NAPIER COMMEMORATIVE LECTURE. 5D 
A 
10’ 
applications of the ratio 1 — 107 bal 
the term 10’ (1——.)" is got from 10’ by n successive 
The number of the ratios for 10’ (1 - Wt " would thus ben. 
With this definition of logarithms, the logarithms of the 
numbers denoted by the Greek letters in the above 
scheme would be the suffixes of the corresponding a’s, 
and the last term in the 69th column of the Third Table 
would have for its logarithm 6,900,000. 
Such a table of logarithms would correspond exactly to | 
Burgi’s Table of Red and Black numbers. But Napier did 
not introduce his logarithms in this way. The method 
which he followed, and the definition which he gave, are 
very remarkable: for they indicate, as we have remarked 
above, that he had already in his mind, though it may be 
very dimly and vaguely, the principles on which Newton, 
some 50 years later, built up the Differential Calculus. It 
will be best to borrow from the Constructio the different. 
steps in Napier’s argument.’ 
“Hitherto we have explained,” he says, ‘‘ how we may most 
easily place in tables sines or natural numbers progressing in 
geometrical proportion.” 
22. “It remains, in the Third Table at least, to place beside the 
sines or natural numbers decreasing geometrically their logarithms. 
or artificial numbers increasing arithmetically.” 
23. “To increase arithmetically is, in equal times, to be aug- 
mented by a quantity always the same.” 
24. “To decrease geometrically is this, that in equal times, first: 
the whole quantity then each of its successive remainders is 
diminished, always by a like proportional part.” 
1 Cf, Macdonald’s English translation of the Constructio, Edinburgh, 
1889. In quoting from this work, Macdonald’s rendering is followed. 
