
• In 1887 they carried out a much more accurate version of Maxwell's experiment.
• How does it know when it is on its journey whether the detector is moving or not? FitzGerald explained the failure of the Michelson-Morley experiment in 1889 by suggesting that a moving object is foreshortened in the direction of travel.
• The amount of this foreshortening for an object moving with velocity v was ͩ0;(1 - v2/c2), where c is the velocity of light.
• Of course this is very close to 1 for velocities v which are small compared to that of light but it explained the results of the experiment by foreshortening the instruments while still allowing there to be an aether through which light travelled at a fixed velocity.
• Lorentz, independently, made a similar suggestion to FitzGerald and worked out the full implications of it in 1904 giving transformations which would describe the way that light would look to observers moving relative to each other.
• In 1915 Einstein published the general theory of relativity which predicted the bending of rays of light passing through a gravitational field.
• In 1919 Eddington made an expedition to Principe Island off the west coast of Africa to observe a solar eclipse and to measure the apparent position of stars observed close to the disk of the eclipsed sun.
• In 1926 Michelson carried out his last and most accurate experiment to determine the velocity of light.
• Another important development in the understanding of light, namely the development of quantum theory, had taken place over this same period of time, roughly 1880 to 1926.
• Josef Stefan discovered experimentally in 1879 that the total radiation energy per unit time emitted from a blackbody, namely a body which absorbs all radiation energy falling on it, is proportional to the fourth power of the absolute temperature of the body.
• In 1896 Wilhelm Wien described the spectrum produced by a blackbody when it radiates.
• Lord Rayleigh made an important contribution to light in 1899 when he explained that the sky is blue, and sunsets are red, because blue light is scattered by molecules in the earth's atmosphere.
• Planck, in 1900, showed that the impossible result could be corrected by assuming that electromagnetic energy can only be emitted in quanta.
• Heinrich Hertz discovered the photoelectric effect, so called because it was caused by light rays, in 1887.
• In 1900 Philipp Lenard, a student of Hertz, showed that the photoelectric effect was caused by electrons, which had been discovered by J J Thomson three years earlier, being ejected from the surface of a metal plate when it was struck by light rays.
• In 1905 Einstein explained the photoelectric effect by showing that light was composed of discrete particles, now called photons, which are essentially energy quanta.
• A fuller model incorporating this feature was discovered by Fowler in 1931.
• Bose published his paper Planck's Law and the Hypothesis of Light Quanta in 1924 which derived the blackbody radiation from the hypothesis that light consisted of particles obeying certain statistical laws.
• In 1927 de Broglie's claim that electrons could behave like waves was experimentally verified and, in the following year, Bohr put forwards his complementarity principle which stated that photons of light (and electrons) could behave either as waves or as particles, but it is impossible to observe both the wave and particle aspects simultaneously.
• In 1927 Heisenberg put forward his uncertainty principle which states that there is a limit to the precision with which the position and the momentum a particle of light can be known.
• By 1930 the interpretation of quantum theory called the Copenhagen interpretation, mainly due to Bohr and his co-workers, was essentially complete.
• 1992).','1]:- .
• The basic concept goes back to Einstein in 1916 when he showed that if an atom is excited so that it moves to a higher energy level, then if light falls on the atom at the instant it is moving to the higher energy level, then it emits radiation that is in phase with the wave that stimulated it and so amplifies that wave.
• C H Townes, J P Gordon and H J Zieger built a device at Columbia University in 1953 which used ammonia to produce a coherent beam, not of light at optical wavelengths, but of microwave radiation.
• In 1958 A L Schawlow and C H Townes described how a device might amplify light by stimulated emission and the first such device was built in 1960 at the Hughes Research Laboratories by T H Maiman using a rod of ruby.

118. References for Real numbers 3
• 50 (1-2) (1997), 131-158.
• 5 (1990), 67-73.
• 1 (1) (1977), 9-20.
• "Nauka", Moscow, 1973), 176-180; 338.
• 23 (1978), 71-76; 357.
• 23 (1978), 56-70; 357.
• (1) (1981), 106-107.
• 25/26 (3-4) (1978), 120; 208.
• 25/26 (2) (1978), 57-69; 111.
• Logic 8 (1) (1987), 25-44.
• 11 (1-2) (1975), 24-27.
• 9 (1-2) 2001/03), 95-113.

119. Galileo's Difesa
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• On 18 August 2009 I [EFR] received an email from the Head of Special Collections at the University of St Andrews Library asking me to write a short essay on Galileo's Difesa di Galileo Galilei ..
• contro alle calunnie & imposture di Baldessar Capra &#9417; for inclusion in a book to be published on the "most interesting and spectacular items" held within the Special Collections.
• Angelo Contarini (1575-1637) was a Venetian ambassador and almost certainly a patron of Galileo.
• As an aside let us mention that Angelo Contarini was sent to London in 1625 to congratulate Charles I on his accession.
• We give more details of this later but, continuing with a chronological account, we next describe a nova seen in 1604.
• A nova had appeared in 1572 and had been observed by Tycho Brahe.
• Galileo first observed the nova on 28 October, just a few days before the star reached its maximum brightness on 1 November.
• Around the time that Galileo began observing the nova again in January-February 1605, a treatise was published by Baldessare Capra entitled Consideratione astronomica circa la stella nona dell'anno 1604 &#9417;.
• Capra had become a student of Simon Mayr in 1602 and the two had observed the nova in October 1604.
• He did so in 1606 publishing the Italian text as Le operazioni del compasso geometrico e militare di Galileo &#9417;.
• Sixty copies of the book, dedicated to Cosimo de Medici on 10 July 1606, were printed on a press owned by Galileo; he distributed them among influential people.
• In early 1607 a Latin version of Galileo's instruction manual entitled Usus et Fabrica Circini Cuiusdam Proportionis &#9417; appeared under Capra's name.
• Perhaps worst of all, he had dedicated Le operazioni del compasso geometrico &#9417; to Cosimo de Medici, one of his patrons, and if Capra was to be believed then Galileo had committed a crime in dedicating something which was not his to give.
• These include one from Giacomo Alvise Cornaro written on 6 April, one from Pompeo di Pannicchi written on 14 April, and one from Paolo Sarpi, a Venetian scientist and Church reformer, who wrote on 20 April 1607:- .
• These came from people such as Giacomo Badovere (on 13 May), Marcantonio Mazzoleni, Galileo's technician, (on 24 May), and Giovanni Francesco Sagredo (on 1 June).
• In the Difesa, Galileo also argues against Capra's claim that he had copied Le operazioni del compasso geometrico &#9417;.
• Galileo did not believe that such a book existed but in fact Capra was referring to one of two books printed in 1604 and 1605 which described a reduction compass attributed to Joost Burgi.
• We can, of course, be certain that nothing was covering the autograph at the time Forbes purchased this copy in 1845.

120. References for Calculus history
• K Andersen, Precalculus, 1635-1665, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 292-307.
• 26 (2) (1995), 323-351.
• 39 (3) (1986), 223-253.
• W Breidert, Berkeleys Kritik an der Infinitesimalrechnung, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 185-191.
• (N.S.) 3 (1950), 542-554.
• Torino 46 (1) (1988), 1-29.
• N Guicciardini, Three traditions in the calculus : Newton, Leibniz and Lagrange, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 308-317.
• N Guicciardini, The Development of Newtonian Calculus in Britain, 1700-1800 (Cambridge, 1989).
• T Guitard, On an episode in the history of the integral calculus, Historia Mathematica 14 (2) (1987), 215-219.
• P Kitcher, Fluxions, limits, and infinite littlenesse : A study of Newton's presentation of the calculus, Isis 64 (221) (1973), 33-49.
• 28 (2) (1991), 117-146.
• A Perez de Laborda, Newtons Fluxionsrechnung im Vergleich zu Leibniz' Infinitesimalkalkul, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 239-257.
• J A van Maanen, Die Mathematik in den Niederlanden im 17.
• Jahrhundert und ihre Rolle in der Entwicklungsgeschichte der Infinitesimalrechnung, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 1-13.
• 23 (1) (1993), 199-218.
• L Pepe, Les mathematiciens italiens et le calcul infinitesimal au debut du XVIIIe siecle, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 192-201.
• 1 (2) (1981), 43-101.
• J Pieters, Origines de la decouverte par Leibniz du calcul infinitesimal, in Cahiers du Centre de Logique 2 (Louvain-la-Neuve, 1981), 1-22.
• A Rosenthal, The history of calculus, The American Mathematical Monthly 58 (1951), 75-86.
• A dialogue between Leibniz and Newton (1675-1677), Archive for History of Exact Sciences 2 (1964), 113-137.
• (3) (1986), 87-93.
• G C Smith, Thomas Bayes and fluxions, Historia Mathematica 7 (4) (1980), 379-388.
• 25 (97) (1975), 304-308.
• D T Whiteside, Patterns of mathematical thought in the later seventeenth century, Archive for History of Exact Sciences 1 (1960), 179-388.

121. Greek sources I
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• i) The year 888 AD seems a long time ago but it is only 1100 years ago while it is 1200 years after the Elements was written.
• 2 (1900), 147-171.','8] and [',' J L Heiberg, Paralipomena zu Euklid, Hermes 39 (1930), 46-74; 161-201; 321-356.','9]).
• Six particularly old fragments (dating from around 225 BC) of what may be parts of the text were found on Elephantine Island in 1906.
• It appears, however, that there is no truth in the story that the Arabs burnt this library, see for example [',' A J Butler, The Arab conquest of Egypt and the last thirty years of the Roman Dominion (Oxford, 1902 reprinted 1978).','1].
• Between 1883 and 1888, Heiberg published an edition of the Elements which was as close to the original as he was able to produce (see [',' J L Heiberg, H Menge and M Curtze (eds.), Euclid Opera Omnia (9 Vols.) (Leipzig, 1883-1916).','5]).
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• Heath's edition of 1908 ([',' T L Heath, The Thirteen Books of Euclid&#8217;s Elements (3 Volumes) (New York, 1956).','4] is a later edition of this work) was based on Heiberg's edition and contains a description of the different manuscripts which have survived.
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• We refer to [',' T L Heath, The Thirteen Books of Euclid&#8217;s Elements (3 Volumes) (New York, 1956).','4] and [',' J L Heiberg, H Menge and M Curtze (eds.), Euclid Opera Omnia (9 Vols.) (Leipzig, 1883-1916).','5] for a detailed description.
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• William of Moerbeke (1215-1286) was archbishop of Corinth and a classical scholar whose Latin translations of Greek works played an important role in the transmission of Greek knowledge to medieval Europe.
• The first of the two Greek manuscripts has not been seen since 1311 when presumably it was destroyed.
• In 1899 an exceptionally important event occurred in our understanding of the works of Archimedes.
• In 1906 Heiberg was able to start examining the Archimedes palimpsest in Istanbul.
• Originally the pages were about 30 cm by 20 cm but when they were reused the pages were folded in half to make a book 20 cm by 15 cm with 174 pages.
• However before publication of Heiberg's new edition [',' J L Heiberg (ed.), Archimedes Opera omnia cum commentariis Eutocii (Leipzig, 1910-15, reprinted 1972).','6] of Archimedes' works incorporating these remarkable new discoveries was complete, the region was plunged into war along with the rest of Europe.
• However, Turkey was declared a sovereign nation in January 1921 but, later that year, the Greek armies made major advances almost reaching Ankara.
• The French collector may have sold it quite recently, but all we know for certain is that the palimpsest appeared at auction in Christie's in New York in 1998 sold on behalf of an anonymous seller.
• Was the palimpsest sold or stolen in 1922? .
• Who owned the palimpsest during the years 1922 to 1998? .
• The palimpsest was seen to have a number of icons on it when displayed by Christie's in New York in 1998 but Heiberg had not mentioned any icons on the work.
• More strangely they are different from those appearing in Heiberg version of On floating bodies in [',' J L Heiberg (ed.), Archimedes Opera omnia cum commentariis Eutocii (Leipzig, 1910-15, reprinted 1972).','6] which has the text from the palimpsest.

122. Longitude2
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A poem written in 1661 described the work going on at Gresham College (see [',' D Howse, Greenwich time and the discovery of the longitude (Oxford, 1980).','6]):- .
• In 1662 the group from Gresham College, which included John Wilkins, John Wallis and Robert Hooke, and other groups of scientists, became the Royal Society of London for the Promotion of Natural Knowledge.
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• Jonas Moore, a mathematics teacher and surveyor who was greatly in favour with Charles II, became a patron of John Flamsteed in 1670 after he met Flamsteed during a visit Flamsteed made to the Royal Society in London.
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• Proposals were being made to solve the longitude problem and in 1673 one based on magnetic declination was proposed by a certain Henry Bond, see [',' E G R Taylor, Old Henry Bond and the Longitude, Mariner&#8217;s Mirror 25 (1939), 162-169.','9].
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• Nevertheless Hooke had lectured in 1664 on 20 different ways to use a spring to make the balance of a clock more uniform and said that he had a few tricks up his sleeve that might let him produce a sufficiently accurate clock.
• to Flamsteed in 1674.
• From his salary of 63;100 he had to pay 63;10 taxes and also provide all his own instruments so that he might:- .
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• The building the Royal Observatory at Greenwich began in 1675 designed by Wren and directed by Hooke.
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• Flamsteed and Halley advised on the requirements for the instruments and observing began in 1676.
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• Flamsteed wrote in a letter in 1677:- .
• The Greenwich Royal Observatory had to provide large amounts of data and Flamsteed spent 15 years from 1689 to 1704 compiling tables of the moon for the lunar distance method of finding the longitude.
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• In the most serious incident in 1707 over 2000 men were lost when four ships ran aground on the Scilly Islands while returning to England.
• One comical proposal, based on the correct understanding that a knowledge of universal time would allow the longitude to be calculated, is described in [',' L A Brown, The Story of Maps (New York, 1951).','1]:- .
• A more serious proposal came from William Whiston and Humphrey Ditton in 1714.
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• They proposed, see [',' L A Brown, The Story of Maps (New York, 1951).','1]:- .
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• Parliament thought the time had come to make a radical move and, on 16 June 1714, they passed an Act:- .
• To understand the value of this prize one has only to remember poor Flamsteed's annual salary of 63;100 to provide both a living and to buy his instruments.
• John Hadley, who was vice-president of the Royal Society, described in a communication to the Society in May 1731, two new instruments which were based on the principle of double reflection.
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• In fact Thomas Godfrey in Pennsylvania had made the same invention at almost exactly the same time as Hadley, and it was later discovered in Halley's papers that Newton had a similar idea in 1700 but Halley had told him it was not practical.
• John Harrison built his first clock in 1715, the year after the Longitude Act was passed.
• By 1727 he had made a very fine clock with a 'gridiron' pendulum which consisted of nine alternating steel and brass rods to eliminate the effects of temperature changes.
• In 1730 Harrison visited London, taking with him his gridiron pendulum and the grasshopper escapement which he had developed, and there he learnt exactly what was required to win the longitude prize.
• Harrison completed the clock, now called H1, in 1735.
• However Harrison was not happy with H1 and he approached the Board of Longitude in 1737.
• Harrison completed H2 in 1739, as promised, but spent two years testing it himself.
• Harrison decided to build a third clock, H3, and wrote to the Board of Longitude in 1741.
• Clearly the Royal Society were impressed with Harrison's work since they awarded him their Copley Medal in 1749, a remarkable event considering the fact that Harrison had no academic background or training.
• Work on H3 did not go as well as Harrison had hoped and he received a number of further advances from the Board of Longitude before he eventually decided in 1757 not to test H3 but to build a much smaller clock.
• H4 was started in 1757 and completed in two years.
• In 1761 Harrison requested a trial at sea for H4.
• Since 1741 he had received £3000 from the Board of Longitude to help him complete his work and they now gave him another £500 to complete adjusting H4.
• James Bradley, who had succeeded Halley as Astronomer Royal in 1742, and Tobias Mayer were convinced that the lunar distance method would lead to the solution of the longitude problem.
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• Mayer had sent his lunar tables to the Board of Longitude in 1756 but the Seven Years War with France had prevented proper trials.
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• In 1761 Maskelyne, another strong believer in the lunar distance method, was sent to St Helena on Prince Henry to test the lunar distance method for the Board of Longitude and in particular to test Mayer's tables.
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• In [',' D Howse, Nevil Maskelyne: The seaman&#8217;s astronomer (Cambridge, 1989).','7] Howse describes Maskelyne's work on the lunar distance method on this voyage:- .
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• Bradley claimed that he and Tobias Mayer would have shared the 63;10000 longitude prize but for Harrison's blasted watch.
• James Bradley and Mayer both died in 1762 but Mayer's widow later received £3000 from the Board of Longitude.
• H4 lost only 54 seconds in the 5 months of the journey and after correction for errors, which Harrison had set out before the journey, the error was reduced to 15 seconds.
• annual chronometer trials took place at Greenwich from 1821, with prizes for the best chronometers submitted.
• Harrison's H1 clock .
• Harrison's H1 clock .
• Harrison's K1 clock .

123. Group theory
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Mobius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group.
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• Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.
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• (2) In 1761 Euler studied modular arithmetic.
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• Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups.
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• In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.
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• In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation.
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• In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.
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• His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations.
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• However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right.
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• Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.
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• Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation.
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• By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental.
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• Galois' work was not known until Liouville published Galois' papers in 1846.
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• Betti began in 1851 publishing work relating permutation theory and the theory of equations.
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• Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations.
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• Holder was to prove it in the context of abstract groups in 1889.
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• Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry.
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• As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy's.
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• In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups.
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• Cayley's papers of 1854 were so far ahead of their time that they had little impact.
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• However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation.
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• Cayley's work prompted Holder, in 1893, to investigate groups of order .
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• Von Dyck, with fundamental papers in 1882 and 1883, constructed free groups and the definition of abstract groups in terms of generators and relations.
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• Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897.
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• The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra &#9417; published in 1895 and 1896 became a standard text.
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124. Coffee houses
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Antony Wood writes in Athenae Oxonienses &#9417; (1691) that the first coffee house opened in Oxford:- .
• By 1663 it is recorded that there were 82 coffee houses in London.
• For example 'The Women's Petition Against Coffee' was set up and it claimed in 1674 that coffee:- .
• In the following year King Charles II tried to rid London of its coffee houses with an edict [',' A Browning (ed.), English Historical Documents 1660-1714, in D C Douglas (ed.), English Historical Documents VIII (Eyre and Spottiswoode, London, 1953).','5]:- .
• It moved to Lombard Street in 1692 and eventually moved into insurance and became Lloyd's of London.
• The second coffee house mentioned in this quote is the Rainbow, the second oldest coffee house in London, opened by James Farr in Fleet Street in 1657.
• This coffee house had many literary customers and in particular Richard Steele who used it as an office for the Guardian which he began to publish in 1713.
• 50">Beginning January 11, 1713-14, a course of philosophical lectures on mechanics, hydrostatics, pneumatics, optics, ..
• Henry Newman wrote a letter to Richard Steele on 10 August 1713 (see for example [',' A I Dale, Most honourable remembrance : The life and work of Thomas Bayes (New York- Berlin- Heidelberg, 2003).','1]):- .
• Slaughter's Coffee House in St Martin's Lane was established in 1692.
• 66">Among the earliest coffee-houses to be established in the West-end of London was that opened by Thomas Slaughter in St Martin's Lane in 1692 and known as Slaughter's.
• It remained under the oversight of Mr Slaughter until his death in 1740, and continued to enjoy a prosperous career for nearly a century longer, when the house was torn down.
• He even produced a book, in some ways the textbook to supplement his course, which he published in 1703 called Description and Uses of the Celestial and Terrestrial Globes and of Collins' Pocket Quadrant.

125. References for Nine chapters
• S S Bai, A re-examination of a ring area problem in the 'Jiu zhang suanshu' (Chinese), Beijing Shifan Daxue Xuebao 30 (1) (1994), 139-142.
• 19 (Russian) (Moscow, 1974), 231-273.
• (2) 4 (2) (1994), 113-137.
• K Chemla, Relations between procedure and demonstration : Measuring the circle in the 'Nine chapters on mathematical procedures' and their commentary by Liu Hui (3rd century), in History of mathematics and education: ideas and experiences (Essen, 1992) (1996), 69-112.
• J W Dauben, The "Pythagorean theorem" and Chinese mathematics : Liu Hui's commentary on the gou-gu theorem in Chapter Nine of the J'iu zhang suan shu', in Amphora (Basel, 1992), 133-155.
• 28 (2) (1994), 263-268.
• 18 (3) (1991), 212-238.
• Practice Theory (3) (1983), 75-79.
• (2) 19 (1992), 200-202.
• (2) 4 (3) (1995), 207-222.
• 2 (Hohhot, 1991), 31-34.
• 47 (1) (1994), 1-51.
• 21 (2) (1991), 1-6.
• 33 (2) (1999), 179-185.
• 19 (2) (2000), 97-113.
• Z C Shang, A comparison of the Elements and Wei Liu's notes on the theory of proportion and its applications in Jiu Zhang Suanshu (Chinese), in Studies on Euclid's Elements (Hohhot, 1992) , 217-233.
• A 17 (1) (2002), 105-112.
• J Song, The historical value of the 'Nine chapters on the mathematical art' in society and the economy, in Chinese studies in the history and philosophy of science and technology 179 (Dordrecht,1996), 261-266.
• 71 (3) (1998), 163-181.
• 12 (1) (1985), 71-73.
• 6 (2) (1979), 164-188.
• 2 (Hohhot, 1991), 35-39.
• Z W Xi and S L Zhang, On the characteristic of the dialectical thought of the 'Jiu zhang suanshu' and Hui Liu's commentary (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 19 (4) (1993), 103-110.
• S C Yang, "Ratio" and "power" in Hui Liu's commentary on the 'Jiu zhang suan shu' ('Arithmetic in nine chapters') (Chinese), Dongbei Shida Xuebao (4) (1990), 39-43.

126. Greek astronomy
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In this work Hesiod writes that (see [',' B Hetherington, A chronicle of pre-telescope astronomy (Chichester, 1996).','5], also [',' A F Aveni, Empires of time : Calendars, clocks and cultures (New York, 1989).','1] and [',' A Pannekoek, A history of astronomy (New York, 1989).','7]):- .
• An early time scale based on 12 months of 30 days did not work well since the moon rapidly gets out of phase with a 30 day month.
• Around 450 BC Oenopides is said to have discovered the ecliptic made an angle of 2476; with the equator, which was accepted in Greece until refined by Eratosthenes in around 250 BC.
• As Berry writes [',' A Berry, A short history of astronomy (New York, 1961).','2]:- .
• Aristyllus was a pupil of Timocharis and in Maeyama [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','23] analyses 18 of their observations and shows that Timocharis observed around 290 BC while Aristyllus observed a generation later around 260 BC.
• Maeyama writes [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','23]:- .
• 26 (2) (1982), 99-113.','34] van der Waerden makes an interesting suggestion related to the other important astronomer who worked in Alexandria around this time, namely Aristarchus.
• 43 (2) (1991), 93-132.','16] attempt to answer the question of why Timocharis and Aristyllus made their accurate observations.
• 26 (2) (1982), 99-113.','34] suggests that the observations were made to determine the constants in the heliocentric theory of Aristarchus.
• 43 (2) (1991), 93-132.','16] make other interesting suggestions.
• As Berry writes in [',' A Berry, A short history of astronomy (New York, 1961).','2]:- .
• As Jones writes in [',' A Jones, The adaptation of Babylonian methods in Greek numerical astronomy, Isis 82 (313) (1991), 441-453.','21]:- .

127. Tait's scrapbook
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• I gave the lecture at 14 India Street, Edinburgh, the birthplace in 1831 of James Clerk Maxwell, and now owned by the James Clerk Maxwell Foundation.
• The house where James Clerk Maxwell was born is at 14 India Street, Edinburgh about a fifteen-minute walk from the railway station which is in the centre of Edinburgh.
• Documents relating to the purchase of the house, which was built in 1820, are in the Display Cabinet and, especially for the occasion, Tait's school medals are on display.
• Many pages relate to The Unseen Universe: or Physical Speculations on a Future State by P G Tait and Balfour Stewart, originally penned anonymously and first published in 1875.
• P G Tait was born in Dalkeith on 28 April 1831, the son of John Tait, secretary to Walter Francis Scott, fifth duke of Buccleuch, and his wife, Mary Ronaldson.
• A tragedy struck the family when his father died in 1840.
• Following this his mother moved from Dalkeith to Edinburgh where Peter attended Circus Place School for a year before entering Edinburgh Academy in 1841.
• There were school prizes open to all pupils and in 1846 Tait came third overall but first in mathematics, while in the following year Maxwell came first in mathematics with Tait second.
• Tait remained at Edinburgh University for only one year before entering Peterhouse, Cambridge in 1848.
• January 31, 1852 .
• Two of his friends at Peterhouse were sons of the Rev James Porter and through them Tait met their sister, Margaret Archer Porter, who he married in Belfast on 13 October 1857.
• The Chair of Natural Philosophy at the University of Edinburgh became vacant in 1859, J D Forbes having moved to the University of St Andrews to become Principal.
• The second extract comes from an article about Tait which appeared in The Evening Dispatch following his death in 1901:- .
• Tait was proud of the University of Edinburgh as is clear in his address to graduates in 1888 (taken from the Scrapbook):- .
• Tait replied in vigorous fashion in his address to graduates in 1888 (taken from the Scrapbook) saying that this:- .
• In the following exchange he is complaining of sewage problems in St Andrews in 1878.
• Tait had solved the puzzle and submitted a paper on the solution when he saw two articles in the American Journal of Mathematics in 1879, one by W W Johnson and one by W E Story.
• Although it has been claimed that no elementary proof was known until A F Archer gave one in 1999 in the American Mathematical Monthly, it appears to me that Tait's proof is completely elementary.
• We note Maxwell's admiration for the work of Tait expressed in a letter of 1871 to Thomson:- .
• on Saturday 27 July 1901." .

128. References for Perfect numbers
• Medizin 24 (1) (1987), 21-30.
• (4) 8 (2) (1990), 239-241.
• M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.
• 13 (4) (1971), 421-424.
• 34 (1988), 1-10.
• 6 (1975), 84-99.
• M L Nankar, History of perfect numbers, Ganita Bharati 1 (1-2) (1979), 7-8.
• 9 (1) (1981/82), 84-79.
• 16 (2) (1989), 123-136.
• 16 (4) (1989), 343-352.
• 5 (2) (1989), 82-89.
• C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid's Elements, Centaurus 20 (4) (1976), 269-275.
• 18 (1952), 122-131.
• A M Vaidya, Comment on : "History of perfect numbers", Ganita Bharati 1 (3-4) (1979), 22.
• Intelligencer 7 (2) (1985), 66-68.

129. Greeks poetry
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Robert Burns (1759-1796): Caledonia - A Ballad.
• William Wordsworth (1770-1850): The Prelude (1850).
• Rudolph Chambers Lehmann (1856-1929): The Death Of Euclid.
• Vachel Lindsay (1879-1931): Euclid.
• Edna St Vincent Millay (1892-1950): Euclid alone.
• Friedrich von Schiller (1759-1805): Archimedes.
• William Wordsworth (1770-1850): The Excursion.

130. Longitude1
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• They were in dispute over the "new world" and, in 1493, Pope Alexander VI who was Spanish, issued the Bull of Demarcation which would settle the dispute.
• In 1514 Johann Werner published a translation of Ptolemy's Geography.
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• Nunes was appointed professor of mathematics in 1529 specifically to try to solve this and related problems.
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• Gemma Frisius, in 1530, proposed a methods of finding the longitude using a clock.
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• When we have completed a journey of 15 or 20 miles, it may please us to learn the difference of longitude between where we have reached and our place of departure.
• First Philip II offered a prize in 1567.
• Soon after Philip III of Spain came to the throne in 1598 he was advised to offer a large prize to .
• He wrote to the Spanish Court in 1616 proposing that the way to measure absolute time, which could be measured at any point on the Earth, was to use the moons of Jupiter.
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• Galileo first observed the moons in 1610 and by 1612 he had tables of their movements which were accurate enough to allow him to predict their positions several months ahead.
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• A long correspondence over a period of 16 years failed to convince Spain of the virtues of the scheme so, when Holland offered a large prize in 1636 .
• A serious proposal, however, came forward from Jean-Baptiste Morin in 1634 and was made to his own country France.
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• Cardinal Richelieu, chief minister to King Louis XIII of France from 1624 to 1642, set up a commission consisting of Etienne Pascal, Mydorge, Beaugrand, Herigone, J C Boulenger and L de la Porte to investigate Morin's claims.
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• Cardinal Richelieu died in 1642 and his successor, Cardinal Mazarin, gave Morin 2000 livres for his efforts in 1645.
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• In 1651 Cardinal Mazarin, who was then the chief political figure in France, was forced to leave Paris during the struggle between the King and the parliament.
• Jean-Baptiste Colbert became Mazarin's agent in Paris and Colbert was rewarded by Mazarin who, on his deathbed in 1661, recommended Colbert to the King, Louis XIV.
• In 1666, at Colbert's instigation, the Academie Royale des Sciences was founded.
• Huygens was particularly important to the Academie Royale des Sciences as he had patented the pendulum clock in 1656 and several of his clocks had been tried, although not very successfully, at sea in an attempt to find the longitude.
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• Having invented the pendulum clock in 1657, Huygens turned his attention to the longitude problem, convinced that the horological approach - to produce a marine timekeeper that would keep accurate and regular time for months on end in any climate, regardless of the motion of the ship - would soon make it possible to discover the longitude.
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• He had claimed to know the secret of the longitude as early as 1662 but refused to divulge his theories.
• The members of the Academie Royale des Sciences made observations of the Moon over the years 1667 to 1669 which convinced them that the mathematics of the position of the Moon was too difficult to make it useful as a solution to the longitude problem.
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• In 1668 one of his timekeepers, which had kept going during both gales and a sea battle, gave a difference of longitude between Toulon and Crete as 20 76; 30' as against the true value of 19 6; 13', an error of only 100 km or so.
• However in 1668 Cassini, working in Italy, published tables of Jupiter's moons which he had compiled over a period of 16 years.
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• In 1669 Picard was assigned the task of making precise measurements of the size of the Earth.
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• After the measurements had all been taken and the results of the survey had been studied it was announced that the diameter of the Earth was about 12554 km, a good result compared with the equatorial diameter now known to be 12756 km.
• Having completed his measurements of the size of the Earth, Picard was sent on an expedition to Cayenne in 1672.
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• In 1681 the Academie Royale des Sciences mounted an expedition to the island of Goree in the West Indies.
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• Brown, in [',' L A Brown, The Story of Maps (New York, 1951).','1], explains Cassini's instructions for one man and two man observing:- .
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131. Newton poetry
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Samuel Bowden was a physician and poet who published two volumes of poetry in 1733-35.
• James Thomson (1700-1748) was a Scottish poet who is, perhaps, best known for writing the words of "Rule, Britannia!".
• Anonymous: published in London Medley (1731).
• Samuel Bowden was a physician and poet who published two volumes of poetry in 1733-35.
• Jane Brereton (1685-1740) was an English poet and frequent contributor to The Gentleman's Magazine: .
• Jane Brereton (1685-1740) was an English poet and frequent contributor to The Gentleman's Magazine: .
• Originally published in Latin, this poem was translated into English by George Canning and published in 1766.
• Henry Jones (1721-1770) was an Irish poet and dramatist who lived in London for much of his life.
• Anonymous: The Vanity of Philsophick Systems: a Poem addressed to the Royal Society (1761).
• William Blake: You don't believe (1800-10).
• Percy Bysshe Shelley: Queen Mab: A philosophical poem (1813).
• Lord Byron: Don Juan (1819-24).

132. References for Fermat's last theorem
• Monthly 101 (1) (1994), 3-14.
• 14 (3) (1975), 219-236.
• C Goldstein, Le theoreme de Fermat, La recherche 263 (1994), 268-275.
• Intelligencer 7 (4) (1985), 40-47; 55.
• 14 (1-3) (1993), 129-135.
• 17 (2) (1994), 1-11.
• 21 (5) (1994), 150-159.
• (2) 29 (1-2) (1983), 165-177.
• P Ribenboim, Fermat's last theorem, before June 23, 1993, in Number theory (Providence, RI, 1995), 279-294.
• (2) 5 (1) (1984), 14-32.
• 20 (2) (1977), 229-242.
• R Schoof, Fermat's last theorem, in Jahrbuch uberblicke Mathematik (Braunschweig, 1995), 193-211.
• Intelligencer 8 (1) (1986), 59-61.

133. Bolzano's manuscripts
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Bernard Bolzano died in 1848.
• He had worked for many years on Grossenlehre &#9417; which was intended to be an introduction to mathematics covering many different areas of mathematics such as numbers, elementary geometry, geometry in general, function theory, methodology, and the ideas of quantity and space.
• Specific parts such as Functionenlehre &#9417;, and Zahlenlehre &#9417; were written, much was in a less complete form with workings and reworkings of parts of his ambitious project.
• However, Zimmermann's interests were more in the area of philosophy and indeed he was appointed to the chair of philosophy in Prague in 1852, only four years after Bolzano's death.
• One of Bolzano's collaborators, Franz Prihonsky, had ensured that not all of Bolzano's mathematical work lay unpublished in Zimmermann's possession, for he published Bolzano's Paradoxien des Unendlichen &#9417;, in 1851.
• In 1909 Charles Peirce wrote that Bolzano's Paradoxien des Unendlichen &#9417; conferred:- .
• He published four papers published between 1920 and 1923 and gave three lectures to the Union of Czech Mathematicians and Physicists on 3 December 1921, 14 January 1922, and 4 December 1922.
• Jasek had discovered that the unpublished Functionenlehre &#9417; contained some important results in analysis showing that Bolzano had made certain discoveries in that topic well before similar results had been discovered by others.
• The Committee set itself the task of publishing the first volume of Bolzano's manuscripts, which they intended to be Functionenlehre &#9417;, in 1925 and then the remaining material over the following five years.
• Functionenlehre &#9417; appeared in 1930, five years behind schedule and at the time when the Committee had expected the whole project to be completed.
• Progress for a while was at least steady with Zahlentheorie &#9417; being published in 1931, and Von dem besten Staate &#9417; in 1932.
• There was a delay until 1935 before the next volume Der Briefwechsel B Bolzano's mit F Exner &#9417; was published.
• This contained letters between Bolzano and the philosopher Franz Exner (1802-1853).
• The fifth and final publication by the Committee was Memoires geometriques &#9417; which did not appear until 1948.
• However, in 1948, Communists took control of the country and in 1952 the Czech Academy of Sciences was disbanded and at the same time the Bolzano Committee was also disbanded.
• The Czechoslovak Academy of Sciences was founded in 1952 but the Bolzano Committee was not at this stage re-established under the new academy.
• This did happen in 1958 but three years later the new Committee was disbanded with no further volumes of Bolzano's manuscripts being published during these three years.
• Before the first volume in the series appeared in 1969 there were a number of related publications.
• Kazimir Vecerka published Bolzano's Anti-Euclid in 1967.
• Bob van Rootselaar published Bolzano's corrections to his Functionenlehre &#9417; in 1969.
• It is, perhaps one of, Bolzano's attempts to correct certain errors in Functionenlehre &#9417;.
• The first volume in the new series Bernard Bolzano-Gesamtausgabe &#9417; published by Friedrich Frommann Verlag and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
• The second volume, which set the scene for the whole series, appeared in 1972.
• This is a detailed catalogue of that part of Bolzano's manuscripts which was donated in 1892 to the then existing Court Library in Vienna by his former disciple Robert Zimmermann (1824-1898).
• Moreover, in the supplement, 19 letters from Bolzano to the philosopher Franz Exner (1802-1853), dealing inter alia with questions of logical semantics, are described.
• The bibliography lists (1) in chronological order all works of Bolzano published in any language until March 1971 and (2) literature on Bolzano in alphabetic order ..

134. References for Egyptian mathematics
• G G Joseph, The crest of the peacock (London, 1991).
• Annual Iranian Mathematics Conference, Isfahan, 1990 (Isfahan, 1992), 1-20.
• 14 (1952), 81-91.
• 7 (1945), 11-15.
• E M Bruins, The Egyptian shadow clock, Janus 52 (1965), 127-137.
• E M Bruins, Egyptian arithmetic, Janus 68 (1-3) (1981), 33-52.
• E M Bruins, Reducible and trivial decompositions concerning Egyptian arithmetics, Janus 68 (4) (1981), 281-297.
• E M Bruins, The part in ancient Egyptian mathematics, Centaurus 19 (4) (1975), 241-251.
• 4 (1966), 339-354.
• 16 (10) (1980), 219-221.
• 12 (3) (1985), 261-268.
• 22 (1959), 247-250.
• 6 (4) (1979), 442-447.
• 12 (1974), 291-298.
• 8 (4) (1981), 456-457.
• M Guillemot, De l'arithmetique egyptienne a l'arithmetique arabo-islamique, in Deuxieme Colloque Maghrebin sur l'Histoire des Mathematiques Arabes, Tunis, 1988 (Tunis, 1990), 95-105.
• 2 Tipaza, 1990 (Algiers, 1998), 125-145.
• R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.
• 7 (2) (1980), 186-187.
• 23 (1978), 181-191; 358.
• Chronicle 10 (1-2) (1981/82), 13-30.
• 1 (1) (1974), 93-94.
• 12 (2) (1985), 107-122.
• G J Toomer, Mathematics and Astronomy, in J R Harris (ed.), The Legacy of Egypt (Oxford, 1971), 27-54.
• Istorii Estestvoznaniya 2 (1948), 426-498.
• Istorii Estestvoznaniya 1 (1947), 269-282.

135. Chrystal and the RSE
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• He was asked to address the Society in 1879, the year of his appointment to Edinburgh, then he was elected a Fellow of the Royal Society of Edinburgh at its meeting on Monday, 2nd February 1880.
• In November of that year he was elected to the Council of the Society for the first of three terms of office: 1880-3; 1884-7; and 1895-1901.
• On the death of Professor Tait in 1901, Chrystal succeeded him as General Secretary of the RSE.
• The RSE had been one of the bodies to propose this building in 1821 and, since the completion of the building, had shared it with the Society of Antiquaries and the Royal Institution for the Encouragement of the Fine Arts.
• For those familiar with Edinburgh today and interested in identifying the Royal Institution, we should say that it has been named the Royal Scottish Academy since 1911.
• By this stage the RSE were the sole occupants of the Royal Institution, the Society for the Promotion of Fine Arts having ceased to exist and the Society of Antiquaries having left in 1892.
• In 1903, through the initiative of Sir John Murray, a scheme was proposed to secure the whole of the Royal Institution building for the RSE.
• In 1906 a Liberal Government was elected and one of its first acts was to introduce the National Galleries of Scotland Bill.
• He looked for support from the Royal Society of London and wrote to Sir Joseph Larmor on 13 November 1906 [',' Letter from Professor G Chrystal to Sir J Larmor, 13 November 1906, Royal Society of London Library (MSS.
• Our annual deficit is already about 63;300 and we are paying for our publications partly out of capital.
• The deputation is fixed for Thursday, 22nd at 12.30 in Dover House.
• The next day Chrystal again wrote to Sir Joseph Larmor, replying to Larmor's letter of 15 November 1906 [',' Letter from Professor G Chrystal to Sir J Larmor, 16 November 1906, Royal Society of London Library (MSS.
• Many thanks for your kind letter of 15th and promise to countenance our demonstration.
• Thursday 22nd at 12.30 in the Scottish Office Whitehall is the hour and place.
• Sir William Turner persuaded [',' N Campbell, R Martin and S Smellie, The Royal Society of Edinburgh 1783-1983 (Edinburgh, 1983).','1]:- .
• the Secretary for Scotland, Lord Pentland, that the Treasury granted the necessary 63;25 000 for purchase of 22-24 George Street and 63;3 000 to cover the cost of removal and equipment.
• Lord Kelvin, the President of the RSE, died in 1907 and Sir William Turner was elected President.
• It is due to Lord Pentland that we should record our sense of his courtesy at our interviews with him, as well as our hearty thanks for the effective advocacy of our claim to obtain the requisite funds from the Treasury, both for the purchase and equipment of our habitation and for an annual grant of 63;600 to assist in the discharge of our scientific work.
• In 1810 the Society purchased No.40 George Street, in which house it was accommodated until 1826, when it removed to the Royal Institution Buildings.

136. test2.html
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The world map (employing Ptolemy's first projection in 1.24) follows the end of Book 7, while the twenty-six regional maps alternate with their respective captions in Book 8.
• 191 (Vatican), a paper codex containing a large corpus of mathematical and scientific writings, copied by numerous hands and assembled about 1296.

137. References for Egyptian Papyri
• G G Joseph, The crest of the peacock (London, 1991).
• Annual Iranian Mathematics Conference, Isfahan, 1990 (Isfahan, 1992), 1-20.
• 14 (1952), 81-91.
• 7 (1945), 11-15.
• E M Bruins, The Egyptian shadow clock, Janus 52 (1965), 127-137.
• E M Bruins, Egyptian arithmetic, Janus 68 (1-3) (1981), 33-52.
• E M Bruins, Reducible and trivial decompositions concerning Egyptian arithmetics, Janus 68 (4) (1981), 281-297.
• E M Bruins, The part in ancient Egyptian mathematics, Centaurus 19 (4) (1975), 241-251.
• 4 (1966), 339-354.
• 16 (10) (1980), 219-221.
• 12 (3) (1985), 261-268.
• 22 (1959), 247-250.
• 6 (4) (1979), 442-447.
• 12 (1974), 291-298.
• 8 (4) (1981), 456-457.
• M Guillemot, De l'arithmetique egyptienne a l'arithmetique arabo-islamique, in Deuxieme Colloque Maghrebin sur l'Histoire des Mathematiques Arabes, Tunis, 1988 (Tunis, 1990), 95-105.
• 2 Tipaza, 1990 (Algiers, 1998), 125-145.
• R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.
• 7 (2) (1980), 186-187.
• 23 (1978), 181-191; 358.
• Chronicle 10 (1-2) (1981/82), 13-30.
• 1 (1) (1974), 93-94.
• 12 (2) (1985), 107-122.
• G J Toomer, Mathematics and Astronomy, in J R Harris (ed.), The Legacy of Egypt (Oxford, 1971), 27-54.
• Istorii Estestvoznaniya 2 (1948), 426-498.
• Istorii Estestvoznaniya 1 (1947), 269-282.

138. References for Black holes
• M Bartusiak, Black Hole: How an idea abandoned by Newtonians, hated by Einstein, and gambled on by Hawking became loved (Yale University Press, New Haven-London, 2015).
• J M Galat, Black Holes and Supernovas (Capstone, 2011).
• S S Gubser and F Pretorius, The Little Book of Black Holes (Princeton University Press, 2017).
• S Hawking, Black Holes: The Reith Lectures (Random House, 2016).
• S Latta, Black Holes: The Weird Science of the Most Mysterious Objects in the Universe (Twenty-First Century Books, 2017).
• D Nardo, Black Holes (Independent Publishing Platform, 2015).
• D J Raine and E G Thomas, Black Holes: An Introduction (Imperial College Press, 2010).
• D M Rau, Black Holes (Capstone, 2015).
• W H Tucker, Chandra's Cosmos: Dark Matter, Black Holes, and Other Wonders Revealed by NASA's Premier X-Ray Observatory (Smithsonian Institution, 2017).
• S Chandrasekhar, The Black Hole in Astrophysics: The Origin of the Concept and its Role, Contemporary Physics 15 (1974), 1-24.
• A Chodos (ed.), 1783: John Michell anticipates black holes, Physics History 18 (10) (2009).
• F Dyson, Chandrasekhar's role in 20th-century science, Physics Today 63 (12) (2010), 44-48.
• J Eisenstaedt, The Early Interpretation of the Schwarzschild Solution, in D Howard and J Stachel (eds), Einstein and the History of General Relativity: Einstein Studies 1 (Birkhauser, Boston, 1989), 213-233.
• https://www.centauri-dreams.org/2011/11/29/cygnus-x-1-a-black-hole-confirmed/ .
• S Schaffer, John Mitchell and Black Holes, Journal for the History of Astronomy 10 (1979), 42-43.
• K Subramanian, Before S Chandrasekhar won the Nobel in 1983 his theories were overlooked because of his race, Business Standard (20 October 2017).

139. References for Pell's equation
• C Brezinski, History of continued fractions and Pade approximants (Berlin, 1991).
• 32 (1986), 39-49.
• A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.
• 29 (1985), 177-189, 347.
• 8 (1) (1991), 23-27.
• Fractions 3 (1994), 4-31.
• I-Kh I Gerasim, On the genesis of Redei's theory of the equation xÛ -DyÛ = -1 (Russian), Istor.-Mat.
• 32-33 (1990), 199-211.
• C Houzel, Introduction a l'histoire de l'analyse diophantienne, in Analyse diophantienne et geometrie algebrique (Paris, 1993), 1-12.
• 62 (4) (1989), 253-259.
• 71 (6) (1996), 490-493.
• (2) (1997), 13-18.
• 5 (1) (1988), 10-15.
• Nauk 31 (5) (191) (1976), 57-70.

140. References for Christianity and Mathematics
• D Alexander, Rebuilding the matrix (Oxford, 2001).
• B Russell, History of Western Philosophy (London, 1961).
• W C Charron and J P Doyle, On the self-refuting statement "There is no truth": a medieval treatment, Vivarium 31 (2) (1993), 241-266.
• S Knuuttila and A I Lehtinen, Change and contradiction: a fourteenth-century controversy, Synthese 40 (1) (1979), 189-207.
• F Krafft, Astronomie als Gottesdienst : Die Erneuerung der Astronomie durch Johannes Kepler, in Der Weg der Naturwissenschaft von Johannes von Gmunden zu Johannes Kepler (Vienna, 1988), 182-196.
• E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo (Cambridge, 1998), 271-347.
• 20 (1-4) (1978), 271-283.
• E Rosen, Calvin's attitude towards Copernicus, Journal of the History of Ideas 21 (1960), 431-441.
• E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 (1975), 225-231.
• E Rosen, Was Copernicus' Revolutions approved by the Pope, Journal of the History of Ideas 36 (1975), 531-542.
• E Rosen, Galileo's misstatements about Copernicus, Isis 32 (1958), 319-330.
• 32 (1981), 190-197.
• 20 (1) (1989), 1-23.
• G Zimmermann, Die Gottesvorstellung des Nicolaus Copernicus, Studia Leibnitiana 20 (1) (1988), 63-79.

141. References for Bourbaki 2
• 35 (1988), 43-49.
• Context 10 (2) (1997), 297-342.
• Intelligencer 15 (1993), 27-35.
• L Baulieu, Bourbaki's art of memory : Commemorative practices in science: historical perspectives on the politics of collective memory, Osiris (2) 14 (1999), 219-251.
• Contemp., Amsterdam, 1998), 75-123.
• L Baulieu, Dispelling a myth : questions and answers about Bourbaki's early work, 1934-1944, in The intersection of history and mathematics, (Birkhauser, Basel, 1994), 241-252.
• Intelligencer 8 (4) (1986), 84-85.
• A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
• 45 (3) (1998), 373-380.
• A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt.
• (1) (1998), 8-15.
• Intelligencer 2 (4) (1979-80), 175-180.
• J Delsarte, Compte rendu de la reunion Bourbaki du 14 janvier 1935, Gaz.
• 84 (2000), 16-18.
• Monthly 77 (1970), 134-145.
• 20 (2) (1975), 66-76.
• 14(47) (1971), 50-61.
• Nauk 28 (3)(171) 1973), 205-216.
• A 77 (1972), 447-460.
• 18 (2) (1969), 13-25.
• H Freudenthal, The truth about Bourbaki (Dutch), Euclides (Groningen) 61 (10) (1985/86), 330.
• Wissensch., XII (Steiner, Wiesbaden, 1985), 607-611.
• Intelligencer 7 (2) (1985), 18-22.
• 49 (1) (2002), 1-10.
• C Houzel, The influence of Bourbaki (Italian), in Italian mathematics between the two world wars (Italian), Milan/Gargnano, 1986 (Pitagora, Bologna, 1987), 241-246.
• K Krickeberg, Comment: "Twenty-five years with Nicolas Bourbaki 1949-1973" by A Borel, Mitt.
• (2) (1998), 16.
• 2 (1) (1980), 16-29.
• Intelligencer 8 (2) (1986), 5.
• 24 (2) (1975/76), 169-187.
• Intelligencer 21 (3) (1999), 16-17.
• O Pekonen, Nicolas Bourbaki in memoriam (Finnish), in In the forest of symbols (Finnish) (Art House, Helsinki, 1992), 55-71.
• 4 (1959), 673-678.
• M Senechal, The continuing silence of Bourbaki - an interview with Pierre Cartier, June 18, 1997, Math.
• Intelligencer 20 (1) (1998), 22-28.
• 6(41) (2001), 100-110; 388.
• (2) 3 (1957), 289-297.
• 3 (1961), 23-35.

142. Weil family
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• She was born in Russia, but life there became hard after Russian Jews were attacked after the assassination of Tsar Alexander II in 1881.
• When she complained to one of Andre's teachers that he may not have a very good grounding in arithmetic, the teacher replied [',' F du Plessix Gray, Simone Weil (Weidenfeld & Nicolson, London, 2001).','2]:- .
• She had, as Andre later wrote [',' F du Plessix Gray, Simone Weil (Weidenfeld & Nicolson, London, 2001).','2]:- .
• In 1914 they moved to a larger home on the Boulevard Saint-Michel.
• However, 1914 saw the start of World War I and Bernard Weil was drafted.
• In 1919 Andre and Simone learnt for the first time that they were Jewish.
• In 1920, at the age of fourteen, he first met Hadamard.
• Andre later wrote [',' F du Plessix Gray, Simone Weil (Weidenfeld & Nicolson, London, 2001).','2]:- .
• When Andre was awarded a school prize he asked Hadamard to help him choose some mathematics books and as a result he became the proud owner of Jordan's Cours d'Analyse &#9417; and Thomson and Tait's Treatise of Natural Philosophy.
• He began his studies at the Ecole Normale Superieure which he entered in 1922 and, once there, asked Sylvain Levi for advice on a book to read in the vacation.
• Levi suggested that Andre read Bhagavad Gita &#9417; saying [',' F du Plessix Gray, Simone Weil (Weidenfeld & Nicolson, London, 2001).','2]:- .
• In 1928 she entered the ecole Normale Superieure.
• Mrs Reinherz, Selma's mother, who lived with the Weils, died of cancer in 1929 at around the time they moved to their new home.
• Bernard died in 1955 and Selma in 1965.

143. Wave versus matrix
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The model provided by Bohr in 1913 had great appeal, and for that matter it still does.
• Max Born expressed that view in 1923:- .
• For example, in 1850 Foucault showed that light travels slower in water than in air.
• In his doctoral thesis written in 1923, de Broglie proposed his wave/particle duality theory in which particles, even electrons, could also behave like waves.
• Schrodinger wrote in 1926:- .
• The theory by Heisenberg to which Schrodinger refers is quantum mechanics which he put forward in 1925.
• Heisenberg wrote in Der Teil und das Ganze &#9417; (Munich, 1969) about a visit of Schrodinger to Copenhagen on 4 October 1926 and he reports in this book details a discussion between Bohr and Schrodinger.
• Bohr repeated these arguments in print in 1927:- .

144. Crimean pictures
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .

145. Statistics index
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .

146. The four colour theorem
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• After practising as a barrister he went to South Africa in 1861 as a Professor of Mathematics.
• Cayley also learnt of the problem from De Morgan and on 13 June 1878 he posed a question to the London Mathematical Society asking if the Four Colour Conjecture had been solved.
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• Shortly afterwards Cayley sent a paper On the colouring of maps to the Royal Geographical Society and in was published in 1879.
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• On 17 July 1879 Alfred Bray Kempe announced in Nature that he had a proof of the Four Colour Conjecture.
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• At Cayley's suggestion Kempe submitted the Theorem to the American Journal of Mathematics where it was published in 1879.
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• He was knighted in 1912.
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• He published two improved versions of his proof, the second in 1880 aroused the interest of P G Tait, the Professor of Natural Philosophy at Edinburgh.
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• The Four Colour Theorem returned to being the Four Colour Conjecture in 1890.
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• In 1896 de la Vallee Poussin also pointed out the error in Kempe's paper, apparently unaware of Heawood's work.
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• In 1898 he proved that if the number of edges around each region is divisible by 3 then the regions are 4-colourable.
• The search for avoidable sets began in 1904 with work of Weinicke.
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• Renewed interest in the USA was due to Veblen who published a paper in 1912 on the Four Colour Conjecture generalising Heawood's work.
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• Franklin in 1922 published further examples of unavoidable sets and used Birkhoff's idea of reducibility to prove, among other results, that any map with ≤ 25 regions can be 4-coloured.
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• Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976.
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• Heesch in 1969 introduced the method of discharging.
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• Now from Euler's formula we can deduce that the sum of the charges over all the vertices must be 12.
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• There were difficulties with his approach since some of his configurations had a boundary of up to 18 edges and could not be tested for reducibility.
• They managed to keep the boundary ring size down to Ͱ4; 14 making computations easier that for the Heesch case.
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• Details of the proof appeared in two articles in 1977.

147. Newton's bucket
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Isaac Newton conducted an experiment with a bucket containing water which he described in 1689.
• One certainly can't physically try this today any more than one could in 1689.
• In 1870 Carl Neumann suggested a similar situation to the bucket when he imagined that the whole universe consisted only of a single planet.
• However, he wrote in 1872 in History and Root of the Principle of the Conservation of Energy:- .
• We quote from an 1883 work by Mach on Newton's bucket:- .
• His theory was published in 1915.
• He did so in a letter which he wrote to Mach in 1913 in which he told Mach that his view of Newton's bucket was correct and agreed with general relativity.
• In 1918 Joseph Lense and Hans Thirring obtained approximate solutions of the equations of general relativity for rotating bodies.
• In 1966 Dieter Brill and Jeffrey Cohen showed that frame dragging should occur in a hollow sphere.
• In 1985 further progress by H Pfister and K Braun showed that sufficient centrifugal forces would be induced at the centre of the hollow massive sphere to cause water to form a concave surface in a bucket which is not rotating with respect to the distant stars.

148. test
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• &#961; .
• &#961; .
• &#961; .
• &#961; .
• &#961; .
• &#961; .

149. The Scottish Book
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• What follows is the preface to a typed document entitled "The Scottish Book" sent by Stan Ulam from Los Alamos to Professor Copson in Edinburgh on January 28 1958.
• The enclosed collection of mathematical problems has its origin in a notebook which was started in Lwow, in Poland in 1935.
• The last date figuring in the book is May 31, 1941.
• A general word of explanation may be here in order: I left Poland late in 1935 but, before the war, visited Lwow every summer in 1936, '37, '38, and '39.
• 34 183-245 and Problem 77(a), R.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• x2 + 10x = 39.
• x2 + 21 = 10x.
• Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum &#9417; published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.
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• In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita &#9417;, now known as the Suma, appeared.
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• 18.m.R.90 .
• (6 + W30;10) .
• Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2.
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• Remarkably, dal Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.
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• News of Tartaglia's victory reached Girolamo Cardan in Milan where he was preparing to publish Practica Arithmeticae &#9417; (1539).
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• In 1545 he published Ars Magna the first Latin treatise on algebra.
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• When solving x3 = 15x + 4 he obtained an expression involving ͩ0;-121.
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• (q2 - 4p3 + 4 pr) + (-16p2 + 8r)y - 20 py2 - 8y3 = 0 .
• The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.
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151. History overview
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637.
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• For example the sign "=" was introduced by Recorde in 1557.
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152. Chinese overview
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan shu shu &#9417; dating from around 180 BC was discovered.
• The next important books of which we have records are a sixteen chapter work Suanshu &#9417; written by Du Zhong and a twenty-six chapter work Xu Shang suanshu &#9417; written by Xu Shang.
• The oldest complete surviving text is the Zhoubi suanjing &#9417; which was compiled between 100 BC and 100 AD (see the article on The Ten Classics).
• The Zhoubi suanjing &#9417; contains a statement of the Gougu rule (the Chinese version of Pythagoras's theorem) and applies it to surveying, astronomy, and other topics.
• Although it is widely accepted that the work also contains a proof of Pythagoras's theorem, Cullen in [',' C Cullen, Astronomy and Mathematics in Ancient China (Cambridge, 1996).','3] disputes this, claiming that the belief is based on a flawed translation given by Needham in [',' J Needham, Science and Civilisation in China 3 (Cambridge, 1959).','13].
• One early 'choren' was Luoxia Hong (about 130 BC - about 70 BC) who produced a calendar which was based on a cycle of 19 years.
• Dong and Yao write [',' Y Z Dong and Y Yao, The mathematical thought of Liu Hui (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 13 (4) (1987), 99-108.','24]:- .
• He found approximations to π using regular polygons with 3 5; 2n sides inscribed in a circle.
• His best approximation of π was 3.14159 which he achieved from a regular polygon of 3072 sides.
• Liu also wrote Haidao suanjing &#9417; (see the article on The Ten Classics) which was originally an appendix to his commentary on Chapter 9 of the Nine Chapters on the Mathematical Art.
• For example Sun Zi (about 400 - about 460) wrote his mathematical manual the Sunzi suanjing &#9417; which on the whole provides little new.
• Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing &#9417; which contains representations of numbers in the decimal notation using positive and negative powers of ten.
• Zhang Qiujian (about 430 - about 490) wrote his mathematical text Zhang Qiujian suanjing &#9417; some time between 468 and 486.
• One of the most significant advances was by Zu Chongzhi (429-501) and his son Zu Geng (about 450 - about 520).
• He wrote the Zhui shu &#9417; in which he proved that 3.1415926 < π < 3.1415927.
• He recommended using 355/113 as a good approximation and 22/7 in less accurate work.
• With his son Zu Geng he computed the formula for the volume of a sphere using Cavalieri's principle (see [',' L Y Lam, and K S Shen, The Chinese concept of Cavalieri&#8217;s principle and its applications, Historia Math.
• 12 (3) (1985), 219-228.','25]).
• He wrote the Jigu suanjing &#9417;, a text with only 20 problems which later became one of the Ten Classics.
• Interpolation was an important tool in astronomy and Liu Zhuo (544-610) was an astronomer who introduced quadratic interpolation with a second order difference method.
• The collection is now called The Ten Classics, a name given to them in 1084.
• Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist.
• He wrote the Meng ch'i pi t'an &#9417; which contains many accurate scientific observations.
• The next major mathematical advance was by Qin Jiushao (1202 - 1261) who wrote his famous mathematical treatise Shushu Jiuzhang &#9417; which appeared in 1247.
• Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians.
• His most famous work is the Ce yuan hai jing &#9417; written in 1248.
• He also wrote Yi gu yan duan &#9417; in 1259 which is a more elementary work containing geometric problems solved by algebra.
• He wrote the Xiangjie jiuzhang suanfa &#9417; in 1261, and his other works were collected into the Yang Hui suanfa &#9417; which appeared in 1275.
• Guo Shoujing (1231-1316), although not usually included among the major mathematicians of the thirteen century, nevertheless made important contributions.
• He produced the Shou shi li &#9417;, worked on spherical trigonometry, and solved equations using the Ruffini-Horner numerical method.
• The last of the mathematicians from this golden age was Zhu Shijie (about 1260 - about 1320) who wrote the Suanxue qimeng &#9417; published in 1299, and the Siyuan yujian &#9417; published in 1303.
• For example Ding Ju published the Ding ju suan fa &#9417; in 1355, He Pingzi published the Xiangming suan fa &#9417; in 1373, Liu Shilong published the Jiu zhang tong ming suanfa &#9417; in 1424, and Wu Jing published the Jiu zhang suan fa bi lei da quan &#9417; in 1450.
• Again Cheng Dawei (1533 - 1606) published the Suanfa tong zong &#9417; in 1592 which is written in the style of the Nine Chapters on the Mathematical Art but provides an even larger collection of 595 problems.
• Xu Guangqi (1562 - 1633) certainly recognised exactly this and offered possible explanations including scholars neglecting practical computational tools and an identification of mathematics with mystical numerology under the Ming dynasty.
• The most famous member of this family was Mei Wending (1633-1721) and his comment on the golden section is typical of the sensible attitude he took towards Western mathematics (see for example [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','9]):- .
• Mei Juecheng (1681-1763), who was Mei Wending's grandson, was asked in 1705 by Emperor Kangxi to be editor-in-chief of the major mathematical encyclopaedia Shuli jingyun &#9417; (1723).
• Mei Juecheng also edited his grandfather Mei Wending's work producing the Meishi congshu jiyao &#9417; in 1761.
• For example Dai Zhen (1724 - 1777) became an editor for the Siku quanshu &#9417; which was a project set up by Emperor Qianlong in 1773.
• Ruan Yuan (1764 - 1849) produced his famous work the Chouren zhuan &#9417; containing biographies of 275 Chinese and 41 Western "mathematicians".
• Li Rui (1768 - 1817) assisted Ruan Yuan.
• His most important work is Lishi suan xue yi shu &#9417;.
• For example Li Shanlan (1811-1882) is important as a translator of Western science texts but he is most famous for his own mathematical contributions.
• There were many other efforts to promote Chinese mathematics, and in particular a mathematics journal, the Suanxue bao, was set up in 1899.
• For example Knopp taught there between 1910 and 1917, and Turnbull between 1911 and 1915.
• Chinese students began to study mathematics abroad and in 1917 Minfu Tah Hu obtained a doctorate from Harvard.
• China was represented for the first time at the International Congress of Mathematicians in Zurich in 1932.
• The Chinese Mathematical Society was formed in 1935.

153. Non-Euclidean geometry
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom.
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• One such 'proof' was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:- .
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• It was produced in 1697 by Girolamo Saccheri.
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• a) The summit angles are > 9076; (hypothesis of the obtuse angle).
• b) The summit angles are < 9076; (hypothesis of the acute angle).
• c) The summit angles are = 9076; (hypothesis of the right angle).
• In 1766 Lambert followed a similar line to Saccheri.
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• Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Elements de Geometrie &#9417;.
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• In trying to show that the angle sum cannot be less than 18076; Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle.
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• D'Alembert, in 1767, called it the scandal of elementary geometry.
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• He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four.
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• However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates.
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• In 1823 Janos Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded ..
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• Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829.
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• He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work.
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• The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary.
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• Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
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• The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900).
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• In 1868 he wrote a paper Essay on the interpretation of non-Euclidean geometry which produced a model for 2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry.
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• Beltrami's work on a model of Bolyai - Lobachevsky's non-Euclidean geometry was completed by Klein in 1871.
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• Klein's work was based on a notion of distance defined by Cayley in 1859 when he proposed a generalised definition for distance.
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154. Cubic surfaces
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1849 Salmon and Cayley published the results of their correspondence on the number of straight lines on a cubic surface.
• He proved in 1856 that:- .
• In 1858 Schlafli became the first to classify the cubic surfaces with respect to the number of real straight lines and tritangent planes on them, finding that there were exactly five types in his classification.
• He divided cubic surfaces into 23 species according to the nature of their singularities in 1863 and he published the classification in his paper On the distribution of surfaces of the third order into species, in reference to the presence or absence of singular points and the reality of their lines.
• In March 1866 Cremona published Memoire de geometrie pure sur les surfaces du troisieme ordre &#9417;.
• In 1869, at Clebsch's suggestion, Christian Wiener constructed plaster of Paris models of cubic surfaces which, together with other models of surfaces he had constructed, were exhibited in London in 1876, Munich in 1893, and Chicago also in 1893.
• Klein investigated cubic surfaces in 1870 and his work shows a special concern for geometric intuition regarding spatial constructions.
• Le Paige spent his whole career at the University of Liege where he worked on the theory of algebraic forms, a topic whose study had been initiated by Boole in 1841 and then developed by Cayley, Sylvester, Hermite, Clebsch and Aronhold.
• Fano studied with Klein in 1893 and did an Italian translation of Klein's Erlanger Program (1872), which gave his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations.

155. Cosmology
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In fact, Einstein might have predicted that the Universe is expanding after he first proposed his theory in 1915.
• The Russian mathematician and meteorologist Friedmann had realised in 1917 that Einstein equations could describe an expanding universe.
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• But a decisive blow was dealt to the Steady State model when in 1965 Penzias and Wilson discovered a cosmic microwave background radiation.
• This was interpreted as the faint afterglow of the intense radiation of a Hot Big Bang, which had been predicted by Alpher and Hermann back in 1949.
• It turns out to be so remarkably uniform, that it was only in 1992 that NASA's Cosmic Background Explorer satellite detected the first anisotropies in this background radiation.

156. Mathematics and Art
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Now in fact Alberti wrote two treatises, the first was written in Latin in 1435 and entitled De pictura &#9417; while the second, dedicated to Brunelleschi, was an Italian work written in the following year entitled Della pittura &#9417;.
• De pictura &#9417; is in three parts, the first of which gives the mathematical description of perspective which Alberti considers necessary to a proper understanding of painting.
• In Trattato d'abaco &#9417; which he probably wrote around 1450, Piero includes material on arithmetic and algebra and a long section on geometry which was very unusual for such texts at the time.
• In fact the third book of Pacioli's Divina proportione&#9417; is an Italian translation of Piero's Short book on the five regular solids.
• By 1500, however, Durer took the development of the topic into Germany.
• He published Unterweisung der Messung mit dem Zirkel und Richtscheit &#9417; in 1525, the fourth book of which contains his theory of shadows and perspective.
• An excellent example of this is in the geometrical shape he sketched in 1524.
• We mention first Federico Commandino who published Commentarius in planisphaerium Ptolemaei &#9417; in 1558.
• Wentzel Jamnitzer wrote a beautiful book on the Platonic solids in 1568 called Perspectiva corporum regularium &#9417;.
• Daniele Barbaro's La Practica della perspectiva &#9417; published in 1569, the year after Jamnitzer's treatise, complained that painters had stopped using perspective.
• Barbaro was interested in perspective in stage sets mainly because he had published an Italian translation of Vitruvius's De architectura &#9417; in 1556 and his interest had been aroused by this work.
• His preface to Le due regole della prospettiva pratica di M Iacomo Barozzi da Vignola &#9417; was published in 1583, three years before his death.
• He produced a work entitled A book containing various studies of mathematics and physics in 1585 which contains a treatise on arithmetic, some other short works and letters on various scientific topics, as well as a short treatise on perspective De rationibus operationum perspectivae &#9417;.
• Del Monte's six books on perspective Perspectivae libri sex &#9417; (1600) contain theorems which he deduces with frequent references to Euclid's Elements.
• In 1636 Desargues published the short treatise La perspective &#9417; which only contains 12 pages.
• Three years later, in 1639, Desargues wrote his treatise on projective geometry Brouillon project d'une atteinte aux evenemens des rencontres du cone avec un plan &#9417;.
• He had written a work on conics in 1673 before he discovered Desargues' Brouillon project.
• In 1679 he made a copy of Desargues' book writing:- .
• In 1685 la Hire published Conic sections which is a projective approach to conics which combines the best of the ideas from his earlier work and also those of Desargues.
• Before discussing the work of Brook Taylor, with which we will end our article, let us mention that of Humphry Ditton who wrote A treatise on perspective, demonstrative and practical in 1712.
• In many ways Brook Taylor's Linear perspective: or a new method of representing justly all manners of objects which appeared three years later in 1715, is similar to Ditton's work in its quality.
• In 1719 Taylor published a much modified second edition New principles of linear perspective.
• The first is by the famous English artist William Hogarth (1697-1764) whose Perspective absurdities formed the frontispiece to J J Kirby's book Dr Brook Taylor's method of perspective made easy in both theory and practice (1754).
• [All M C Escher works © 2001 Cordon Art - Baarn - Holland.

157. Quantum mechanics history
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It is hard to realise that the electron was only discovered a little over 100 years ago in 1897.
• The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859.
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• In 1879 Josef Stefan proposed, on experimental grounds, that the total energy emitted by a hot body was proportional to the fourth power of the temperature.
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• The same conclusion was reached in 1884 by Ludwig Boltzmann for blackbody radiation, this time from theoretical considerations using thermodynamics and Maxwell's electromagnetic theory.
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• In 1896 Wilhelm Wien proposed a solution to the Kirchhoff challenge.
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• In 1901 Ricci and Levi-Civita published Absolute differential calculus.
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• It had been Christoffel's discovery of 'covariant differentiation' in 1869 which let Ricci extend the theory of tensor analysis to Riemannian space of n dimensions.
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• In 1905 Einstein examined the photoelectric effect.
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• By 1906 Einstein had correctly guessed that energy changes occur in a quantum material oscillator in changes in jumps which are multiples of v where is Planck's reduced constant and v is the frequency.
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• Einstein received the 1921 Nobel Prize for Physics, in 1922, for this work on the photoelectric effect.
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• In 1913 Niels Bohr wrote a revolutionary paper on the hydrogen atom.
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• Arthur Compton derived relativistic kinematics for the scattering of a photon (a light quantum) off an electron at rest in 1923.
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• In fact Rutherford had introduced spontaneous effect when discussing radio-active decay in 1900.
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• In 1924 Einstein wrote:- .
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• Schrodinger in 1926 published a paper giving his equation for the hydrogen atom and heralded the birth of wave mechanics.
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• Also in 1926 Born abandoned the causality of traditional physics.
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• Heisenberg wrote his first paper on quantum mechanics in 1925 and 2 years later stated his uncertainty principle.
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• Also in 1927 Bohr stated that space-time coordinates and causality are complementary.
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• Pauli realised that spin, one of the states proposed by Bose, corresponded to a new kind of tensor, one not covered by the Ricci and Levi-Civita work of 1901.
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• However the mathematics of this had been anticipated by Eli Cartan who introduced a 'spinor' as part of a much more general investigation in 1913.
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• Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity.
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• He devised a challenge to Niels Bohr which he made at a conference which they both attended in 1930.
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• In 1932 von Neumann put quantum theory on a firm theoretical basis.
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158. Water-clocks
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• 424-427.','10] This process was easily inverted so that water flowed out of small tanks rather than into them.
• 441-452.','11] This water clock was used as a timer to prevent trials and speeches from going too long.
• 157-168.','12] .
• xvi, 333 p : ill ; 28 cm.','13] .
• 441-452.','11] .
• Maccius Plautus (250-184 B.C.) .

159. U of St Andrews History
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Mathematics in St Andrews to 1700 .
• His charter of incorporation is dated 28 February 1412 (1411 according to the Scottish calender which had a year start of 25 March until 1600) and he set up the University partly for prestige but mainly so that students could be educated for the Church.
• Albertus had written a Realist commentary on Aristotle which was widely used but the Faculty of Arts in St Andrews banned Albertus's book in 1418.
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• The 4 year degree in Arts in the 15th Century took in pupils of 13 years of age and their main study was Aristotle.
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• John Maior, renowned for his work in philosophy, logic and in particular on infinity, lectured in theology at St Andrews from 1531 until 1534 when he became the Provost of St Salvator's, a post he held until his death at the age of 80 in 1550.
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• Napier entered the University of St Andrews in 1563 at the age of 13.
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• Reforms in 1574 meant that specific duties were allocated to professors in each college and a master in St Mary's College was to act as Professor of Mathematics.
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• In 1579 further reforms, led by Andrew Melville a Protestant extremist, were brought in mainly with the motive of making the University a Presbyterian showpiece.
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• Mathematical research at the University of St Andrews really began in 1668.
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• James Gregory, who had been elected a Fellow of the Royal Society on 11 June 1668 and was a friend of Moray, was appointed the first holder of the chair.
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• The building of the library had begun in 1612 but the work was only completed in 1643.
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• In February 1671 he discovered Taylor's theorem (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671.
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• The notes Gregory made in discovering this result still exist written on the back of a letter sent to Gregory on 30 January 1671 by an Edinburgh bookseller.
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• The clock, made by Joseph Knibb of London, was purchased in 1673 and originally hung on the library wall.
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• Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock.
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• In 1674 Gregory cooperated with colleagues in Paris to make simultaneous observations of an eclipse of the moon and he was able to work out the longitude for the first time.
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• In 1673 the university allowed Gregory to purchase instruments for the observatory, but (things have not changed much!) told him he would have to make applications and organise collections for funds to build the observatory.
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• On 19 July 1673 Gregory wrote to Flamsteed, the Astronomer Royal, asking for advice and contemplated using St Rule's tower for some of his instruments.
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• In fact Gregory left St Andrews for Edinburgh in 1674.
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• William Sanders was appointed to the Chair of Philosophy at St Andrews in 1672.
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• He actively supported Gregory in his plans for the new Observatory and, when Gregory left for Edinburgh, Sanders was appointed in 1674 to the Regius Chair of Mathematics.
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• Sanders, although in no way of Gregory's stature, was at least also a believer in the new scientific ideas as he demonstrated in a book published in in 1674.
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• During his period as Regius Professor he also published The Elements of Geometry in 1686.
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• Sanders resigned the Regius Chair in 1688 to become a schoolmaster in Dundee.
• It is unclear why he should have made such a move and it is equally unclear why James Fenton should have been appointed to the Chair in 1689.
• He is thought to have been a graduate of St Andrews but failed to bring distinction since he appears to have been sacked in 1690 and expelled from the University one year after being appointed.
• In 1707 Charles Gregory, the son of James Gregory's brother David, was appointed to the Regius Chair.
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• In 1714 the University Senate introduced a new course 'Experimental Philosophy' which was to teach the .
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• The observatory set up by James Gregory was dismantled in 1736 and does not seem to have been used by Charles.
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• Charles Gregory held the Regius Chair until his son was old enough to succeed him, which he did in 1739.
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160. Hirst's diary
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• (1858) With respect to Bertrand I am still in doubt whether his harsh, forbidding, arrogant exterior is a true index of his character or merely a cloak to a better nature.
• (16 May 1864) I had much more conversation with Bertrand than ever before.
• (18 Nov 1857) I went to hear Chasles' first lecture on geometry, and was far from satisfied with it.
• (16 May 1864) Chebyshev called on me and left me some of his papers.
• (15 June 1862) [De Morgan] had no better remark to make than "How did you come across that Problem?" There was such an immense variety of similar questions.
• (13 Oct 1852) He is a rather tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf: it was early, he was unwashed, and unshaven (what of him required shaving), with his "schlafrock", slippers, cup of coffee and cigar.
• (31 Oct 1852) Dirichlet cannot be surpassed for richness of material and clear insight into it: as a speaker he has no advantages - there is nothing like fluency about him, and yet an eye and understanding make it indispensable: without an effort you would not notice his hesitating speech.
• (12 Aug 1852) Personally he is a venerable, fine old fellow, with a contented manly expression.
• (18 Nov 1857) He is a pleasant, chatty little man with whom I soon felt at perfect ease.
• (18 May 1879) A little shrivelled gouty old man [Liouville] has become and very garrulous.
• (24 March 1861) [Maxwell is] talkative with a Scotch brogue.
• (16 Oct 1859) On Monday having received a letter from Sylvester I went to see him at the Athenaeum Club.
• (10 Oct 1872) Sylvester's animus against me was disagreeably manifest.
• (15 June 1863) It was the first time I had been introduced to Thomson.

161. References for Zero
• G G Joseph, The crest of the peacock (London, 1991).
• R Mukherjee, Discovery of zero and its impact on Indian mathematics (Calcutta, 1991).
• 4 (1) (1984), 25-63.
• R C Gupta, Who invented the zero?, Ganita-Bharati 17 (1-4) (1995), 45-61.
• 12 (2) (1977), 225-231.
• (2) 1 (1) (1991), 59-62.
• 29 (5) (1998),729--744.
• 13 (1980), 7-20.
• 12 (1959), 393-420.

162. References for function concept
• 33 (2) (2002), 107-112.
• Student 63 (1-4) (1994), 153-166.
• Studies 56 (2) (1970), 3-21.
• R Cantoral, Formation of the notion of analytic function (Spanish), Mathesis, Mathesis 7 (2) (1991), 223-239.
• 32-33 (1990), 34-39.
• J Dhombres, Un texte d'Euler sur les fonctions continues et les fonctions discontinues, veritable programme d'organisation de l'analyse au 18e siecle, Cahiers du seminaire d'histoire des mathematiques 9 (Univ.
• 27 (2) (2000), 107-132.
• 25 (3) (1998), 290-317.
• 2 (2) (1993), 183-209.
• Y Komatu, The change in the concept of functions(Japanese), in Essays in celebration of the 100th anniversary of Chuo University (Chuo Univ., Tokyo, 1985), 67-81.
• A Kopackova, Phylogenesis of the concept of a function (Czech), in Mathematics throughout the ages II (Czech), (Prometheus, Prague, 2001), 46-80.
• 25/26 (1) (1978), 5-32; 56.
• Monthly 105 (1) (1998), 59-67.
• Monthly 105 (3) (1998), 263-270.
• 20 (1975), 232-245; 380.
• Univ., Moscow, 1973), 153-158.
• Sciences 24 (94) (1974), 29-50.
• 57 (1955), 117-120.
• Intelligencer 6 (4) (1984), 72-77.
• 4 (1965), 70-103.
• America, Washington, DC, 1995), 105-121.
• doch es gibt ihn nicht : Der Begriff der reellen Funktion im 19.
• Tech., Berlin, 2000), 182-215.
• Tech., Berlin, 2000), 128-181.
• A P Youschkevitch, La notion de fonction chez Condorcet, in For Dirk Struik (Reidel, Dordrecht, 1974), 131-139.
• 17 (1966), 123-150.
• "Nauka", Moscow, 1974), 158-166, 301.
• 16 (1) (1976/77), 37-85.

163. References for Cartography
• T Campbell, Early maps (New York, 1981).
• T Campbell, The earliest printed maps, 1442-1500 (London, 1987).
• H Stevens, Ptolemy's Geography : A brief account of all the printed editions down to 1730 (London, 1908).
• J N Wilford, The mapmakers : the story of the great pioneers in cartography from antiquity to the space age (New York, 1981).
• A Ahmedov and B A Rozenfel'd, "Cartography" - one of Biruni's first essays to have reached us (Russian), in Mathematics in the East in the Middle Ages (Russian) (Tashkent, 1978), 127-153.
• K Andersen, The central projection in one of Ptolemy's map constructions, Centaurus 30 (2) (1987), 106-113.
• v Shkole (3) (1988), i; 81.
• 32-33, (1990), 95-120.
• H Kautzleben, Carl Friedrich Gauss und die Astronomie, Geodasie und Geophysik seiner Zeit, in Festakt und Tagung aus Anlass des 200 Geburtstages von Carl Friedrich Gauss, Berlin, 1977 (Berlin, 1978), 123-136.
• C Lardicci, Geometric aspects of cartography (Italian), Archimede 34 (1-2) (1982), 23-42.
• 49 (3) (1995), 271-284.
• O Neugebauer, Ptolemy's Geography, book VII, chapters 6 and 7, Isis 50 (1959), 22-29.
• 13 (1999/00), 1-55.
• 9 (1-2) (1991), 31-43, 131-129.

164. References for Mathematics and Architecture
• M S Bulatov, Geometric harmonization in Central Asian architecture in the 9th-15th centuries, Historical-theoretic research Nauka (Moscow, 1988).
• 20 (1988), 194-201.
• Artes Barcelona 48 (10) (1989), 23.
• 6 (3) (1975), 149-184.
• Y Dold-Samplonius, Calculation of arches and domes in 15th century Samarkand, in Nexus III : architecture and mathematics, Ferrara, June 4-7, 2000 (Pisa, 2000), 45-55.
• 15 (1) (1995), 85-132.
• II 203 (1994), 101-116.
• 26 (3) (1995), 253-274.
• 168 (2) (1997), 161-178.
• 118 (5-6) (1984), 325-338.
• L Pepe, Architecture and mathematics in Ferrara from the thirteenth to the eighteenth centuries, Nexus III : architecture and mathematics, Ferrara, 2000 (Pisa, 2000), 87-104.
• M A Reynolds, A new geometric analysis of the Pazzi Chapel in Santa Croce, Florence, in Nexus III : architecture and mathematics, Ferrara, June 4-7, 2000 (Pisa, 2000), 105-121.
• 12 (2) (1985), 107-122.
• 1 (1979), 13-25; 73.
• Z Sagdic, Ottoman architecture: relationships between architectural design and mathematics in architect Sinan's works, in Nexus III : architecture and mathematics, Ferrara, June 4-7, 2000 (Pisa, 2000), 123-132.
• 1 (3) (1999), 9 pp.
• S R Wassell, Art and mathematics before the Quattrocento : a context for understanding Renaissance architecture, in Nexus III : architecture and mathematics, Ferrara, June 4-7, 2000 (Pisa, 2000), 157-168.

165. References for Bakhshali manuscript
• G G Joseph, The crest of the peacock (London, 1991).
• M N Channabasappa, Mathematical terminology peculiar to the Bakhshali manuscript, Ganita Bharati 6 (1-4) (1984), 13-18.
• 11 (2) (1976), 112-124.
• M N Channabasappa, The Bakhshali square-root formula and high speed computation, Ganita Bharati 1 (3-4) (1979), 25-27.
• R C Gupta, Centenary of Bakhshali manuscript's discovery, Ganita Bharati 3 (3-4) (1981), 103-105.
• 21 (1) (1986), 51-61.
• R Sarkar, The Bakhshali manuscript, Ganita Bharati 4 (1-2) (1982), 50-55.

166. References for Infinity
• E Maor, To infinity and beyond : A cultural history of the infinite (Princeton, NJ, 1991).
• 5 (1996), 55-73, 170.
• Bologna 10 (1988/89), 117-134.
• F C M Bosinelli, Uber Leibniz' Unendlichkeitstheorie, Studia Leibnitiana 23 (2) (1991), 151-169.
• 29 (1985), 37-49.
• A A Davenport, The Catholics, the Cathars, and the concept of infinity in the thirteenth century, Isis 88 (2) (1997), 263-295.
• P Dugac, La theorie des fonctions analytiques de Lagrange et la notion d'infini, in Konzepte des mathematisch Unendlichen im 19.
• 37 (2) (1990), 153-156.
• 5 (3-4) (1971), 311-318.
• (2) 4 (3) (1995), 207-222.
• 22 (1987), 3-23.
• C G Huang, The notion of infinity for 2500 years (Chinese), Tianjin Jiaoyu Xueyuan Xuebao Ziran Ban (1) (1993), 3-7.
• (Siwan) 15 (1) (1981), B13-B19.
• 54 (2) (1999), 87-99.
• W R Knorr, Infinity and continuity in ancient and medieval thought (Ithaca, N.Y., 1982), 112-145.
• Werk und Wirkung 1 (Berlin, 1985), 518-542.
• Magdeburg 33 (2) (1989), 61-62.
• 272 (4) (1995), 112-116.
• E Naert, Double infinite chez Pascal et Monade, Studia Leibnitiana 17 (1) (1985), 44-51.
• I, SCIAMVS 2 (2001), 9-29.
• 26 (90-3) (1993), 271-279.
• R Rucker, The actual infinite, Speculations in Science and Technology 3 (1980), 197-208.
• D D Spalt, Die Unendlichkeiten bei Bernard Bolzano, in Konzepte des mathematisch Unendlichen im 19.
• Jahrhundert (Gottingen, 1990), 189-218.
• 6 (2) (1986), 109-132.
• 26 (2) (1985), 171-204.
• 6 (4) (1979), 430-436.

167. References for Classical time
• D A Anapolitanos, Time and Continuum, Neusis 3 (1995), 87-96; 226.
• 26 (2) (1995), 323-351.
• G Bierhalter, Zyklische Zeitvorstellung, Zeitrichtung und die fruhen Versuche einer Deduktion des Zweiten Hauptsatzes der Thermodynamik, Centaurus 33 (4) (1990), 345-367.
• A D Chernin, The physical conception of time from Newton to the present (Russian), Priroda (8) (1987), 27-37.
• (3) 3 (1-2) (1993), 85-115.
• J Ehlers, Concepts of time in classical physics, in Time, temporality, now, Tegernsee, 1996 (Berlin, 1997), 191-200.
• (5) 73 (10) (1987), 394-404.
• 24 (4) (1987), 452-468.
• 23 (1) (1993), 199-218.
• (5) 73 (11) (1987), 465-476.
• 26 (1) (1995), 133-153.
• 26 (2) (1995), 295-321.

168. References for Mathematics and Art
• K Andersen, Desargues' method of perspective : its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry, Centaurus 34 (1) (1991), 44-91.
• K Andersen, Perspective and the plan and elevation technique, in particular in the work by Piero della Francesca, in Amphora (Basel, 1992), 1-23.
• K Andersen, Stevin's theory of perspective: the origin of a Dutch academic approach to perspective, Tractrix 2 (1990), 25-62.
• R Bkouche, La naissance du projectif : de la perspective a la geometrie projective, in Mathematiques et philosophie de l'antiquite a l'age classique (Paris, 1991), 239-285.
• 24 (3) (1981), 165-194.
• J W Dauben, The art of Renaissance science: Galileo and perspective, a video-cassette (Providence, RI, 1991).
• 51 (1988), 190-196.
• 48 (1985), 71-99.
• 2 (2) (1987), 3-40.
• J V Field, Perspective and the mathematicians : Alberti to Desargues, in Mathematics from manuscript to print, 1300-1600, Oxford, 1984 (New York, 1988), 236-263.
• 10 (2) (1995), 509-530.
• 4 (2) (1989), 31-118.
• A Flocon, Wentzel Jamnitzer : Perspectiva corporum regularium, in Sciences of the Renaissance, Tours, 1965 (Paris, 1973), 143-151.
• P Freguglia, De la perspective a la geometrie projective : le cas du theoreme de Desargues sur les triangles homologiques, in Entre mecanique et architecture/Between mechanics and architecture (Basel, 1995), 89-100.
• Monthly 58 (1951), 597-606.
• W R Knorr, On the principle of linear perspective in Euclid's Optics, Centaurus 34 (3) (1991), 193-210.
• 10 (2001), 63-76.
• 40 (3) (1995), 130-150.
• 53 (1990), 14-41.
• 13 (1) (1998), 265-298.
• K Veltman, Piero della Francesca and the two methods of Renaissance perspective, in Between art and science : Piero della Francesca (Italian), Arezzo-Sansepolcro, 1992 (Venice, 1996), 407-419.

169. References for Non-Euclidean geometry
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170. References for Indian Sulbasutras
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171. References for Prime numbers
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172. References for Topology history
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173. References for Orbits
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174. References for Mayan mathematics
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175. References for Quadratic etc equations
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176. References for Water-clocks
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177. References for Weather forecasting
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178. References for Longitude1
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179. References for Squaring the circle
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180. References for Mathematical games
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181. References for Longitude2
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182. References for Trigonometric functions
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183. References for General relativity
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184. References for Inca mathematics
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185. References for Mental arithmetic
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• J Fauvel and P Gerdes, African slave and calculating prodigy : bicentenary of the death of Thomas Fuller, Historia Mathematica 17 (2) (1990), 141-151.
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• 119 (1944), 180-191.
• 23 (3-5) (1987), 68-71.
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186. References for Maxwell's House
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187. References for The four colour theorem
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188. References for Science in the 17th century
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189. References for 20th century time
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• 3 (2) (1988), 101-147.
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• (2) 6 (1) (1991), 53-64.
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• 34 (1993), 103-128.
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• 26 (2) (1993), 189-207.
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190. References for Sundials
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191. References for Cosmology
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192. References for Trisecting an angle
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193. References for Special relativity
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194. References for Fund theorem of algebra
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195. References for Group theory
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196. References for Burnside problem
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197. References for Indian numerals
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198. References for Doubling the cube
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199. References for Greek astronomy
• A Berry, A short history of astronomy (New York, 1961).
• 26 (4) (1991), 383-387.
• J L Berggren, The relation of Greek spherics to early Greek astronomy, in Science and philosophy in classical Greece (New York, 1991), 227-248.
• (N.S.) 29 (1) (1992), 7-33.
• G L Geison, Did Conon of Samos transmit Babylonian observations, Isis (3) (193) 58 (1967), 398-401.
• B R Goldstein, The obliquity of the ecliptic in ancient Greek astronomy, in Theory and observation in ancient and medieval astronomy (London, 1985), 12-23.
• 33 (110) (1983), 3-14.
• B R Goldstein and A C Bowen, A new view of early Greek astronomy, Isis 74 (273) (1983), 330-340.
• 43 (2) (1991), 93-132.
• 22 (73 pt 2) (1989), 129-150.
• Jones, A Greek Saturn table, Centaurus 27 (3-4) (1984), 311-317.
• A Jones, Babylonian and Greek astronomy in a papyrus concerning Mars, Centaurus 33 (2-3) 1990), 97-114.
• A Jones, On Babylonian astronomy and its Greek metamorphoses, in Tradition, transmission, transformation (Leiden, 1996), 139-155.
• A Jones, The adaptation of Babylonian methods in Greek numerical astronomy, Isis 82 (313) (1991), 441-453.
• 29 (104) (1979), 68-94.
• Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.
• 12 (2) (1977), 120-126.
• K P Moesgaard, The full moon serpent : A foundation stone of ancient astronomy?, Centaurus 24 (1980), 51-96.
• 67 (1906), 2-17.
• 7 (2) (1976), 109-123.
• D Rawlins, Eratosthenes' geodest unraveled : was there a high-accuracy Hellenistic astronomy, Isis 73 (1982), 259-265.
• 38 (1944), 143-153.
• 38 (2) (1994), 207-248.
• 32 (2) (1985), 95-104.
• 18 (4) (1977/78), 359-383.
• 26 (2) (1982), 99-113.
• 29 (2) (1984), 125-130.
• 4 (2) (1973/74), 165-172.

200. References for Abstract linear spaces
• T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.
• T W Hawkins, Another look at Cayley and the theory of matrices, Archives Internationales d'Histoire des Sciences 26 (100) (1977), 82-112.
• T W Hawkins, Weierstrass and the theory of matrices, Archive for History of Exact Sciences 17 (1977), 119-163.
• T Hawkins, The theory of matrices in the 19th century, Proceedings of the International Congress of Mathematicians 2 (Montreal, Que., 1975), 561-570.
• 13 (5) (1977), 117-123.
• E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
• E Knobloch, Zur Vorgeschichte der Determinantentheorie, Theoria cum praxi : on the relationship of theory and praxis in the seventeenth and eighteenth centuries IV (Hannover, 1977), 96-118.
• J Tvrda, On the origin of the theory of matrices, Acta Historiae Rerum Naturalium necnon Technicarum (Prague, 1971), 335-354.

201. References for Brachistochrone problem
• Band 2 : Der Briefwechsel mit Pierre Varignon : Erster Teil : 1692-1702, Die Gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli (Basel, 1988).
• 40 (3) (1994), 459-475.
• Basel 70 (1959), 81-146.
• 20 (5) (1999), 299-304.
• T Koetsier, The story of the creation of the calculus of variations : the contributions of Jakob Bernoulli, Johann Bernoulli and Leonhard Euler (Dutch), in 1985 holiday course : calculus of variations (Amsterdam, 1985), 1-25.
• 19 (1971), 61-74; 108.
• 19 (6) (1988), 575-585.
• 7 (3) (1999), 311-341.
• 27 (1997), 257-276.
• (2) 34 (3-4) (1988), 365-385.
• 15 (6) (1980), 930-934.

202. References for Harriot's manuscripts
• J W Shirley (ed.), A Source book for the study of Thomas Harriot (New York, 1981).
• R C H Tanner, Henry Stevens and the associates of Thomas Harriot, in Thomas Harriot: Renaissance scientist (Oxford, 1974), 91-106.
• Harriot's will, History of science 6 (Cambridge, 1967), 1-16.

203. References for Arabic mathematics
• and trans.), The Algebra of Omar Khayyam (1931, reprinted 1972).
• and trans.), The Algebra of Abu Kamil (1966).
• and trans.), The Algebra of Mohammed ben Musa (1831, reprinted 1986).
• Catholique Louvain, Louvain, 1972 (Louvain, 1976), 87-100.
• 2 (1) (1978), 66-100.
• 18 (3) (1977/78), 191-243.
• Nauk USSR (2) (1967), 3-14.
• as found in anonymous manuscripts.(Arabic), in Deuxieme Colloque Maghrebin sur l'Histoire des Mathematiques Arabes, Tunis 1988 (Tunis, 1990), A97-A109.

204. References for Modern light
• A Einstein, Lens-like action of a star by the deviation of light in the gravitational field, Science 84 (1936), 506-507 .
• J Eisenstaedt, Dark bodies and black holes, magic circles and Montgolfiers : light and gravitation from Newton to Einstein, in Einstein in context (Cambridge, 1993), 83-106.
• J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
• 37 (1) (1980), 59-79.
• (2) 10 (2) (2000), 163-176.
• A postscript to Einstein's 1936 Science paper: "Lens-like action of a star by the deviation of light in the gravitational field", Science 275 (5297) (1997), 184-186.
• J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein: the formative years, 1879-1909 (Boston, MA, 2000), 231-251.

205. References for Matrices and determinants
• T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.
• T W Hawkins, Another look at Cayley and the theory of matrices, Archives Internationales d'Histoire des Sciences 26 (100) (1977), 82-112.
• T W Hawkins, Weierstrass and the theory of matrices, Archive for History of Exact Sciences 17 (1977), 119-163.
• T Hawkins, The theory of matrices in the 19th century, Proceedings of the International Congress of Mathematicians 2 (Montreal, Que., 1975), 561-570.
• 13 (5) (1977), 117-123.
• E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66.
• E Knobloch, Zur Vorgeschichte der Determinantentheorie, Theoria cum praxi : on the relationship of theory and praxis in the seventeenth and eighteenth centuries IV (Hannover, 1977), 96-118.
• J Tvrda, On the origin of the theory of matrices, Acta Historiae Rerum Naturalium necnon Technicarum (Prague, 1971), 335-354.

206. References for Alcuin's book
• Mental Exercises from the Realm of Charlemagne (Anaconda Verlag, 2010).
• Alcuin, Metrodoros, Abu Kamil (Czech) (Prometheus, Prague, 2001).
• Soc., Providence, RI, 1995), 1-30.
• 116 (6) (1978), 13-80.
• M Folkert, Die Alkuin Zugeschriebenen Propositiones ad acuendoes iuvenes, in Paul Leo Butzer and Dietrich Lohrmann (eds.), Science in Western and Eastern civilization in Carolingian times (Birkhauser, Basel, 1993), 273-281.
• J Hadley and D Singmaster, Problems to Sharpen the Young, The Use of the History of Mathematics in the Teaching of Mathematics, The Mathematical Gazette 76 (475) (1992), 102-126.
• I Pressman and D Singmaster, The jealous husbands and the missionaries and cannibals, The Mathematical Gazette 73 (464) (1989), 73-81.
• D Singmaster, The history of some of Alcuin's Propositiones, in P L Butzer, H Th Jongen and W Oberschelp (eds.), Charlemagne and his Heritage 1200 Years of Civilization and Science in Europe 2 (Brepols, Turnhout, 1998), 11-29.

207. References for Forgery 2

208. References for ETH history
• G Frei and U Stammbach, Hermann Weyl und die Mathematik an der ETH Zurich 1913-1930, (Birkhauser, Basel, 1992) .
• G Guggenbuhl, Geschichte der Eidgenossischen Technischen Hochschule in Zurich, in: Eidgenossische Technische Hochschule 1855-1955.
• (Ecole Polytechnique Federale, Buchverlag der Neuen Zurcher Zeitung, Zurich, 1955, 1-257) .
• http://www.hls-dhs-dss.ch/textes/d/D10396.php .

209. References for Chinese numerals
• D C Cheng, The use of computing rods in China, Archiv der Mathematik und Physic 32 (1925), 492-499.

210. References for Voting
• F Abeles, C L Dodgson and apportionment for proportional representation, Gadnita Bharati 3 (3-4) (1981), 71-82.
• 111, (1990), 7-43.
• 7 (2) (1990), 99-108.
• 18 (3) (2001), 571-600.
• 111 (1990), 45-59.

211. References for Kepler's Laws
• A E L Davis, 'Kepler's Road to Damascus', Centaurus 35 (1992) pp.156-157; .
• A E L Davis, Kepler's unintentional ellipse - a celestial detective story, Mathematical Gazette 82, no.493 (1998), p.42; .
• A E L Davis: 'The Mathematics of the Area Law: Kepler's successful proof in Epitome Astronomiae Copernicanae (1621)', Archive for History of Exact Sciences 57, 5 (2003), especially Section 6.

212. References for Golden ratio
• Monthly 2 (1895), 260-264.
• Monthly 25 (1918), 232-237.
• 225-231.
• L Curchin and R Herz-Fischler, De quand date le premier rapprochement entre la suite de Fibonacci et la division en extreme et moyenne raison?, Centaurus 28 (2) (1985), 129-138.
• R Fischler, On applications of the golden ratio in the visual arts, Leonardo 14 (1981), 31-32; 262-264; 349-351.
• 19 (1981), 406-410.
• 20 (2) (1982), 146-158.
• 169 (2-3) (1998), 145-178.
• G Sarton, When did the term golden section or its equivalent in other languages originate, Isis 42 (1951), 47.
• 17 (4-6) (1989), 613-638.
• (4) 17 (2) (1999), 229-245.

213. References for Decimal time

214. References for Chinese overview
• D Bodde, Chinese Thought, Society and Science : The Intellectual and Social Background of Science and Technology in Pre-Modern China (Honolulu, 1991).
• B Qian, History of Chinese mathematics (Chinese) (Peking, 1981).
• K Chemla, Reflections on the world-wide history of the rule of false double position, or : How a loop was closed, Centaurus 39 (2) (1997), 97-120.
• Y Z Dong and Y Yao, The mathematical thought of Liu Hui (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 13 (4) (1987), 99-108.
• 12 (3) (1985), 219-228.
• (2) 4 (2) (1994), 103-111.
• 19 (2) (2000), 97-113.
• 38 (4) (1988), 285-305.
• 71 (3) (1998), 163-181.

215. References for Elliptic functions
• R Ayoub, The lemniscate and Fagnano's contributions to elliptic integrals, Archive for History of Exact Sciences 29 (2) (1984), 131-149.
• R Fricke, Elliptische Funktionen, in Encyklopadie der mathematischen Wissenchaften 2 (3) (1913), 177-348.
• R M Porter, Historical development of the elliptic integral (Spanish), Congress of the Mexican Mathematical Society (Mexico City, 1989), 133-156.
• Monthly 88 (1981), 387-395.
• E I Slavutin, Euler's works on elliptic integrals (Russian), History and methodology of the natural sciences XIV: Mathematics (Moscow, 1973), 181-189.
• 27 (1983), 163-178.
• J Stillwell, Mathematics and history (New York, Berlin, Heidelberg, 1989), 152-167.

216. 2000 places of Pi
• 3.14159265358979323846264338327950288419716939937510 .
• 58209749445923078164062862089986280348253421170679 .
• 82148086513282306647093844609550582231725359408128 .
• 48111745028410270193852110555964462294895493038196 .
• 44288109756659334461284756482337867831652712019091 .
• 45648566923460348610454326648213393607260249141273 .
• 72458700660631558817488152092096282925409171536436 .
• 78925903600113305305488204665213841469519415116094 .
• 33057270365759591953092186117381932611793105118548 .
• 07446237996274956735188575272489122793818301194912 .
• 98336733624406566430860213949463952247371907021798 .
• 60943702770539217176293176752384674818467669405132 .
• 00056812714526356082778577134275778960917363717872 .
• 14684409012249534301465495853710507922796892589235 .
• 42019956112129021960864034418159813629774771309960 .
• 51870721134999999837297804995105973173281609631859 .
• 50244594553469083026425223082533446850352619311881 .
• 71010003137838752886587533208381420617177669147303 .
• 59825349042875546873115956286388235378759375195778 .
• 18577805321712268066130019278766111959092164201989 .
• 38095257201065485863278865936153381827968230301952 .
• 03530185296899577362259941389124972177528347913151 .
• 55748572424541506959508295331168617278558890750983 .
• 81754637464939319255060400927701671139009848824012 .
• 85836160356370766010471018194295559619894676783744 .
• 94482553797747268471040475346462080466842590694912 .
• 93313677028989152104752162056966024058038150193511 .
• 25338243003558764024749647326391419927260426992279 .
• 67823547816360093417216412199245863150302861829745 .
• 55706749838505494588586926995690927210797509302955 .
• 32116534498720275596023648066549911988183479775356 .
• 63698074265425278625518184175746728909777727938000 .
• 81647060016145249192173217214772350141441973568548 .
• 16136115735255213347574184946843852332390739414333 .
• 45477624168625189835694855620992192221842725502542 .
• 56887671790494601653466804988627232791786085784383 .
• 82796797668145410095388378636095068006422512520511 .
• 73929848960841284886269456042419652850222106611863 .
• 06744278622039194945047123713786960956364371917287 .
• 46776465757396241389086583264599581339047802759010 .

217. References for Ring Theory
• Van der Waerden, B.L., Moderne Algebra (2 Vols) (Berlin 1930, 1931).
• Weber, H., Lehrbuch der Algebra, (Braunschweig, 1895-1896).

218. References for Newton poetry
• R C Archibald, Mathematicians, and Poetry and Drama II, Science 89 (2299) (1939), 46-50.
• William Powell Jones, Newton Further Demands the Muse, Studies in English Literature, 1500-1900 3 (3) (1963), 287-306.

219. References for Greek sources I
• T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
• J L Heiberg, H Menge and M Curtze (eds.), Euclid Opera Omnia (9 Vols.) (Leipzig, 1883-1916).
• J L Heiberg (ed.), Archimedes Opera omnia cum commentariis Eutocii (Leipzig, 1910-15, reprinted 1972).
• J Gray, Sale of the century?, The Mathematical Intelligencer 21 (3) (1999), 12-15.
• 2 (1900), 147-171.
• J L Heiberg, Paralipomena zu Euklid, Hermes 39 (1930), 46-74; 161-201; 321-356.

220. References for Greeks poetry
• R C Archibald, Mathematicians, and Poetry and Drama, Science 89 (2298) (1939), 19-26.

221. References for Neptune and Pluto
• S Drake and C T Kowal, Galileo's sighting of Neptune, Scientific American 243 (6) (1980) 52-59.
• N Foster, John Couch Adams, the astronomer, Astronomy Now 3 (1989), 34-37.
• J E Littlewood, A Mathematician's Miscellany (London, 1953), 116-134.
• 34 (2) (1992), 295-299.
• W Sady, The rational reconstruction of discovery of Neptune planet, Grenzfragen zwischen Philosophie und Naturwissenschaft (Vienna, 1989), 106-109.
• 16 (2) (1981), 118-129.
• W M Smart, John Couch Adams and the Discovery of Neptune, Occasional Notes of the Royal Astronomical Society 2 (1947), 33-88.
• Intelligencer 18 (2) (1996), 6-11.
• C Tombaugh, The search for the ninth planet, Pluto, Astronomical society of the Pacific Leaflet 209 (1946).

222. References for Size of the Universe

223. References for Fractal Geometry
• Science, New Series 156 3775 (May 5, 1967): 636-638.

224. References for Coffee houses
• A Browning (ed.), English Historical Documents 1660-1714, in D C Douglas (ed.), English Historical Documents VIII (Eyre and Spottiswoode, London, 1953).

225. References for Planetary motion
• The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609.
• They were validated in his later work: Epitome of Copernican Astronomy, Book V, Frankfurt 1621.
• Bernard Cohen: The Birth of a New Physics, Norton 1985 (up-dated), p.166.

226. References for Ptolemy Manuscript
• Isis 22(2) (1935) 533-539.
• The American Journal of Philology 62(2) (1941) 244-246.
• Classical Philology 35(3) (1940) 333-336.

227. References for Egyptian numerals
• G G Joseph, The crest of the peacock (London, 1991).

228. References for Greek sources II
• T L Heath, A history of Greek mathematics I, II (Oxford, 1921).

229. References for Greek numbers

230. References for Chrystal and the RSE
• N Campbell, R Martin and S Smellie, The Royal Society of Edinburgh 1783-1983 (Edinburgh, 1983).

231. References for Weil family
• F du Plessix Gray, Simone Weil (Weidenfeld & Nicolson, London, 2001).

232. References for Knots and physics
• P M Harman (ed.), James Clerk Maxwell, The scientific letters and papers of James Clerk Maxwell 1862-1873 II (Cambridge, 1995).
• C G Knott, Life and Scientific Work of Peter Guthrie Tait (Cambridge, 1911).
• 62 (4) (1989), 219-232.
• Topological notions in 19th-century natural philosophy, Arch.
• 52 (4) (1998), 297-392.
• M Epple, Geometric aspects in the development of knot theory, in I M James (ed.), History of topology (Amsterdam, 1999), 301-358.
• C Nash, Topology and physics - a historical essay, in I M James (ed.), History of topology (Amsterdam, 1999), 359-416.

233. References for U of St Andrews History

234. References for Euclid's definitions
• 20 (2) (1993), 180-192.

235. References for The number e
• Monthly 57 (1950), 591-602.

236. References for Pi chronology
• D H Bailey, J M Borwein, P B Borwein, and S Plouffle, The quest for Pi, The Mathematical Intelligencer 19 (1997), 50-57.

237. References for Babylonian numerals
• G G Joseph, The crest of the peacock (London, 1991).
• J Hoyrup, Babylonian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 21-29.
• Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), 277-318.

238. References for Chandrasekhar Eddington
• M Bartusiak, Black Hole: how an idea abandoned by Newtonians, hated by Einstein, and gambled on by Hawking became loved (Yale University Press, New Haven-London, 2015).
• K C Wali, Chandra: A biography of S Chandrasekhar (University of Chicago Press, Chicago, 1991).
• S Chandrasekhar, The maximum mass of ideal white dwarfs, Astrophysical Journal 74 (1931), 81-82.
• S Chandrasekhar, The highly collapsed configurations of a stellar mass (Second Paper), Monthly Notices of the Royal Astronomical Society 95 (1935), 207-225.
• S Chandrasekhar, Stellar configurations with degenerate cores, Monthly Notices of the Royal Astronomical Society 95 (1935), 226-260.
• S Chandrasekhar, The highly collapsed configurations of a stellar mass, Monthly Notices of the Royal Astronomical Society 91 (1931), 456-466.
• Mit 3 Abbildungen, Zeitschrift fur Astrophysik 5 (1932), 321-327.
• R H Daltiz, Some recollections of S Chandrasekhar, in K C Wali (ed.), S Chandrasekhar: The man behind the legend (Imperial College Press, London, 1997), 142-155.
• P A M Dirac, R Peierls and M H L Pryce, On Lorentz invariance in the quantum theory, Mathematical Proceedings of the Cambridge Philosophical Society 38 (2) (1942), 193-200.
• A S Eddington, On 'relativistic degeneracy', Monthly Notices of the Royal Astronomical Society 95 (1935), 146-206.
• F Dyson, Chandrasekhar's role in 20th-century science, Physics Today 63 (12) (2010), 44-48.
• G Gamow and M Schoenberg, Neutrino Theory of Stellar Collapse, Physical Review 59 (1941), 539-547.
• F Hoyle, W A Fowler, G R Burbridge and E M Burbridge, On relativistic astrophysics, Astrophysics Journal 139 (1963), 909-928.
• W Israel, Dark Stars: the evolution of an idea, in S W Hawking and W Israel (eds.), 300 years of gravitation (Cambridge University Press, Great Britain, 1989), 199-276.
• H Kragh, On Arthur Eddington's Theory of Everything, Cornell University Library (2015).
• P Ledoux, 'An introduction to the study of stellar structure' de S Chandrasekhar, Ciel et Terre 55 (1939), 412-415.
• E A Milne, The analysis of stellar structure, Monthly Notices of the Royal Astronomical Society 91 (1930), 4-55.
• C Moller and S Chandrasekhar, Relativistic degeneracy, Monthly Notices of the Royal Astronomical Society 95 (1935), 673-676.
• J R Oppenheimer and H Snyder, On Continued Gravitational Contraction, Physical Review 56 (455) (1939), 455-459.
• R Peierls, Note on the derivation of the equation of state for a degenerate relativistic gas, Monthly Notices of the Royal Astronomical Society 96 (1936), 780-784.
• M Rotondo, On the concept of degenerate stars: the case of white dwarfs, Atti del XXXV Convegno annuale SISFA, Arezzo, 2015 (Pavia University Press, Pavia, 2016), 273-279.
• B Stromgren, Review: An Introduction to the Study of Stellar Structure, by S Chandrasekhar, Popular Astronomy 47 (1939), 287-289.
• K C Wali, Chandrasekhar vs Eddington an unanticipated confrontation, Physics Today 35 (10) (1982), 33-40.
• K C Wali, Chandra: A biographical portrait, Physics Today 63 (12) (2010), 38-43.
• S Weart, Interview of Subrahmanyan Chandrasekhar on 17 May 1977, Niels Bohr Library, American Institute of Physics.

239. References for Measurement
• A Favre, Les origines du systeme metrique (Paris, 1931).
• E F Cox, The metric system : A quarter-century of acceptance, 1831-1876, Osiris 13 (1959), 358-379.
• M Crosland, The Congress on definitive metric standards, 1798-1799 : The first international scientific conference?, Isis 60 (1969), 226-309.
• 116 (9) (1979), 144-165.
• H R Jenemann, Zur Geschichte der Substitutionswagung und der Substitutionswaage, Technikgeschichte 49 (2) (1982), 89-131; 176.
• L L Kulvecas, Two dates in the history of the development of the metric system of measures (Russian), in Problems in the history of mathematics and mechanics (Kiev, 1977), 109-115; 133.
• P Redondi, The French Revolution and the history of science (Russian), Priroda (7) (1989), 82-91.

240. References for Arabic numerals
• G G Joseph, The crest of the peacock (London, 1991).
• D E Smith and L C Karpinski, The Hindu-Arabic numerals (Boston, 1911).

241. References for The Scottish Book
• R D Mauldin, The Scottish Book, Mathematics from the Scottish Cafe (1981) .

242. References for Gravitation

243. 10000 digits of e
• 2.718281828459045235360287471352662497757247093699959574966967627724076630353 .
• 547594571382178525166427427466391932003059921817413596629043572900334295260 .
• 595630738132328627943490763233829880753195251019011573834187930702154089149 .
• 934884167509244761460668082264800168477411853742345442437107539077744992069 .
• 551702761838606261331384583000752044933826560297606737113200709328709127443 .
• 747047230696977209310141692836819025515108657463772111252389784425056953696 .
• 770785449969967946864454905987931636889230098793127736178215424999229576351 .
• 482208269895193668033182528869398496465105820939239829488793320362509443117 .
• 301238197068416140397019837679320683282376464804295311802328782509819455815 .
• 301756717361332069811250996181881593041690351598888519345807273866738589422 .
• 879228499892086805825749279610484198444363463244968487560233624827041978623 .
• 209002160990235304369941849146314093431738143640546253152096183690888707016 .
• 768396424378140592714563549061303107208510383750510115747704171898610687396 .
• 965521267154688957035035402123407849819334321068170121005627880235193033224 .
• 745015853904730419957777093503660416997329725088687696640355570716226844716 .
• 256079882651787134195124665201030592123667719432527867539855894489697096409 .
• 754591856956380236370162112047742722836489613422516445078182442352948636372 .
• 141740238893441247963574370263755294448337998016125492278509257782562092622 .
• 648326277933386566481627725164019105900491644998289315056604725802778631864 .
• 155195653244258698294695930801915298721172556347546396447910145904090586298 .
• 496791287406870504895858671747985466775757320568128845920541334053922000113 .
• 786300945560688166740016984205580403363795376452030402432256613527836951177 .
• 883863874439662532249850654995886234281899707733276171783928034946501434558 .
• 897071942586398772754710962953741521115136835062752602326484728703920764310 .
• 059584116612054529703023647254929666938115137322753645098889031360205724817 .
• 658511806303644281231496550704751025446501172721155519486685080036853228183 .
• 152196003735625279449515828418829478761085263981395599006737648292244375287 .
• 184624578036192981971399147564488262603903381441823262515097482798777996437 .
• 308997038886778227138360577297882412561190717663946507063304527954661855096 .
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• 782303809656788311228930580914057261086588484587310165815116753332767488701 .
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• 577354293018673943971638861176420900406866339885684168100387238921448317607 .
• 011668450388721236436704331409115573328018297798873659091665961240202177855 .
• 885487617616198937079438005666336488436508914480557103976521469602766258359 .
• 905198704230017946553679 .

244. References for Poincaré - Inspector of mines
• M Roy and R Dugas, Henri Poincare, Ingenieur des Mines, Annales des Mines 193 (1954), 8-23.

245. References for Ten classics

246. Ptolemy Source
• The world map (employing Ptolemy's first projection in 1.24) follows the end of Book 7, while the twenty-six regional maps alternate with their respective captions in Book 8.

Societies etc

1. South African Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Attempts had been made in 1951 and 1952 to found a South African mathematical society but these were not successful.
• The people who led the way in founding the Society in 1957 were Hendrik Stefanus Steyn and Johann van der Mark.
• in statistics from the University of Edinburgh in 1947 for his thesis On Multivariate Sampling With and Without Replacement supervised by Alex Aitken, had been appointed to the Statistics Department in the University of Pretoria in 1949.
• In 1967 the name was changed from 'The South African Mathematical Association' to the present 'The South African Mathematical Society'.
• The Society grew steadily in size from a total of 73 members in 1957 to 218 in 1977 and in 1993 there were over 300 full members.
• The American Mathematical Society agreed a reciprocity arrangement in 1972 after ensuring that the South African Society had no discriminatory rules but, after protests, they cancelled the agreement in 1974.
• He wrote [',' P Hilton, Reflections on a visit to South Africa, Focus 1 (4) (1981), 1-2.','1]:- .
• Towards the end of 1978 I was approached by the South African Mathematical Society about the possibility of making a lecture tour in South Africa in the summer of 1981.
• Hilton told about his experiences, writing in [',' P Hilton, Reflections on a visit to South Africa, Focus 1 (4) (1981), 1-2.','1]:- .
• However, he also explains why he went to South Africa [',' P Hilton, Reflections on a visit to South Africa, Focus 1 (4) (1981), 1-2.','1]:- .
• The journal Quaestiones Mathematicae began publication in 1976.
• The Notices of the South African Mathematical Society began publishing in 1969 with Niko Sauer as editor.
• The first Congress of the Association in 1958 in Pietermaritzburg consisted of a Council Meeting, three plenary lectures of one hour each by Professors Hyslop, Isaacs and Van der Merwe, seven short lectures of thirty minutes each, an AGM on the Monday evening attended by 27 members, followed by a popular lecture by Prof D B Sears.
• 1957-1959 J M Hyslop .
• 1959-1961 J H van der Merwe .
• 1961-1963 A P Burger .
• 1963-1965 H J Schutte .
• 1965-1967 H Rund .
• 1967-1969 K O Househam .
• 1969-1971 J H van der Merwe .
• 1971-1973 H J Schutte .
• 1973-1975 A P Burger .
• 1975-1977 H S P Grasser .
• 1977-1979 G J Hauptfleisch .
• 1979-1981 D H Jacobson .
• 1981-1983 K A Hardie .
• 1983-1984 N Sauer .
• 1984-1985 N Sauer .
• 1985-1986 D H Martin .
• 1986-1987 A P J van der Walt .
• 1987-1988 J Swart .
• 1988-1989 R I Becker .
• 1989-1990 J J Grobler .
• 1990-1991 A R Meijer .
• 1991-1993 W J Kotze .
• 1993-1995 C H Brink .
• 1995-1997 N Sauer .
• 1997-1999 J Persens .
• 1999-2001 T G Schultz .
• 2001-2003 E Bruning .
• 2007-2015 H Siweya .

2. Women in Mathematics Association
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Association for Women in Mathematics was founded in January 1971.
• There was a joint meeting of the American Mathematical Society, the Mathematical Association of America and the Association for Symbolic Logic in Atlantic City, New Jersey, USA, from 21 to 24 January 1971.
• There was a group of women mathematicians already in existence organised by Linda Rothschild and Alice Schafer at Wellesley College, a women's college in Wellesley, Massachusetts founded in 1875.
• Carol Wood, president in 1991-93, said [',' L Blum, A Brief History of the Association for Women in Mathematics: The Presidents&#8217; Perspectives, Amer.
• Notices 38 (7) (1991), 738-774.','2]:- .
• Mary Gray produced the first Newsletter of the Association, still with its original title of 'Association of Women Mathematicians', in May 1971.
• Here are two short pieces she wrote for this first Newsletter [',' M Gray, Association of Women Mathematicians Newsletter 1 (1) (May 1971).','4]:- .
• The second Newsletter appeared in September 1971, still written almost entirely by Mary Gray.
• In it the change of name is recorded [',' M Gray, Association for Women in Mathematics Newsletter 1 (2) (September 1971).','3]:- .
• She wrote [',' L Blum, A Brief History of the Association for Women in Mathematics: The Presidents&#8217; Perspectives, Amer.
• Notices 38 (7) (1991), 738-774.','2]:- .
• In the summer of 1976 at Toronto, Mary Gray talked about Sophie Germain, Linda Keen talked about Sonya Kovalevskaya, and Martha Smith about Emmy Noether.
• At Atlanta in 1978 they held "Black Women in Mathematics" sessions.
• By 1981 the Association had over 1000 members from sixteen different countries.
• Note that Radcliffe College, where this meeting was held, was at that time a women's college but it became a part of Harvard University in 1999.
• Notices 46 (1) (1999), 27-38.','5] write:- .
• By 1991 the Association was a widely respected organization with a large influence internationally: AWM had a professional newsletter, an extensive program at the January Joint Meetings, and various projects for encouraging younger women to study mathematics.
• Beginning in 1991 the Association for Women in Mathematics began regular workshops at each winter Joint Meeting and each summer meeting of the SIAM.
• The website of the Association for Women in Mathematics states [',' Association for Women in Mathematics website.','1]:- .
• The website of the Association for Women in Mathematics states [',' Association for Women in Mathematics website.','1]:- .
• The website of the Association for Women in Mathematics states [',' Association for Women in Mathematics website.','1]:- .
• The website of the Association for Women in Mathematics states [',' Association for Women in Mathematics website.','1]:- .
• The website of the Association for Women in Mathematics states [',' Association for Women in Mathematics website.','1]:- .
• The Schafer Prize was established in 1990 by the Executive Committee of the Association for Women in Mathematics and is named for the Association for Women in Mathematics former president and one of its founding members, Alice Turner Schafer, who contributed a great deal to women in mathematics throughout her career.
• Finally, we list the Presidents of the Association for Women in Mathematics [',' Association for Women in Mathematics website.','1]: .
• 1971-1973 Mary W Gray .
• 1973-1975 Alice T Schafer .
• 1975-1979 Lenore Blum .
• 1979-1981 Judith Roitman .
• 1981-1983 Bhama Srinivasan .
• 1983-1985 Linda P Rothschild .
• 1985-1987 Linda Keen .
• 1987-1989 Rhonda J Hughes .
• 1989-1991 Jill P Mesirov .
• 1991-1993 Carol Wood .
• 1995-1997 Chuu-Lian Terng .
• 1997-1999 Sylvia M Wiegand .
• 1999-2001 Jean E Taylor .
• 2001-2003 Suzanne Lenhart .
• 2009-2011 Georgia Benkart .
• 2011-2013 Jill Pipher .
• 2013-2015 Ruth Chrney .
• 2015-2017 Kristin Lauter .

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Statistical Society of Canada was founded in 1972.
• Before that Canada had been served by Chapters of the American Statistical Association which opened its first Canadian Chapter in Montreal in 1955 followed by Chapters in Toronto and Ottawa in 1968.
• The Canadian Mathematical Congress (now the Canadian Mathematical Society) held a meeting in Vancouver in the summer of 1968.
• 14 (1) (1999), 80-125.','1]:- .
• However, when Roger Fischler (University of Toronto) wrote a final report as secretary of the Committee, dated 29 March 1971, he suggested that:- .
• A number of people were keen to push ahead and organised a statistics conference in Montreal from 31 May to 2 June 1971.
• Plans for a Canadian Society were, however, opposed by some who were involved in the annual meeting of the American Statistical Association which was planned for Montreal in 1972.
• 14 (1) (1999), 80-125.','1]:- .
• The Society was set up and held its first Annual Meeting on 16 September 1972.
• Disagreements followed and eventually a rival Canadian Statistical Society was set up on 1 February 1974.
• Despite there now being two rival organisations, the Statistical Science Association of Canada went ahead with its next annual conference held in Toronto, Ontario from 30 May to 1 June 1974.
• The rival Canadian Statistical Society held its first annual meeting in Edmonton, Alberta on 13 August 1974.
• However the two rival Canadian Statistical Societies merged in 1977 to form the Statistical Society of Canada.
• 14 (1) (1999), 80-125.','1] which gives full details of the two rival Societies and the political moves which eventually led to their coming together in 1977.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Lithuanian Academy of Sciences was established on 16 January 1941 but there is a long history of the development of science in Lithuania which led up to the founding of the Academy.
• In our article below we follow [',' History of the Lithuanian Academy of Sciences, Lithuanian Academy of Sciences.','1] closely making some changes and giving additional material: .
• As a system, science in the Grand Duchy of Lithuania began developing after the founding of Vilnius University - Academia et Universitas Vilnensis - in 1579.
• In 1773, a group of professors of Vilnius University led by the prominent astronomer Martynas Po .
• obutas (1728-1810) was born in Grodno and was educated at the Jesuit College there.
• He established the Department of Applied Mechanics in the University of Vilnius in 1780.
• The Polish-Lithuanian Commonwealth had existed from the 16th century but in 1768 it became a protectorate of the Russian Empire.
• In 1803 Vilnius University was reorganized and the number of courses taught here rose significantly to include mechanics, technology, probability theory, agronomy, statistics, and diplomacy.
• In 1832, the tsarist government closed Vilnius University because its professors and students participated in the November 1831 uprising in the partitioned Republic (Rzeczpospolita) against the Russian Empire.
• Vilnius Academy of Medicine and Surgery was closed in 1842, and the Theological Academy was moved to St Petersburg.
• This process was further accelerated after the lifting of the ban on the Lithuanian press in 1904.
• In 1907, Lietuvis&#371; mokslo draugija (the Lithuanian Society for Science) was founded with Dr Jonas Basanavičius as its chairperson.
• Basanavičius (1851-1927) had studied medicine in Moscow and worked as a doctor in Bulgaria from 1880 to 1905.
• Returning to Lithuania in 1905 he was elected chairman of the major assembly, the Great Seimas of Vilnius, held in December of that year which demanded autonomy for Lithuania within the Russian Empire.
• The Polish scientific organization, Towarzystwo Przyjacioł Nauk w Wilnie (Society of Friends of Science in Vilnius) was founded in 1907.
• In 1925, the YIVO Institute for Jewish Research was opened in Vilnius, and it was a significant addition to the existing and well developed system of Jewish cultural and educational institutions in Vilnius.
• After the First World War, in 1918, Lithuania declared its independence and a number of institutions of learning and research were either reopened or newly established.
• Among these, mention should be made of the re-opened Vilnius University and the Higher Courses in Kaunas opened in 1920.
• In 1922, the Higher Courses were reorganized into the University of Lithuania which was renamed as Vytautas Magnus University in 1930.
• The Agricultural Academy was founded in Dotnuva, a town 50 kilometres to the north of Kaunas, in 1924, and the Veterinary Academy opened in Kaunas in 1936.
• However, it was difficult to establish an organization like this without tangible support from the country's government, and it was only on 1 September 1938 that the Institute of Lithuanian Studies was established.
• On 1 September 1940, the Wroblewski State Library was transferred to the Institute of Lithuanian Studies which took over its holdings and property.
• Owing to historical circumstances, the Academy of Sciences was established on 16 January 1941, that is, after Lithuania had already lost its independence.
• ius (1882-1954) was a poet, writer and novelist who was appointed as Prime Minister of Lithuania in June 1940.
• It was especially stimulated by Professor Juozas Matulis, a great authority in electrochemistry who served as the fourth President of the Academy of Sciences from 1946 to 1984.
• Matulis (1899-1993) graduated from the University of Lithuania in 1929 and had spent time at the University of Leipzig before being appointed to the University of Vilnius.
• He was made a corresponding member of the Academy of Sciences of the USSR in 1946.
• The first public meeting of the steering group of the S&#261;jkūdis, or National Reform Movement, was held in the Academy's conference hall on 9 June 1988; the Green movement, which started at approximately the same time, was also born in the Academy of Sciences.
• In as early as 1989, the Lithuanian Academy of Sciences declared its independence from the USSR Academy of Sciences.
• At that time the prominent physicist Academician Juras Požela was President of the Lithuanian Academy of Sciences (he served in this role from 1984 to 1992).
• The Academy of Sciences was structured as a network of 17 scientific research institutes, and a number of auxiliary scientific and industrial enterprises.
• On 12 February 1991, the law of the Republic of Lithuania on Research and Studies was passed which defined the status of the Lithuanian Academy of Sciences within the system of the country's scientific institutions and defined its relations with the State.
• Following this law, a new Statute of the Academy of Sciences was drafted and was approved by the Parliament of the Republic of Lithuania on 18 March 2003.
• Academician Jūras Banys, the present President of the Academy, was elected on 1 February 2018.
• Much information about mathematics at the Lithuanian Academy of Sciences during its first 50 years is given in [',' S Skerus, Mathematics at the Lithuanian Academy of Sciences, The Lithuanian Journal of Science for Lithuania and the World 1 (1) (Lithuanian Academy of Sciences, 1993), 81-92.','3].
• One of them was Jonas Kubilius who graduated from the post-graduate courses at St Petersburg University and defended his thesis for a Candidate's degree in physics and mathematics in 1961.
• In 1962 he began his career at the Institute of Physics and Technology at the newly established sector of physics, mathematics and astronomy.
• With the application of probabilistic methods in the theory of numbers, in the course of several years he obtained valuable results and pooled them in his Doctor's thesis "Investigations into the Probabilistic Theory of Numbers", defended in 1958.
• J Kubilius' monograph "Probabilistic Methods in the Theory of Numbers" (published in 1959) initiated a new branch of mathematical sciences - a probabilistic theory of numbers.
• On October 1, 1956 the Institute of Physics and Mathematics was founded.
• Due to his active scientific and organizational work J Kubilius was nominated rector of Vilnius University in 1958.

5. Singapore Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Singapore had become a British Crown Colony in 1946 and in 1948 the Federation of Malaya was created which eventually (in 1957) gained independence.
• During this time he was active in planning to create the University of Malaya in Singapore and, in 1949, he became Dean of the Faculty of Arts of the University of Malaya.
• When the Mathematical Society of Malaya and Singapore was founded in 1952, Oppenheim became the first president.
• In 1953 the Mathematical Society of Malaya and Singapore began publication of the Bulletin of Malayan Mathematical Society.
• This was published from 1953 to 1959 and then in 1960 the name was changed to NABLA.
• The publication NABLA stated [',' Singapore Mathematical Society website.','1]:- .
• Daniel Pedoe was appointed as head of the Mathematics Department at the University of Singapore in 1959.
• He became the second president of the Mathematical Society of Malaya and Singapore in 1960 following after Oppenheim.
• Let us go back to 1956 for in that year the Mathematical Society of Malaya and Singapore launched its first Inter-school Mathematical Competition.
• We will say a little more about this Competition below but first let us note that in 1967 the Mathematical Society of Malaya and Singapore was renamed the Singapore Mathematical Society, the name by which the Society is known today.
• Returning to our discussion of the Mathematical Competition, after several changes of name, in 1995 it was renamed as the Singapore Mathematical Olympiad [',' Singapore Mathematical Society website.','1]:- .
• The current name Singapore Mathematical Olympiads (Junior, Senior and Open) started in 1995.
• The Society takes its Olympiad materials into Primary Schools [',' Singapore Mathematical Society website.','1]:- .
• The Southeast Asian Mathematical Society was founded in 1972.
• The Singapore Mathematical Society joined the Southeast Asian Mathematical Society as a founding member society in 1972.
• Kumkum Kumar Sen, the President in 1973, wrote [',' Singapore Mathematical Society website.','1]:- .
• In 1975 the Society joined both the International Mathematical Union, where it represented Singapore, and in the same year the Society also became an Institutional Member of the Singapore National Academy of Science.
• The Society initiated its Distinguished Visitor Programme in 1998.
• However, the programme was much broader than this for it also had distinguished mathematicians visiting schools in Singapore to interact with both mathematics teachers and their pupils [',' Singapore Mathematical Society website.','1]:- .
• Experienced mathematics professors from Singapore universities go into schools with the following aim [',' Singapore Mathematical Society website.','1]:- .
• The Society also organises Lectures and Workshops [',' Singapore Mathematical Society website.','1]:- .
• 1952-1959 Alexander Oppenheim .
• 1961 Jayaratnam Eliezer .
• 1962-1972 P H Diananda .
• 1972-1973 Kumkum Kumar Sen .
• 1976-1977 Kumkum Kumar Sen .
• 1978-1979 Chew Kim Lin .
• 1980-1982 Peng Tsu Ann .
• 1985-1986 Lam Lay Yong .
• 1988-1990 Louis Chen Hsiao Yun .
• 1991-1993 Leonard Y H Yap .
• 1994-1995 Chong Chi Tat .
• 1996-1997 Koh Khee Meng .
• 2001-2005 Tan Eng Chye .
• 2009-2012 Zhu ChengBo .
• 2013-2016 Ling San .

6. American Academy of Arts and Sciences
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The American Academy of Arts and Sciences was established in 1779 by a group of Harvard College graduates.
• The Academy was founded largely to rival the American Philosophical Society which had been established earlier in Philadelphia in 1743.
• 1780-1790 James Bowdoin .
• 1814-1820 Edward Augustus Holyoke .
• 1820-1829 John Quincy Adams .
• 1829-1838 Nathaniel Bowditch .
• The original series had four volumes published between 1785 and 1821; a second series contained nineteen volumes published between 1833 and 1946; no volumes appeared between 1947 and 1956, then in 1957 one further volume, called Series 3, Volume 24 was published.
• Volume 1 (Adams and Nourse, Boston, 1783).','1]:- .
• The Proceeding of the American Academy of Arts and Sciences began publication in 1848 with the first volume recording the proceedings of the Academy from May 1846 to May 1848.
• At the meeting on 12 August 1846 Benjamin Peirce gave a lengthy report on the astronomical observations which had been made by the Cambridge Observatory and communicated to him by William Cranch Bond, the Director.
• At the 12 August meeting there was an interesting report on "Henry Safford, the young Vermont mathematician." Safford (1836-1901) was a calculating prodigy who was ten years old at the time.

7. Serbian Society of Mathematicians and Physicists
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A 'Mathematical Club' led by Anton Bilimovic had been founded by the mathematicians at the University of Belgrade in 1926 [',' Mathematical Society of Serbia website.','1]:- .
• Also in 1926 the mathematics students from Belgrade set up the 'Union of Mathematics Students' which ran a library, held meetings and published textbooks, working in collaboration with similar unions in Zagreb and Ljubljana.
• In 1937 the Mathematical Club became a broader society when mathematicians, physicists and astronomers from universities of Belgrade, Zagreb and Ljubljana formed the Yugoslav Mathematical Society, based in Belgrade and with the Belgrade mathematician Tadija Pejovic as its President.
• He attended the gymnasium in Kragujevac but his education was interrupted by the Balkan wars and then when World War I broke out in 1914 he was put into the military.
• He served with distinction throughout the war and even after it ended in 1918 he remained in Dubrovnik as commander of a railway station until mid-June 1919.
• He graduated in 1921 and taught mathematics in a high school as well as becoming an assistant at the University.
• He was awarded a doctorate in 1923 for his thesis on the generalized Riccati differential equation which he defended on 6 February of that year.
• When World War II affected Serbia in 1941 he became a lieutenant colonel in the Serbian army but was captured by the Germans and spent four years in German prisoner of war camps.
• After his release in 1945 he was able to return to Belgrade and, after a year negotiating with the authorities, he was able to resume his work at the University.
• When the Society of Mathematicians and Physicists of Serbia was founded in 1948 Serbia was part of Yugoslavia, as it had essentially been since 1919 when the Kingdom of Serbs, Croats, and Slovenes was formed.
• Following the German invasion in 1941, Serbia was partitioned between Germany, Hungary, Bulgaria, and Italy but, following World War II, Serbia became part of the Federal People's Republic of Yugoslavia.
• The report of this meeting by Dobrivoje Mihajlovic, at which the Society of Mathematicians and Physicists of Serbia was founded, set out the aims of the new Society [',' Mathematical Society of Serbia website.','1]:- .
• Tadija Pejovic was President of the Society from 1948 to 1952, being re-elected in October 1948; February 1950; November 1950; and October 1951.
• Mathematics competitions for secondary school pupils began to be held in the 1950s with the first in Belgrade being in 1958.
• The Society of Mathematicians and Physicists of Serbia became the Society of Mathematicians, Physicists and Astronomers of Serbia, then in 1981 it split into three separate societies.
• The 1st Congress, Bled, 8-12 November 1949; .
• The 5th Congress, Ohrid, 14-19 September 1970; .
• The 7th Congress, Budva-Becici, 6-11 October 1980; .
• The 10th Congress, Belgrade, 21-24 January 2001; .
• The Society of Mathematicians and Physicists of Serbia had three publications prior to its split in 1981.
• These were Matematicki Vesnik, a scientific journal which began publication in 1949; Nastava matematike, a journal intended for the use of primary and secondary school teachers, which began publication in 1952; and Matematicki list za ucenike osnovne skole, a popular journal for primary school pupils, which began publication in 1966.
• We now give further information on these three publications (the information is taken from [',' V Micic, Z Kadelburg and B Popovic, The Mathematical Society of Serbia - 60 years, The Teaching of Mathematics 11 (1) (2008), 1-19.','2] so refers to the position in 2008).
• The first issue of the journal was published in the beginning of 1949.
• The first issue for 1951 contained the first article written in French.
• Up to 1953 all the authors of articles were Serbian but, in that year, Einar Hille published in the journal.
• The Society began publication of this journal in 1952.
• From the year 1954 the title of the journal was Nastava matematike i fizike and in 1974, after 20 years of this title, a new series named Nastava matematike began having dropped the physics connection.
• The 'Mathematical Newsletter for Primary Schools' was first published in 1967.
• However, it soon turned into a journal for children between the ages of 10 and 15 used across the whole of Yugoslavia.
• Two further publications began after the Society of Mathematicians, Physicists and Astronomers of Serbia split into three separate societies in 1981.

8. Spitalfields Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Spitalfields Mathematical Society was founded in 1717 by Joseph Middleton who taught mathematics to sailors who required mathematical skills for navigating.
• In 1725 it moved to the White Horse in Wheeler Street, then in 1735 to the Ben Johnson's Head in Woodseer Street.
• The original rules of the Society limited the number of members to "the square of eight" but clearly this proved hard to maintain for by 1735 it had been reduced to "the square of seven".
• Basically it operated as a working men's club and we know that the members of 1744 [',' J W S Cassels, The Spitalfields Mathematical Society, Bull.
• 11 (1979), 241-258.','1]:- .
• Records of the Society written in 1784 give some information about its history (see for example [',' J W S Cassels, The Spitalfields Mathematical Society, Bull.
• 11 (1979), 241-258.','1]):- .
• The Mathematical Society flourished to the extent that in 1783 it increased its membership back to the square of eight.
• In 1793 the Society moved into a permanent room in Crispin Street.
• The Society produced a notebook in 1804 which gives the following information about public lecture the Society started to give [',' J W S Cassels, The Spitalfields Mathematical Society, Bull.
• 11 (1979), 241-258.','1]:- .
• However the case had a large effect since (according to the Minutes of the Society) the [',' R A Sampson, The decade 1840-1850, in J L E Dreyer and H H Turner (eds.), History of the Royal Astronomical Society (London, 1923), 83-109.','3]:- .
• produce of the lectures delivered in 1799 - 1800 had been very materially diminished by the effect of the information lodged against several of the members by the Gang of Informers, who have occasioned so much trouble and expense to the Society during the past year.
• By 1804 the number of members was changed to the square of nine and now it started to look more like other mathematical societies in that it elected a president, secretary, treasurer, and six trustees.
• More ambitious lecture programmes were started and, for example, in 1821 a course was put on [',' R A Sampson, The decade 1840-1850, in J L E Dreyer and H H Turner (eds.), History of the Royal Astronomical Society (London, 1923), 83-109.','3]:- .
• In 1825 the Minutes record that:- .
• Membership dropped, being 54 in 1839, 30 in 1841 and 19 by 1845.
• The Society wrote to the Royal Astronomical Society who responded on 10 May 1845:- .
• A meeting of the Council of the Royal Astronomical Society took place yesterday, and I brought forward the suggestions contained in your recent letters to me relating to the venerable Mathematical Society of London, and the Council were unanimous in regretting that this ancient Society of 130 years standing should be on the eve of dissolution and decline.

9. Serbian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Mathematical Society of Serbia was founded in 1981 when the Society of Mathematicians, Physicists and Astronomers of Serbia split into three separate societies.
• The website [',' Mathematical Society of Serbia website.','1] gives the following information about the Mathematical Society of Serbia:- .
• The journal "Nastava matematike" started in 1952 and it became the official bulletin of the Society, obtained by all members.
• "Matematicki list za ucenike osnovnih skola" started in 1967 and it immediately became the most popular publication of the Society.
• The journal "Tangenta", intended for high school pupils, started in 1995.
• The article [',' V Micic, Z Kadelburg and B Popovic, The Mathematical Society of Serbia - 60 years, The Teaching of Mathematics 11 (1) (2008), 1-19.','2] gives the following information:- .
• By 2008 the Society had a membership of 1800.
• When Society split from the Society of Mathematicians and Physicists of Serbia, there were already three publications Matematicki Vesnik, a scientific journal which began publication in 1949; Nastava matematike, a journal intended for the use of primary and secondary school teachers, which began publication in 1952; Matematicki list za ucenike osnovne skole, a popular journal for primary school pupils, which began publication in 1966.
• After the Mathematical Society of Serbia split off in 1981, two further publications were added: Tangenta, a journal for mathematics and computer science, intended for secondary school students, which began publication in 1995; and The Teaching of Mathematics, a journal which publishes research works in mathematical education, which began publication in 1998.
• We now give further information on these two publications (the information is taken from [',' V Micic, Z Kadelburg and B Popovic, The Mathematical Society of Serbia - 60 years, The Teaching of Mathematics 11 (1) (2008), 1-19.','2] so refers to the position in 2008).
• To fulfill this need, in 1998 the Society began publishing The Teaching of Mathematics.

10. Cyprus Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Cyprus Mathematical Society was founded in 1983.
• The Society states its aims, and the ways it attempts to achieve those aims, in its statutes [',' The Cyprus Mathematical Society website.','1]:- .
• This is written in Greek and began publication in 1984.
• As an example of its contents, let us look at Volume 8 No 1 (2009) which was given over to publishing papers by plenary speakers from two previous conferences in the series 'Symposium on Elementary Maths Teaching' (SEMT).
• Here is an extract from the Introduction [',' J Novotna and D Pitta-Pantazi (eds.), Introduction, Mediterranean Journal for Research in Mathematics Education 8 (1) (2009), i-ii.','3]:- .
• The 9th Symposium being in August 2007, we can easily calculate that the 1st SEMT took place in August 1991.
• Conceived in 1990, SEMT was born in 1991 as the only conference focusing on teaching and learning of elementary mathematics.
• 1991: The teaching of mathematics to elementary mathematics pupils.
• 1991: The changing face of elementary mathematics.
• 2001: What is meant by the competence and confidence of people involved in the teaching of elementary mathematics.

11. Hamburg Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Hamburg Mathematical Society (Mathematische Gesellschaft in Hamburg), founded in 1690, is the oldest mathematical society in the world which still exists today.
• It was not, however, the only society founded in Hamburg in the 17th century for, after the Thirty Years War (1618-1648), various citizens of Hamburg got together and formed societies of people with similar interests.
• For example, before the mathematical society, a medical society was founded in 1644 and a music society in 1660.
• The need for such a mathematical Society was shown by the quotes in [',' J F Bubendey, Beitrage zur Geschichte der Mathematischen Gesellschaft in Hamburg 1690-1790, Mitteilungen der Mathematischen Gesellschaft in Hamburg 1 (1) (1881), 8-16.','1]:- .
• To overcome these problems, the Society was founded as the 'Kunstrechnungsliebende Societat in Hamburg' by Heinrich Meissner (20 April 1644 - 1 September 1716) and Valentin Heins.
• Heinrich Meissner was a pupil at the Knackenruggesche school in Hamburg, becoming a teacher there in 1669.
• In 1688 he became a teacher at the St Jacobi Church school where he taught writing and calculating until shortly before his death.
• Valentin Heins, the son of a linen weaver, taught arithmetic from 1651 in Hamburg, then he studied theology from 1658 to 1659 at Jena and Leipzig.
• For example Heinrich Meissner published Stern und Kern der Algebrae in 1692 which begins with a discussion of basic arithmetic including the extraction of square and cubic roots.
• Valentin Heins (15 May 1637 - 17 November 1704) published Tyrocinium Mercatorio-Arithmeticum (1694) which was a much more conventional commercial arithmetic.
• Meissner, who died in 1716, was the last of the founding members.
• After that the Society only had three members from Hamburg and by 1717 only two remained.
• It certainly did not flourish so the Society revised its statutes in 1774, and again in 1789-90 to distinguish itself from other societies that were starting up aiming to include scholars, lawyers and merchants wishing to improve trade and commerce.
• For example in 1819 the Society published a famous handbook of navigation by Reinhard Woltman (1757-1837), the Handbuch der Schiffahrtskunde.
• One important person in the development of the Society was Hermann Schubert who moved to Hamburg in 1876 to take up an appointments as a teacher at the Johanneum, the renowned humanistic school.
• This turned into the publication of its journal, the Mitteilungen der Mathematischen Gesellschaft in Hamburg, with volume 1 being published in 1881.
• Plans to found the university had been made much earlier but had to be shelved when World War I broke out in 1914.
• Only after the war ended in 1918 was it possible to continue with the plan.
• With the opening of the university in 1919, Wilhelm Blaschke was appointed to the chair in the University of Hamburg.
• For four years the Society was unable to operate but from around 1950, under the leadership of Werner Burau (1906-1994), the Society was rebuilt and by 1963 it had returned to have as many members as there had been in the 1920s.

12. Montenegran Society of Mathematicians and Physicists
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Montenegro had been the Kingdom of Montenegro from 1910 to 1918 and it became part of Yugoslavia when it was created following World War I.
• The Yugoslav Mathematical Society was formed in Belgrade in 1937 and this served mathematicians in Yugoslavia until the break-up of that country.
• Many parts of Yugoslavia declared independence in 1991-92 but Montenegro remained a part of Serbia and Montenegro.
• The Society of Mathematicians and Physicists of Serbia had been founded on 4 January 1948 and it had split into two societies, mathematics and physics, in 1981 and at that time the Mathematical Society of Serbia was founded.
• Milojica Jacimovic was born in Kostenica, a village in the Bijelo Polje Municipality in northern Montenegro, on 1 March 1950.
• He graduate with a mathematics degree from the University of Belgrade in 1973 and was appointed as an assistant at the University of Montenegro in Podgorica in November of that year.
• He was awarded his doctorate in 1980 by the University of Belgrade for his thesis Iterative regularisation of a minimisation method.
• He was promoted to assistant professor at the University of Montenegro in 1980, associate professor in 1986 and full professor in 1991.
• In July 2010 it applied for membership of the International Mathematical Union and it was admitted as a member on 1 September 2010.
• (vi) In recent years mathematicians living in Montenegro published 5-8 papers in respected mathematical journals and about 10-15 in other mathematical journals.
• 58 Secretariat - International Mathematical Union (July 2010).','1]:- .
• Some of the activities performed by our Society are as follows: the Society organised the Congress of Mathematicians, Physicists and Astronomers of Yugoslavia in Becici, Montenegro; it also twice organised, in 1995 and 2004, the Congress of Mathematicians and Physicists of Serbia and Montenegro in Petrovac, Montenegro.
• 58 Secretariat - International Mathematical Union (July 2010).','1]:- .
• The first issue of the journal was in 1993.

13. Ramanujan Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 2014 the Society described itself in the following terms [',' Ramanujan Mathematical Society website.','1]:- .
• The Journal of the Ramanujan Mathematical Society first appeared in 1986 with the first volume containing papers from Indian, American, French and English mathematicians such as David A Brannan, Growth rates of subharmonic functions in the plane, Ashok K Agarwal and George E Andrews, On Asai's polynomials related to the twisting operators of Finite Classical Groups, and Bernard Malgrange, Deformations of Differential Systems, II.
• K S Padmanabhan, the first editor-in-chief, continued in this role until 1991.
• V Kannan succeeded him as editor-in-chief of the Journal in 1992 and continued in this role till 1996.
• The next editor-in-chief was Kumar Murty who took on this position in 1997.
• The Society stated in 2013 [',' Ramanujan Mathematical Society website.','1]:- .
• In 2013-14 the Society ran a large number of Undergraduate Teachers Enrichment programmes: Group-actions in Aurangabad, Group-actions in Nanded, Group-actions in Latur, Number Theory and Cryptography in Lady Shri Ram College for Women (New Delhi), Multivariable Calculus in Pune, Complex Analysis in Vivekananda Mahavidyalaya, Burdwan, Finite Group theory and applications in Deshbandhu College (New Delhi), Graph theory and Operations research in Holy Cross College, Tiruchirapalli, Euclidean Geometry from a Group-theoretic Viewpoint, North Maharashtra University, Jalgaon, Differential Geometry in Kuvempu, Euclidean Geometry from a Group-theoretic Viewpoint, and Differential Equations in Rani Channamma University, Belgaum, Rings and Modules in D A V College, Jalandhar, and Finite Group theory in Punjabi University, Patiala.
• They described them as follows [',' Ramanujan Mathematical Society website.','1]:- .
• Also beginning in 2014, the Society organised Compact Courses for Post-graduate Students [',' Ramanujan Mathematical Society website.','1]:- .
• (i) The Prof C S Venkitaraman Memorial Lectures are named for C S Venkitaraman (1918-1994).
• (ii) The Prof W H Abdi Memorial Lecture's named for Wazir Hasan Abdi (1922-1999).
• He became the founder Professor and Head of the Department of Mathematics and Statistics of University of Cochin in 1977.
• He published the book 'Toils and Triumphs of Srinivasa Ramanujan - The Man and the Mathematician' in 1992.
• A memorial lecture has been given at the annual conference of the Ramanujan Mathematical Society since 2001.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Estonian Academy of Sciences was founded in 1938.
• On 13 April the President-Regent approved the first twelve Members of the Estonian Academy of Sciences.
• There were six in the Humanities Section, namely Edgar Kant, Oskar Loorits, Julius Mark, Hendrik Sepp, Gustav Suits and Juri Uluots, and six in the Natural Sciences Section, namely the biologist Hugo Kaho (1885-1964), the chemist Paul Nikolai Kogermann (1891-1951), the medical scientist and military officer Aleksander Paldrok (1871-1944), the surgeon Ludvig Puusepp (1875-1942), the microbiologist Karl Schlossmann (1885-1969) and the astronomer and astrophysicist Ernst Opik (1893-1985).
• On 19 December 1938 a Plenary Session of the Academy was held which agreed to register the Estonian Naturalists' Society and the Estonian Learned Society as being attached to the Estonian Academy of Sciences.
• On 15 May, the botanist Teodor Lippmaa (1892-1943) who was President of the Estonian Naturalists' Society, was elected to the Natural Sciences Section of the Academy.
• A decision was taken on 27 October to publish an annual yearbook, the Annales Academiae Scientiarum Estonicae, with the first issue to appear in 1940.
• Soviet troops invaded Poland on 17 September and by 29 September Poland was partitioned between Russia and Germany.
• The Soviets invaded Finland in November 1939 with the Baltic States remaining neutral but on 17 June 1940 the Red Army invaded the Baltic States, setting up a puppet government.
• For a month Konstantin Pats was forced to sign decrees, one of which was the "Estonian Academy of Sciences Liquidation Act" signed on 17 July which declared the Academy terminated from 20 July.
• A committee of six were appointed to set up the reinstated Academy, headed by the historian and founding member of the Estonian Communist Party Hans Kruus (1891-1976).
• On 13 September 1945 he submitted to the Chairman of Council of People's Commissars a proposed structure of the Academy of Sciences of the Estonian Soviet Socialist Republic, a list of statutes for the Academy and a list of individuals who might be considered for membership of the learned councils of the Academy's institutes and those who might be invited to be members or corresponding members of the Academy.
• The Council of People's Commissars agreed to the Academy being set up in Tallinn on 23 January 1946 and, on 5 April of that year the proposed statutes, structure, full members and corresponding members were approved by the Council [',' Estonian Academy of Sciences website.','1]:- .
• A decision was taken in 1950, and approved by the General Assembly on 18 July of that year, to dismiss Hans Kruus from the office of President.
• He was removed as a full member of the Academy in 1951.
• The reason for Kruus being dismissed was that he fell from favour and was designated a "bourgeois nationalist." Johan Eichfeld was elected President on 13 September 1950.
• In 1956 the Council of People's Commissars reinstated Hans Kruus as a full member of the Academy.
• March 1990 elections saw a big majority for independence which was declared in August 1991 and was recognized by the Soviet Union in September of that year.
• Since its founding in 1946, the Academy of Sciences of the Estonian Soviet Socialist Republic had not been considered as a continuation of the Estonian Academy of Sciences which had been founded in 1938.
• We note that in fact there had been a steady change of structure from 1946 up to this time, the details of which are given in [',' Estonian Academy of Sciences website.','1].
• He was born on 11 October 1957, attended Kohila Secondary School, the University of Tartu, and was awarded his candidate's degree in mathematics and physics by Moscow State University in 1984.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Japan Academy was founded on 15 January 1879 with the aim of advancing education and science in Japan.
• In 1906 the Tokyo Academy was renamed the Imperial Academy and it was again renamed in December 1947 when it adopted its present name of the 'Japan Academy'.
• The setting up of the Academy came a few years after the end of the Edo Period (1615-1868) [',' The Edo Period in Japanese History, Victoria and Albert Museum.','3]:- .
• By 1720 Edo had more than a million inhabitants.
• The period of self-imposed national isolation came to a dramatic end in 1853 when four American battleships arrived in Edo Bay.
• In 1868 external pressure combined with growing internal unrest and led to the overthrow of the Tokugawa shogun and the restoration of the Meiji Emperor.
• This college became part of the University of Tokyo in 1877.
• The Bansho Shirabesho, the Institute for Researching Foreign Books, was founded in 1857.
• It was renamed YMsho shirabesho, the Institute for the Study of Western books, in 1862 and, like the Shoheiko, it became part of the University of Tokyo in 1877.
• The beginning of the Meiji Period in 1868 saw the emperor move to Tokyo which became the capital of Japan.
• At this time Japan was looking at the European models of education and learning, leading them to follow the European Academy model in establishing the Tokyo Academy in 1879.
• The Tokyo Academy had a maximum membership of 40 when it was established in 1879.
• The first President of the Tokyo Academy was Fukuzawa Yukichi (1835-1901), an author and teacher who was an advocate of educational reform.
• He founded Keio University in central Tokyo in 1858 which was modelled completely on Western universities.
• After the opening of Japanese trade in the 1850s, he went on a diplomatic mission to the United States in 1860.
• Fukuzawa Yukichi was President for less than a year for, in June 1879, Nishi Amane (1829-1897) became the second President.
• Kato Hiroyuki (1836-1916), also a founder member of the Meiji 6 Society, served as President between Nishi Amane's two terms, and later served as President for all but two of the 23 years from June 1886 to June 1909.
• In addition to providing reports and proposals in response to government inquiries, in 1911 the Academy established the Imperial Prize to recognize and encourage superb creative research.
• In 1920 the Imperial Academy created the National Research Council [',' B C Dees, The Allied Occupation and Japan&#8217;s Economic Miracle: Building the Foundations of Japanese Science and Technology 1945-52 (Routledge, 2013).','1]:- .
• In May 1925 the maximum membership of the Imperial Academy was increased from 60 to 100 and, during World War II, in March 1942, the first part of the first volume of the Transactions of the Imperial Academy was published in Japanese.
• During the war the National Research Council became the body to advise on war related research [',' B C Dees, The Allied Occupation and Japan&#8217;s Economic Miracle: Building the Foundations of Japanese Science and Technology 1945-52 (Routledge, 2013).','1]:- .
• The fascinating discussions are fully described in [',' B C Dees, The Allied Occupation and Japan&#8217;s Economic Miracle: Building the Foundations of Japanese Science and Technology 1945-52 (Routledge, 2013).','1].
• In January 1949 the reorganisation compatible with the wishes of the Allies was complete and it became an institution, with maximum membership of 150, attached to the Science Council of Japan.
• In 1974, the Academy's newly constructed Assembly Hall was inaugurated.

16. European Women in Mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The International Congress of Mathematicians was held in Berkeley, United States, from 3 August to 11 August 1986.
• Bodil Branner writes [',' S Munday and E Resmerita (eds.), European Women in Mathematics Newsletter 18 (1) (2011).','2]:- .
• The second meeting of the European Women in Mathematics was held in Technical University of Denmark, Copenhagen, Denmark, in 1987.
• Bodil Branner describes that meeting [',' S Munday and E Resmerita (eds.), European Women in Mathematics Newsletter 18 (1) (2011).','2]:- .
• The second meeting of European Women in Mathematics took place in 1987 at my university, the Technical University of Denmark, in Copenhagen.
• Capi Corrales Rodriganez from the Departamento de Algebra, Facultad de Matematicas, Universidad Complutense de Madrid, writes [',' S Munday and E Resmerita (eds.), European Women in Mathematics Newsletter 18 (1) (2011).','2]:- .
• 1991 Marseille, France .
• 2001 Malta .
• 2011 Barcelona, Spain .
• Now with the exception of 1989, this list shows that between 1986 and 1991 there was a meeting each year.
• However, at the 1991 meeting, and at each meeting since, a standing committee and a convenor was elected and it was decided to hold the meetings every second year.
• By 1993 European Women in Mathematics had obtained legal status, established an office in Helsinki and was registered as a legal organisation in Finland.
• The main reason for choosing Finland was the fact that the European Mathematical Society had been registered there in 1990.
• Sara Munday writes [',' S Munday and E Resmerita (eds.), European Women in Mathematics Newsletter 18 (1) (2011).','2]:- .
• I did not think much more about it until the International Mathematical Congress in 1986 in Berkeley.

17. Malaysian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Malaysian Mathematical Society of was founded in 1970 by a group of mathematicians at the University of Malaya to replace the Mathematical Society of Malaya and Singapore which had served both the mathematicians of Malaya and Singapore since 1952.
• During this time he was active in planning to create the University of Malaya and, in 1949, he became Dean of the Faculty of Arts of the University of Malaya.
• When the Mathematical Society of Malaya and Singapore was founded in 1952, Oppenheim became the first president.
• In 1953 the Mathematical Society of Malaya and Singapore began publication of the Bulletin of Malayan Mathematical Society.
• This was published from 1953 to 1959 and then in 1960 the name was changed to NABLA.
• Now Daniel Pedoe was appointed as head of the Mathematics Department at the University of Singapore in 1959.
• He became the second president of the Mathematical Society of Malaya and Singapore in 1960 following after Oppenheim.
• The Malaysian Mathematical Society has undergone a number of changes of name over the years since its founding in 1970.
• The Southeast Asian Mathematical Society was founded in 1972.
• When the Malaysian Mathematical Society was founded in 1970 it set out its aims as follows (see [',' Bulletin of the Malaysian Mathematical Sciences Society.','1]): .
• However, in 1996 the length of the term was changed so that elections only took place biennially.
• When it began in 1970 the Society had 42 members, almost all from the University of Malaya.
• The Malaysian Mathematical Society Bulletin began publication in 1978 and the last volume under this title was Volume 22 in 1999.
• When the Society changed its name in 1999 the Bulletin began a second series under a slightly different name, namely the Bulletin of the Malaysian Mathematical Sciences Society.
• The author of the article [',' A N Zainab, Internationalization of Malaysian Mathematical and Computer Science Journals, Malaysian Journal of Library & Information Science 13 (1) (2008), 17-33.','3] looked at all issues of the Bulletin of the Malaysian Mathematical Sciences Society from 2000 to 2007.
• Often the author, A N Zainab, comments on both journals but in these cases we have changed the text to just refer to the Bulletin of the Malaysian Mathematical Sciences Society [',' A N Zainab, Internationalization of Malaysian Mathematical and Computer Science Journals, Malaysian Journal of Library & Information Science 13 (1) (2008), 17-33.','3]:- .

18. South-Eastern Europe Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Mathematical Society of South-Eastern Europe (MASSEE) was established on 1 March 2003.
• However, the Society might be considered as a continuation of the Balkan Mathematical Union founded in 1937 and re-founded in 1966.
• Less than six months after the Mathematical Society of South-Eastern Europe was founded, their first Congress was held in Borovetz, Bulgaria, 15-21 September 2003.
• The following few paragraphs are a slightly modified version of an article found on [',' Mathematical Society of South Eastern Europe (MASSEE) website.','1].
• The first Balkan mathematical journal Revue Mathematique de l'Union Interbalkanique appeared in 1936 and in 1938.
• The first contacts and discussions aimed at reviving the Union started in 1956.
• The 70's were the golden era for the Balkan Mathematical Union: the Fourth Congress (Istanbul 1971), the Fifth Congress (Belgrade 1974), several conferences and symposiums, Balkaniads for university students and for young researchers, scientific sessions during the meetings of the Executive Council of Balkan Mathematical Union, etc.
• It was published by the Union of Mathematicians, Physicists and Astronomers of Yugoslavia and edited by Academician Djuro Kurepa (Volume 1 appeared in 1971 with Volume 6 published in 1980).
• In accordance with the decision of the Executive Council of the Balkan Mathematical Union (taken in July 1984) in 1987, the National Committee for Mathematics of Bulgaria and the Bulgarian Academy of Sciences started the publication of Mathematica Balkanica - New Series, edited by Academician Blagovest Sendov.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Chinese Academy of Sciences was established on 1 November 1949 in Beijing.
• On 1 October 1949 Mao Zedong declared that mainland China was now the People's Republic of China.
• The Chinese Academy of Sciences was quickly established and the quick establishment was possible since it was built on the Academia Sinica which had been founded in 1927 to promote and undertake scholarly research in sciences and humanities.
• In fact the beginning of strong scientific policy in China goes back to 1911 when the last emperor gave way to the establishment of the Republic.
• Sun Yat-sen (1866-1925), the first leader of Republican China, was a physician and philosopher who believed strongly in science and this led to the founding of learned societies, scientific journals and science departments in several universities.
• Academia Sinica, first proposed in 1922, was actually established in July 1927 but did not hold its first meeting until 9 June 1928.
• Four Institutes were established in January 1928, five more in 1929 and by 1947 it had thirteen research institutes.
• From its founding in 1949, the Chinese Academy of Sciences built on these earlier academies and initially developed following the model of the USSR Academy of Sciences.
• In 1956, the central government asked the Chinese Academy of Sciences to oversee preparation of the country's first 12-year national programme for science and technology development, which propelled China's drive for modernisation of science and technology.
• The Chinese Academy of Sciences proposals have resulted in the launch of a number of key national scientific programmes including the "863 Program" in 1986, which has propelled China's overall high-tech development, and the "973 Program", or National Basic Research Program, in 1997, which called for the development of science and technology in various fields.
• If the reader wants more details about the Academy in general we refer them to the 276 page overview given by [',' 2014 Guide to CAS, Chinese Academy of Sciences.http://english.cas.cn/institutes/cas_guide/201409/U020141124419569240513.pdf ','1].
• The Institute of Mathematics was organised in March 1941 under the directorship of Li-Fu Chiang as part of the Academia Sinica before the founding of the Chinese Academy of Sciences.
• At this time Lifu Jiang (1890-1978) helped found this Institute and became its first director.
• The other major figure in founding the Institute was Shiing-shen Chern who had returned to China from the United States in 1946.
• He briefly acted as director of the Institute of Mathematics but left China for the United States in 1948 when life became difficult due to the civil war.
• After the Chinese Academy of Sciences was founded, the Institute of Mathematics was founded as part of the new Academy in 1952 [',' History, The Academy of Mathematics and Systems Science.','4]:- .
• In January 1951, the State Council approved the appointment of Hua Luogeng as the director of the Institute of Mathematics yet to be founded.
• On 1 July 1952, the Institute was established and located on the campus of Tsinghua University.
• In the first half of 1953, there were 32 researchers in the institute [among whom] 17 were electednacademicians of the Chinese Academy of Sciences.
• In 1956, the state formulated the 12-Year Program for the Development of Science and Technology.
• In 1957, Professor Xiong Qinglai, who returned from France, founded the research group of the theory of functions.
• The Institute was temporarily moved to the Xiyuan Hotel in 1957.
• In the next year, it was moved to Zhongguancun (the northern building of the Institute of Computing Technology).nIn 1961, research divisions were set up according to subjects, including "four branches" (number theory, algebra, geometry, topology), function theory, differential equations, functional analysis, mathematical logic, theoretical physics, probability and statistics.
• After the "cultural revolution" started in 1966, all research stopped.
• In 1972, Premier Zhou Enlai gave instruction on intensifying basic research, so mathematical research started again.
• In 1973, Chen Jingrun made public his result on the Goldbach conjecture.
• In 1975, Yang Le and Zhang Guanghou published their result on the value distribution theory.
• After the "gang of four" was smashed in 1976, especially after the National Science Congress was held in 1978, China greeted a springtime of science.
• In 1998, however, they were brought together in an Academy, the Academy of Mathematics and Systems Science, still part of the Chinese Academy of Sciences [',' The Academy of Mathematics and Systems Science, 2014 Guide to CAS, Chinese Academy of Sciences.http://english.cas.cn/institutes/cas_guide/201409/U020141124419569240513.pdf ','3]:- .
• The Academy of Mathematics and Systems Science was founded in 1998 upon the merger of the Institute of Mathematics (founded in 1952), the Institute of Applied Mathematics (founded 1979), the Institute of Systems Science (founded 1979) and the Institute of Computational Mathematics and Scientific/Engineering Computing (founded in 1995).
• The Academy of Mathematics and Systems Science is authorised to confer advanced degrees in 18 academic disciplines.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Croatian Academy of Sciences and Arts was founded in Zagreb on 29 April 1861.
• This, of course, looks a little confusing today since one immediately thinks, "How could there be a Yugoslavian Academy in 1861 when Yugoslavia did not exist until the 20th century?" The answer is that perhaps a better translation of 'Jugoslavenska akademija znanosti i umjetnosti' would be 'South Slavic Academy of Sciences and Arts'.
• Croatia became part of the Habsburg Monarchy in 1527 but was largely conquered by the Ottomans.
• In 1849 Croatia became a Habsburg crown territory under the emperor Franz Joseph I.
• Josip Juraj Štrossmajer (1815-1905) was born into a Croatian family in Osijek.
• After studying theology and being awarded a doctorate from Budapest in 1835 and, after being a priest in Petrovaradin, he obtained a second doctorate in Vienna.
• He entered politics as leader of the People's Party, consisting of Illyrianist reformers, in 1860.
• In 1860 he became part of the Croatian legislative assembly and on his recommendation Josip Šokčević was appointed as ban, the head of the assembly, on 19 June 1860.
• He also wrote in a letter to Šokčević that the Academy should [',' Founding of the Academy, Croatian Academy of Sciences and Arts.','1]:- .
• On 29 April 1861, at a session of the Croatian legislative assembly, Štrossmajer formally proposed the setting up of the Academy.
• Bishop Štrossmajer was elected patron and Franjo Rački (1828-1894) was elected the first president.
• He became a member of the Croatian legislative assembly in 1861.
• For them South Slavs, or Yugoslavians, would have included Bulgarians but although the Academy was named Yugoslav Academy of Sciences and Arts, it was firmly based in Zagreb and was a Croatian Academy [',' Founding of the Academy, Croatian Academy of Sciences and Arts.','1]:- .
• Rad began publication in 1867 and all the departments of the Academy contributed to this publication until 1882 when some departments began their own publications.
• Another early publication was Ljetopis (Annals) which began publication in 1887; it also continues to be published.
• Much of the early work of the Academy involved Croatian history and numerous publications on this topic are detailed in [',' Founding of the Academy, Croatian Academy of Sciences and Arts.','1].
• On 1 December 1918 the Kingdom of Serbs, Croats and Slovenes came into existence.
• In order to combat nationalism within the Kingdom, it was named Yugoslavia in 1929.
• When World War II broke out in 1939, Yugoslavia declared itself neutral.
• This, however, did not stop Germany invading in 1941 and they occupied and partitioned the country.
• Playing on the Croats desire for independence, Germany and Italy set up an independent Croatia in April 1941.
• From 1941 to 1945 there was fighting between Serbs and Croats and a great many lost their lives but with the end of World War II in 1945 Croatia became a republic within the Socialist Federal Republic of Yugoslavia.
• With the break-up of the Soviet Union beginning in 1989, Croatia held elections in 1990 resulting in a right-wing nationalist government.
• On 25 June 1991 Croatia declared independence and the Academy again changed its name, this time back to the Croatian Academy of Sciences and Arts.

21. Columbian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Colombian Mathematical Society was founded in 1955.
• The National University of Columbia (Universidad Nacional de Colombia) was established in 1867 in Bogata.
• These were Carlo Federici (1906-2004) and Janos Horvath (1924-2015) and since their role in the founding of the Colombian Mathematical Society is vital, we give some details of them.
• He obtained a doctorate in physics from the University of Genoa in 1928 for his thesis Su un ds2 einsteiniano, then in 1930 a doctorate in mathematics for his thesis Sulle congruenze binomie.
• He became an assistant to Alessandro Padoa, working on mathematical logic at the University of Genoa from 1932 to 1942, then became professor of mathematical logic at the Cristoforo Colombo Gymnasium 1942-1948.
• As a Communist member of an anti-fascist group, he was arrested and imprisoned in 1945 and following this episode decided to emigrate to Columbia.
• He studied at the University of Budapest working under Lipot Fejer and Frigyes Riesz for his doctorate which he was awarded in 1947.
• He arrived in Bogota in 1951 to strengthen the area of mathematics in the departments of engineering, architecture and economics at the newly founded Universidad de los Andes.
• He founded the Revista de Matematicas Elementales (Journal of Elementary Mathematics) in 1952 which was a joint publication of the National University of Columbia and the Universidad de los Andes.
• He moved to the United States in 1958.
• The Colombian Mathematical Society was founded on 10 August 1955, at a meeting held at 7 o'clock in the evening in the house of Julio Carrizosa Valenzuela (1895-1974).
• In addition to Julio Carrizosa Valenzuela, Carlo Federici Casa and Juan Horvath, the founding members of the Society were: Antonio Maria Gomez (1913-1979); Dario Rozo (1881-1964); Erwin Von Der Walde (1927-2016); Gabriel Poveda Ramos (1931); Guillermo Castillo Torres (1923-2000); Gustavo Perry Zubieta (1912-1986); Henry Yerly (1901-1984); Jorge Acosta Villaveces (1891-1965); Jose Ignacio Nieto (1930); Leopoldo Guerra Portocarrero (1911-1964); Luciano Mora Osejo (1928-2016); Luis De Greiff Bravo (1908-1967); Luis Ignacio Soriano (1903-1973); Michel Valero (1928-2008); Otto De Greiff (1903-1995); and Pablo Casas Santofimio (1927-1983).
• Julio Carrizosa Valenzuela was elected as President of the new Society and he served from 1955 to 1957.
• The statutes of the Society began as follows [',' C H Sanchez, Homage to the Colombian Mathematical Society on the fortieth anniversary of its foundation (Spanish), Lecturas Matematicas 16 (2) (1995), 231-243.','1]:- .
• On 13 May 1999 the Society revised its statutes.
• It had its origin in the Journal of Elementary Mathematics created by Juan Horvath in 1952 and published by the National University of Columbia and the University of the Andes.
• Lecturas Matematicas is the official newsletter of the Colombian Mathematical Society created in 1980 with the purpose of providing a platform for the Colombian community to publish works from the elementary to the advanced level.
• It has been awarded every second year since 2011 and the presentation is made at the Colombian Mathematics Congress.
• The first award was made in 2017 to Mauricio Fernando Velasco, professor in the Mathematics Department of the Universidad de los Andes in Bogota.
• Having graduated with a mathematics degree from the Universidad del Valle in 1977, he obtained a master's degree at the Institute of Pure and Applied Mathematics in Rio de Janeiro in 1979 and a Ph.D.
• from the University of California at Berkeley in 1986, after overcoming a serious health problem.
• 1955-1957 Julio Carrizosa Valenzuela .
• 1957-1963 Gustavo Perry Zubieta .
• 1963-1967 Carlos Lemoine Amaya .
• 1967-1968 Ricardo Losada Marquez .
• 1968-1970 Jaime Lesmes Camacho .
• 1970-1971 Otto Raul Ruiz .
• 1971-1973 Jairo Charris Castaneda .
• 1973-1975 Carlos Ruiz Salguero .
• 1975-1983 Alonso Takahashi Orozco .
• 1983-1987 Jaime Lesmes Camacho .
• 1987-1990 Myriam Munoz de Ozac .
• 1990-1993 Victor Albis Gonzalez .
• 1993-1998 Ernesto Acosta Gempeler .
• 2003-2017 Carlos H Montenegro .

22. Colombian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Colombian Mathematical Society was founded in 1955.
• The National University of Colombia (Universidad Nacional de Colombia) was established in 1867 in Bogata.
• These were Carlo Federici (1906-2004) and Janos Horvath (1924-2015) and since their role in the founding of the Colombian Mathematical Society is vital, we give some details of them.
• He obtained a doctorate in physics from the University of Genoa in 1928 for his thesis Su un ds2 einsteiniano, then in 1930 a doctorate in mathematics for his thesis Sulle congruenze binomie.
• He became an assistant to Alessandro Padoa, working on mathematical logic at the University of Genoa from 1932 to 1942, then became professor of mathematical logic at the Cristoforo Colombo Gymnasium 1942-1948.
• As a Communist member of an anti-fascist group, he was arrested and imprisoned in 1945 and following this episode decided to emigrate to Colombia.
• He studied at the University of Budapest working under Lipot Fejer and Frigyes Riesz for his doctorate which he was awarded in 1947.
• He arrived in Bogota in 1951 to strengthen the area of mathematics in the departments of engineering, architecture and economics at the newly founded Universidad de los Andes.
• He founded the Revista de Matematicas Elementales (Journal of Elementary Mathematics) in 1952 which was a joint publication of the National University of Colombia and the Universidad de los Andes.
• He moved to the United States in 1958.
• The Colombian Mathematical Society was founded on 10 August 1955, at a meeting held at 7 o'clock in the evening in the house of Julio Carrizosa Valenzuela (1895-1974).
• In addition to Julio Carrizosa Valenzuela, Carlo Federici Casa and Juan Horvath, the founding members of the Society were: Antonio Maria Gomez (1913-1979); Dario Rozo (1881-1964); Erwin Von Der Walde (1927-2016); Gabriel Poveda Ramos (1931); Guillermo Castillo Torres (1923-2000); Gustavo Perry Zubieta (1912-1986); Henry Yerly (1901-1984); Jorge Acosta Villaveces (1891-1965); Jose Ignacio Nieto (1930); Leopoldo Guerra Portocarrero (1911-1964); Luciano Mora Osejo (1928-2016); Luis De Greiff Bravo (1908-1967); Luis Ignacio Soriano (1903-1973); Michel Valero (1928-2008); Otto De Greiff (1903-1995); and Pablo Casas Santofimio (1927-1983).
• Julio Carrizosa Valenzuela was elected as President of the new Society and he served from 1955 to 1957.
• The statutes of the Society began as follows [',' C H Sanchez, Homage to the Colombian Mathematical Society on the fortieth anniversary of its foundation (Spanish), Lecturas Matematicas 16 (2) (1995), 231-243.','1]:- .
• On 13 May 1999 the Society revised its statutes.
• It had its origin in the Journal of Elementary Mathematics created by Juan Horvath in 1952 and published by the National University of Colombia and the University of the Andes.
• Lecturas Matematicas is the official newsletter of the Colombian Mathematical Society created in 1980 with the purpose of providing a platform for the Colombian community to publish works from the elementary to the advanced level.
• It has been awarded every second year since 2011 and the presentation is made at the Colombian Mathematics Congress.
• The first award was made in 2017 to Mauricio Fernando Velasco, professor in the Mathematics Department of the Universidad de los Andes in Bogota.
• Having graduated with a mathematics degree from the Universidad del Valle in 1977, he obtained a master's degree at the Institute of Pure and Applied Mathematics in Rio de Janeiro in 1979 and a Ph.D.
• from the University of California at Berkeley in 1986, after overcoming a serious health problem.
• 1955-1957 Julio Carrizosa Valenzuela .
• 1957-1963 Gustavo Perry Zubieta .
• 1963-1967 Carlos Lemoine Amaya .
• 1967-1968 Ricardo Losada Marquez .
• 1968-1970 Jaime Lesmes Camacho .
• 1970-1971 Otto Raul Ruiz .
• 1971-1973 Jairo Charris Castaneda .
• 1973-1975 Carlos Ruiz Salguero .
• 1975-1983 Alonso Takahashi Orozco .
• 1983-1987 Jaime Lesmes Camacho .
• 1987-1990 Myriam Munoz de Ozac .
• 1990-1993 Victor Albis Gonzalez .
• 1993-1998 Ernesto Acosta Gempeler .
• 2003-2017 Carlos H Montenegro .

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It is [',' Allahabad Mathematical Society website.','1]:- .
• He was born on 12 January 1899 in the Azamgarh District of Uttar Pradesh, India.
• from the Banaras Hindu University in 1921.
• He then undertook research at the Department of Mathematics of the Banaras Hindu University where he was appointed as an assistant lecturer in 1922.
• He was given leave to travel to the UK in 1929 and, after spending a while at Edinburgh, he went to the University of Liverpool where he completed a Ph.D.
• in 1931 supervised by E C Titchmarsh.
• He then went to Paris where he worked under Arnaud Denjoy and was awarded a Docteur es Science in 1932 for his thesis Contribution a l'etude de la series conjuguee d'une serie de Fourier.
• The Indian Journal of Mathematics, which is devoted to original research papers in different branches of mathematics and mathematical statistics, began publication in 1958 with Part 1 of Volume 1 appearing in December of that year.
• The Journal has appeared annually with three parts per year since Volume 11 in 1969.
• Volume 9 (1967) B N Prasad Memorial Volume.
• Volume 20 (1978) P L Bhatnagar Memorial Volume.
• Volume 22 (1980) C T Rajgopal Memorial Volume.
• Volume 28 (1986) S R Sinha Memorial Volume.
• Volume 29 (1987) Ramanujan Centenary Volume.
• Volume 32 (1990) Hansraj Gupta Memorial Volume.
• Volume 33 (1991) U N Singh Memorial Volume.
• Volume 41 (1999) B N Prasad Birth Centenary Commemoration Volume I.
• Volume 52 (2010) T Pati Memorial Volume.
• Volume 56 (2014) Special Volume Dedicated to Professor Billy E Rhodes .
• We note that Volume 33 is dedicated to U N Singh who was the President of the Allahabad Mathematical Society from 1976 to 1986.
• The first paper in that Volume is [',' B S Yadav, U N Singh: His life and Work, Indian Journal of Mathematics 33 (1991), i-xxiv.','3], namely B S Yadav, U N Singh: His life and Work, Indian Journal of Mathematics 33 (1991), i-xxiv.
• The Bulletin of the Allahabad Mathematical Society was first published in 1986.
• Volume 25 (2010) T Pati Memorial Volume .
• The Society also publishes a Lecture Note Series [',' Allahabad Mathematical Society website.','1]:- .
• The first conference of the Allahabad Mathematical Society was held in Allahabad on 18 November 1967.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Pontifical Academy of Sciences was founded in 1936 but its origins go back more than 300 years before this date.
• The Accademia dei Lincei was founded in 1603 by Federico Cesi, the son of the Duke of Acquasparta, and a member of an important family from Rome.
• Cesi died in 1630 and the Academy closed down.
• In 1745 a group of scientists in Rimini refounded the Academy, but it only functioned for a very short time.
• Padre Feliciano Scarpellini founded a private Academy in Rome in 1795 which he named the 'Lincei'.
• Pope Gregory XVI suggested in 1838 refounding the Academy as 'Accademia Pontificia dei Nuovi Lincei' (The Pontifical Academy of the New Lynxes), but this did not happen until 1847 when Pope Pius IX revived the Academy.
• Revolts occurred in the Papal States in 1849 and a short-lived Roman Republic was established.
• However, the Papal States of Emilia, Umbria, and Marche voted to join the Italian kingdom after Austria's defeat in 1859 then, when French troops withdrew from Rome in 1870, Italian forces took the area around the Vatican.
• Its headquarters was moved to the Casina Pio IV villa in the Vatican Gardens in 1922.
• This villa, designed by Pirro Ligorio, had been completed in 1561 as a summer residence for Pope Pius IV and surrounded by the trees and lawns of the Vatican gardens.
• The Casina Pio IV is a well-preserved treasury of 16th century frescoes, stucco reliefs, mosaics and fountains which was extended in the 1930s [',' Casina Pio IV, Pontifical Academy of Sciences.','2]:- .
• On 20 December 1931, the then President of the Accademia Pontificia dei Nuovi Lincei, Giuseppe Gianfranceschi, announced the plans for the enlargement of the Casina.
• Pope Pius XI was able to inaugurate the new extension, comprising a gallery and the great hall where the Plenary Sessions of the Academy are held, on 17 December 1933.
• Pope Pius XI renewed and reconstituted the Academy in 1936, and gave it its present name of Pontifical Academy of Sciences.
• The statutes drawn up in 1936 state that the aim of the Academy is:- .
• The number to date (2018) appears to be 43, including Max Planck, Niels Bohr, Werner Heisenberg, Paul Dirac and Erwin Schrodinger, all of whom have biographies in this archive.
• Of course there is another point here in that all academies strive to be independent from political pressure and it might appear that the Pontifical Academy of Sciences would be controlled by the Roman Catholic Church but Pius XII stressed the independence of the Academy in an address he gave in 1940 [',' Discourses of the Popes from Pius XI to John Paul to the Pontifical Academy of Sciences (Pontifical Academy of Sciences, Vatican City, 1986).','1]:- .
• The first President of the Academy was Agostino Gemelli (1878-1959) who was appointed to this role on 28 October 1936.
• These include: Giuseppe Armellini (24 October 1887 - 16 July 1958), Professor of Astronomy at the University of Rome and Director of the Astronomy Observatory in Rome; Emilio Bianchi (26 September 1875 - 11 September 1941), Professor of Astronomy and Geodesic Science at the University of Milan and Director of the Astronomy Observatory in Milan; Ugo Amaldi (18 April 1875 - 11 November 1957), Professor of Algebraic and Infinitesimal Mathematical Analysis at the University of Rome; Marcello Boldrini (9 February 1890 - 5 March 1969), Professor of Statistics at the University of Rome; and Enrico Pistolesi (2 December 1889 - 29 February 1968), Professor of Mechanics Applied to Machines and Aeronautical Construction, University of Pisa.
• A famous mathematician nominated for the Council on 5 April 1940 was Francesco Severi (13 April 1879 - 8 December 1961), President of the Istituto Nazionale di Alta Matematica and Professor of Higher Geometry at the University of Rome.
• On 22 November 1951 Pope Pius XII received members of the Pontifical Academy of Sciences for a week long meeting on seismology [',' G Tanzella-Nitti, Interdisciplinary Encyclopedia of Religion & Science','8]:- .
• According to a widely circulated version of events today, Pope Pius XII supposedly claimed in a discussion held at the Pontifical Academy of Sciences in November 1951 that the recent astronomical discoveries confirmed the initial page of the Book of Genesis when the latter describes the creation of the universe as a Fiat lux.
• The address that Pope Pius XII gave to the Academicians at this time was almost certainly drafted by Edmund Whittaker who had been inducted into the Academy by Pope Pius XI when it was founded in 1936.
• The first President Agostino Gemelli died on 15 July 1959 and, on 19 March 1960, the second President was installed.
• On the occasion of the centenary of Albert Einstein's birth, Pope John Paul II addressed the Academy and quoted words by Lemaitre from his time as President (see [',' Discourses of the Popes from Pius XI to John Paul to the Pontifical Academy of Sciences (Pontifical Academy of Sciences, Vatican City, 1986).','1]):- .
• In this address Pope John Paul II said [',' Discourses of the Popes from Pius XI to John Paul to the Pontifical Academy of Sciences (Pontifical Academy of Sciences, Vatican City, 1986).','1]:- .
• The Pope then again quoted words by Lemaitre from his time as President (see [',' Discourses of the Popes from Pius XI to John Paul to the Pontifical Academy of Sciences (Pontifical Academy of Sciences, Vatican City, 1986).','1]):- .
• Georges Lemaitre died on 20 June 1966 and the third President of the Academy was appointed on 15 January 1968.
• This was Daniel Joseph Kelly O'Connell who had studied mathematics and physics at the University of Dublin and, after being ordained and studying astronomy at Harvard College Observatory with Harlow Shapley, was appointed to the Riverview Observatory in Sydney, Australia, in 1933.
• In 1952 he was appointed as director of the Vatican Observatory.
• He retired as director of the Vatican Observatory in 1970 but continued to serve as President of the Academy until 15 January 1972.
• In 1976 Pope Paul VI updated the statutes of the Academy and, with minor exceptions, these still hold today.
• The minor exceptions relate to the number of members, set at 70 in 1976 but increased to 80 by Pope John Paul II on 8 January 1986.
• The Pius XI Medal, first awarded in 1962, is presented by the Pontifical Academy of Sciences every two years to a young scientist under the age of 45, chosen by the Academy for his or her exceptional promise.
• To illustrate this we note that Stephen Hawking was named a member of the Academy by Pope Paul VI in 1968.
• Hawking was presented with the Academy's Pius XI Gold Medal in 1975.
• These include: Hermann A Bruck (15 August 1905 - 4 March 2000), Professor of Astronomy, University of Edinburgh, Scotland, nominated 5 April 1955; Louis de Broglie (15 August 1892 - 19 March 1987), Honorary Professor of Physics, Faculte des Sciences, Paris, France and Honorary Perpetual Secretary, Academy of Sciences, Paris, nominated 5 April 1955; Daniel Joseph Kelly O'Connell (25 July 1896 - 15 October 1982), Vatican Observatory, Vatican City, nominated 24 September 1964; Stanislaw Lojasiewicz (9 October 1926 -13 November 2002), Professor of Mathematics, Jagiellonian University, Cracow, Poland, nominated 27 January 1983; Ennio de Giorgi (8 February 1928 - 25 October 1996), Professor of Mathematical Analysis, Scuola Normale Superiore, Pisa, Italy, Nominated 12 May 1981; Paul Marie Germain (28 August 1920 - 26 February 2009), Professor Emeritus of Mechanics at the University of Pierre et Marie Curie and Secretaire perpetuel honoraire of the Academy of Sciences, Paris, nominated 9 June 1986, Martin John Rees (23 June 1942 -), Professor of Astronomy, University of Cambridge, England, nominated 25 June 1990; and Luis Angel Caffarelli (8 December 1948 -), Professor of Mathematics, Institute for Advanced Study, Princeton, USA, nominated 2 August 1994.

25. Turkish Women in Mathematics Association
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The first summer school for graduate students took place from 20 June to 1 July 2016 at the Middle East Technical University in Ankara.
• In addition the Association organised the conference 'Women and Mathematics: Algebras', on 11 March 2017 at the Istanbul Centre for Mathematical Sciences.

26. Japanese Association of Mathematical Sciences
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Japanese Association of Mathematical Sciences was founded by Professor Tatsujiro Shimizu in 1948.
• Let us give a few details about Tatsujiro Shimizu (1897-1992) modifying material taken from [',' H Sugiyama, A sketch of the life of Dr Shimizu, Math.
• 40 (1) (1994), 1-15.','2]:- .
• In 1924, Tatsujiro Shimizu graduated from the Department of Mathematics, School of Science, Tokyo Imperial University and continued his work at the same department as a member of staff.
• In 1932, he became professor of Osaka Imperial University and contributed to the establishment of the Department of Mathematics in the School of Science.
• Actually, he started the publication of the journal "Mathematica Japonicae" using his own funds in 1948.
• In 1949, he left Osaka University and became professor at Kobe University.
• In 1951, he moved again to professor of mathematical sciences at the University of Osaka Prefecture and continued the publication of Mathematica Japonicae.
• In 1961, he became professor at the Science University of Tokyo.
• As noted above, Tatsujiro Shimizu established Mathematica Japonicae in 1948 and it was set up to publish papers in both pure and applied mathematics.
• The Association also published nine issues over three volumes of the journal Scientiae Mathematicae in 1998, 1999, and 2000.
• In 2001 the two journals were merged to form the Scientiae Mathematica Japonicae.
• The numbering, however, was retained from Mathematica Japonicae so that the first volume of Mathematica Japonicae which appeared in 2001 was Volume 53.
• On a personal note, let me [EFR] add that I reviewed one of the papers in the first issue to appear in 2001.
• The Byelaws of the Society, amended in 2005, state that [',' International Society for Mathematical Sciences website.','1]:- .

27. Mexican Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It was officially founded on 30 June 1943 for a period of 50 years and, on 1 July 1993, this was extended by a further period of 50 years.
• The person who was very influential in the setting up of the Society, despite the fact that he died seven years before the Society was founded, was Sotero Prieto Rodriguez (1884-1935).
• The son of the mining engineer and professor of mathematics Raul Prieto Gonzalez Bango and Teresa Rodriguez de Prieto, Prieto attended high school in Mexico City from 1897 to 1901, and then studied civil engineering at the National School of Engineers from 1902 to 1906.
• He made a special study of higher mathematics and in 1932 founded the Mathematics Section of the National Academy of Sciences "Antonio Alzate", now the National Academy of Sciences of Mexico.
• Several of those who were part of this group of mathematicians became founders of the Mexican Mathematical Society, but the person who perhaps was the most influential of these was Alfonso Napoles Gandara (1897-1992).
• He was born in Cuernavaca, Morelos on 14 October 1897 and, from 1910, studied at the National Preparatory School.
• In 1916 he entered the National School of Engineers where he was taught by Sotero Prieto.
• He replaced Sotero Prieto as professor in that school in 1921.
• In 1930 he was awarded a Guggenheim Foundation Scholarship to study at the Massachusetts Institute of Technology where, between 1930 and 1932, he attended fourteen courses.
• He was part of Sotero Prieto's group of enthusiasts for higher mathematics and, after Sotero Prieto's death in 1935, he was the main figure to keep alive the idea of forming a group of mathematicians to promote the subject.
• The Bulletin of the Mexican Mathematical Society (Boletin de la Sociedad Matematica Mexicana) began publication in 1943 with one volume per year and two parts per volume.
• Alfonso Napoles Gandara was the editor of this first series which continued publication until volume 11 in 1954.
• The Bulletin was not published in 1955, then the second series began publication in 1956 with Jose Adem and Emilio Lluis Riera as editors.
• A third series of the Bulletin began publishing in 1995, with one volume consisting of two issues each year, and this series continues to be published.
• The Mexican Mathematical Society began publishing Miscelanea Matematica in 1972.
• Ana Meda Guardiola coordinates the present editorial board which consists of 13 Mexican mathematicians.
• Examples are the International Symposium of Algebraic Topology held in 1956 and the International Symposium of Differential Equations held in 1959.
• The Organizing Committee of the Mexican Mathematics Olympiad, established in 1987, looks for mathematically talented young Mexicans.
• The "Sotero Prieto Prize" was established in 1991.
• The following quote is from [',' The Mexican Mathematical Society website.','1]:- .

28. Bulgarian Mathematicians' Union
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The original Society was named the Sofia Physical and Mathematical Society when it was founded on 14 February 1898.
• The University of Sofia is the oldest establishment of higher education in Bulgaria and had been set up in Sofia in 1888.
• This followed the ending of Ottoman rule in Bulgaria in 1878, and Sofia being designated the Bulgarian capital in the following year.
• The aims of the original Society are given in [',' S Grozdev, Union of Bulgarian Mathematicians, European Mathematical Society Newsletter 31 (March, 1999), 20-21.','1].
• It recognised this by changing its name to the Bulgarian Physical and Mathematical Society in 1938.
• Grozdev writes in [',' S Grozdev, Union of Bulgarian Mathematicians, European Mathematical Society Newsletter 31 (March, 1999), 20-21.','1]:- .
• For example the student section in the town of Stara Zagora had 120 members in 1939, while the one in the region of Rajkovo had 200 members in 1940.
• On 17 October 1971 the formal split occurred, with the Society of Bulgarian Mathematicians serving the mathematicians and computer scientists, while the Society of Bulgarian Physicists was also created at this time.
• Alipi Mateev was the first President of the Society of Bulgarian Mathematicians from its separate creation in 1971.
• Grozdev writes [',' S Grozdev, Union of Bulgarian Mathematicians, European Mathematical Society Newsletter 31 (March, 1999), 20-21.','1]:- .
• In 1998, the year of its centenary, the Union of Bulgarian Mathematicians has about 5000 members - teachers in Mathematics and Computer science, University lecturers, scholars and specialists from all parts of the country.

29. Argentina Mathematical Union
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Argentina Mathematical Union (Union Matematica Argentina) was founded in 1936 for [',' Union Matematica Argentina website.','4]:- .
• They served from 1936 to 1938 with Alberto Gonzalez Dominguez and Yanny Prenkel as secretaries and Raquel Simonetti as treasurer.
• In the first issue, Volume 1 No 1 (1936), they set out the aims of the Argentina Mathematical Union [',' Union Matematica Argentina website.','4]:- .
• Information about all the early volumes is given in [',' Alphabetical index of authors (1936-1996), Rev.
• Argentina 43 (2002), 113-181.','1] and [',' Index of Volumes (1936-1996), Rev.
• Argentina 43 (2002), 49-112.','2].
• In 1945 the Revista became the Revista de la Union Matematica Argentina y de la Asociacion Fisica Argentina (Journal of the Argentina Mathematical Union and of the Argentina Physics Association).
• In 1968 it went back to its original title of Revista de la Union Matematica Argentina.
• For example Volume 41 has four parts, two in 1998, and one in each of 1999 and 2000.
• 1936-1938: .
• 1938-1940: .
• 1940-1941: .
• 1942-1943: .
• 1943-1944: .
• 1944-1945: .
• 1945-1947: .
• 1947-1949: .
• 1949-1951: .
• 1951-1953: .
• 1953-1955: .
• 1955-1957: .
• 1957-1963: .
• 1963-1965: .
• 1965-1967: .
• 1967-1968: .
• 1968-1970: .
• 1970-1972: .
• 1972-1974: .
• 1974-1976: .
• 1976-1978: .
• 1978-1980: .
• 1980-1982: .
• 1982-1984: .
• 1984-1986: .
• 1986-1989: .
• 1989-1991: .
• 1991-1993: .
• 1993-1995: .
• 1995-1997: .
• 1997-2001: .
• 2001-2005: .
• 2009-2011: .
• 2011-2013: .
• 2013-2015: .
• 2015-2017: .

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Georgian Academy of Sciences was founded on 10 February 1941 and states in the preamble to its statutes that it "is the successor to the Old Georgian Academies - Gelati and Iqalto." We should, therefore, say a few words about these very early academies but first let us give an account of the even earlier events which led to the founding of these two 12th century academies [',' T V Gamkrelidze, The Georgian National Academy of Sciences (GNAS) and its First President, Georgian National Academy of Sciences.','1]:- .
• King David the Builder began constructing the monastery and academy in 1106 as a grand tribute to his victory over the Turks.
• He received both his primary and higher education in Byzantium, at Mangana Monastery, which had been founded by the Byzantine emperor Constantine IX Monomachus (1042-1055).
• In 1114 King Davit the Restorer summoned Arsen back to Georgia, to the Gelati Academy in the west.
• Two major contributions to the founding of this Academy came from the founding of Tbilisi State University in February 1918 due to the efforts of Ivane Javakhishvili (1876-1940) and the setting up of the Georgian branch of the Academy of Sciences of the USSR in Tbilisi in the 1930s.
• For example the Institute of Mathematics and Mechanics of Tbilisi University was set up on the initiative of Nikoloz Muskhelishvili in 1933 and he was the driving force in the founding of the Mathematical Institute of the Georgian branch of the Academy of Sciences of the USSR in 1935.
• Nikoloz Muskhelishvili was one of the professors at Tbilisi State University who pressed for the setting up of the Georgian Academy of Sciences and, on 10 February 1941, the Georgian Government passed legislation setting up the Academy.
• Sixteen scholars became founding members of the Academy which held its first session on 26 February 1941 when Muskhelishvili was unanimously elected to serve as President.
• The President Muskhelishvili gave his inaugural address on the following day in which he [',' T V Gamkrelidze, The Georgian National Academy of Sciences (GNAS) and its First President, Georgian National Academy of Sciences.','1]:- .
• In August 1939 Russia and Germany made a secret pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them and, on 1 September 1939, Germany invaded Poland with Soviet troops attacking Poland from the east some days later.
• On 22 June 1941, however, Germany broke the non-aggression pact and invaded the Soviet Union.
• In fact the country was renamed the Republic of Georgia on 14 November 1990 before the dissolution of the Soviet Union on 9 April 1991.
• &#8217;&#8217;On the Georgian National Academy of Sciences&#8217;&#8217;, Georgian National Academy of Sciences.','3]:- .
• &#8217;&#8217;On the Georgian National Academy of Sciences&#8217;&#8217;, Georgian National Academy of Sciences.','3] the Academy details how it intends to achieve the above stated objectives.

31. Dutch Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Dutch Mathematical Society, the Wiskundig Genootschap, was founded in 1778.
• The Society he founded in 1778 was designed to provide financial support for his periodicals which did prove the commercial success that he had hoped.
• Rather its membership consisted of amateurs and [',' P P Bockstaele, Mathematics in the Netherlands from 1750 to 1830, Janus 65 (1-3) (1978), 67-95.','5]:- .
• (3) 26 (1) (1978), 177-205.','2]:- .
• The young Society was small and rather insignificant during the first years of its existence; it had 95 members by 1782.
• In [',' D J Beckers, &#8217;&#8217;Mathematics our goal!&#8217;&#8217; Dutch mathematical societies around 1800, Nieuw Arch.
• (4) 17 (3) (1999), 465-474.','3] Beckers points out that the gap between the social classes in the Netherlands was not as large as in other regions of Europe which helped the Society to make the necessary changes to provide a link between amateur and professional mathematicians.
• The Society published from shortly after it was founded with the Wiskundig Genootschap appearing in 1782.
• In 1856 the Archief first appeared and it 1875 it became the Nieuw Archief voor Wiskunde and was distributed to all members of the Society.
• Pieter Hendrik Schoute was an editor of this journal from 1898 until his death in 1923.
• He was also a founding editor of another Dutch Mathematical Society Journal Revue semestrielle des publications mathematique from 1893, when the journal was founded, again until his death in 1923.
• In more recent times L E J Brouwer, J A Schouten and T J Stieltjes have been important figures, and the Society published Stieltjes' Collected works in two volumes in 1914 and 1918, and Brouwer's Collected works in two volumes in 1975 and 1976.
• A major event for the Society was hosting the International Congress of Mathematicians in Amsterdam in 1954 at which Kunihiko Kodaira and Jean-Pierre Serre were awarded Fields Medals.
• By 1965 the monthly meetings of the Society were becoming less well attended despite the increased membership of the Society.
• They were discontinued and yearly Dutch Mathematical Congresses were organized [',' P P Bockstaele, Mathematics in the Netherlands from 1750 to 1830, Janus 65 (1-3) (1978), 67-95.','5]:- .
• The award of the medal was instituted in 1970 with the first award being made to Rene Thom.

32. Belgium Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• On 14 March 1921 a preliminary meeting took place which decided to set up a (according to their minute translated in [',' L Lemaire, A brief history of the Belgium Mathematical Society, European Mathematical Society Newsletter 29 (September, 1998), 24-25.','1]):- .
• In November 1921 the statutes for the Belgium Mathematical Circle were adopted and, in the following January, the 'Circle' decided to change their name to Belgium Mathematical Society.
• The statutes state that [',' L Lemaire, A brief history of the Belgium Mathematical Society, European Mathematical Society Newsletter 29 (September, 1998), 24-25.','1]:- .
• As Lemaire notes in [',' L Lemaire, A brief history of the Belgium Mathematical Society, European Mathematical Society Newsletter 29 (September, 1998), 24-25.','1], the Society was clearly not that certain that the Bulletin would continue publication since they did not number this first volume.
• Guy Hirsch was elected deputy secretary of the Society in 1947 and he ran it almost single-handed for many years.
• In 1977 the Bulletin split into two series, with Hirsch remaining the sole editor of one of the two series until 1993.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Mexican Academy of Sciences was founded on 12 August 1959, originally called the Mexican Academy of Scientific Research.
• The motivation behind the founding came from Alberto Sandoval Landazuri (1918-2002) and Guillermo Haro Barraza (1913-1988) so we will give brief details of these two men.
• He studied chemistry at the National Autonomous University of Mexico and in April 1941 became a research assistant at the new Institute of Chemistry where he wrote a Ph.D.
• After talking with the Mexican astronomer Luis Enrique Erro (1897-1955) in 1937 he became fascinated with astronomy and, despite having no formal qualifications in astronomy, later he was appointed by Erro as an assistant astronomer at the Observatorio Astrofisico de Tonantzintla.
• In 1947 he moved to the Observatorio de Tacubaya of the National Autonomous University of Mexico.
• He became the first Mexican elected to the Royal Astronomical Society in 1959.
• The Mexican Academy of Scientific Research was renamed the Mexican Academy of Sciences in 1996 [',' About the Mexican Academy of Sciences, Mexican Academy of Sciences.','1]:- .
• The Funny Mathematics Competition was founded in 1998 for students under the age of 12.
• The Spring Mathematics Competition was founded in 1996 and is aimed at two categories of Junior High School students: under thirteen and under fifteen.
• This programme was founded in 1995 with the help of the USA-Mexico Foundation for Science.
• The Scientific Research Summer, which started in 1991, aims at building college students' interest in scientific activity within the fields of physics, mathematics, biomedicine, chemistry, social studies, the humanities, engineering and technology.
• Since the Scientific Research Summer began in 1991 the Academy has ensured that it would be open and widely shared, permitting students from all areas of knowledge and every state in Mexico to participate.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Before we look at the founding of this Academy we should mention thenScientific and Technological Research Council of Turkey which was founded in 1963.
• I [EFR] would like personally to thank the Scientific and Technological Research Council of Turkey for funding my research visit to the University of Cukurova, Adana, Turkey, in 1992.
• The political situation in Turkey, with military coups in 1960, 1971 and 1980, meant that funding for science was difficult throughout this period.
• An important step in developing science research in Turkey was the Five Year Development Plan for the years 1979-1983 which laid the foundation for a much more significant step forward produced in the document "Turkish Science and Technology Policy: 1983-2003".
• 497 on 2 September 1993 [',' About us, Turkish Academy of Sciences.','1]:- .
• The Academy was founded with the following vision [',' About us, Turkish Academy of Sciences.','1]:- .
• The Statutes were set out to require the Academy to operate as follows [',' About us, Turkish Academy of Sciences.','1]:- .
• Changes to the Turkish Academy of Sciences were brought about in 2011.
• In August 2011 the method of election of members changed with one third being assigned by the Council of Ministers, one third assigned by the Scientific and Technological Research Council of Turkey, and one third elected by the Academy.
• He resigned from the Turkish Academy of Sciences in 2011 and gave his reasons in [',' M Ali Alpar, Bilim Akademisi - the New Science Academy in Turkey, American Physical Society.','2] from which we quote the first paragraph:- .
• In August 2011 the Turkish Government issued a decree to bring in government appointments to the Turkish Academy of Sciences.

35. Kharkov Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Kharkov Mathematical Society was founded in 1879 on the initiative of V G Imshenetskii who served as its first President.
• A charter set the aims of the Society [',' I V Ostrovskii, Kharkov Mathematical Society, European Mathematical Society Newsletter 34 (December, 1999), 26-27.','1]:- .
• Imshenetskii left Kharkov and went to St Petersburg where he succeeded in founding the St Petersburg Mathematical Society in 1890.
• However Aleksandr Mikhailovich Lyapunov, a former student of Chebyshev, moved from St Petersburg to Kharkov in 1885 and he remained there for 17 years.
• During this time he played a leading role in running the Society and presented 27 reports to the monthly meetings [',' I V Ostrovskii, Kharkov Mathematical Society, European Mathematical Society Newsletter 34 (December, 1999), 26-27.','1]:- .
• When Lyapunov left Kharkov in 1902, his former student Vladimir Andreevich Steklov became chairman of the Mathematical Society.
• The chairmanship of the Kharkov Mathematical Society was filled by Dmitrii M Sintsov who then held the position for forty years until 1946 [',' I V Ostrovskii, Kharkov Mathematical Society, European Mathematical Society Newsletter 34 (December, 1999), 26-27.','1]:- .
• The mathematical activity of the Society was strongly influenced by Bernstein during the years from 1906 to 1933 that he spent in Kharkov.
• Naum Akhiezer was encouraged by Bernstein to leave Kiev and join the Kharkov School of function theory in 1933.
• Akhiezer filled the position of Director of the Institute of Mathematics vacated by Bernstein in 1933 and after Sintsov's death in 1946 he also became Chairman of the Kharkov Mathematical Society.
• In 1950 the Institute of Mathematics in Kharkov was closed by the Government, but the Mathematical Society continued to survive, largely thanks to the support of the newly opened Institute for Low Temperature Physics and Engineering.

36. Royal Astronomical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Minutes taken that day record (see [',' H H Turner, The decade 1820-1830, in J L E Dreyer and H H Turner (eds.), History of the Royal Astronomical Society (London, 1923), 1-49.','3]):- .
• There are records of Pearson proposing an Astronomical Society in 1812, or earlier, and certainly Baily's recommendation that such a Society be formed appears in print in a 1819 article.
• There was considerable activity in the days following the initial meeting on 12 January, mainly concerning the address that Herschel was drawing up.
• Other aims of the Society which were listed in Herschel's address were [',' H H Turner, The decade 1820-1830, in J L E Dreyer and H H Turner (eds.), History of the Royal Astronomical Society (London, 1923), 1-49.','3]:- .
• In June 1831 the possibility of rooms in Somerset House in the Strand was suggested.
• Seven further associate members were proposed in the first year and elected in January 1821.
• Members of the Society read papers at the meetings and by 1821 the Memoirs of the Astronomical Society had been proposed which published these papers beginning in 1822.
• The Monthly Notices began publication in 1827 but prior to this there had been reports of meetings of the Philosophical Magazine.
• In May the President Sir James South petitioned the King and, on 15 December 1830 the King signed the book as Patron of the Society.
• A charter was eventually agreed and signed by the King in March 1831.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Krzemie'nska writes in [',' B Krzemie&#8217;nska, The founding of the Academy of Sciences and Arts in St Petersburg in 1724-1725 (some remarks on the 250th anniversary of the Academy of Sciences in the USSR) (Czech), DVT - Dejiny Ved.
• a Techniky 7 (1974), 131-147.','7]:- .
• In [',' M D Gordin, The importation of being earnest : the early St Petersburg Academy of Sciences, Isis 91 (1) (2000), 1-31.','3] Gordin:- .
• The name varied over the years, becoming The Imperial Academy of Sciences and Arts 1747-1803), The Imperial Academy of Sciences (1803- 1836), and finally, The Imperial Saint Petersburg Academy of Sciences (from 1836 and until the end of the empire in 1917).
• Following the Revolution in 1917 it was renamed the Russian Academy of Sciences.
• In 1934 it moved from Leningrad (which is what St Petersburg had been renamed) to Moscow.
• In 1991 its name of the Russian Academy of Sciences was reinstated.

38. Kaiser Wilhelm Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Kaiser Wilhelm Society was founded on 11 January 1911 and renamed the Max Planck Society for the Advancement of Science in 1948.
• Adolf Harnack (1851-1930), a German Lutheran theologian and historian, was the twin brother of the mathematician Axel Harnack.
• In 1909 Adolf Harnack made a proposal to Kaiser Wilhelm II to reform German science by setting up independent research institutes to complement the work done in the universities.
• In 1910 the University of Berlin celebrated its 100th anniversary and, on that day, Kaiser Wilhelm II announced the creation of the new Society.
• On 11 January 1911, 83 members of the new Society attended the meeting to mark its founding in the Berlin Academy of Arts.
• The Kaiser Wilhelm Institute of Physics in Berlin had been proposed in 1914 but was not officially founded until 1 October 1917 with Albert Einstein becoming its director.
• Max Planck had become a committee member of the Kaiser Wilhelm Society in 1916 and, as secretary of the Society, succeeded in persuading Einstein to accept the role of director.
• World War I ended with the Armistice on 11 November 1918.
• Kaiser Wilhelm II abdicated, but the Kaiser Wilhelm Society decided in 1919, however, not to change the Society's name.
• At this stage even the statute with Kaiser Wilhelm II as patron remained in place but his name as patron was removed from the statutes in 1921.
• The Society continued to thrive and in 1922 moved into new headquarters, the Berlin Palace, which, with the end of the monarchy, had to find a new role.
• By 1927, however, the Society was extending its reach outside Berlin with the holding of its annual general meeting in Dresden.
• In 1930 Max Planck was elected as President of the Society.
• He had received the Nobel Prize for Physics in 1918 and, by the time he was elected President, there were seven Nobel Prize winners who were members of the Society.
• The impact of this Law on the Kaiser Wilhelm Society was clear to the President, Max Planck, so he met with Adolf Hitler in an attempt to have him reverse the policy [',' J C O&#8217;Flaherty, Max Planck and Adolf Hitler, American Association of University Professors Bulletin 42 (3) (1956), 437-444.','3]:- .
• Hitler, as one might expect, totally rejected Planck's plea to allow top Jewish scientists to continue their work [',' J C O&#8217;Flaherty, Max Planck and Adolf Hitler, American Association of University Professors Bulletin 42 (3) (1956), 437-444.','3]:- .
• Planck's unsuccessful meeting with Hitler was followed by dismissals [',' History of the Kaiser Wilhelm Society, Max Planck Society.','1]:- .
• The Kaiser Wilhelm Society dismissed a total of 126 staff members, 104 of them scientists.
• Haber resigned before being sacked and died in 1934.
• The Kaiser Wilhelm Society was one of the organisations behind a commemoration of Haber which was arranged in 1935.
• By 1941 the President of the Society was appointed by the Nazi government.
• As Germany faced defeat in 1945 the President of the Kaiser Wilhelm Society, Albert Vogler, a Nazi appointment and sympathiser, committed suicide.

39. Turkish Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Turkish Mathematical Society was founded In 1948.
• On its website the Society indicates the activities that it undertakes [',' Turkish Mathematical Society website.','1]: .
• In 1960 the Society became a full member of the International Mathematical Union.
• The magazine has been published since 1991 and, to illustrate the contents, we note that the first issue in 1991 contains articles on the following: Plane Geometric Angles and Measures; Four Colour Problems; Drawings that cannot be made with ruler and compass; The Extraordinary Features of Infinite Cardinals; 1, 2, 3, Endless Or Rapid Disaster!; Mathematics Teaching in the World.

40. New Zealand Royal Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Royal Society of New Zealand was formally founded in 1867 when, on 16 August, William Travers introduced a private members bill into Parliament named the New Zealand Institute Act.
• The first European settlers arrived in New Zealand in 1840.
• The New Zealand Society was founded in Wellington by the governor George Grey in 1851.
• Similar organisations were founded in Canterbury in 1862 and Auckland in 1867.
• The person considered as the founding father of the Royal Society of New Zealand is the Scottish geologist, naturalist and surgeon James Hector (1834-1907) who arrived in New Zealand in 1862 to conduct a geological survey of Otago province [',' Our History, Royal Society of New Zealand.','2]:- .
• In 1865, he is appointed to the Geological Survey in Wellington for three years.
• Hector was elected a fellow of the Royal Society of London in 1865.
• In the following year the geologist and biologist Frederick Wollaston Hutton (1836-1905) arrived in New Zealand.
• The bill introduced by William Travers on 16 August 1867 to set up the New Zealand Institute did not pass easily through parliament since money was tight and many argued that the country could not afford to fund such a body.
• The first volume was published in 1869 with the Preface, written by James Hector, dated 5 May 1869.
• The Governor was Sir George Ferguson Bowen (1821-1899) an author and professional administrator.
• George Malcolm Thomson (1848-1933), a naturalist, teacher and politician, led the moves for reform [',' E Y Speirs, Thomson, George Malcolm, Dictionary of New Zealand Biography (1933).','4]:- .
• Disillusionment with the dominance of the New Zealand Institute by James Hector, its manager and the director of the Geological Survey and Colonial Museum, had led Thomson to launch a quarterly, the 'New Zealand Journal of Science', in 1882.
• If we look at Volume 25 of the Transactions and Proceedings of the New Zealand Institute published in 1892 there are 30 papers in Section I: Zoology, 15 papers in Section II: Botany, 7 papers in Section III: Geology, 2 papers in Section IV: Chemistry, and 22 papers in Section V: Miscellaneous.
• The second of these papers, perhaps the most of any papers in the volume to be related to physics, was Analogy between light and sound: Are they Convertible? read at a meeting of the Otago Institute on 11 October 1892 by Miss Annette Wilson [',' Transactions and Proceedings of the New Zealand Institute Volume 25.','6]:- .
• The lack of mathematics or physics papers changed in 1894 when Ernest Rutherford published his first paper in the Transactions and Proceedings of the New Zealand Institute journal, volume 27, entitled The Magnetization of Iron by High Frequency Discharges.
• In 1903 the reforms for which Thomson argued were formalised by an Act of Parliament [',' Our History, Royal Society of New Zealand.','2]:- .
• The eventual new Act of Parliament in 1903 makes special mention of this desire to link the Institute's governance more closely with affiliated societies.
• James Allan Thomson, George M Thomson's son, compiled a report in 1919 on the state of science in New Zealand and in it he drew attention to the problems of the New Zealand Institute, particularly financial problems arising from the fact that the government grant of £500 per year, originally made in 1867, had never been increased.
• In 1930, the President of the New Zealand Institute proposed that its name be changed to the Royal New Zealand Institute.
• This never happened but in 1933 the New Zealand Institute changed its name to the Royal Society of New Zealand.
• For example the New Zealand Science and Technology Gold Medal, renamed the Rutherford Medal in 2000, was awarded to Vaughan Jones in 1991.
• Awarded since 1912, the award was won by Rod Downey in 2011 for his influential and innovative work in mathematical logic:- .
• The 2011 Hector Medal for an outstanding contribution to the advancement of mathematical and information sciences: awarded too Rodney Graham Downey for his outstanding, internationally acclaimed work in recursion theory, computational complexity, and other aspects of mathematical logic and combinatorics.
• This work attracted the attention and involvement of several leading complexity theorists worldwide, and culminated in the publication of a large monograph in 1999.
• I have written two joint papers with Marston, one in 1992 and one in 1994.

41. Portuguese Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• A new constitution was put in place in 1933 which set up the National Assembly, all seats in which were filled by government supporters.
• ','1]:- .
• As part of its aims to promote mathematics, Mathematics Clubs were set up in schools, the first in 1942.
• After this B Caraca took over, followed by A Ferreire de Macedo in 1945 and M Zaluar Nunes in 1947.
• ','1]:- .
• The political situation changed in 1974.
• The Portuguese Mathematical Society became a legal entity in 1977 and once again could begin to function in the way it wished to promote the mathematical sciences at all levels throughout Portugal.
• The Society also began the work of establishing Olympiad competitions in 1980, the first nations such competitions being three years later.
• The first Portuguese team participated in the International Olympiad competition for the first time in 1989.

42. Danish Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Danish Mathematical Society, named Dansk Matematisk Forening, was founded in 1873.
• The original aim of the Society, as drawn up by the Board, was to (see for example [',' F D Pedersen (ed.), Dansk Matematisk Forening, 1923-1973 (Copenhagen, 1973).','1]):- .
• All the three Board members were professors at the University of Copenhagen at the time the Society was created: Zeuthen had appointed as an extraordinary professor of mathematics in 1871; Petersen held the chair of mathematics; and Thiele was professor of astronomy.
• The first foreign speaker to the Society was Mittag-Leffler in 1900.
• The number of foreign speakers then grew and from 1921 an invitation was given every second year to a leading mathematician to give a series of lectures.
• The first series was given in 1921 by Hilbert.
• In this same year of 1921 the Society discussed whether it should join the International Mathematical Union, but there was strong opposition since the Union was actively discriminating against mathematicians from countries on the losing side in World War I.
• Matematisk Tidsskrift was founded in 1859 and Zeuthen became an editor in 1871.
• Shortly after he ended his editorial duties, the journal split into two parts in 1890.
• The Mathematical Society took over the publication of these journals in 1919.
• This Union came about in September 1951.
• The first meeting of the editorial board was in May and the journal was first published in 1953.
• Although negotiations for the joint publication of Nordisk Matematisk Tidskrift took a little longer to finalise, it first appeared in 1953.
• The Danish Mathematical Society was established in 1952 and the statutes changed from those of the Mathematical Society.
• The new statutes are given in full in [',' F D Pedersen (ed.), Dansk Matematisk Forening, 1923-1973 (Copenhagen, 1973).','1].
• In [',' B Branner, Danish Mathematical Society : Dansk Matematisk Forening, European Mathematical Society Newsletter 35 (March, 2000), 14-15.','3] Branner writes of the changes to the activities of the Society in the years following 1954 when more mathematical seminars were organised by individual Danish university mathematics departments, and fewer by the Danish Mathematical Society:- .

43. Swiss Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The three mathematicians most involved in the founding of the Society were: Rudolf Fueter, who had been appointed as professor of mathematics at the University of Basel in 1908; H Fehr, who was at the University of Geneva; and Marcel Grossmann, who had been appointed as professor of descriptive geometry at the Eidgenossische Technische Hochschule in Zurich in 1907.
• This project had been suggested at the time of the International Congress of Mathematicians which took place in Zurich in 1897.
• The proposal had been made in 1883 (on the centenary of Euler's death) by Ferdinand Rudio, who was at that time a lecturer in mathematics at the Eidgenossisches Polytechnikum (later called the Eidgenossische Technische Hochschule) in Zurich, and repeated by him at the Congress fourteen years later.
• Karl Geiser, from Zurich, was the President of the 1897 Zurich Congress and in 1907 he joined Rudio and others in approaching the Swiss Academy of Natural Sciences with the proposal to publish Euler's complete works.
• This, and other factors, led to the foundation of the Society in 1910 to bring pure mathematics into the foreground and to support its national and international promotion.
• Fueter was appointed as the first editor but the its aim was not at first to be international, rather (see [',' U Stammbach, Swiss Mathematical Society, European Mathematical Society Newsletter 33 (September, 1999), 18-20.','1]):- .
• The Swiss Mathematical Society played a major role in organising the International Congress of Mathematicians which took place in Zurich in 1932.
• In 1930 the Society set up a Committee to look after the archive of material left by Steiner and in 1937 the same committee was also given the responsibility for the archive left by Schlafli.
• The Committee was given the task of (see [',' U Stammbach, Swiss Mathematical Society, European Mathematical Society Newsletter 33 (September, 1999), 18-20.','1]):- .
• In 1975 the Swiss Mathematical Society took over responsibility for publishing Elemente der Mathematik which had existed since 1946.
• The Society also went on the organise a third International Congress of Mathematicians which took place in Zurich in 1994.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Heidelberg Academy of Sciences and Humanities was founded in 1909.
• The Kurpfaelzische Akademie is much older, dating back to 1763 when it was founded in Mannheim by Charles Theodore (1724-1799).
• Later, in 1777, he was also prince-elector and Duke of Bavaria.
• This new role led to him moving the Kurpfaelzische Akademie to Munich in 1778.
• This Academy was closed in 1803 as a result of the Napoleonic wars.
• Karl Wilhelm Konstantin Philipp Lanz (1873-1921) was an engineer and industrialist who was well known for his technical innovations.
• He took over his father's business in 1905 and donated large sums to various projects, for example for aviation projects to support German experiments.
• The Heidelberg Academy of Sciences and Humanities was founded with the patronage of Friedrich II (1857-1928), the Grand Duke of Baden.
• Friedrich II had become Grand Duke of Baden in 1907 following the death of his father Friedrich I and ruled until 1918 when the German monarchies were abolished.
• He was also awarded an honorary degree by the Ruprecht-Karls-Universitat Heidelberg in 1909.
• In 1918 the German monarchies were abolished and the residence of the Grand Duke of Baden on Karlsplatz (a spacious square named after the Grand Duke Karl Friedrich) in Heidelberg became vacant.
• In 1920 the Heidelberg Academy of Sciences and Humanities moved into the former residence of the Grand Duke of Baden on the Karlsplatz.
• The building had been constructed in 1710 as a city palace for the Privy Councillor and chief clerk Carl Philipp Freiherr von Hundheim.
• It is situated immediately below the castle and, after the castle was struck by lightning in 1767 and become uninhabitable, the building served as a residence for the Palatine Elector and later for the Grand Duke of Baden.
• The Academy website states [',' The Heidelberg Academy of Sciences and Humanities website.','1]:- .
• The organisation of the Academy is described on its website [',' The Heidelberg Academy of Sciences and Humanities website.','1]:- .
• Mathematical-Scientific Section; full member elected 1953; corresponding member since 1958; full member since 1961.
• Mathematical-Scientific Section; full member elected 1941; deleted from the list of members 1947? .

45. Zurich Scientific Research Society
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• It was founded in 1746 as the Physical Society (Physicalische Societat).
• Johannes Gessner (1709-1790) showed a particular liking for mathematics and botany from his school days in Zurich.
• At the age of seventeen he went to the University of Leyden where he studied mathematics under Willem 'sGravesande and other science subjects with the anatomist Bernhard Siegfried Albinus (1697-1770) and the botanist Herman Boerhaave (1668-1738).
• After visiting Paris in 1727 he went to Basel in the following year where he studied mathematics with Johann Bernoulli and medicine with other scholars.
• When he founded the Physical Society in Zurich in 1746 he brought together many scholars living in Zurich with wide ranging scientific interests.
• The Physical Society soon built up a scientific library, a collection of mathematical and physical instruments, natural history collections, a botanical garden, and an astronomical observatory built in 1759.
• The Society collected weather data from the time of its founding in 1746 and when the Meteorological Central Institute was founded in 1880, all the Society's weather data was given to them [',' Grundung und Zweck, Naturforschende Gesellschaft in Zurich.','1]:- .
• The Eidgenossische Polytechnikum (opened in 1855) (from 1911 Eidgenossische Technische Hochschule ETH) benefited, among other things, from the observatory, which had already been established in 1759.
• The scientific lectures were published in three volumes of the "Essays" which appeared between 1761 and 1766.
• The Society published the journal the Bericht uber die Verhandlungen der Naturforschende Gesellschaft in Zurich (1826-1837) which later was known as the Mitteilungen der Naturforschende Gesellschaft in Zurich (1847-1856).
• In 1856 it changed its name to the Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich and it published four times a year.
• Rudolf Wolf had become a member of the Society in 1839 and joined the committee in 1856.
• By 1859 the Vierteljahrsschrift was still edited by Rudolf Wolf, but with the designation Professor of Astronomy in Zurich.
• By 1861 Rudolf Wolf has again become Professor of Mathematics in Zurich, while in 1862 he is again Professor of Astronomy in Zurich.
• He continued as editor until his death in 1893 and he was still the editor of the 1893 volume.
• The next editor of the Vierteljahrsschrift was the mathematician Ferdinand Rudio and the first volume he edited in 1894 began with a 64-page obituary of Rudolf Wolf.
• Rudio continued as editor until 1911.
• The 1875 volume of the Vierteljahrsschrift contains several papers by Wilhelm Fiedler who had become a member of the Society in 1867 and joined the committee in 1871.
• This volume contains a list of members in 1875 and, in addition to Wilhelm Fiedler, these include the mathematicians: Jacob Amsler, joined 1851; Elwin Christoffel, joined 1862; Theodor Reye, joined 1863; Hermann Schwarz, joined 1869, became a committee member in 1871 and Vice-President in 1874; Heinrich Weber, joined 1870 and became a committee member in 1872; Heinrich Suter, joined 1871; and Rudolf Clausius, joined 1869.
• The annual publication, the Neujahrsblatt der Naturforschenden Gesellschaft in Zurich, was established in 1799 with a first volume giving details of the purpose of the Society, described the Society's collections and promoted the benefits of the natural sciences.
• Switzerland was invaded by the French in 1798 who took over the country and renamed it the Helvetic Republic.
• The second volume of the Neujahrsblatt der Naturforschenden Gesellschaft in Zurich, published in 1800, was entitled "The devastation of the country by the warlike events of the year 1799." .
• The Naturforschende Gesellschaft in Zurich is one of the oldest scientific associations in Switzerland, celebrating its 250th anniversary in 1996.

46. Nepal Mathematical Society
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• Tribhuvan University in Kirtipur, Kathmandu, Nepal is the oldest and largest university in Nepal; it was established in 1959.
• The first group of students was small, about six in total, and they graduated with Master's degrees in 1961.
• The Department published a half-yearly report The Nepali Mathematical Sciences Report containing research papers, research notes, expository articles and survey articles beginning in 1975.
• After 15 years the dream of the first group of enthusiasts at Tribhuvan University was realised and the Nepal Mathematical Society was legally founded on 19 January 1979.
• Beginning in 2004, the Society has organised its anniversary NMS-Day each year on 13 or 14 May with a formal programme followed by a "One day mathematics seminar".

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• The Romanian Academy was founded on 13 April 1866 under the name Societatea Literară Romană (Romanian Literary Society).
• It changed its name to Societatea Academică Romină (Romanian Academic Society) in 1867, and finally to Academia Romană (Romanian Academy) in 1879.
• Modern Romania came into existence in 1859 when the principalities of Wallachia and Moldavia were united under Prince Alexandru Ioan Cuza who was the ruling prince of both Principalities.
• He was overthrown in 1866 and replaced by Carol I, king of Romania from 1866 to 1881.
• Both Wallachia and Moldavia had schools of higher learning, one in Bucharest founded around 1689 became the University of Bucharest and one founded in Iasi in 1707 eventually became the University of Iasi.
• These schools did not operate as academies, but there were societies founded which operated as academies in Brasov (1821), Bucharest (1844), Sibiu (1861), and Cernauti (1862).
• Two people who played an important role in creating this national academy were Constantin Alexandru Rosetti (1816-1885) and Ion Heliade Rădulescu (1802-1872) so let us give a little information about these two men.
• He served at Minister of Education and Religion under Carol I, leader of united Romania from 1866 to 1881, and it was in his capacity as Minister of Education that he was the leading founder of the Romanian Academy.
• He was also a leading figure in the 1848 revolution and elected to the Chamber of Deputies in 1866.
• A founder member of the Romanian Academy, he became its first President serving from 1867 to 1870.
• The Society founded in 1866 had 21 founding members who were not all from Romania but came from a much wider area.
• Although the Society was founded in 1866 as a literary society, it soon broadened its areas to include the sciences.
• The statutes of 1867 established three sections: (i) Philology and Literature (which included the visual arts), (ii) History and Archaeology, and (iii) Natural Sciences.
• The first president to be trained as a mathematician was Ion Ghica (1816-1897) who became the fourth President of the Romanian Academic Society when he took on that role in 1876.
• He was born in Bucharest and studied mathematics in Paris from 1837 to 1840.
• After the union of Wallachia and Moldavia in 1859 he became the first prime minister of Romania.
• He served four times as President of the Romanian Academy, serving in 1876-1882, 1884-1887, 1890-1893, and 1894-1895.
• The President of the Academy, Ionel-Valentin Vlad, said [',' Romanian Academy hosts solemn session to celebrate the December 1, 1918 Greater Union, nineoclock (28 November 2016).','6]:- .
• During World War II, Romania was forced to cede territory to the Soviet Union in 1940, then in 1941 joined Germany in an attempt to recover that territory.
• In 1944 the Red Army entered Romania which changed sides and declared war on Germany.
• During this period an Institute with the name "The Institute of Mathematical Sciences" was founded on 22 December 1945 and it was formally recognised in 1946.
• In 1947 king Michael I was forced to abdicate by the Communist regime and Romania was declared a People's Republic.
• The much changed Academy set up many research institutes, one of the first being the "Institute of Mathematics of the Academy of the People's Republic of Romania" in 1949.
• This Institute was based on "The Institute of Mathematical Sciences" which had been established in 1945.
• Pompeiu, however, was unwell so although he remained as director until his death in 1954, the Institute was run from the beginning by Stoilow.
• After Pompeiu died in 1954, Stoilow became the director, holding that position until 1961.
• In the first two decades of the communist regime, the Academy and its scientific network grew considerably, from 7 research facilities with nearly 400 scientific collaborators in 1948 to 56 institutes or centres with about 2,500 employees in 1966.
• Things changed however, beginning in 1969.
• Further institutes were lost in 1970 and in 1974 the Romanian Academy had its byelaws changed so that it was under the control of the National Council for Science and Technology.
• In fact in 1973 the Institute of Mathematics was taken away from the Academy and became part of the Ministry of Education.
• Ceauescu had run the State Council since 1967 but when the position of President of Romania was created in 1974, Ceauescu assumed that role and was then able to rule by decree.
• A Revolution in 1989 saw Ceaușescu executed on 25 December 1989 and Ion Iliescu became President.
• For example, the Institute of Mathematics was refounded in 1990 with Gheorghe Gussi as its director.
• The Academy's description of its current position is given in [',' Academy today, Romanian Academy.','1] which begins as follows:- .
• The President of the Section of Mathematical Sciences of the Romanian Academy is Romulus Cristescu (born Ploiesti, Romania, 1928) who was elected to the Academy in 1990.
• The President of the Institute of Mathematics is Viorel Barbu (born Deleni, Vaslui, Romania, 1941) who has been a full member of the Academy since 1991.
• He was Vice-President of the Romanian Academy from 1998 to 2002 and President of the Iasi Branch of the Romanian Academy since 2001.

48. Italian Society of Applied and Industrial Mathematics
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• The Italian Society of Applied and Industrial Mathematics, in Italian, Societa italiana di matematica applicata e industriale (SIMAI), was founded by a deed dated 20 December 1988 although its date of foundation is usually given as 1989.
• It was set up in the Istituto per le Applicazioni del Calcolo "Mauro Picone" in the Italian National Research Council in Rome, which had been founded in 1923 with Volterra as a president.
• 1996: III Congresso Nazionale della Societa Italiana di Matematica Applicata e Industriale, Salice Terme, PV, 27-31 May.
• 2002: VI Congresso Nazionale della Societa Italiana di Matematica Applicata e Industriale, Grand Hotel Chia Laguna, 27-31 May.
• 2008: IX Congresso Nazionale della Societa Italiana di Matematica Applicata e Industriale, Rome, 15-19 September.
• 2014: XIII Congresso Nazionale della Societa Italiana di Matematica Applicata e Industriale, Hotel Villa Diodoro, Taormina, 7-10 July.
• 2016: XIV Congresso Nazionale della Societa Italiana di Matematica Applicata e Industriale, Politecnico di Milano, Milan, 13-16 September.
• The website of the journal states [',' Communications in Applied and Industrial Mathematics, Walter de Gruyter (2018).','1]:- .
• The following information about the 'SEMA-SIMAI Book Series' appears on the Springer page [',' SEMA-SIMAI Springer Series, Springer International Publishing (2018).','3]:- .

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• The Serbian Academy of Sciences was founded on 1 November 1886 as the Serbian Royal Academy.
• However this was established following the suspension of an earlier Society and we should go back to the Society of Serbian Letters founded in 1842 to understand the events which led to the foundation of the Academy.
• In 1830 the Ottoman government granted Serbia full autonomy but internal dissension followed.
• He was forced to abdicate in 1839 but gangs of bandits and lawlessness continued to create widespread problems.
• It was in these difficult circumstances that Jovan Sterija Popovic and Atanasije Nikolic began in September 1841 to work towards founding the Society of Serbian Letters which formally existed from 31 May 1842.
• In 1842 the national assembly elected Alexander as ruler.
• In 1844 a series of laws were introduced concerning the administration and the education system in Serbia and the Society was able to recommence its work in August 1844.
• Alexander was deposed in 1859 and Milos Obrenovic was brought back but he died in 1860 and was succeeded by his son Mihailo Obrenovic.
• In 1863 the Society of Serbian Letters wanted to make Garibaldi a member and the resulting dispute between the Society and the Minister of Education led to Mihailo Obrenovic suspending it on 27 January 1864.
• The Serbian Learned Society, like the previous Society, became involved in a dispute with the Minister of Education and was suspended on 13 May 1886.
• The Serbian Royal Academy was created by a law passed on 1 November 1886 and the Academy was given the library, collections, and property of the Serbian Learned Society.
• The Minister of Education saw that the only solution was to merge the two, which he did in 1892.
• In 1954 new laws concerning scientific activity altered the way the Academy operated.

50. Indian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• On 16 May 1907, V Ramaswami Aiyar wrote to the Mathematical Gazette as follows [',' An Indian Mathematical Society, The Mathematical Gazette 4 (65) (1907), 106-107.','1]:- .
• The annual subscription is 25 rupees (63;1.
• A full account of V Ramaswami Aiyar founding of the Society is given in his presidential address of 1926, extracts of which are given at THIS LINK.
• B Hanumantha Rao was the first President of the Society, serving in that role from 1907 to 1912 and seeing it through its initial stages.
• In 1910, when the revised rules and constitution were adopted, the Society acquired its present name, the 'Indian Mathematical Society'.
• The first conference of the Society was held at Madras in 1916.
• The second conference was held at Bombay in 1919, the third in Lahore in 1921 and the fourth in Pune in 1924.
• From that time on, a conference was held every two or three years until 1951 when it was decided to hold the conferences annually.
• The Twenty Fifth conference of the Society, which was held at Allahabad in 1951, was inaugurated by Pandit Jawaharlal Nehru, the first Prime minister of India.
• The P L Bhatnagar Memorial Award Lecture (instituted in 1987).
• The Srinivasa Ramanujan Memorial Award Lecture (instituted in 1990).
• The V Ramaswamy Aiyar Memorial Award Lecture (instituted in 1990).
• The Hansaraj Gupta Memorial Award Lecture (instituted in 1990).
• The Ganesh Prasad Memorial Award Lecture (instituted in 1993 and delivered every alternate year).
• At these celebrations the Society decided to publish a second journal, namely 'The Mathematics Student', which first appeared in 1933.
• Ramanujan met with Ramaswami Aiyar at his office in 1910, and brought one of his famous notebooks to show Ramaswami Aiyar.
• Ramanujan published a paper consisting of questions in the 1911 volume of the Journal of the Indian Mathematical Society as well as a fifteen page paper entitled "Some properties of Bernoulli Numbers".
• 1907-1912 B Hanumantha Rao .
• 1912-1915 R N Apte .
• 1915-1915 E W Middlemast .
• 1915-1917 R Ramachandra Rao .
• 1917-1921 A C L Wilkinson .
• 1921-1926 H Balakram .
• 1926-1930 V Ramaswamy Aiyar .
• 1930-1932 M T Naraniengar .
• 1932-1934 P V Sheshu Iyer .
• 1934-1936 H G Gharpure .
• 1936-1940 R P Paranjape .
• 1940-1942 R Vaidyanathswam .
• 1942-1947 F W Levy .
• 1947-1949 M R Siddiqui .
• 1949-1951 A Narasinga Rao .
• 1951-1953 T Vijayraghavan .
• 1953-1957 Ram Behar .
• 1957-1959 V Ganapathy Iyer .
• 1959-1960 B S Madhav Rao .
• 1960-1961 B N Prasad .
• 1961-1962 B S Madhav Rao .
• 1962-1963 C N Srinivasienger .
• 1963-1964 Hansaraj Gupta .
• 1964-1966 P L Bhatanagar .
• 1966-1968 R S Verma .
• 1968-1969 P L Bhatanagar .
• 1969-1970 R P Bambah .
• 1970-1971 M Venkatraman .
• 1971-1973 J N Kapoor .
• 1973-1974 K G Ramanathan .
• 1974-1975 V Krishnamurthy .
• 1975-1977 P C Vaidya .
• 1977-1979 U N Singh .
• 1979-1981 K Venkatchelienger .
• 1981-1982 V V Naralikar .
• 1982-1984 R S Mishra .
• 1984-1985 R P Agarwal .
• 1985-1986 S D Chopra .
• 1986-1987 H C Khare .
• 1987-1988 V Singh .
• 1988-1989 M K Singal .
• 1989-1990 M P Singh .
• 1990-1991 V M Shah .
• 1991-1992 D K Sinha .
• 1992-1993 V Kannan .
• 1993-1994 U P Singh .
• 1994-1995 H P Dixit .
• 1995-1996 N K Thakare .
• 1996-1997 S Bhargava .
• 1997-1998 A R Singal .
• 1998-1999 B K Lahiri .
• 2000-2001 Satya Deo .
• 2001-2002 P V Arunachalam .
• 2009-2010 Peeyush Chandra .
• 2010-2011 R.
• 2011-2012 P K Banerji .
• 2012-2013 Huzoor H Khan .
• 2013-2014 Geetha S Rao .
• 2014-2015 S G Dani .
• 2015-2016 A M Mathai .

51. New Zealand Mathematical Society
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• The Royal Society of New Zealand began its existence as the New Zealand Institute which was set up by an act of parliament in 1867 for the study of science, art, philosophy and literature.
• The name was changed to Royal Society of New Zealand in 1933 with royal assent and an Act of Parliament, and in 1965 the Society became more science oriented, adding Social Sciences and Technology.
• This was discussed at the first two New Zealand Mathematics Colloquia in 1966 and 1967 when it was decided to approach the Royal Society of New Zealand to set up the necessary committee.
• The first meeting of the committee took place in Wellington on 15 December 1967 when Simon Bernau was elected as chairman, and a constitution was agreed.
• C J Seelye, Head of the Mathematics Department at Victoria, wrote to the Royal Society of New Zealand in a letter dated 4 September 1972 passing on a recommendation from a staff meeting [',' M Carter, The National Committee for Mathematics (1967-1996), Newsletter of the New Zealand Mathematical Society 88 (August 2003).','1]:- .
• The Royal Society of New Zealand forwarded Seelye's letter to Cecil Segedin, acting chairman of the National Committee for Mathematics, who asked David Vere-Jones to present a detailed proposal to the Committee for a New Zealand Mathematics Society together with a draft constitution [',' M Carter, The National Committee for Mathematics (1967-1996), Newsletter of the New Zealand Mathematical Society 88 (August 2003).','1]:- .
• Kevin Broughan takes up the story at this point, with the National Committee for Mathematics report being passed to the Colloquium Business Meeting held at the University of Waikato in 1973.
• A working party consisting of David Vere-Jones, Donald Joyce and Kevin Broughan was elected to make a specific proposal which was considered and approved at the next meeting in 1974.
• He was elected an Honorary Life Member at the first AGM in 1975.
• At the first meeting in 1974, the new Society did elect two Honorary Life Members, Henry George Forder and Jim Campbell.
• The publishing situation was somewhat complicated by the fact that the Mathematical Chronicle had been founded in 1969 by some members of the University of Auckland Department of Mathematics and was already playing a major role in publishing mathematical research papers in New Zealand.
• As mentioned above, the Mathematical Chronicle had been published from 1969 onwards but in 1992, the University of Auckland Department of Mathematics and the New Zealand Mathematics Society came to an agreement to jointly publish the New Zealand Mathematical Journal.
• There is the Research Award, first awarded in 1990, the Early Career Research Award first awarded in 2006, the Kalman Prize for Best Paper first awarded in 2016, and the Aitken Student Prize first awarded in 1995.
• 1974-1975 David Vere-Jones .
• 1975-1976 John C Butcher .
• 1976-1978 Gordon M Petersen .
• 1978-1979 Graeme C Wake .
• 1979-1980 John C Turner .
• 1980-1981 W Dean Halford .
• 1981-1982 David B Gauld .
• 1982-1983 Jim H Ansell .
• 1983-1984 William Davidson .
• 1984-1985 Michael R Carter .
• 1985-1987 Ivan L Reilly .
• 1987-1989 Brian Woods .
• 1989-1991 Gillian Thornley .
• 1991-1993 Derek Holton .
• 1993-1995 Marston D E Conder .
• 1995-1997 Douglas Bridges .
• 1997-1999 Rob I Goldblatt .
• 1999-2001 Graeme Wake .
• 2001-2003 Rod Downey .
• 2009-2011 Charles Semple .
• 2011-2013 Graham Weir .
• 2013-2015 Winston Sweatman .
• 2015-2017 Astrid an Huef .
• 2017-2019 Vivien Kirk .

52. Max Planck Society for Advancement of Science
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• The Kaiser Wilhelm Society was founded on 11 January 1911 and renamed the Max Planck Society for the Advancement of Science in 1948.
• The Max Planck Society quickly gained prestige, particularly when the physicist and Society member Walther Bothe won the Nobel Prize for Physics in 1954.
• The statutes of the Max Planck Society were revised in 1964 and at that time the following sentence was added:- .
• This came about in 1992, although the administrative headquarters which has been in Munich remained in that city.
• In 1997, Max Planck Society President Hubert Markl appointed an independent commission of historians to study the history of the Kaiser Wilhelm Society during the National Socialist era.
• This Institute was founded in Bonn in 1980 by Friedrich Hirzebruch in 1980.
• Until he retired in 1995, Hirzebruch was director of the Institute.
• The Max Planck Institute for Mathematics preprint series was established in 1983 shortly after the institute itself.
• The Hirzebruch Collection is a media archive that collects documents, images, videos, and other resources related to the work and life of the Max Planck Institute for Mathematics' founding director Professor Dr Friedrich Hirzebruch (1927-2012).
• The institute was founded in Leipzig on 1 March 1996 and it works closely with the University of Leipzig [',' Max Planck Institute for Mathematics in the Sciences.','5]:- .

53. Edinburgh Royal Society
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• The Royal Society of Edinburgh was founded in 1783.
• The Society for the Improvement of Medical Knowledge was founded in Edinburgh in 1731 and Maclaurin became one of its members.
• In 1737 the broader Society was formed with the full title "Edinburgh Society for Improving Arts and Sciences and particularly Natural Knowledge".
• This Club was founded in 1716 nearly ten years before Maclaurin was appointed to the University of Edinburgh, and it was a Club which suited Maclaurin with its mixture of congenial fellowship and the aim of its members in pursuing knowledge.
• Shapin describes the events which led to the founding of the Royal Society of Edinburgh in [',' S A Shapin, The Royal Society of Edinburgh : A study of the social context of Hanoverian science (Doctoral Thesis, University of Pennsylvania, 1971).','2], see also his paper [',' S A Shapin, The Royal Society of Edinburgh, British J.
• 7 (1974), 1-.','4].
• The authors of [',' N Campbell, R Martin and S Smellie, The Royal Society of Edinburgh 1783-1983 (Edinburgh, 1983).
• ','1] write:- .
• The beginning of the Royal Society of Edinburgh was described in the first volume of the Transactions of the Royal Society of Edinburgh published in 1788.
• In 1810 the Society purchased 42 George Street, and it occupied this building until 1826 when the Royal Institution Building on Princes Street was completed.
• William Thomson, later Baron Kelvin of Largs, was President from 1873 to 1878 and again from 1886 to 1890.
• From 1890 to 1895 Thomson was President of the Royal Society of London, then, this time as Lord Kelvin, he was President of the Royal Society of Edinburgh for a third time from 1895 until his death in 1907.
• Two others from our archive held the office of President: D'Arcy Thompson from 1934 to 1939, followed by Edmund Whittaker from 1939 to 1945.
• Three mathematicians from our archive served the Royal Society of Edinburgh as General Secretaries: Playfair from 1798 to 1819, Tait from 1879 to 1901, and Chrystal from 1901 to 1912.
• Aitken was Vice-President for six years, namely from 1948 to 1951 and then again from 1956 to 1959.
• Joseph Wedderburn was elected a Fellow in 1903 when he was 21 years of age making him one of the youngest Fellows ever elected.
• It is worth noting that Joseph Wedderburn's brother, Sir Ernest Wedderburn, served the Society as Treasurer for ten years from 1937 to 1947.

54. Dublin Trinity College
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Trinity College Dublin Mathematical Society was founded following a meeting on 12 December 1923.
• Charles Henry Rowe (1893-1943) had been born in Cork, studied for his bachelor's degree at Cork, then received his M.A.
• from Trinity College, Dublin in 1917.
• He won a fellowship of Trinity College, Dublin, in 1920 and held this until his death.
• In 1921 he was appointed Donegal Lecturer in Mathematics at Trinity College and after a while, the Erasmus Smith Chair of Mathematics being vacant, he was made acting professor.
• Charles Henry Rowe became Erasmus Smith Professor of Mathematics at Trinity College, Dublin, in 1926.
• One of the students at this inaugural meeting was Ernest Thomas Sinton Walton (1903-1995).
• Walton, born in Abbeyside, Dungarvan, County Waterford, won a scholarship to study mathematics and physics at Trinity College, Dublin, and began his studies there in 1922.
• After graduating with a Master's Degree in 1927 he studied for his doctorate in physics at Trinity College, Cambridge.
• He was awarded his doctorate in 1931 and remained at Cambridge until 1934 when he returned to Ireland as a fellow of Trinity College, Dublin.
• He was appointed Erasmus Smith Professor of Natural and Experimental there in 1946.
• He was awarded the Nobel Prize in Physics in 1951 for his work on particle-accelerators giving experimental verification of atomic structure.
• In 1931 the Dublin University Mathematical Society moved into its own rooms in House No.
• 7 and later, in 1944, it moved into No.
• 39 to the room which it occupied until August 1991 when the Society moved, with the Department, to Westland Row.
• 20 (1) (1945), 57-58.','1]:- .
• He entered Trinity College, Dublin, in 1881 and obtained a Fellowship in 1889.
• In 1910 he was appointed Professor of Natural Philosophy.
• He retired from the Chair in 1926 when he was succeeded by his former pupil, Dr J L Synge.
• In 1909 he married Margaret Barker, daughter of the late Col.
• The 2016-17 Trinity College Dublin Mathematical Society booklet for the 94th Session of the Society, gives the following information about the Society:- .

55. Spanish Society of Applied Mathematics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Spanish Society of Applied Mathematics, in Spanish 'La Sociedad Espanola de Matematica Aplicada' (SEMA or SeMA) gives as its date of foundation the year 1991.
• In some senses this date is too early since the Society was not formally registered until 1993 but, on the other hand, the date is somewhat too late in the sense that moves to found the Society began in 1988.
• A number of decisions were taken, the most significant being the decision that the next Congress, which was to be in 1989, would be XI Spanish Congress of Differential Equations and Applications - I Congress of Applied Mathematics known by XI CEDYA - I CMA.
• The XI CEDYA - I CMA meeting was held in September 1989 in Fuengirola (Malaga) organised by the Universidad de Malaga, After the XI CEDYA - I CMA meeting, Antonio Valle wrote to the members of the Commission suggesting that various applied mathematics research groups should be integrated into the next congress which would be XII CEDYA - II CMA to be held in September 1991 in Oviedo, organised by the Universidad de Oviedo.
• On 2 September 1991 a letter, signed by A Bermudez, J I Diaz, A Linan, J M Sanz Serna, C Simo and A Valle was widely circulated inviting everyone who was interested to join their initiative of founding 'La Sociedad Espanola de Matematica Aplicada' (SEMA).
• Included with the letter was a registration form for signing up to be a member of the Society, with a deadline for registration given as 15 October 1991.
• This is considered the date of founding of the Society and explains the given date of 1991 as the date of its founding.
• He was awarded his doctorate in 1965 by the Universidad Complutense de Madrid for his thesis Problemas de control optimo en ecuaciones diferenciales abstractas de evolucion.
• Antonio Valle Sanchez [',' C Pares, C Vazquez and F Coquel, Preface, Advances in Numerical Simulation in Physics and Engineering (Springer, 2014), viii.','1]:- .
• was the founder of a very large Spanish community of Numerical Methods in Partial Differential Equations that grew up from the three universities in which he was a Professor: Santiago de Compostela (1967-1973), Sevilla (1973-1984), and Malaga (1984-).
• The French government made Antonio Valle a knight of the National Order of Merit in 1997.
• It was renamed SeMA Journal with the numbering continuing, so the first with the new name was Issue No 53 in 2011.
• The following information about the 'SEMA-SIMAI Book Series' appears on the Springer page [',' SEMA-SIMAI Springer Series, Springer International Publishing (2018).','4]:- .
• The award was established in 1998, but it was only named for Antonio Valle from 2013 onwards.
• 1993-1994: Antonio Valle, Universidad de Malaga.
• 1995-1996: Mariano Gasca, Universidad de Zaragoza.
• 2006-2010: Carlos Vazquez Cendon, Universidad de la Coruna.
• 2010-2012: Pablo Pedregal Tercero, Universidad de Castilla-La Mancha.
• 2012-2016: Rafael Bru Garcia, Universidad Politecnica de Valencia.
• 2016-2018: Rosa M Donat Beneito, Universidad de Valencia .

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Chermiss writes [',' H Chermiss, The riddle of the early Academy (New York, London, 1980).','1]:- .
• What then was Plato's Academy? Chermiss writes [',' H Chermiss, The riddle of the early Academy (New York, London, 1980).','1]:- .

57. Indonesian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Indonesian Mathematical Society was founded on 15 July 1976 in Bandung.
• In the late 1960s it ceased to function for both Singapore and Malaya and it became the Singapore Mathematical Society with the Malaysian Mathematical Society being founded in 1970.
• The Southeast Asian Mathematical Society was founded in 1972.
• The next was held in Penang in 1974 followed by Bandung in 1976.
• There had been the "Mathematics Study Group", followed by the "Mathematics Learning Group" and then, in 1973, the "Mathematics Interest Association" had been founded by Slamet Dajono.
• Slamet Dajono (born in Tulungagung on 24 June 1927) did his doctoral studies in mathematics in 1962-64 and became a lecturer at the State University of Surabaya.
• Discovering that there was a "Mathematics Interest Association" in Surabaya encouraged Lee Peng Yee who returned to Bandung and encouraged those at the Bandung Institute of Technology to organise the first Indonesian National Mathematics Conference there in 1976 which was held in parallel with the Southeast Asian Mathematical Society Conference.
• 1991 6th National Mathematics Conference.
• In 2013 the Indonesian Mathematical Society made proposals to improve the quality of training for teachers of mathematics [',' B Kaur, O Nam Kwon and Y Hoong Leong (eds), Professional Development of Mathematics Teachers: An Asian Perspective (Springer, 2016).','1]:- .
• The Indonesian Mathematical Society Team (2013) recommended that the curriculum for mathematics and mathematics education must consist of at least seven strands of study, they are: (1) general field of study (for example, science, humanities or knowledge subjects), (2) mathematics content field of study (for example, real analysis, abstract algebra, complex numbers, etc), (3) school mathematics field of study (for example, school mathematics topics such as number, geometry, algebra, etc), (4) mathematics education field of study (for example, learning theories of mathematics, use of teaching aids to develop conceptual knowledge, etc.), (5) pedagogy field of study (for example, psychology of teaching and learning, managing students, etc.), (6) additional skill field of study (for example, enrichment courses like academic writing, public communication, etc.), and (7) special field of study (this depends on the expertise of the university and their prime focus).

58. Mathematical Society of Japan
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Mathematical Society of Japan was founded in 1877 as the Tokyo Mathematics Society (Tokyo Sugaku Kaisha).
• In 1884, seven years after the Society was founded, it broadened itself by becoming the 'Tokyo Mathematical and Physical Society', still at this stage not considering itself as a national society.
• However, in 1918 it did recognise its status as a national society and again changed its name to become the 'Physico-Mathematical Society of Japan'.
• The present form of the 'Mathematical Society of Japan' came into existence in 1946 when the 'Physico-Mathematical Society of Japan' split into two separate societies, one whose primary interest was mathematics, namely 'The Mathematical Society of Japan' and one whose primary interest was physics, named 'The Physical Society of Japan'.
• The Prize was established in 1987 funded from donations by the family of Yasuo Akizuki and others.
• These prizes are named after Katahiro Takebe (1644-1739), a disciple of Seki Takakazu, who was noted for his creation of tables of trigonometric functions.
• Fujioka City, which is associated with Seki Takakazu (1642-1708), awards certificates of merit and bronze statues of Seki to the winners of this prize.
• The Takagi Lectures are named for Teiji Takagi (1875-1960), the creator of Class Field Theory and the person considered to be the father of modern mathematical research in Japan.
• The Mathematical Society of Japan International Research Institute was inaugurated in 1993 and one or two were held most years until 2006 when its scope was expanded and it was renamed the Mathematical Society of Japan Seasonal Institute and held annually.
• This journal, founded by the Mathematical Society of Japan in 1924, produces two issues per year.
• This journal was founded in 1948 and publishes research articles across a wide range of mathematical topics.
• It began publication in 1996.
• This is a book series published by the Mathematical Society of Japan, the first volumes appearing in 1983.
• The following is taken from [',' M Kotani, Mathematics in the advanced information age, The Mathematical Society of Japan.','1]:- .

59. Norwegian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• It was not the first time a national mathematical society had been proposed for Norway, however, for Lie had started moves to create such a Society in 1885.
• The idea was brought up again in 1918 when Arnfinn Palmstrom, who became professor of actuarial mathematics in the following year, persuaded Norwegian insurance companies to put up the necessary finance.
• Heegaard had been professor of mathematics at Copenhagen University, where he edited the Danish Mathematical Journal, but resigned in 1917 because of a heavy work load and disagreements with colleagues.
• The Norsk Matematisk Tidsskrift (The Norwegian Mathematical Journal) first appeared in 1919.
• Birkeland writes [',' B Birkeland, The Norwegian Mathematical Society, European Mathematical Society Newsletter 41 (September, 2001), 17-18.','1]:- .
• In 1950 there was an initiative by certain Danish mathematicians to create two Scandinavian journals of mathematics, one for research level mathematics and the other to cover more elementary mathematics.
• 93 (2003), 5-18.','3]:- .
• The first meeting of the editorial board was in May and the journal was first published in 1953.
• Although negotiations for the joint publication of Nordisk Matematisk Tidskrift took a little longer to finalise, it first appeared in 1953.
• Friederich Engel and Poul Heegaard were appointed as editors and they published the first volume in 1922.
• However it was a major task which faced various problems, partly financial, and last volume, namely the seventh, only appeared in 1960.
• Birkeland [',' B Birkeland, The Norwegian Mathematical Society, European Mathematical Society Newsletter 41 (September, 2001), 17-18.','1] writes of other activities of the Society:- .
• For many years starting in 1922, Crown Prince Olav awarded a prize for the best solutions to a series of problems posed in the Journal.

60. Southeast Asian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Southeast Asian Mathematical Society was founded in 1972.
• Much of the information given below is taken from [',' P Y Lee, Ten years of SEAMS, Southeast Asian Bulletin of Mathematics 7 (1983), 10-15.','1] which is the Presidential Address by Lee Peng Yee to the Sixth Biennial General Meeting of the Society at Manila on 5 June 1982.
• The series came into existence due to a meeting between Yukiyoshi Kawada, Secretary of the International Congress on Mathematical Education from 1975 to 1978, and Lee Peng Yee, as the representative of the Southeast Asian Mathematical Society.
• They met in Tokyo in 1976 and discussed setting up the Southeast Asian Conference on Mathematics Education and then, on his return journey, Lee Peng Yee met Bienvenido Nebres, the president of the Mathematical Society of the Philippines, in Manila and over a dinner they agreed to hold the Southeast Asian Conference on Mathematics Education in 1978 [',' P Y Lee, Ten years of SEAMS, Southeast Asian Bulletin of Mathematics 7 (1983), 10-15.','1]:- .
• This conference series continued, one being held every three years in a Southeast Asian country: 1981 Kuala Lumpur, Malaysia; 1984 Hat Yai, Thailand; 1987 Singapore; 1990 Bandar Seri Begawan, Brunei; 1993 Surabaya, Indonesia; 1996 Hanoi, Vietnam; 1999 Manila, the Philippines; 2002 Singapore.
• The 2002 conference merged with the East Asian Regional Conference on Mathematics Education and this merged conference continued to be held under the name of East Asian Regional Conference on Mathematics Education: 2005 Shanghai, China; 2007 Penang, Malaysia; 2010 Tokyo, Japan; 2013 Phuket, Thailand; 2015 Cebu, the Philippines.
• It began publication in 1977 and continues to publish research papers in all areas of mathematics with the primary aim of disseminating original research from mathematicians in Southeast Asia to both the regional and international scientific community.
• 1973-1974 Wong Yung Chow (University of Hong Kong) .
• 1975-1976 Teh Hoon Heng (Nanyang University) .
• 1977-1978 Bienvenido F Nebres (Ateneo de Manila University) .
• 1979-1980 Tan Wang Seng (Universiti Sains Malaysia) .
• 1981-1982 Lee Peng Yee (Nanyang University) .
• 1983-1984 Virool Boonyasombat (Chulalongkorn University, Thailand) .
• 1985-1987 Lim Chong Keang (University of Malaya) .
• 1988-1989 Kar Ping Shum (Chinese University of Hong Kong) .
• 1990-1991 MariJo Ruiz (Ateneo de Manila University, Philippines) .
• 1992-1993 Modin Mokta (Malaysia) .
• 1994-1995 Suwon Tangmanee (Suranaree University of Technology, Thailand) .
• 1998-1999 Polly W Sy (University of the Philippines) .
• 2000-2001 Do Long Van (Institute of Mathematics, Hanoi) .
• 2010-2011 Fidel R Nemenzo (University of the Philippines) .
• 2012-2013 Le Tuan Hoa (Vietnam Institute for Advanced Study in Mathematics) .
• 2014-2015 Edy Tri Baskoro (Institut Teknologi Bandung) .
• 2016-2017 San Ling (Nanyang Technological University, Singapore) .
• 2018-2019 Jose Maria P Balmaceda (University of the Philippines Diliman) .

61. Pakistan Mathematical Society
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• The Pakistan Mathematical Society was founded on 26 June 2001.
• The initiative to form such a body was made by mathematicians from the Quaid-e-Azam University who issued an invitation to those who were interested to meet in the Best Western Hotel in Islamabad on 15 May 2001.
• The Quaid-e-Azam University, established in Islamabad in 1967, began offering postgraduate degrees of Ph.D.
• The first meeting of the Society was held on 26 June 2001 at the Allama Iqbal Open University.
• This university, based in Islamabad, was founded in 1974 and was modelled on the British Open University providing university level education to those who cannot take years out to study at a conventional university.
• Mushtaq, born in Sheikhupura, Pakistan, in 1954, studied for an M.Sc.
• Before the conference ended, it was decided to hold the 2nd Pure Mathematics Conference in Islamabad in August 2001.
• At the Society's first meeting in the Allama Iqbal Open University in June 2001 the constitution, which had already been drawn up following the May meeting, was approved by those present.
• The meeting, the Society's first AGM, then elected the officers of the Society to serve for the two years 2001-2002.
• Although it was founded on 26 June 2001, it was still necessary for it to be officially registered [',' The Pakistan Mathematical Society website.','1]:- .
• The PakMS was registered on 16th November 2001.
• The Society had to apply again for registration under the ACT XXI of 1860 to meet the requirement of the Pakistan Science Foundation for registration of the society with it.
• It was finally registered under this Act on 17th February 2003.
• This is the Society legitimacy and legal protection under the ACT XXI of 1860 of Societies.
• The Pakistan Mathematical Society summarises its activities as follows [',' The Pakistan Mathematical Society website.','1]:- .

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Royal Academy of Sciences and Arts of Barcelona was founded on 18 January 1764 with the name "Physico-Mathematical Conference".
• The result of the two competing claims was the Spanish War of Succession (1701-1714).
• In 1706 Catalonia broke its allegiance to Philip and recognised Charles as King of Spain.
• Victory in the Spanish War of Succession by Philip became complete when Barcelona surrendered on 11 September 1714.
• Philip then moved to punish Catalonia and in 1716 abolished much of Catalonia's independent structure.
• In particular, he suppressed all Catalan universities and, in 1717, the University of Barcelona moved to Cervera.
• One of the people who actively worked towards founding an academy in Barcelona was the mathematician Tomas Creda (1715-1791).
• Creda studied the French translation of Colin Maclaurin's Treatise of fluxions before moving to Barcelona in 1757 where he taught at the Jesuit College of Cordelles until 1764.
• Charles, the fifth son of Philip V, became king of Spain in 1759 taking the title Charles III.
• On 17 December 1765, Charles III signed a Royal Charter making the "Physico-Mathematical Conference" the "Royal Physics Conference".
• In 1786 Charles III was still king of Spain; in fact he continued in that role until his death in 1788.
• It [',' R Pascual, The Royal Academy of Sciences and Arts of Barcelona, Europhysics News (2013).','2]:- .
• Shortly before this, the Academy had been given an official role in standardising local Barcelona time [',' R Pascual, The Royal Academy of Sciences and Arts of Barcelona, Europhysics News (2013).','2];- .
• In 1886, in order to standardize the local time and disseminate it throughout the city of Barcelona, the Academy accepted the mission to define the time in Barcelona and in 1891 that time was declared the 'official time' for the City.
• In 1895, the City Council declared the Academy also responsible of the accuracy of the clocks of the Cathedral and the City Hall and, later on, other clocks around the city.
• The building on La Rambla was not the only one designed for the Academy by Domenech i Estapa at this time, for he also designed the Fabra Observatory which was sited in the Tibidabo with work commencing in 1902.
• It was opened on the 7 April 1904 by King Alfonso XIII [',' R Pascual, The Royal Academy of Sciences and Arts of Barcelona, Europhysics News (2013).','2]:- .
• Furthermore, Comas was the first person to observe and describe the presence of an atmosphere on Titan, the largest satellite of Saturn, on the night of 13 August 1907.
• The Academy has other properties in addition to those just mentioned [',' History of the Royal Academy of Sciences and Arts of Barcelona.','1]:- .

63. Trinity Cambridge Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Trinity Mathematical Society was founded in 1919 by a group of Cambridge undergraduates, supported by G H Hardy.
• Theory Series B 25 (1978), 240-243.','2]).
• Henry Dudeney produced a puzzle in 1902 which asked for a square to be dissected into squares, but there was a rectangle in this solution.
• Several mathematicians examined the problem of dissecting squares into squares including Max Dehn in 1903.
• Nikolai Luzin, in 1930, conjectured that it was impossible to dissect a square into a finite number of squares all of different sizes.
• The problem was solved by four members of the Trinity Mathematical Society, namely Rowland Leonard Brooks (1916-1993), Cedric Austin Bardell Smith (1917-2002), Arthur Harold Stone (1916-2000) and William Thomas Tutte (who has a biography in this archive).
• They all met in 1936 as follows [',' W T Tutte, Squaring the square, in Mathematical games, Scientific American (November 1958).','5]:- .
• However he managed to dissect a 176 by 177 rectangle into 11 unequal squares.
• Rowland Brooks, Cedric Smith, Arthur Stone and Bill Tutte published their results in the paper The dissection of rectangles into squares (1940), see [',' R L Brooks, C A B Smith, A H Stone and W T Tutte, The dissection of rectangles into squares, Duke Mathematical Journal 7 (1940), 312-340.','1].
• 2 (1950), 197-209.','4].
• Let us note that Rowland Brooks and Cedric Smith were successive Presidents of the Trinity Mathematical Society, Brooks in 1937 and Smith in 1938.
• The smallest possible solution was discovered in 1978 and so the present logo of the Society is at THIS LINK.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Wilson brought the bill to the Senate on February 20, 1863, where it was passed on March 3.
• A report on the young Academy, written five years after its founding, states [',' Meeting of the National Academy Of Sciences, The College Courant 3 (9) (1868), 134.','2]:- .
• The National Academy of Sciences, the representative in this country of the Institute of France, and the Royal Society of Great Britain, was chartered by Congress in 1863.
• In 1866, it met at Northampton, last year in Hartford, and in 1868, at Northampton again.
• Not all reports of the early meetings of the Academy were positive, however, and a report of the 1866 meeting at Northampton, mentioned in the above quote, contains some critical comments [',' National Academy of Sciences, Scientific American 15 (9) (1866), 131.','5]:- .
• It is interesting to see that the Academy's own annual report of 1866 recommended the adoption of the decimal system in the United States:- .
• The report begins [',' National Academy of Sciences, Science 1 (12) (1883), 323-324.','3]:- .
• The report of the 1906 meeting begins [',' W H Hale, National Academy of Sciences, Scientific American 95 (23) (1906), 419.','1]:- .
• In 1916 the Academy established the National Research Council which was designed to coordinate the activities of scientists and engineers in a wide variety of different situations such as universities, industry, and government.
• Although set up because of the war, after the World War I ended in 1918, it was seen to provide a valuable service in times of peace as well as in times of war and so the National Research Council continued to exist.
• In 1956 and again in 1993 the remit of the National Research Council was broadened.

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• The Indonesian Academy of Sciences was founded on 13 October 1990.
• After years of domination by the Dutch East India Company, when the Company became bankrupt in 1800 the Netherlands established the Dutch East Indies under Dutch control.
• During World War II the Japanese invaded and took control from 1942 to 1945.
• Formed on 1 May 1948, this organisation was an Indonesian-Dutch collaboration.
• On 17 August 1950 the Republic of Indonesia was proclaimed, being officially recognised by the Dutch and all other nations.
• A new constitution was set up for the country, markedly different to that proposed in 1945 when they first declared independence.
• This renamed organisation published a newsletter in English in 1950 in which it drew up plans for establishing the Indonesian Academy of Sciences.
• Andrew Goss writes [',' A Goss, The Floracrats: State-Sponsored Science and the Failure of the Enlightenment in Indonesia (University of Wisconsin Press, 2011).','1]:- .
• The Indonesians had inherited the OSR from the Dutch, which in 1951 received an Indonesian name (Organisasi Penyelidikan Ilmu Pengetahuan Alam).This organization kept the building on the Koningsplein Zuid, now renamed Jalan Merdeka Selatan, and in November of 1951 it organized a small workshop in Bogor.
• In 1951 the minister of education and culture established a committee charged with investigating the founding of an Indonesian council of sciences.
• By the middle of 1954, the committee had a basic outline of the future council.
• A draft proposal was agreed with the Indonesian Ministry of Justice in 1956 to create the Indonesian Academy of Sciences but it did not progress further as nothing was put before the Indonesian government.
• In 1966 the Provisional People's Consultative Assembly annulled the law establishing the Madjelis Ilmu Pengetahuan Indonesia (Indonesian Council for Science, MIPI).
• Although the Indonesian Institute of Sciences put forward proposals to establish the Indonesian Academy of Sciences in 1969, again there was no government follow-up and no further action was taken until 1983 when a new committee was set up to draw up statutes for an Indonesian Academy of Sciences.
• This took longer than many hoped but, on 13 October 1990, the Academy formally came into being.
• The Academy of Sciences of Indonesia, established under the Law of the Republic of Indonesia (No.8 / 1990), as an independent institution: (i) to provide opinions, suggestions and considerations on matters relating to science and technology to the Government and to society.
• Bacharuddin Jusuf Habibie was an engineer who was the third President of the Republic of Indonesia in 1998 but at the time the Academy was founded he was Secretary of State for Research and Technology.
• Bambang Hidayat was elected to the Indonesian Academy of Sciences in 1991.
• Bambang Hidayat was born on September 18, 1934 in Kudus, Central Java, Indonesia.
• He obtained his PhD from Case Institute of Technology in Cleveland, Ohio in 1965 in the field of Astrophysics, where he worked with Professors McCuskey and Blanco.
• Professor Hidayat became a Full Professor in 1976, Associate Professor in 1974 and Assistant Professor in 1968.
• In 1968, he was appointed Director of the Bosscha Observatory, a post he held for over 15 years.
• He was the Chairman of the Indonesian-Dutch Astronomy Programme in 1982, Chairman of the Indonesian-Japan Astronomy Programme from 1980 until 1994 and the Vice-President of the International Astronomical Union from 1994-2000.
• We note that he is also a member of the International Astronomical Union; the American Astronomical Society; the Royal Astronomical Society; the Indonesian Astronomical Society (he was the founder in 1978); and the Indonesian Physics Society (he was a co-founder).
• My PhD degree was earned from School of Mathematics, the University of New South Wales in 1992.

66. Jagiellonian University Mathematics Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• After operating for three years, there followed three years of inactivity, and then in 1900 Antoni Hoborski became President.
• Another rather remarkable task that the Society took on was the production of textbooks from the lecture courses they were attending [',' K Ciesielski, 100th Anniversary of the Jagiellonian University Students&#8217; Mathematics Society, Math.
• Intelligencer 17 (4) (1991), 42-46.','1]:- .
• The German invasion of Poland on 1 September 1939 followed by the Russian invasion on 17 September put a stop to any possible activities by the Society.
• In June 1941 Germany attacked the Soviet Union, putting Poland into a very strange position.
• One day in 1950 Society members turned up for one of the regular meetings to discover that the room they used had been sealed.
• In 1959 rules were relaxed and the Society could again operate openly.
• It renewed its activities of meetings, lectures, problem sessions, and joint meetings with other Polish student mathematics societies started up again (but stopped in 1976).

67. International Society for Mathematical Sciences
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Byelaws of the Society, amended in 2005, state that [',' International Society for Mathematical Sciences website.','1]:- .

68. German Society for Applied Mathematics and Mechanics
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• In 1923 Reissner was elected as vice president and these three were continually re-elected to their respective positions until 1933.
• The meetings from 1922 to 1933 (inclusive) were held in Leipzig, Marburg, Innsbruck, Dresden and Danzig (two meetings), Zurich, Bad Kissingen, Hamburg, Prague, Berlin, Bad Elster, Berlin, and Wurzburg.
• By 1933 the German Society for Applied Mathematics had around 450 members.
• In fact this worked somewhat to their advantage when the political climate changed in 1933.
• After the resignation of Reissner and von Mises in 1933, they were replaced by Trefftz and C Weber, respectively while Prandtl continued as president.
• For example he wrote to the Ministry on 15 June 1938 [','','1]:- .
• This viewpoint, which, since the breakthrough of 1933, has fallen somewhat in the background, must generally be helped to become again valid if Germany does not wish to suffer damage.
• There was then a break from 1944 to 1949 when it was impossible to organise the annual meeting, but they recommenced in 1950 with a meeting in Darmstadt.
• Richard Grammel, from Stuttgart, had been elected vice-president of the Society in 1937 and held the post until the end of World War II in 1945.
• The Society essentially ceased to exist until operations restarted in 1950 and at this time Grammel was elected president.
• The structure of the Society changed in 1973 with posts of President, Vice-president, Secretary, Vice-secretary, and Treasurer.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Louis Napoleon Bonaparte (1778-1846), a younger brother of Napoleon Bonaparte, ruled over the Kingdom of Holland from 1806 to 1810 as Louis I.
• The Royal Decree signed by Louis I stated that the new Institute was to [',' 1808 - Signed: Louis, History of the Academy, Koninklijke Nederlandse Akademie van Wetenschappen.','10]:- .
• The annual general meeting was held in the left half of the Trippenhuis Building which was owned by Cornelis Roos, an art collector, poet, and member of the History and Fine Arts Section of the Academy who had made the purchase in 1797.
• When Napoleon Bonaparte threatened to take Amsterdam by force in 1810 Louis Napoleon Bonaparte resigned as King and his son Napoleon Louis Bonaparte became ruler, known as Louis II.
• Members of the Institute felt that they now had a building worthy of their prestigious body [',' 1812 - The Trippenhuis Building: The Academy&#8217;s Permanent Headquarters, History of the Academy, Koninklijke Nederlandse Akademie van Wetenschappen.','11]:- .
• Cornelis Roos, who had owned the left half of the Trippenhuis Building, sold it and in 1814 this part of the building became the National Museum.
• Now the French had left in 1813 and after a period when different plans for the area were proposed, in 1815 the United Kingdom of the Netherlands was formed with William I as king.
• In 1825 Adolphe Quetelet became a corresponding member of the Royal Institute of Science, Letters and Fine Arts and two years later he was elected as a full member.
• Also elected to the Institute on 11 October 1827 was Wilhelm Bessel.
• There was unhappiness, however, between Belgium and the other provinces and in 1839 Belgium left the United Kingdom of the Netherlands.
• In 1840 William I abdicated with his son William II becoming king.
• The Netherlands, however, had failed to match Belgium in development and following the unrest across all of Europe in 1848, William II was persuaded to allow liberal and democratic reform.
• The king asked Johan Rudolf Thorbecke (1798-1872) to draft a new constitution which came into force on 3 November 1848.
• William II died in 1849 and was succeeded by William III who, reluctantly, chose Thorbecke to head the government.
• Members of the Institute put up with the 1849 budget but when the government only allocated a trivial sum in 1850 they began to fight to save the Royal Institute.
• Guillaume Groen van Prinsterer (1801-1876) was an historian who was a member of the Royal Institute and also a member of the government.
• On 26 October 1851, Thorbecke issued a Decree closing the Royal Institute of Science, Letters and Fine Art putting in its place a Royal Academy of Sciences.
• On 19 April 1853 Thorbecke was forced to resign after it was claimed he had sympathies with the Roman Catholic Church and Floris Adriaan van Hall (1791- 1866) became head of the government [',' 1851 - Institute Closed, Academy Founded, History of the Academy, Koninklijke Nederlandse Akademie van Wetenschappen.','12]:- .
• Adolphe Quetelet, already a member of the Royal Institute, became a member of the newly formed Royal Academy of Sciences as a foreign member when it was created on 26 October 1851.
• John Herschel was elected as a foreign member of the Academy on 1 May 1858.
• Hendrik Lorentz was elected a member on 10 May 1881.
• Charles Hermite was elected as a foreign member on 10 May 1890.
• J Willard Gibbs was elected as a foreign member on 10 May 1892.
• Arthur Cayley was elected as a foreign member on 12 May 1893.
• Ludwig Boltzmann, Felix Klein and Henri Poincare were elected as foreign members on 11 May 1897.
• Simon Newcomb was elected as a foreign member on 13 May 1898.
• Gaston Darboux was elected as a foreign member on 10 May 1901.
• In 1902 [',' 1902 The Second Golden Age, History of the Academy, Koninklijke Nederlandse Akademie van Wetenschappen.','13]:- .
• In 1938 the Academy adopted the name by which it is known today, namely the Royal Netherlands Academy of Arts and Sciences.
• When World War II broke out in 1939 the Netherlands claimed neutrality.
• This, however, did not stop the German armies entering the country in the spring of 1940 and soon the country was occupied by German troops.
• From that time until the end of the war in 1945, the Academy was known as the Netherlands Academy of Arts and Sciences, but as soon as Germany was defeated they restored their Royal name.
• The duties, activities, and mission of the Royal Netherlands Academy of Arts and Sciences are given in [',' Academy Duties, Koninklijke Nederlandse Akademie van Wetenschappen.','21]:- .

70. American Mathematical Society
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• The American Mathematical Society started its existence as the New York Mathematical Society which was founded in 1888.
• The New York Mathematical Society certainly become a national society in name on 1 July 1894 when it renamed itself the American Mathematical Society.
• At first, of course, little changed as the Society was firmly based in New York and it held its first meeting as the American Mathematical Society in Brooklyn in 1894.
• He was vice-president of the Society from 1898 to 1901, an editor of the Transactions of the American Mathematical Society from 1899 to 1905, and President of the Society from 1903 to 1904.
• He was appointed professor at Columbia University in 1895, and in the following year he was appointed as Secretary of the American Mathematical Society, a post he held until 1920.
• This was not his only work for the American Mathematical Society for in 1897 he was appointed as editor-in-chief of the Bulletin of the American Mathematical Society, holding this position until just before his death in 1926.
• The first was held in Springfield, Massachusetts in the summer of 1895, the second at Buffalo, New York in the summer of 1896.
• The summer meeting at Buffalo in 1896 is memorable for the first colloquium of the Society.
• White's suggestion was supported by the Society and the first Colloquium speaker was James Pierpont of Yale who lectured at the summer meeting at Buffalo in 1896.
• By 1901 the American Mathematical Society had 357 members and in the previous year 112 papers were read at Society meetings.
• By 1902 156 papers were read and the Society had elected its first President who was not from the North East, namely Eliakim Moore.
• Matters came to a head in 1911 when the Chicago section complained bitterly about the place of the meeting at which Bocher would give his Presidential address.

71. Norwegian Statistical Association
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• The journal began publication in 1974 and the journal has the following overview, aims and scope: .
• It was founded in 1974 by four Scandinavian statistical societies.
• It is named for Erling Sverdrup who was the professor of mathematical statistics and insurance mathematics in the Department of Mathematics at Oslo University from 1953 to 1984.

72. Latvian Mathematical Society
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• The Latvian Mathematical Society was founded on 15 January 1993 with 66 founding members.
• Uldis Eaitums was elected as the first Chair in 1993 and held this position until 1997, Alexander Sostak was elected as the second Chair in 1997 and held this position until 2000, Andreja Reinfelds was elected as the third Chair of the Society.
• The aims of the Society are set out in [',' A Sostak, The Latvian Mathematical Society after 10 years, European Mathematical Society Newsletter 48 (June, 2003), 21-25.
• ','1]:- .

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• The Slovenian Academy of Sciences and Arts (Slovenska Akademija znanosti in umetnosti) was founded in 1938 but there is a long history going back many hundreds of years relating to the founding, or attempts to found, earlier academies in Ljubljana.
• Let us therefore give some details going back to the 17th century, beginning with the 'Academia Operosorum Labacensium' founded in 1693.
• Following the example of the Accademia dell'Arcadia in Rome, dei Gelati in Bologna and other learned Italian societies, the chronicler, historian and lawyer Janez Gregor Dolničar (Thalnitscher) (1655-1719) and the cathedral provost, Janez Krstnik Prešeren (1656-1704) became leaders of Ljubljana's intellectual elite, founding a similar society in 1693 called 'Academia Operosorum Labacensium' (The Workers' Academy of Ljubljana).
• One of the main aims of the 13 lawyers, 6 theologians and 4 doctors who joined the society at its official convening in 1701 was, as written in the academy's charter, to publish "the Ljubljana academy's learned discussions on theology, jurisprudence, medicine, civics ..
• ." The number of Academy members grew, reaching 42 by 1714, including respected foreigners such as the Italian poet and literary historian, Giovanni Mario Crescimbeni, and Valvasor's colleague, the Croatian writer and scholar, Pavao Ritter Vitezovič.
• In 1701 Florijančič saw a great future in the Academy saying in a speech:- .
• The Academy closed for a variety of reasons in 1725.
• In 1779, however, there was a serious attempt to revive the academy by the scholar and linguist Blaž Kumerdej (1738-1805).
• In 1797 French armies led by Napoleon occupied the region of Slovenia including Ljubljana but this was only for a couple of months before they withdrew.
• Another short occupation by the French in 1805 was again only for a couple of months but in 1809 they invaded for a third time establishing the Illyrian Provinces in October 1809.
• By 1813 the French armies began to withdraw and Austrian forces moved in.
• was founded in 1864 with the voluntary contributions of educators, traders and entrepreneurs in order to print more demanding works from different fields in Slovene, raise the level of education and knowledge, create Slovenian terminology for various professions, etc.
• After World War I, in 1918, the Kingdom of Serbs, Croats and Slovenes came into existence and gained international recognition as Yugoslavia on 13 July 1922 at a Paris conference.
• The University of Ljubljana was founded in 1919 but [',' P Štih, V Simoniti and P Vodopivec, A Slovene history, Inštitut za novejšo zgodovino (2008).','4]:- .
• In fact a great many Societies were set up in Slovenia such as Catholic societies which were part of the Educational Union, the National Gallery of Slovenia established in 1918, the French Institute in Ljubljana established in 1921, the Scientific Society for Humanities, and the oldest of all the Slovene Society.
• In 1927, the National Gallery published a promotional booklet "Our most important cultural task: Academy of Sciences and Arts" (Slovenian) which made a strong case for founding an Academy in which:- .
• In 1938 the first eighteen members of the Academy were nominated and a meeting was held.
• On 4 January 1939, Rajko Nahtigal (1877-1958) was nominated as the first President of the Academy.
• In 1942 Milan Vidmar (1885-1962) became the second President.
• Vidmar was an electrical engineer, famous as a chess Grandmaster being among the top dozen chess players in the world from 1910 to 1930.
• The post-war period saw the Communist authorities force the Academy to follow their requirements [',' History, Slovenian Academy of Sciences and Arts.','1]:- .
• On the other hand, under the presidency of France Kidrič; (1945 to 1950), a literary historian, and for a short period under the linguist Fran Ramovš; (1950 to 1952), the institution was characterised by structural expansion and the enlargement of Academy's ranks.
• Yet, quite soon this organizational set-up of the Academy, particularly from 1955 to 1958, began to change, with the separation of large technical institutes, which became independent ..
• On the basis of the laws passed in 1948 and 1949, the autonomy of the Academy was not merely limited - the Slovenian assembly abolished it.
• On 25 June 1991 Slovenia declared itself an independent country.
• Bernik, a historian and writer, had been elected a corresponding member of the Academy in 1983 and a full member in 1987.
• In 1994 the Academy gained its independence from political control and two years later it reinstated several members who had been expelled by the Communists after World War II.

74. Hong Kong Mathematical Society
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• He held a meeting in Hong Kong in June 1970 which prepared the way to the founding of the Southeast Asian Mathematical Society in 1972.
• The Hong Kong Mathematical Society was founded in 1979 with Wong Yung Chow elected as its first president.
• The Society website states [',' Hong Kong Mathematical Society website.','1]:- .
• Our institutional members have increased from 4 to 14 and have included all tertiary institutions in Hong Kong.
• The Society is responsible for editing of the Bulletin of South East Asian Mathematical Society, and in 1997 the Society began to publish its own Bulletin of the Hong Kong Mathematical Society.
• Among the major conferences which the Society assisted in organising was the South East Asian Mathematical Society Conference in 1980 and the First Asian Mathematical Conference in 1990.
• They assisted in organising the First International Conference on Scientific Computing and Partial Differential Equations at the Honk Kong Baptist University, 12-15 December 2002.
• The Second in this series was held in conjunction with the First East Asian SIAM Symposium at the Honk Kong Baptist University, 12-16 December 2005.
• Four further conferences in the Scientific Computing and Partial Differential Equations series were held, also in the Honk Kong Baptist University, the Third conference, 8-12 December 2008, the Fourth conference, 5-9 December 2011, the Fifth conference, 8-12 December 2015, and the Sixth conference, 5-8 June 2017.
• They also organised the International Mathematical Olympiad in 1994.
• In 1974, the Northcote College of Education held the first Hong Kong Inter-school Mathematics Olympiad, which is now known as the Hong Kong Mathematical Olympiad.
• In 1986, the International Mathematical Olympiad Hong Kong Committee was founded which paved the way for the International Mathematical Olympiad being held in Hong Kong in 1994.
• This was held at the Lam Woo Conference Centre of Hong Kong Baptist University from 13 to 16 December 2000.
• The Chinese Mathematical Society wrote to the Hong Kong Mathematical Society in 1996 asking them to support the Chinese Mathematical Society's application to host the International Congress of Mathematicians in Beijing in 2002.

75. Chilean Mathematical Society
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• The Mathematical Society of Chile (Sociedad de Matematica de Chile) (known by the abbreviation SOMACHI) was founded in 1975 when mathematicians met informally but the Society was only formally registered in 1983.
• When it was registered it had the following mission [',' The Mathematical Society of Chile website.','1]:- .
• This journal was founded in 1994 with the aim of contributing to the training of teachers, allowing them to maintain a continuing study of mathematics and, above all, to make contributions to lifting their spirits by learning new things that motivate them and so encourage them to enthusiastically convey to their students new ideas and challenges.

76. German Academy of Scientists Leopoldina
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• The Academia Naturae Curiosorum was founded in the Free Imperial City of Schweinfurt on 1 January 1652.
• The statutes of 1662 give the Academy's purpose in the following terms:- .
• The Academy began to publish a journal, Ephemeriden, in 1670, which they dedicated to the Holy Roman emperor Leopold I.
• In 1677 Leopold awarded the Academy the title 'Sacri Romani Imperii Academia Naturae Curiosorum'.
• The reputation of the Academy continued to grow and in 1687 Leopold gave it numerous privileges as wll as yet a further new title 'Sacri Romani Imperii Academia Caesareo- Leopoldina Naturae Curiosorum'.
• After this return to Halle in 1878 (the first period in Halle had been 1745-1769), this city became its permanent home.
• The Academy library was built in Halle in 1904.
• The next President Emil Abderhalden reorganised the Sections of the Academy in 1932 and introduced the series Lebensdarstel-lungen deutscher Naturforscher (Biographies of German Natural Scientists).
• In 1949 the Soviet Occupation Zone became the GDR while the other three Occupation Zones were merged to form West Germany.
• This, however, failed after the construction of the Wall in 1961.
• In December of 1990 elections were held within the new united country.
• A resolution of 5 April 1991 registered the Academy and its statutes (which have been modified in minor ways in the succeeding years) established that:- .
• Founded in Schweinfurt in 1652, and vested with privileges by Emperor Leopold I in 1687 that were confirmed by Emperor Karl VII in 1742, the Academy is identical with and constitutes the uninterrupted continuation of its predecessor, the 'Imperial Leopoldina Carolina German Academy of Natural Scientists'.

77. Israel Mathematical Union
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• Also in 1954 Moshe Jarden recalls attending a meeting of the Union with his father Dov Jarden.
• both Mathematics and the Hebrew language resulted in 1946 in establishing a journal in Mathematics entitled "Riveon LeMathematika" (i.e.
• I still remember that when I was 12, that is in 1954, my father took me to a meeting of the Israel Mathematical Union that took place at the Institute of Mathematics of the Hebrew University.
• Although Agmon is 95 years old today, he just joined the Institute of Mathematics at the Hebrew University in 1952.
• However, he told me, he believed that the Union had been established after the war of independence (which ended in 1949), so may be some time after 1950.
• The Israel Mathematical Union maintains contacts with other mathematical associations worldwide: it is a corporate member of the European Mathematical Society [since 1991], an associate member of the International Council for Industrial and Applied Mathematics, and has a reciprocity membership agreement with the American Mathematical Society, and with the Society for Industrial and Applied Mathematics.
• This prize was first awarded in 1977 and was, at that time, known as the Erdős Prize.
• The name was changed in 1996 to the Anna and Lajos Erdős Prize reflecting the original wish of Paul Erdős.
• Charlotte Levitzki (nee Ascher) was born in Berlin in 1910.
• She became a librarian and worked in a large bookstore until the Nazis led by Hitler came to power in 1933.
• In 1939 she married Jacob Levitzki and the family settled in Jerusalem.
• She rose quickly through the ranks of the Hebrew University National Library and became head of the acquisition department, a role she held until she retired in 1972.
• From that time, almost until her death in 1997, she volunteered in various institutions, including Hadassah hospital.
• Before describing the prize, let us give some information about Haim Nessyahu following the biography in [',' Israel Mathematical Union website.','1]: .
• His formal education began in 1970 in the "Gavrieli" school, in Tel Aviv.
• Then, in 1973, Haim joined a newly formed class of gifted children, the first of its kind in Israel.
• In 1982, Haim joined the military academic reserve, in the framework of which he studied towards a B.Sc.
• He graduated in 1984, Summa Cum Laude.
• After resigning from the army, in 1989, he joined Professor Tadmor at NASA Langley Research Center, in Hampton Virginia, as a graduate fellow, where he continued his mathematical research.
• He completed his doctoral dissertation in 1994 and was accepted for a post-doctoral position as Assistant Professor of Computational and Applied Mathematics at the University of Los Angeles.

78. Alphabetical List of Mathematical Societies and Academies
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79. French Statistical Society
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• The Societe Francaise de Statistique (French Statistical Society) was founded in 1997.
• However, this date gives a totally wrong impression of the history of the Society since it really dates back to 1860 when the 'Societe de Statistique de Paris' (Statistical Society of Paris) was founded.
• It was also built on the 'Association pour la Statistique et ses Utilisations' (Association for Statistics and its Uses) which was itself originally the 'Association des Statisticiens Universitaires' (Association of University Statisticians) founded in 1969 and became the 'Association for Statistics and its Uses' in 1987.
• Let us give some details here of the French Statistical Society since its founding in 1997.
• The SFdS is a learned society founded in 1997, specialising in statistics, whose mission is to promote the use of statistics and its understanding and to promote its methodological developments.
• These were begun by the Association of University Statisticians, the first being held in Lyon in 1970.
• The SFdS has organised these workshops since 2001 for professional statisticians.

80. List of societies by date of foundation
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• 1751 Gottingen Academy of Sciences .
• 1831 British Association for the Advancement of Science .
• 1861 Croatian Academy of Sciences .
• 1891 Janos Bolyai Mathematical Society .
• 1900 to 1939 .
• 1911 German Statistical Society .
• 1911 Kaiser Wilhelm Society .
• 1911 Spanish Mathematical Society .
• 1921 Belgium Mathematical Society .
• 1921 Petrograd Physico-Mathematical Society .
• 1931 Catalan Society for Physics .
• 1940 to 1969 .
• 1941 Georgian Academy of Sciences .
• 1941 Lithuanian Academy of Sciences .
• 1971 Iranian Mathematical Society .
• 1971 Korean Statistical Society .
• 1971 Women in Mathematics Association .
• 1981 Serbian Mathematical Society .
• 1991 Armenian Mathematical Union .
• 1991 Bulgarian Statistical Society .
• 1991 Spanish Society of Applied Mathematics .
• 2001 Pakistan Mathematical Society .

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• Skoropadsky was descended from an 18th-century Cossack hetman and, proud of the fact, he took the title "Hetman of Ukraine".
• He had been kept in power by German-Austrian support and when they were defeated in November Skoropadsky's regime could not survive and he resigned on 14 December.
• Nikolai Krylov was appointed chairman of the physics and mathematics department in 1922.
• In 1936 it was renamed the Academy of Sciences of the Ukrainian SSR which was its name until 1991.
• Russia and ten other former Soviet republics declared themselves independent on 21 December 1991 and founded the Commonwealth of Independent States.
• The USSR legally ceased to exist on 31 December 1991.
• From 1991 to 1993 the Academy reverted to its original name, the Academy of Sciences of Ukraine, then in 1994 it adopted its current name which is the National Academy of Sciences of Ukraine.
• The Mechanics Institute in Kiev was founded in 1919:- .
• The Hydromechanics Institute in Kiev was founded in 1926, joining the Ukrainian Academy in 1934:- .
• Outlines of its development (Ukrainian) (Kiev, 1997).','1].
• The Institute was founded in Kiev in 1934 from several existing commissions and Grave served as its first Director from its foundation until his death in 1939.
• From 1939 to 1941 and then again from 1944 to 1949, Lavrentev was the Director of the Institute in Kiev.
• Gnedenko became Director of the Institute of Mathematics in 1949.
• Mytropolshy moved to the Institute of Mathematics in 1951 and was made Director in 1958 after Gnedenko left for Moscow.
• The Applied Mathematics and Mechanics Institute was set up in Donetsk in 1965 with I I Daniliuk as its first director:- .
• The Applied Problems of Mechanics and Mathematics Institute in Lvov was founded in 1978:- .

82. Göttingen Mathematical Society
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• The Gottingen Mathematical Society was founded in 1892 by Felix Klein and Heinrich Weber.
• William Henry Young, who was at Gottingen from 1899 to 1908, described the working of the young Mathematical Society (see [',' V Peckhaus, Ernst Zermelo in Gottingen, History and Philosophy of Logic 11 (1990), 19-58.','1]):- .
• Caratheodory, who went to Gottingen as a research student in 1902, described it as resembling:- .

83. Hellenic Mathematical Society
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• The Hellenic Mathematical Society was founded in 1918 [','','1]:- .
• The first President of the Society was Nikolaos Hatzidakis who served in this role from 1918 to 1925.
• He undertook research into differential geometry and when he became a founder member of the Hellenic Mathematical Society he had been a professor at the University of Athens since 1901.
• He undertook research in function theory and had been appointed as professor of Higher Mathematical Analysis at the University of Athens in 1912 and he had also been appointed to the Technical University of Athens in 1916.
• The Society began publication of the Bulletin of the Greek Mathematical Society in 1919 and Remoundos was a member of the editorial board.
• Remoundos was President from 1925 to 1927 and then Konstantinos Maltezos was President in 1927.
• Maltezos, who worked on mechanics and theoretical physics, had been dismissed by the University of Athens in 1920 by the royalist Minister of Education after the exiled King Constantine I had been restored to his throne.
• In 1931, under Sakellariou's Presidency, the Society organised the first Panhellenic Mathematical Competition.
• In particular he interfered in the running of the Society and N Kritikos resigned from the executive committee of the Hellenic Mathematical Society in 1936 due to political interference in the affairs of the Society.

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• The Academy was formally constituted on 14 May 1847 when the statutes were approved.
• This was not the first attempt to found an Academy in Austria, for in fact the idea goes back to 1713 when Gottfried Leibniz suggested establishing an Academy of Sciences in Vienna, quoting the Royal Society in London and the Academy of Sciences in Paris as models to use.
• Nothing came of Leibniz's proposals, nor did later proposals made J C Gottsched in 1750 meet with any greater success.
• The successful proposal for an Austrian Academy of Sciences came in 1837 in a petition submitted by twelve scholars.
• It still took many years of negotiation before the Academy formally came into being and, as we notes above, the foundation in 1846 was formalised by an Imperial Patent on 14 May 1847.
• The first President of the Academy was Joseph von Hammer-Purgstall, an orientalist, and Christian Doppler was an early member of the Academy being elected in 1848.
• The Academy moved into permanent buildings in 1857.
• For instance, it was responsible for founding the Central Office for Meteorology and Geomagnetism in 1851, the establishment of the observatories on the peaks of the Sonnblick and the Obir mountains, and in 1909 founded the Institute for Radium Research in Vienna.
• In the years 1879-1914 the Academy was expanded several times into a "universal research centre".
• The legal basis of the Austrian Academy of Sciences is now the "Federal Law of 14 October 1921, concerning the Academy of Sciences in Vienna", only slightly modified by an Act of 9 May 1947.
• The "Anschluss" to the German Reich on 12 March 1938 had inevitable effects on the Academy.
• The statutes of the Academy were replaced in 1938 by a "provisional statute", which led to some changes in the organization.
• In 1945, the 1938 statute was replaced by the original statute of 1921.
• The historian Heinrich von Srbik was elected on 1 April 1938 becoming the new president of the Academy.
• At the Academy's meetings he identified with Hitler's war policy; in 1940 he spoke of the "struggle of the German people for self-assertion," and in 1943 he repeated his "firm confidence of victory." He tried to preserve as far as possible the independence of the Vienna Academy from Berlin central offices.
• He had great respect for the academy members, so that in 1941, after the end of his term of office, he was re-elected as president.
• After the war ended in 1945, the 1921 statutes were put back in place, members who had been excluded had mostly left Austria so were given corresponding membership.
• In 1947 the Academy celebrated its centenary and at this time changed its name to the 'Austrian Academy of Sciences'.
• In 1954, the Austrian Academy of Sciences was awarded the Karl Renner Prize of the City of Vienna.
• In 1973 the Academy established its own publishing house.
• The Prize was set up in 1991 and is awarded every second year.
• The first awarded in 1992 was made to Michael Drmota, the second in 1994 to Johannes Schoissengeier, the third in 1996 to Gerhard Larcher, and the fourth in 1998 to Monika Ludwig.

85. Icelandic Mathematical Society
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• He completed his matriculation examination from Laerða School in Reykjavik in 1897 and, later that year, entered the University of Copenhagen to study mathematics.
• He was awarded his Master's Degree in 1904 and, returning to Iceland, he applied for a vacant teacher position at the High School in Reykjavik where he had himself been a pupil a few years earlier.
• In 1909 he submitted his thesis to the University of Copenhagen and was awarded a doctorate, becoming the first Icelander to be awarded a doctorate in mathematics.
• But it was something he had been thinking about for a very long time for the first fundamentals of this material is found in an article he wrote in the same journal as a twenty-two year old in 1900.
• Danielsson agreed that the Society would participate in publishing these journals in 1951 and the two journals appeared in 1953.
• The minutes of a meeting of the Society in 1952 record that it had received a formal request from the Danish Mathematical Society to be involved in publishing Mathematica Scandinavica.
• The Statutes of the Icelandic Mathematical Society were revised at the Society's Annual General Meeting on 12 January 2010.
• Leifur Asgeirsson (born 25 May 1903, died 19 August 1990) graduated from the University of Reykjavik in 1927 and then undertook research at the University of Gottingen with Richard Courant as his supervisor.
• He graduated from Gottingen in 1933, and returned to Iceland where he was head of the district school at Laugum in Reykjadalur from 1933 to 1943.
• He was appointed as a lecturer in mathematics at the University of Iceland in 1943 and became the first full-time professor of mathematics at the University of Iceland in 1945.
• He became a full professor at Princeton in 1960.
• He was made an Honorary member of the Icelandic Mathematical Society in 1997.
• Sadly no complete record of the winners of this award exists despite strenuous efforts by the Society in 1989 to find the names of all the winners.
• The Society played a major role in organising the Nineteenth Nordic Mathematical Congress held in Reykjavik in 1984.
• They organised another major conference in 1990 which was to honour Bjarni Jonsson on his 70th birthday.

86. Spanish Royal Academy of Sciences
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• This was, however a reconstitution of the Academy, the origins of which go back to the reign of Alfonso el Sabio in the 13th century, to the Madrid Mathematical Academy of Phillip II, and also to the Academia Naturae curiosorum founded in 1657.
• The Madrid Mathematical Academy (Academia de Matematicas de Madrid) was founded in 1582 (some sources give the date 1575) by Phillip II of Spain.
• The Academy, which ceased to exist in 1625, is examined in detail in [',' P Garcia-Barreno, The Madrid Mathematical Academy of Phillip II, Boll.
• 20 (2000), 87-188.
• ','1].
• The Royal Academy of Exact, Physical and Natural Sciences of 1847 was largely due to Jorge Juan y Santacilia (1713-1773) who was an expert on navigation.
• Madrid 67 (1973), 11-26.','2] for details).

87. Czech Mathematicians and Physicists Union
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• Professor Ernst Mach was also very supportive and, in 1868, he suggested that the Society might wish to meet in one of his lecture theatres and have the use of one of his physics laboratories in which Society members could carry out any experimental work.
• In 1869 the Society was officially registered but now it was under the name Union of Czech Mathematicians.
• Publishing a journal was also high on the agenda and by the time the 10th anniversary of the Union came round in 1872 they could celebrate with launching the publication The Journal for the Cultivation of Mathematics and Physics.
• The Union continued to expand its operations outside Prague and it opened a number of branches, in particular one in Brno in 1911 and one in Bratislava in 1929.
• In 1919 it went into the business of publishing in a big way obtaining a licence to print, publish, and sell books.
• A National Assembly was formed and a new democratic constitution was adopted in 1920.
• In 1921 the Union of Czech Mathematicians changed its name to reflect its involvement in both mathematics and physics and renamed itself the Union of Czech Mathematicians and Physicists.
• The Union purchased a house in 1930 and erected a new building to the rear of the house which was their bookshop and later the library.
• The Union moved into its new headquarters in 1938.
• Hitler's armies invaded on 14 March 1939 and Hitler installed his representative in Prague to run the country.
• In 1952 the Czechoslovak Academy of Sciences was founded in Prague and a number of specialised institutions were attached to it including the Union of Czechoslovak Mathematicians and Physicists as the Union was now called.
• Reforms in Czechoslovakia in the 1960s came to an end in the spring of 1968 when Soviet troops entered the country.
• The Union of Slovak Mathematicians and Physicists split off from the old Union in 1969.
• The modern Czech Republic came into being on 1 January 1993, when the union with Slovakia was ended.

88. Spanish Mathematical Society
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• The Spanish Mathematical Society, the Sociedad Matematica Espanola (later the Royal Spanish Mathematical Society, the Real Sociedad Matematica Espanola) was founded in 1911.
• For example Zoel Garcia de Galdeano (1846-1924) was professor of mathematics at various schools and institutes throughout Spain.
• In 1891 he founded the first Spanish mathematics journal El progresso Matematico.
• It stopped publishing in 1895 but was restarted in 1899, failing again after only one year.
• In 1907 the Board for Advanced Studies and Scientific Research, the Junta para Ampliacion de Estudios e Investigaciones (JAE), was founded.
• In 1908 the Spanish Association for the Advancement of Science, the Asociacion Espanola para el Progreso de la Ciencia (AEPC), was founded.
• At the first Congress of the AEPC it was proposed by Manuel Benitez y Parodi (1845-1911), a military general, that a mathematics society be founded and this led to the founding of the Spanish Mathematical Society in 1911.
• Manuel Benitez, Jose Echegaray y Eizaguirre (1832-1916) who was a civil engineer, and Julio Rey Pastor played a major role in establishing the Society.
• In May 1911 the Society published the first part of its journal, the Revista de la Sociedad Matematica Espanola.
• However, the journal suffered from the same problems that had forced the closure of El progresso Matematico in 1900.
• The problems came to a head in 1915 when Rey Pastor criticised senior members of the Society and argued that it was necessary for Spanish mathematics to become a part of the Europe wide mathematics scene.
• Rey Pastor had certainly played his part in supporting the journal, particularly in the period 1911-13, with a series of papers.
• Our discussion of the Revista has taken us up to 1917 but we should go back to 1915 for in that year, at the suggestion of the Spanish Mathematical Society, the Mathematical Laboratory and Seminar of the Board for Advanced Studies and Scientific Research was created.
• In 1916 Zoel Garcia de Galdeano who had founded the first failed Spanish mathematical journal became the second president of the Spanish Mathematical Society.
• It must have been a great sadness to him to see this second Spanish mathematical journal, the Revista, fail in 1917.
• With the Society in crisis, it was Rey Pastor who revitalised it in 1919.
• A new journal, the Revista Matematica Hispano-Americana, was founded in 1919 and this was mentioned by G A Miller in an article in Science in 1919.
• The second one was a result of the civil war, 1936-39, and in 1941 the Society was again re-established, this time by Franco's dictatorship.
• A third crisis in 1961 was followed by a fourth, very deep crisis, which began in 1990 and continued until the Society was again re-established in 1996.
• Revista Matematica Hispano-Americana which was founded by Rey Pastor in 1919 has continued to be published but it changed its name to Revista Matematica Iberoamericana in 1985 when it began again from Volume 1.
• Starting with Volume 28 (2012), Revista Matematica Iberoamericana has been printed and distributed by the Publishing House of the European Mathematical Society.
• The Society published the journal Gaceta Matematica beginning in 1949.
• La Gaceta de la Real Sociedad Matematica Espanola began publication in 1998, with 3 issues per volume per year.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Swiss Academy of Natural Science, or Schweizerische Naturforschende Gesellschaft, was founded in 1815.
• The first attempt to found a Swiss Academy occurred in 1797 but the turmoil caused by the Napoleonic wars meant that it did not succeed.
• Another attempt was made in 1802 but again political turmoil caused by the French conquests under Napoleon prevented the formation of an Academy.
• Geneva is, of course, now in Switzerland but in 1815 it was part of the Kingdom of Sardinia-Piedmont.
• It also did not want to be an academy consisting of only top scientists so it aimed at being a very different organisation to other academies which had been set up [',' P Kupper and B C Schar, &#8217;&#8217;Einfache und Anspruchslose Organisation&#8217;&#8217;.
• Ein Streifzug durch 200 Jahre SCNAT, Swiss Academy of Sciences.','1]:- .
• The Konigsberg astronomer Friedrich Wilhelm Bessel, therefore, proposed a new, cross-border network of triangles, a project that the Prussian general Johann Jacob Baeyer took up in 1861.
• This commission participated in the process of reaching an agreement between the participating States on the standardization of methods and measures, which was negotiated between 1864 and 1912 at 17 conferences.
• In 1886, when Mexico, Chile, Argentina, the USA and Japan joined, it became the "International Earth Measurement".
• In 1869 Schwarz was appointed to a chair at Zurich but in 1875 he left to take up a chair at Gottingen.
• The Swiss Mathematical Society, however, was founded in 1910 as a section of the Swiss Academy of Natural Science.
• Remarkably, at a time when academies were for men, the finances of the Swiss Academy was run by a woman, Fanny Custer (1867-1930), for over 40 years [',' P Kupper and B C Schar, &#8217;&#8217;Einfache und Anspruchslose Organisation&#8217;&#8217;.
• Attempts after 1945 to modernise Switzerland's scientific research structure were led by the Academy, but after the Swiss National Science Foundation was set up in 1952, it had the effect of diminishing the importance of the Academy.
• In 1988 it adopted the name of 'academy' with the 'Swiss Academy of Natural Science', slightly changing it in 2004.
• It is named for Alexander Friedrich Schlafli (1832-1863) from Burgdorf who died on 6 October 1863 in Bagdad.
• In 1998 it was awarded to Viviane Baladi for the work "Periodic orbits and dynamical spectra", in 2007 to Christian Wuthrich for "Self-points on elliptic curves" and, also in 2007, to Tatiana Mantuano for "Laplacians in Riemannian Geometry: a Spectral Comparison through Discretization".

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• Armenia became a constituent republic of the Soviet Union in 1936 and the Armenian Academy of Sciences was founded on 10 November 1943 as a branch of the USSR Academy of Sciences.
• Orbeli served as president from 1943 to 1947 when he was succeeded by the theoretical astrophysicist Victor Amazaspovich Ambartsumian who held this position until 1993.
• The modern Republic of Armenia became independent in 1991 after the break-up of the Soviet Union and in 1993 the Academy became the 'National Academy of Sciences of the Republic of Armenia'.
• Studies in contemporary mathematics in Armenia date back to 1944, when a Section for Mathematics and Mechanics was created within the newly born Armenian Academy of Sciences.
• The section later developed into the Institute of Mathematics and Mechanics of Armenian Academy of Sciences whose first Director was Academician Artashes Shahinyan (1906-1978), a mathematician well known for his results in complex analysis.
• Shahinyan, the son of the physicist Aram Shahinyan, became a professor in 1944, a corresponding member of the Soviet Academy of Sciences in 1945 and a full member of the Soviet Academy of Sciences in 1947.
• The Institute of Mathematics of Armenian Academy of Sciences separated from the combined mathematics and mechanics Institute in 1971.
• This Institute, founded in 1971, has the following main areas of interest: .
• This Institute, founded in 1955, has the following main areas of interest: .
• This Section, founded in 1992, has the following main areas of interest: .
• It was founded in 1945 on the initiative of Victor Ambartsumian, who became its first director.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The State of Israel was founded in 1948 but prior to this the British Mandate of Palestine had existed from 1917 when the British Prime Minister Lloyd George declared that:- .
• On 14 May 1948 the last British troops left Palestine and the Jewish People's Council met in Tel Aviv and declared the establishment of a Jewish State to be known as the State of Israel.
• Ben-Zion Dinur (1884-1973) was a born in what is now Ukraine and became a lecturer at the University of Odessa before emigrating to the British Mandate of Palestine in 1921.
• In 1936 he was appointed lecturer in modern Jewish history at the Hebrew University and was promoted to professor in 1948.
• He was elected to the first Knesset, Israel's parliament, when it was set up in 1949 and served as Minister of Education and Culture from 1951 to 1955.
• In this role in 1954, he led a committee of scientists who proposed setting up two independent Academies in Israel, a Natural Sciences Academy and a Humanities Academy.
• David Ben-Gurion (1886-1973) was the first Prime Minister of the State of Israel and he reacted to the proposal of two academies by preferring a single Academy.
• One of the leaders in the Preparatory Committee for the Establishment of the National Academy of Sciences and Humanities, set up by Government resolution dated 9 November 1958, was Martin Buber (1878-1965) [',' W J Morgan, Martin Buber, philosopher of dialogue and resolution of conflict, British Academy Review 10 (2007), 11-14.','4]:- .
• Following his removal by the Nazis from the Chair of Philosophy of Religion at the University of Frankfurt, he became the Director of Jewish adult education programmes, until his final departure for Palestine in 1938, when he took up the Chair of Social Philosophy at the Hebrew University of Jerusalem.
• Although this was a single Academy, it retained some of the structure of the original proposal by Ben-Zion Dinur in 1954 in that it had a Humanities Section and a Science Section, each Section electing a Chairperson of the Section.
• The first meeting of the Academy's General Assembly, chaired by Prime Minister Ben-Gurion, elected Martin Buber as the first President of the Academy [',' Historical Background, Israel Academy of Sciences and Humanities.','1]:- .
• Israel's parliament, the Knesset, approved the Israel Academy of Sciences and Humanities Law in June 1961.
• Two mathematicians were founder members of the Academy, becoming members in 1959.
• Dvoretzky was elected president of the Israel Academy of Sciences and Humanities in 1974.
• Markus Reiner (1886-1976) [',' Obituary: Marcus Reiner, Physics Today 29 (9) (1976), 70-71.','5]:- .
• He was awarded the ingenieur degree from the Technische Hochschule in Vienna in 1909 and a Doctor of Technology degree just before World War I.
• Following service as a lieutenant in the Austrian Army, he immigrated to Palestine in 1922.
• Reiner was awarded the Israel Prize in Exact Science in 1958.
• The only mathematician elected in 1960, the first year of the Academy's existence, was Abraham Fraenkel.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The National Academy of Sciences of Italy (the "Academy of Forty") was founded in 1782 as the 'Societa Italiana' by the mathematician Antonio Mario Lorgna.
• Two of these are Boscovich and Malfatti while the other two are Carlo Barletti (1735-1800) and Lazzaro Spallanzani (1729-1799).
• Over the following ten years discussions went on and by 1776 Lorgna and Malfatti were discussing Lorgna's "idea of forming an Academy for all Italian scholars".
• On 1 March 1781, Lorgna went ahead with his proposal sending out a circular letter to leading scientists.
• He published the first part of the 'Memorie', the Society's Memoirs, in 1782 the year the Society was founded in Verona.
• The Society, he wrote, "belongs to the whole of Italy, not just to a single city." Lorgna was the first President of the Societa Italiana, holding this role until his death in 1796.
• It is worth realising that this "Italian Society" was proposed by Lorgna about 100 years before the unification of Italy which took place in 1861.
• Originally set up in Verona, the Society relocated to Milan, then to Modena, and in 1875, when Rome was declared the capital of Italy, the headquarters of the Society moved there where it remains to this day.
• Here is a list of 28 Italian scientists who became members of the Societa Italiana, the "Academy of Forty", when it was founded in 1782.
• Carlo Barletti (1735-1800).
• Teodoro Bonati (1724-1820).
• Ruggero Giuseppe Boscovich (1711-1787).
• Sebastiano Canterzani (1734-1818).
• Angelo Cesaris (1749-1832).
• Felice Fontana (1730-1805).
• Gregorio Fontana (1735-1803).
• Michele Girardi (1731-1797).
• Marsilio Landriani (1751-1815).
• Antonio Mario Lorgna (1735-1796).
• Vincenzo Malacarne (1744-1816).
• Gianfrancesco Malfatti (1731-1807).
• Carlo Lodovico Morozzo (1743-1804).
• Pietro Moscati (1739-1824).
• Pietro Paoli (1759-1839).
• Tommaso Perelli (1704-1783).
• Ermenegildo Pini (1739-1825).
• Giordano Riccati (1709-1790).
• Giuseppe Angelo Saluzzo (1734-1810).
• Antonio Scarpa (1752-1832).
• Giuseppe Slop De Cadenberg (1740-1808).
• Made extensive studies of Uranus after its discovery in 1781.
• Lazzaro Spallanzani (1729-1799).
• Giuseppe Toaldo (1719-1797).
• Giuseppe Torelli (1721-1781).
• Alessandro Volta (1745-1827).
• Leonardo Ximenes (1716-1786).
• Eustachio Zanotti (1709-1782).
• He observed the 1761 transit of Venus in Bologna and made many observations of the moon, the sun and the planets.
• Giovanni Verardo Zeviani (1725-1808).
• Giovanni Arduino (1714-1795).
• Antonio Cagnoli (1743-1816).
• Leopoldo Marco Antonio Caldani (1725-1813).
• Giovanni Francesco Cigna (1734-1790).
• Domenico Cirillo (1739-1799).
• Domenico Cotugno (1736-1822).
• Luigi Lagrange (1736-1813).
• Paolo De Langes (-1810).
• Pietro Ferroni (1745-1825).
• Vittorio Fossombroni (1754-1844).
• Barnaba Oriani (1752-1832).
• He worked at the Observatory of Brera in Milan, becoming its director in 1802.
• Pietro Rossi (1738-1804).
• An entomologist who was professor of logic at Pisa, becoming professor of natural history there in 1801.
• Leonardo Salimbeni (1752-1823).
• Simone Stratico (1733-1824).
• Giuseppe Vairo (1741-1795).
• Giovanni Battista Venturi (1746-1822).

93. Austrian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Mathematical Society of Vienna (Mathematische Gesellschaft in Wien) was founded in 1903 by Ludwig Boltzmann, Gustav von Escherich and Emil Muller.
• Gustav von Escherich (1849-1935) was born in Mantua, now in Italy but at that time part of the Austrian Empire, and after studying at the University of Vienna, went on to become Professor of Mathematics at the University of Graz.
• He was appointed to the University of Vienna in 1884 and in 1903-04 he was president of the University.
• Before his role in founding the Mathematical Society of Vienna, he had founded the journal Monatshefte fur Mathematik und Physik in 1890 in collaboration with Emil Weyr.
• Emil Muller (1861-1927) was born in Lanskron, now in the Czech Republic but at that time part of the Austrian Empire, and he studied at the University of Vienna.
• He habilitated at the University of Konigsberg and worked there from 1898 to 1902.
• Ludwig Boltzmann was appointed to the chair of theoretical physics at the University of Vienna in 1894 but left Vienna in 1900 because of a dispute with Ernst Mach.
• After Mach left Vienna, Boltzmann returned to the chair of theoretical physics at the University of Vienna in 1902.
• The first meeting of the new Society took place on 14 January 1904 and a report of the meeting appeared in [',' Mathematische Gesellschaft in Wien, Jahresberichte der Deutsche Mathematiker-Vereinigung 13 (1904), 135.','3]:- .
• - On 14 January 1904, a mathematical society was formed with the aim of cultivating pure and applied mathematics through lectures, presentations, etc.
• At the first few meetings of the new Society, the following lectures were given (see [',' Mathematische Gesellschaft in Wien, Jahresberichte der Deutsche Mathematiker-Vereinigung 13 (1904), 135.','3]): .
• Rudolf Inzinger formally re-registered the Society on 10 August 1946.
• The Society continued to operate under the same rules that had been in place in 1938 and at this first general meeting they elected the committee.
• In the autumn of 1948, at the General Assembly on 29 October, the Mathematical Society of Vienna formally changed its name to the Austrian Mathematical Society.
• As we noted above, the Monatshefte fur Mathematik und Physik had been founded in 1890 by Gustav Ritter von Escherich (1849-1935) and Emil Weyr.
• Due to World War II, it stopped publishing in 1941 with Volume 50.
• The decision to award this prize was made at the board meeting on 18 November 1955 at the request of Hans Hornich.

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• The Hungarian Academy of Sciences was named the Hungarian Scholarly Society when it was founded in Pest in 1825.
• The Society published an Hungarian Mathematical Dictionary in 1834 but it failed to achieve the aims we have described [',' B Szenassy, History of Mathematics in Hungary until the 20th Century (Berlin-Heidelberg-New York, 1992).
• ','1]:- .
• The publication of Euclid's Elements in Hungarian by the Society in 1832 was probably more significant.
• The first such foreign member was Babbage, elected in 1833, followed by Gauss and Poncelet in 1847 and John Herschel and Quetelet in 1858.
• The journal was split into two in 1859, then three covering different areas.
• The Academy hit real problems in 1849 following the Hungarian War of Independence.
• In that year the Hungarians had defeated the Habsburgs and declared Hungarian independence on 14 April.
• Following this a combined force from Russia and Austria retook the country and the Hungarian army surrendered on 13 August.
• This allowed the Academy to begin again to support ventures to strengthen Hungarian scholarship and, in 1860, a committee was set up to give special support to mathematics and natural science.
• In the Compromise of 1867 the Hungarian Kingdom and the Austrian Empire became independent states within the Austro-Hungarian Monarchy.
• The Compromise led to rising standards of education in mathematics and the sciences and after pressure from the Academy, the Technical University of Budapest was set up from the polytechnic school in 1871.

95. Warsaw Scientific Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• With the Polish education system controlled by the Russian rulers, Dickstein decided to do what he could to promote a Polish education and he directed his own private school for ten years beginning in 1878.
• In 1884 he was one of the two founders of a series of mathematics and physics textbooks which were written in Polish.
• He also continued publication of Circle of Polish Mathematicians which had begun publishing in St Petersburg in 1880.
• Kuratowski, thinking about the development of Polish mathematics, notes in [',' K Kuratowski, Half a century of Polish mathematics (Warsaw, 1973).','1] the importance of the publications:- .
• In 1903 Dickstein was a founder of the Warsaw Scientific Society and he was important in the development of the Polish Mathematical Society.
• Despite the fact that the Polish Mathematical Society was set up in 1919, the Warsaw Scientific Society continued to play a major role in mathematics.
• For example in 1928 Sierpinski was elected vice-chairman of the Warsaw Scientific Society.

• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Australian Academy of Science was founded on 16 February 1954 by Queen Elizabeth II.
• In 1951 a meeting of leading scientists, technologists and industrialists was held in Canberra to discuss the future of science and technology in Australia.
• A council of 10 was set up with Professor Oliphant as the first President.
• Thomas Cherry was the third President of the Academy, serving in that role from 1961 to 1964.
• This building, now called the Shine Dome [',' Australian Academy of Science website.','1]:- .
• 14 (3) (1987), 59-62.','5]).
• The Moran Medal is named for Patrick Alfred Pierce Moran (1917-1988), an Australian statistician who made significant contributions to probability theory and its application to population and evolutionary genetics.
• The Hannan Medal is named for Edward James Hannan (1921-1994), an Australian statistician who is the co-discoverer of the Hannan-Quinn information criterion.
• For example it sponsored the First International Conference on the Theory of Groups held in 1965 and the Second International Conference on the Theory of Groups held in Canberra in August 1973.

97. Chinese Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The reason is that the Chinese Society for Mathematical Sciences was founded in 1929 and carried out the same functions as the Chinese Mathematical Society until 1936.
• The first President of the Society was Hu T-F who appointed on the foundation of the Society in 1935.
• At first the Society prospered and in 1936 it began publication of two journals, one a research publication, the other a low level popular one.
• By 1938 the country was divided between the part controlled by Japan and the remaining Free China which refused to submit.
• The Chinese Mathematical Society was no longer able to operate and although in some sense it continued to exist, from 1938 to 1945 it could not function.
• Japan surrendered in 1945 but China's problems did not end for the country suffered a civil war for four years between Nationalists and Communists.
• Attempts were made to refound the Chinese Mathematical Society and in 1948 it began again to function in support of mathematics throughout China.
• Hu T-F, the first President ended his term of office in 1948.
• The Communist victory led to the creation of the People's Republic of China on 1 October 1949.
• The Chinese Mathematical Society rapidly expanded under the next President Hua Loo-Keng who was appointed in 1951 and served until 1983.

98. Australian Statistical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Statistical Society of Australia was founded in 1962:- .
• B) (1988), 99-109.','2]) which was founded in 1947.
• Henry Oliver Lancaster was one of the founders of this Society and his contributions are detailed in [',' E Seneta, In memoriam Emeritus Professor Henry Oliver Lancaster, AO FAA (1 February 1913-2 December 2001),','3].
• In 1998 the Statistical Society of Australia and the New Zealand Statistical Association amalgamated their two journals and from that time jointly published The Australian and New Zealand Journal of Statistics.

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• The Norwegian Academy of Science and Letters (Det Norske Videnskaps-Akademi) was founded at Oslo in 1857.
• It should not be confused with the Royal Norwegian Society of Sciences and Letters (Det Kongelige Norske Videnskabers Selskab) which is a much older society founded in Trondheim in 1760.
• The first university in Norway was the Royal Frederick University which was established in Christiania (later renamed Oslo) in 1811.
• The first serious attempt to establish an academy in Christiania was made in 1841 but there were difficulties such as a lack of financial support and a general feeling that at the time Norway did not have a sufficiently broad scientific base to merit the founding of an academy.
• A scientific meeting which took place in Christiania in 1844, however, proved an important step [',' G Hestmark, &#8217;&#8217;A primitive country of rocks and people&#8217;&#8217; - R I Murchison&#8217;s Silurian campaign in Norway, 1844, Norsk Geologisk Tidsskrift 88 (2) (2008), 117-141.','5]:- .
• The meeting, held in Christiania 11-18th July 1844, was the first scientific congress ever to take place in Norway.
• This congress gave Christiania the confidence to move forward and the Professor of Medicine, Frantz Christian Faye (1806-1890), came up with both the initiative and the finance to found the 'Videnskabsselskabet i Christiania' which was inaugurated on 3 May 1857.
• Later 'Christiania' was changed to 'Kristiania' after the city made the change in 1877.
• In the 20th century the name was changed again, this time to 'The Norwegian Academy of Science and Letters of Kristiania', and in 1924 'Kristiania' was removed from the name, only shortly before the city of Kristiania was renamed Oslo.
• The first volume was published in 1881.
• It was the geologist Waldemar Christofer Brøgger (1851-1940) who turned round the finances of the Academy.
• His Royal Society of London obituary states [',' C E Tilly, Waldemar Christopher Brogger, 1851-1940, Biographical Memoirs of Fellows of the Royal Society 3 (10) (1941), 502-517.','7]:- .
• Beyond the realm of science his wide interests and public spirit were extended in the service of his colleagues and countrymen as the first Rector of Oslo University (1907-1911) and as member of the Storthing (1907-1909), and it is due largely to his personality, initiative and outstanding executive ability that numerous public funds were established for education and research.
• Hans Rasmus Astrup (1831-1898) was a highly successful businessman who became the Minister of Labour in the Norwegian Government in 1885.
• The Villa was completed in 1887 and after Astrup's death in 1898, two of his daughters, Ebba and Elisabeth, inherited it.
• Brøgger acquired the Villa for the Academy from the daughters in 1911 thus consolidating its identity as a cultural institution and driving force for Norwegian research development.
• It was through pressure from the Academy that the basic research-oriented Norwegian Research Council was set up in 1949.
• The establishment of the Centre for Basic Research at the Norwegian Academy of Sciences in 1992 was an important step.
• The Kavli Prize for each of the three scientific prizes consists of 1 million dollar and is awarded every two years.

100. Slovak Mathematicians and Physicists Union
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• Reforms in Czechoslovakia in the 1960s came to an end in the spring of 1968 when Soviet troops entered the country.
• The Union of Slovak Mathematicians and Physicists split off from the old Union in 1969 at the fifth congress of the Union of Czechoslovak Mathematicians and Physicists.
• The following description of the Union is taken from [',' Union of Slovak Mathematicians and Physicists website.','1].
• It was founded in 1969 as part of the "Union of Czechoslovak Mathematicians and Physicists", established in 1921 from the original 'Association for Free Lectures in Mathematics and Physics', founded in 1862.
• Following this, they have been involved in the organisation of the Conference on Differential and Difference Equations held in 2006, 2008, 2010, 2012, 2014 and 2017.
• The first was held in Prague in 1962 and the 26th was held in Bratislava, Slovakia, in July 2017.
• The following is taken from the report made in 2011 [',' D Velichova, Report from Slovakia, Meeting of Representatives of European Organisations for Women in Mathematics (Barcelona, 4 September 2011.','2]:- .
• There are about 61 active mathematicians (15 female) working at the Mathematical Institute of Slovak Academy of Sciences and about 500 mathematicians (200 female) working as university lecturers at 10 universities in Slovakia.

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• In 1797 the first volume of the Academy's journal appeared.
• This first volume was named Memoirs of the Royal Academy of Sciences of Lisbon and contained papers written from the founding of the Academy up to 1788.
• The second volume of the Memoirs appeared in 1799 but was renamed Memoirs of Mathematics and Physics of the Royal Academy of Sciences of Lisbon.
• For example, Spain invaded in 1801 and a short war followed, then a more serious conflict occurred in 1807 when Napoleon invaded.
• The Academy of Sciences of Lisbon managed to operate by 1812 when the third volume of the Memoirs appeared.
• After a Civil War in 1832-34, a period of stability in Portugal allowed educational reform in 1836 but the 1846-47 Civil War again led to turbulence.
• In May 1851 a military putsch led to longer term political stability.
• The Academy of Sciences of Lisbon was reorganised in 1851 and this led to a marked increase in mathematical research in the country.
• The 1851 Statutes required the foundation of two new publications, a bulletin for the proceedings of the meetings of the Academy and a journal which would contain scientific papers which could not be published in the Academy's Memoirs.
• The Annaes de Sciencias e Lettras was established in 1857 but was quickly discontinued in 1858.
• The Jornal de Sciencias Mathematicas, Physicas e Naturaes was established in 1866 and continued to be published until 1923.

102. French Applied and Industrial Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The French Applied and Industrial Mathematical Society, Societe de Mathematique Appliquees et Industrielle, was founded in 1983 [',' C Picard and M Martin-Deschamps, SMAI (France), European Mathematical Society Newsletter 40 (June, 2001), 18-19.','1]:- .
• One might, however, see the creation of the Society being ten years earlier for, in 1973, the Groupe pour l'Advancemant des Methodes Numerique de l'Ingenieure was founded.
• When the French Applied and Industrial Mathematical Society was founded in 1983 the Groupe pour l'Advancemant des Methodes Numerique de l'Ingenieure became part of the broader organisation.

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• The National Academy of Sciences of Belarus was founded in Minsk on 1 January 1929 (as the Belarusian Academy of Sciences), the date being chosen as it was the tenth anniversary of the founding of the Byelorussian Soviet Socialist Republic.
• An earlier organisation, the Institute of Belarusian Culture, had been established in 1922 and a decree made on 13 October 1928 set up the Belarusian Academy of Sciences by reorganising the Institute.
• In 1936 the name of the Academy was changed to the Academy of Sciences of Byelorussian SSR.
• Russia and ten other former Soviet republics declared themselves independent on 21 December 1991 and founded the Commonwealth of Independent States.
• The USSR legally ceased to exist on 31 December 1991.
• In 1991 the Academy of Sciences of Byelorussian SSR became the Academy of Sciences of Belarus, adopting its present name in 1997.
• The Academy had a staff of 128 when it was first established but by 1941 this had risen to about 750 with nine Institutes.
• By the end of the war the staff was less than half of the 1941 total with around 360 staff in post.
• Following the end of World War II the Academy was rebuilt, with eight Institutes beginning work in 1945 increasing to 29 by 1951.
• The Institute of Mathematics of the Academy was founded in Minsk in 1959.
• Seryya Fizika-Matematychnykh Navuk in1992.
• Seriya Fiziko-Matematicheskikh Nauk in 2001.

104. Estonian Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Estonian Mathematical Society can trace its origins back to 1926.
• Although Estonia had been dominated by Russia for a long period, it had gained its independence in 1917 only to come under German control.
• In 1920 Estonia was again able to assert its independence.
• Tamme writes in [',' M Abel, 75 Years of the Estonian Mathematical Society, European Mathematical Society Newsletter 41 (September, 2001), 18-19.','1]:- .
• On 17 June 1940 Soviet troops occupied the whole of Estonia and on 21 July the Estonian government, having little choice, adopted a resolution to join the USSR.
• Over 10,000 people were deported on the night of 13 June 1941 alone.
• Political events again intervened, for on 22 June 22 1941 Germany attacked the USSR, Estonians then attacked the occupying Soviet forces, largely defeating them before the German army took control of Estonia.
• These suggestions came to nothing, as did a more serious attempt to establish a mathematical society in 1983.
• A Soviet liberalization campaign in the late 1980s provided an opportunity and, in 1987, mathematicians achieved their aim.
• The first meeting of the Estonian Mathematical Society was held on 17 September in the same Festival Hall that the Academic Mathematical Society had held its first meeting in 61 years before.
• The 1987 meeting was attended by 118 members of the new Society together with 52 guests.

105. Kosovar Mathematical Society
• function win1(file,h,v) {if (v > screen.height-80) {v = screen.height-80;} .
• The Yugoslav Mathematical Society was formed in Belgrade in 1937 and this served mathematicians in Yugoslavia until the break-up of that country.
• Many parts of Yugoslavia declared independence in 1991-92 but Kosovo remained a part of Serbia and Montenegro.
• The Society of Mathematicians and Physicists of Serbia had been founded on 4 January 1948 and it had split into two societies, mathematics and physics, in 1981 and at that time the Mathematical Society of Serbia was founded.
• The report [',' Prime Minister Thaci meets the winners of the Mathematical Olympiad of Kosovo, The Republic of Kosovo, Office of the Prime Minister (26 May 2009).','1] gives details of the reception for the winners in May 2009:- .
• He was born on 4 June 1984 and, while at secondary school, won First Prize in the Kosovo Mathematical Olympiad held in Shtimje, Kosovo in 1997.
• He studied at the University of Prishtina, making a study visit to Paris in 2001 funded by the World University Service, Austria.
• In addition to his university position, in 2011 he was appointed Foreign Affairs Advisor to the President of the Republic before becoming, successively, a member of the Board of Directors of the Pro Credit Kosovo Bank, the Kosovo-American Education Fund and the State Council on Quality.
• In fact Qendrim Gashi, the Society's first President was also Chief Coordinator of the Kosovo Mathematical Olympiad in 2009-2012.
• He attended both elementary and secondary school in Gjakove and began his university studies at the Higher Education Institution in Prishtina in 1967.
• This Higher Education Institution became the University of Prishtina in 1970.
• He graduated in 1971 and in that year began his postgraduate studies at the University of Zagreb.
• He was awarded a Master's Degree in 1974 and was appointed as an assistant at the University of Prishtina while continuing to undertake research for his doctorate at Zagreb advised by Sibe Mardesic.
• He was awarded the degree in 1981 for his thesis Shape fibrations for topological spaces.
• He was steadily promoted at Prishtina becoming a docent in 1982, an extraordinary professor in 1987 and a full professor in 1994.
• Sentenced to ten years in prison, he was held in various prisons until March 2001 when he, along with many others held by the Serbs, was released.
• An important aspect of Haxhibeqiri's contribution to the scientific community has been his authorship of university textbooks (among which is the book "Topology" published in 1989).

106. Estonian Statistical Society
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• She attended school in Tallinn, graduating in 1952, and then studied at the University of Tartu, being awarded her first degree in 1957.
• She studied for her doctorate advised by Gunnar Kangro (1913-1975), who was an expert in summation theory and held the Chair of Mathematical Analysis at the University of Tartu.
• She was awarded her doctorate in 1963.
• In 1995 a new Council was elected: Ene-Margit Tiit was re-elected President for three years; Ulo Randaru, Helina Vigla and Reet Malbe were elected Vice-Directors of the Statistical Office; Villem Tamm, Tonu Kollo and Ebu Tamm were elected as Ordinary Council Members.
• Here is the details of some of these conferences in 1995-97.
• 17-18 April 1997, in Rakvere.
• The Society began publication of the Eesti Statistikaseltsi Teabevihik (Journal of Estonian Statistical Society) in 1993.
• Ene-Margit Tiit writes [',' E-M Tiit, Experiences in Publishing a Statistics Journal in Estonia, Proceedings International Conference on Teaching Statistics 5 (1998), 502-504.','2]:- .
• the Estonian Statistical Society was founded [in 1992] and since then has publish a journal twice a year.
• 15-16 April 2008, in Tallinn.
• 13-14 April 2010, in Tartu.
• 20-21 April 2011, in Tallinn.
• 12-13 November 2013, in the National Library of Estonia.
• Here is the first paragraph of one from the Lithuanian Statistical Society [',' Estonian Statistical Society website.','1]:- .
• He received a licentiate in statistics from Lund University in 1954 and he was appointed to the Department of Statistics there.
• in 1961 for his thesis Contributions to the theory of estimation from grouped and partially grouped samples.
• He became the first professor of statistics at Umea University in 1965.
• He was elected to the International Statistical Institute in 1968 and served in several roles including president in 1989-91.
• After the Baltic countries became independent in 1991 he made visits leading to exchange programmes in statistics between Nordic and Baltic countries.
• 2007-2010 Kalev Parna .
• 2010-2013 Imbi Traat .
• 2013-2016 Kaja Sostra .

107. Irish Mathematical Society
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• The Irish Mathematical Society (Cumann Matamaitice na heireann) came into being on 14 April 1976 when a constitution, based on that of the Edinburgh Mathematical Society, was accepted by a meeting held in Trinity College Dublin following a symposium organised by the Dublin Institute for Advanced Studies.
• This informal group, which met in the late 1960's and consisted of J Kennedy (University College Dublin), F Holland (University College Cork), D McAlister (Queens University Belfast), and T West (Trinity College Dublin), decided in 1967 to circulate the Irish mathematical community, north and south, to determine the possibility of organised research symposia or summer schools on selected topics.
• The success of this Summer School led to further Summer Schools each year from 1971.
• These were 'Complex Function Theory' (1971), 'Numerical Analysis' (1972), 'Group Theory and Computation' (1973), 'Spectral Theory' (1974) and a symposium on 'Harmonic Analysis and Topological Algebra' in December 1975 at the Dublin Institute for Advanced Studies.
• In [',' T C Hurley, Report of the First Year&#8217;s Activities, Newsletter of the Irish Mathematical Society 1 (1978), 6-7.','3] the events of the first year of the Society are recorded:- .
• The first was a 'Group Theory' conference held at University College Galway 12-13 May 1978.
• In the Newsletter of 1979 T C Hurley writes [',' T C Hurley, Report on the Activities of the Society, Newsletter of the Irish Mathematical Society 2 (1979), 3-5.','4]:- .
• In 1986 it was given the new title of the Bulletin of the Irish Mathematical Society which continues to appear twice a year.
• The conference is held in September, the first being held in 1988 at Trinity College Dublin.
• In 2016 the annual conference was held in Trinity College Dublin on 15-16 April to commemorate the 40th anniversary of the founding of the Irish Mathematical Society.
• 1981 J J H Miller .
• 1991 R Timoney .
• 2001 E Gath .
• 2011 S Wills .

108. Catalan Mathematical Society
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• The Institute of Catalan Studies (Institut d'Estudis Catalans) was founded in 1907 and the Catalan Society for Physics, Chemistry and Mathematics was founded in 1931 within its Science Section.
• The Mathematics section with that Society expanded so rapidly that in 1986 the four branches of the Catalan Society for Physics, Chemistry and Mathematics each became a separate Society.
• The Society set out its aims in its statutes which include a mission [',' S Xambo-Descamps, The Catalan Mathematical Society, European Mathematical Society Newsletter 36 (June, 2000), 3.','1]:- .
• The Society publishes two journals in Catalan [',' J de Sola-Morales, The Catalan Mathematical Society and Mathematics in Catalonia, European Mathematical Society Newsletter 93 (2014), 47-48.','2]:- .

109. Mathematical Association of America
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• The Mathematical Association of America was set up to manage publication of the American Mathematical Monthly which had begun publication in 1894.
• Herbert E Slaught who edited the Monthly from 1913 to 1916 approached the American Mathematical Society asking if it would take over support for the Monthly.
• He received 450 positive responses, and on 30-31 December 1915 Earl R Hedrick chaired a founding meeting of 104 people.
• Hedrick was elected president, Huntington was elected vice president, and Council consisting of 12 members was appointed.
• Straley writes in [',' T H Straley, A Brief History of the Mathematical Association of America.','1]:- .
• 1921 George A Miller .
• 1931-32 Eric T Bell .
• 1941-42 Raymond W Brink .
• 1951-52 Saunders Mac Lane .
• 1961-62 Albert W Tucker .
• 1971-72 Victor Klee .
• 1981-82 Richard D Anderson .
• 1991-92 Deborah Tepper Haimo .
• 2001-02 Ann E Watkins .

110. Croatian Mathematical Society
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• The Croatian Mathematical Society was founded in 1990.
• However the Society has a much longer history than that date would suggest since it was on 18 April 1990 that the Croatian Society of Mathematicians and Physicists made the decision to split into two separate societies, namely the Croatian Mathematical Society and the Croatian Physics Society.
• The Croatian Society of Mathematicians and Physicists had been founded in 1949 but long before that there had been a mathematics section of the Croatian Natural History Society, itself founded in 1885, which was the beginnings of the Mathematical Society.
• The Croatian Natural History Society was founded in 1885 in Zagreb with a view to developing and popularising the natural sciences.
• From 1886 to 1938, the Society also published Glasnik Hrvatskog naravoslovnog društva, in which research papers on mathematics appeared.
• The Croatian Natural History Society founded the Mathematical-Physical Section in 1945 and a year later, the Department of Mathematics and Physics in the University of Zagreb was established, which started organising workshops for teachers and popular lectures on mathematics and physics.
• The Croatian Society of Mathematicians and Physicists was founded in 1949 and existed until 1990.
• When the Society was founded in 1949 Croatia was a federal unit in Yugoslavia so the Croatian Society became a member of the Alliance of Mathematicians, Physicists and Astronomers of Yugoslavia.
• In 1953, summer courses in mathematics and physics for high school teachers began to be organised, which later developed into the present 'Seminars for Elementary and High School Teachers'.
• The journal Glasnik matematički began publication in 1966 and has continued following the split and the foundation of the Croatian Mathematical Society.
• Now before looking at the events of 1990, we should note that the Croatian Society of Mathematicians and Physicists was most active in Zagreb.
• However the Rijeka Society of Mathematicians and Physicists was founded in 1951 as a branch of the Croatian Society of Mathematicians and Physicists.
• on 18 April 1990 in the Mathematical Lecture Theatre in the Faculty of Science on Marko Maruli Square in Zagreb.
• You can read a version of the minutes [',' S Martinović, XXXIX Annual General Assembly of the Society of Mathematicians and Physicists of Croatia, Glasnik matematički (3) 25 (45) (1990), 430-431.','2] taken at that discussion at THIS LINK.
• It was attended by 187 participants and had 24 invited addresses.
• 2000: Second Croatian Mathematical Congress, 15-17 June, University of Zagreb.
• 2004: Third Croatian Mathematical Congress, 16-18 June, University of Split.
• 2012: Fifth Croatian Mathematical Congress, 18-21 June, University of Rijeka.
• 2016: Sixth Croatian Mathematical Congress, 14-17 June, University of Zagreb.
• This journal was taken over by the Croatian Mathematical Society when the Croatian Society of Mathematicians and Physicists split in two in 1990.

111. Polish Mathematical Society
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• The Polish Mathematical Society began its existence in Krakow as the Mathematical Society in 1917.
• Steinhaus lived in Krakow during 1916 and he relates in [',' H Steinhaus, Reminiscences (Polish) (Cracow, 1970).','1] how, despite the war, it was safe to walk in Krakow:- .
• The informal Mathematical Society, established in 1917, became officially constituted after the end of World War I.
• on 2 April 1919 a meeting was held in the Philosophy Seminar at 12 St Anne Street, Krakow, at which the Mathematical Society was constituted.
• The session was introduced by Zorawski who said [',' J Piorek, Polish Mathematical Society, European Mathematical Society Newsletter 32 (June, 1999), 17-18.','7]:- .
• However, after some months of deliberation, five leading Warsaw mathematicians joined the Mathematical Society in Krakow on 19 September 1919.
• It was announced at that meeting of the Krakow Society that an Extraordinary Meeting would be held on 29 September [',' J Piorek, Polish Mathematical Society, European Mathematical Society Newsletter 32 (June, 1999), 17-18.','7]:- .
• Other leading mathematicians served the Society in the period between the two wars including Sierpinski who was elected president of the Polish Mathematical Society in 1928 and Mazurkiewicz who was president in 1933-35.
• In 1939, just before the start of World War II, Banach was elected as President of the Society.
• In 1936 a committee had been set up by the Polish Academy of Learning to look at the way forward for Polish science.
• They prepared a report "On the present state and needs of mathematics in Poland", completed in 1937, which formulated future requirements.
• The Polish Mathematical Society was unable to operate during the war but was reborn in 1945.

112. Luxembourg Mathematical Society
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• The Luxembourg Mathematical Society began its existence in 1970 under the name Seminaire de Mathematique de Luxembourg.
• In 1988 the members of the Seminaire de Mathematique de Luxembourg planned to create the Societe Mathematique du Luxembourg, or in English "The Luxembourg Mathematical Society".
• Let us give a few details about Jean-Paul Pier (1933-2016).
• He taught mathematics at the Lycee de Garcons in Esch-sur-Alzette from 1956 to 1980 and he was a professor at the Centre universitaire de Luxembourg (now the University of Luxembourg) from its founding in 1974 until he retired in 1998.
• He founded the Mathematics Seminar at the University in 1971.
• Vingt-cinq siecles de mathematiques (1996) and Mathematical analysis during the 20th century (2001).
• After the present Society was founded in 1989 it has continued to organise symposia, for example The Development of Mathematics 1900-1950 (1992), Developments in Mathematics at the Eve of 2000 (1998), Conference on Harmonic Analysis (2002), Conference on Poisson Geometry (2005).
• Seminar sessions take place at Campus Belval on Tuesdays [',' Luxembourg Mathematical Society website.','1]:- .
• The first volume of the j