Page:A History of Mathematics (1893).djvu/182

 $$\scriptstyle{y=BC}$$, then $$\scriptstyle{AC=a-y}$$. The velocity of the point C is $$\scriptstyle{{d(a-y) \over {dt}}=y}$$; this gives — nat. log $$\scriptstyle{y=t+c}$$. When $$\scriptstyle{t=0}$$, then $$\scriptstyle{y=a}$$ and $$\scriptstyle{c=-\text{nat. log }a}$$. Again, let $$\scriptstyle{{dx \over dt}=a}$$ be the velocity of the point F, then $$\scriptstyle{x=at}$$. Substituting for t and c their values and remembering that $$\scriptstyle{a=10^7}$$ and that by definition $$\scriptstyle{x=\text{Nap. log }y}$$, we get

It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases. He took the logarithm of $$\scriptstyle{\sin{90}=0}$$; i.e. the logarithm of $$\scriptstyle{10^7=0}$$. The logarithm of $$\scriptstyle{\sin{\alpha}}$$ increased from zero as $$\scriptstyle{\alpha}$$ decreased from 90°. Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes, Stifel, and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables. Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio. In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute.

Henry Briggs (1556–1631), in Napier's time professor of geometry at Gresham College, London, and afterwards professor at Oxford, was so struck with admiration of Napier's book, that he left his studies in London to do