John Napier

“ Now (by 55) the ratio a k, half radius, to e f, a sine of 45 degrees, is likewise the ratio of e g, also a sine of 45 degrees, to e i, now radius. Consequently (by 37) double the logarithm of the sine of 45 degrees is equal to the logarithms of the extremes, namely radius and its half. But the sum of the logarithms of both these is the logarithm of half radius only, because (by 27) the logarithm of radius is nothing. Necessarily, therefore, the double of the logarithm of an arc of 45 degrees is the logarithm of half radius. ”

“ IT is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided ”

“ Draw e i perpendicular to a ic, then e i is the sine of the arc a d e. Draw a e; its half, f e, is the sine of the arc d e, the half of the arc a d e. Draw e c; its half, e g, is the sine of the arc e h, and is therefore the sine of the complement of the are de. Finally, make a k half the radius a b. Then as a k is to e f, so is e g to e i. For the two triangles c e a and c i e are equi-angular, since i c e or a c e is common to both ”