To improve on the accuracy of a number found from the Chiliades by the use of proportional parts

We have shown by these methods how we will be able to find the logarithms of all numbers, prime and composite, whole and fractional. Of these, we have described thirty Chiliads¹, being comprised of two columns. Of these, the first on the left-hand side contains the increasing series of the natural numbers, from unity to 20,000: then, this series having been interrupted, from 90,000 to 100,000. The other column adjoining the first, has the logarithms of the absolute numbers, and the differences of these: Which not only reveal an error, if these by chance have sneaked in; but also provide proportional parts, as anything we may seek more accurately from this, because it is contained in the same table. However that part from proportionality, which is obtained in addition from these differences, is not absolutely perfect; but always departs a little from the accurate truth we look for. Even with whole numbers wherever they are increasing uniformly, but where the logarithms are greater they have a smaller increment from that [increase]: if for an absolute number, the logarithm of the next smaller number, shall have been increased by a proportional part, that part by the intermediate difference being required, will be always less than it should have been. But on the other hand, if through the logarithm given the whole number is being sought, that by the proportional part will be regularly enlarged more than is right². The same inconvenience happens with the Sines, Tangents, and Secants; and the same with all tables of numbers, where the differences from another part are equal, from unequal remainders. Nevertheless, when a smaller [number] is of unequal differences, with that smaller number will be the difference from the truth. For this reason, the part of proportionality recedes further from the truth in the first of the Chiliads, than with those in the later part: besides we shall be able to bring these [intermediate logarithms] out. (As often as the work demands it, and there could have been time for leisure, to find the number being sought with more accuracy.)

Being given the logarithm (of which the absolute is being sought) in the first place the characteristic is removed, (by which the number being sought can not be changed in any way, but can only be moved in steps higher to a place more removed from unity). Then being taken away from the logarithm of ten, 1,00000,00000,0000 leaving what may be not inconveniently called the Arithmetical complement. This remainder (by adding first the characteristic that we see will have been the most useful) we look within the Chiliad, and the nearest logarithm that is less being taken, and (with the absolute number of the same selected at the time) being added to the given logarithm. The final total in the same Chiliad is necessarily increased, from which the same corresponding number is taken, that we first required. All of this can be summed up in an example. The mean proportional between unity and 1200 will be found, of which the logarithm is 3,07918,12460,4762. On the other hand, the logarithm of the mean being sought is half of that, by the definition of the logarithm, Ch.1 and 2.Axiom 2,Ch.2, which is of course, 1,53959,06230,2381. Having removed the characteristic and being taken from 1,00000,00000,0000 leaves 0,46040,93769,7619. The nearest logarithm to this, if we ignore the characteristic, is found in the 29^th Chiliad 3,46029,63267,5746. And this corresponding to the region of whole numbers set aside for 2886, being added to that given, the total except the characteristic will be 0,99988,69497,8127. Then I look in the 100th Chiliad, where the nearest lesser number I discover 4,99988,27246,5701. And that from (the corresponding whole number) 99973, which being increased by the fraction proportion to 99973,97261,3004. The quotient of this number (being divided by the number 28860 before being separated out) will be ³ 34641,01615,1422.

All these are demonstrated by Axiom 2, Ch.2. It is for .1200 the other now remaining factor 2886, of which the logarithms being added gives 0,99988,69497,8127 equalling the logarithm of the product, undoubtedly of the number 99973,97261,3004. This product being divided by the given factor 2886 will give in the quotient remaining, the factor in question 34641,01615,1422, which is almost equal to the mean proportional sought 34 641016151377546.

But as with this method by division, which being made by a number with four places the working is excessive, other composite numbers should be thought of being selected, which are shown on the preceding page [Table 11-1]. The numbers on the left are factors, from which by continued multiplication these become composite. To the right are the logarithms of the same. For if a number is being divided by a composite, with the place of the division being taken by the same factors, which find the required quotient more easily. But nevertheless it should be shown with an example. The mean proportional between 1 and 10800 is required, of which the logarithm is 4,03342,37554,8695. The logarithm of the mean sought is the half of that, 2,01671,18777,4347 of which the complement is 0,98328,81222,5653. The logarithm being found in the table nearest to this description is 0,98227,12330,3957. (from the position of 9600), which added to that given gives 0,99898,31107,8304. The logarithm of the number 99766, which by the proportional part corrected, is 99766,12651, 6521. This being divided by 9600 (or continually by 2, 6, 8 as Lemma 2, Ch. 5 advises), the quotient will be 1039230484547.

You can examine how this method works:

Here finally we consider the characteristic, hitherto for the whole working almost being ignored. For since the logarithm of the number A has 2 for a characteristic, the first three places for the root being sought signify whole numbers, and subsequently the numbers express a fractional part being added on to that given, as before in Ch.4. Between this characteristic and the rest of the working, there is so much difference, because of the nearest complementary number, which was being sought formerly amongst the Chiliades; this actually being sought in the table of composite numbers described here: that divisor may be a prime number; here this will always be a composite number, all the divisors being substituted have been put in that location, which allow the quotient to be found more easily. As for the divisor 9600: 8, 6, and 2 are substituted, of which the first divides the given number, the second the given quotient, the third that second quotient.

¹ A Chiliad represents a thousand. Briggs is describing the actual tables of logarithms, which are not presented here.

² Briggs is coming to terms with the non-linear nature of the log function, which has a decreasing gradient as the argument is increased.

³ Let us indicate some aspects of the underlying analysis being used by Briggs in interpolating between adjoining numbers and their logarithms from the log tables:
Note initially Mercator's series expansion for the natural logarithm (ln):

ln|1 + x| = x - x²/2 + x³/3 - x⁴/4 +... + (-1)^n-1xⁿ/n + ... , valid for -1 < x 1.     [11.1]

log|1 + x| = log(e).ln|1 + x| = log(e). ( x - x²/2 + x³/3 - x⁴/4 + ... + (-1)^n-1xⁿ/n + ...)      [11.2]

log|10^m + x| = log(10^m|1 + x/10^m|) = m + log|1 + x/10^m| = m + log(e).ln|1 + x/10^m|     [11.3],

Now consider the logarithm A in Table 11-2, corresponding to the unknown positive number N, or 1200, of which the best estimate is required. A complementary number M is sought such that MN is just less than the largest power of ten available (10⁵), for which the corresponding absolute numbers have the most significant places. The complement of A is found without the characteristic and the tables inspected until a suitable number M (2886 in this case) is located, with a logarithm just less than A's complement; and the product M N gives a number just less then 100,000, with a logarithm as close to 1 as possible. The sum of the logs A + C i.e. 4,99988,69497,8127 corresponds to this product, for which the absolute number M₁ is required. The rest follows by proportion, for M₁ lies between the two numbers 99973 and 99974, with logs 4,99988,27246,5701 and 4,99988,70687,5301 respectively; and with a difference 0,00000,43440,9600. Thus, the linear approximation gives:
M₁ = 99973 + 422512426/434409600 = 99973.972613004.
Then N = M₁/M = 99973.9726129/2886 = 34.641016151422
(The true value being 34.6410161513775), as Briggs now asserts.