The line being said to be cut in this way according to the third definition of the sixth book of Euclid, as the ratio of the whole will be the same to the larger segment, as the larger segment to the smaller segment. In what manner this section ought to be done, being shown in Prop. 11, Book 2, and Prop. 30, Book 6 of Euclid. and Ramus, El.3, Book 14, has shown just how the line can be cut proportionally. So thus very briefly and not improperly, the same is indicated that Euclid set forth with the greatest detail. These segments are, as of the whole, so between themselves asymmetric. and therefore, if the number of the whole length of the line shall be given, none of these segments in any way will be able to be expressed with absolute numbers, either with integers or fractions with accuracy; but with irrationals, we will be able to be the closest of the parts being sought yet, and to reach the lengths being sought. Nevertheless they will be able to be defined carefully with the numbers from Algebra, the common use of which will be possible with hardly any difficulty, except with the usual custom of being reduced to the absolute. For let AB be of 10 parts, which being bisected in E, and the line EC being drawn to the opposite angle C of the square, and EF, EC are made equal: the line BF or BH will be the larger and CH the smaller segment of the line BC or AB, being cut proportionally. And BC, BH, HC will be continued proportionals: because BG the square of the larger segment being equal to the rectangle DH, being described by the whole and the smaller segment¹.


If we wish to express this with numbers, the square of the line EB is 25, and the square of the line BC 100: therefore the square of the line EC or EF will be 125: with which no side with the numbers given will be possible. Therefore, it [i.e. the true value] will be more than the segment BF, .125 - 5. and if we wish to reduce this number to the absolute², it will be .125, 11180339887499 almost, from which if taken EB, 5: there will remain the larger segment 6180339887499 and the smaller CH 3819660112501,. For these lines of the two approximations being added; or, being continued on either side, by subtraction of the smaller from the larger, as far as it will have been seen; the same ratio will be maintained always, as you see here, by numbers accurate by algebra, near the true values.
But it is permissible for these absolute numbers to be somewhat lacking from the truth: nevertheless we will be able with whole numbers, with the continual addition of two nearby, to arrive at that which we wish to be near. With any two being taken for the start: with the difference of the square of the middle and the oblong comprising the first and third, serving the same everywhere for the whole series³; it is permitted that the numbers themselves be increased the most. and therefore any three being close, very little departing from that proportion which we seek⁴.



The important use of these sections is in Geometry.
If the radius of a circle is cut proportionally, the larger segment is the side of a decagon inscribed in the same circle. 8.e.18 of Ramus. and the side [i.e. diagonal] of a pentagon is the larger segment being subtended by two sides of a pentagon. and if a dodecahedron and an icosahedron are being inscribed in a cube: the side of the dodecahedron is the smaller segment, the side of the icosahedron the larger segment for the side of the cube.

If any given number to be cut proportionally: you desire to know the larger segment, from the logarithm of the given number, by taking away the difference of the ratio of the extreme and the mean: the remainder will be the logarithm of the larger segment. For let the given number be 55:

And if we wish to know some other more distant term of the series in continued proportion, with neither the larger nor smaller being given; the said difference4 being multiplied by the number of the intervals, between the given number and the sought; the product, with the logarithm of the given number having been added, will give the logarithm of the larger number being sought; the same having been subtracted will leave the logarithm of the smaller. For let the Given be 55 and the third larger from the given being sought, the [log of the ] difference triplicated will be 062696292074994; which being added to logarithm of the given, gives 236732561024418, the logarithm of the number 23298373876244; the same having taken away from the given, it has left 11133996874430, to which corresponds⁵ 1298373876249. Consult Ch. 17 about this.
With these numbers, and in general with all numbers (integers, or fractions: absolute or algebraic) which have arisen by the addition of the two nearest numbers that have come before ; if whatever nearest being taken, of which the number is odd, and the ratio to the fourth power⁶, as five, nine, thirteen the totals [i.e. logs] of the extremes; with the remainders, as seven, eleven, fifteen will be the various differences of the mean.

¹ i.e. the length BH is required that satisfies the relation BC/BH = BH /CH ; if BH is called x, and CH is taken to be 1, then (1 + x)/x = x/1, and the positive value of the ratio is determined to be (1+5)/2 , the 'golden ratio', often denoted , associated with the Fibonacci sequence. Briggs defines log as the difference of the extreme (BC) and mean (BH) values of the ratio.

²(Briggs' explanation is elaborated on a little). Therefore, it will be more than the segment BF, .125 - 5. and if we wish to reduce this number to the absolute, it will be .125, 11180339887499 almost, from which if taken EB, 5: there will remain the larger segment 6180339887499 and the smaller CH 3819660112501, being BF²/BC. These lines by addition are seen to be close to the two required segments, see rows 7 and 8 of Table 27-2A, in which BC is taken as 1. This process can be continuing on either side, using , *, and in the table, because it will have been seen; the same ratio will be maintained always, as you see here, by numbers accurate by algebra, near the true values.


³ If p, q, r are 3 successive numbers, we have pr - q² = 1.
In terms of the Fibonacci sequence as defined by f₀ = f₁ = 1; f_n+1 = f_n + f_n-1, for n > 1; it can be shown that f_n+1.f_n-1 - f_n² = 1, and hence for large n, f_n/f_n+1 ~ f_n-1/f_n ~ 1/, the approximation improving as n increases. Hence, the larger length BH of the section in the unit segment BC is ^-1 and the smaller length CH is 1 - ^-1 = 2 - .

⁴We may write the ratio in the form 55^-3/55 = 55/55³, or 13 233 55².

⁵Briggs talks about 'superposing' the ratio four times on itself: a continuation along the same lines.

⁶If we consider successive powers of ⁴, for a given number N in the Fibonacci sequence, then the various powers : ... , N^-15, N^-11, N^-7, N^-3, (N¹), N⁵, N⁹, N¹³, ... give the possible terms, on making the approximation.