Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/47

The discovery of logarithms

One great invention made early in the seventeenth century must be specially referred to; that of logarithms by John Napier (1550—1617). The special importance of this invention in relation to our subject is due to the fact of that essential connection between the numbers $\pi$ and $e$ which, after its discovery in the eighteenth century, dominated the later theory of the number $\pi$. The first announcement of the discovery was made in Napier's Mirifici logarithmorum canonis descriptio (Edinburgh, 1614), which contains an account of the nature of logarithms, and a table giving natural sines and their logarithms for every minute of the quadrant to seven or eight figures. These logarithms are not what are now called Napierian or natural logarithms (i.e. logarithms to the base e), although the former are closely related with the latter. The connection between the two is

$L=10^{7}\log _{e}{10^{7}}-10^{7}.l$, or $e^{l}=10^{7}e^{-{\frac {L}{10^{7}}}}$,

where $l$ denotes the logarithm to the base $e$, and $L$ denotes Napier's logarithm. It should be observed that in Napier's original theory of logarithms, their connection with the number $e$ did not explicitly appear. The logarithm was not defined as the inverse of an exponential function ; indeed the exponential function and even the exponential notation were not generally used by mathematicians till long afterwards.

Approximate constructions

A large number of approximate constructions for the rectification and quadrature of the circle have been given, some of which give very close approximations. It will suffice to give here a few examples of such constructions.

(1) The following construction for the approximate rectification of the circle was given by Kochansky (Acta Eruditorum, 1685).

Let a length $DL$ equal to 3 . radius be measured off on a tangent to the circle ; let $DAB$ be the diameter perpendicular to $DL$.