Page:A History of Mathematics (1893).djvu/181

 invented by John Napier, Baron of Merchiston, in Scotland (1550–1617). It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure, Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers. What, then, was Napier's line of thought?

Let AB be a definite line, DE a line extending from D indefinitely. Imagine two points starting at the same moment; the one moving from A toward B, the other from D toward E. Let the velocity during the first moment be the same for both: let that of the point on line DE be uniform; but the velocity of the point on AB decreasing in such a way that when it arrives at any point C, its velocity is proportional to the remaining distance BC. While the first point moves over a distance AC, the second one moves over a distance DF. Napier calls DF the logarithm of BC.

Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines; the line AB was the sine of 90° and was taken $$\scriptstyle{= 10^7}$$; BC was the sine of the arc, and DF its logarithm. We notice that as the motion proceeds, BC decreases in geometrical progression, while DF increases in arithmetical progression. Let $$\scriptstyle{AB=a=10^7}$$, let $$\scriptstyle{x=DF}$$,