Letter to the Editor

Dear Editor

I was delighted to see in Volume 29, Issue 3, of the Bulletin (pp. 154–166), an article by Tom Whiteside, to which I turned eagerly. It is a tantalising and stimulating article, with various suggestive, but unexplained allusions. Readers who have enjoyed the article may be interested in an expansion of two of these.

Napier’s way (which was not just by computing powers of 1 – …). (From the Abstract)

In the first section of Napier’s Constructio Napier says that ‘a logarithmic table is picked out from numbers progressing in continuous proportion’. Throughout the Constructio he consistently describes the numbers for which he calculated logarithms as sines. So he is looking for a geometrical progression within which he can identify sines (typically 7 digit integers), varying from sin down to sin , in steps of 1 minute. In sections 14 and 15 Napier proposes various ratios with which he could easily compute terms of a geometrical progression by subtraction to effect multiplication by a number less than, but close to, 1. In section 16, he calculates the first 100 terms of a progression using as common ratio. This is a good choice because starting from the first term of , the steps down (after the first) are less than 1 unit apart, so that if this were continued it would get near enough to every sine. Why then does he stop after 100 steps and not go on? In fact it would take rather more than 81,000,000 subtractions to obtain the geometric progression that he needed by this method, and my estimate of the time required to do this with 14 digit numbers is getting on for two centuries. All the other devices which he introduces are to enable the process to be completed in his lifetime.

Only Kepler, whose own 1624 table is a hybrid born of both Napier and Briggs, would dream of constructing logarithms Napier’s way. (p. 163, line -7)

Kepler learnt of Napier’s logarithms from a 1618 publication of Ursinus, and unaware of Napier’s death, wrote Napier an appreciative letter. Kepler’s logarithms were similar to Napier’s in two respects: firstly Kepler’s logarithms started from log 100000 = 0, and increased down to log 100 = 690770, in a thousand steps, 100 apart; secondly Kepler established and used two theorems from Napier’s Constructio which give upper and lower bounds firstly for a single logarithm, and secondly for the difference between two logarithms. Because Kepler focused on three digit numbers it was particularly easy for him to apply this second theorem to consecutive integers.

Briggs’ account of logarithms was not published until 1624, the same year as Kepler’s. Briggs and Keple did correspond, but little of this survives. An English translation of much of Kepler’s account of logarithms is given in Charles Hutton’s Mathematical Tables currently available on Amazon, and it is quite different from Briggs’ in both style and content. There are however interesting similarities. Briggs’ first publication of his logarithms was of numbers from 1 to 1000 in 1617. Kepler’s tables are of numbers from 1 to 1000, each multiplied by 100. Both Briggs and Kepler make good use of the prime factors of the numbers whose logarithms they seek. Both insist on working with a single geometric progression, needed to support Napier’s claim that ‘the logarithms of similarly proportioned sines are equidifferent’, the key to the power of logarithms, which had been recognized by Chuquet in 1484, and by a succession of sixteenth-century mathematicians, in the matching of terms of a GP with those of an AP. Napier illustrates this claim within his table 1, a single geometric progression. But he applies it to the terms in his tables 2 and 3, which contain no terms of the table 1 progression. Both Briggs and Kepler fill this gap by interpolating within one progression with geometric means. Kepler explains how it is possible to interpolate more than 1 billion terms of a geometric progression between 7 and 10, with 30 square roots. Briggs worked with up to 54 square roots.