Numbers: Their History and Meaning by Graham Flegg

Author:Graham Flegg
Language: eng
Format: epub
Publisher: Dover Publications, Inc
Published: 2013-03-26T16:00:00+00:00

P moves throughout with this constant speed which is numerically equal to the length of AB and CD; Q moves with a decreasing speed always numerically equal to QD. At any given moment, AP is defined to be the logarithm of QD.

Strictly speaking, we have here a situation where there is a continuous change of speed, and we might think that the techniques of the calculus are needed. However, if we consider small enough constant intervals of time and assume that during each interval the speed of Q is constant and equal to its velocity at the beginning of the interval, then successive lengths of QD, corresponding to the beginnings of these intervals will be in geometric progression. Since P travels with constant speed, the corresponding successive lengths AP will be in arithmetic progression.

We can obtain an indication of the way in which Napier calculated his logarithms by looking at the extract from his table below. Here, we start with zero in the arithmetic progression, taking differences of 1, and with 10 000 000 in the geometric progression and a common ratio of

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