THE GENESIS OF LOGARITHMS 



151 



readily be calculated. The angles which these chords or sides 

 subtend at the centre of the circle are also known and clearly 

 half the length of any such chord will give the sine of half the 

 corresponding angle subtended by the chord. Thus in the 

 following table, taking radius as io r , we have 



Now, knowing the sine of any angle, the sine of the half- 

 angle can be calculated by means of some such theorem as : 



" The sum of the squares of the sine and versed sine equals 

 the square of double the sine of half the arc." 



This is, of course, simply the equivalent of 



sin 2 6 + (1 - cos Of = 4 sin 2 - 



£1 







and gives sin - in terms of sin 6. And knowing the sine of any 



angle, we arrive at the sine of the complementary angle by the 

 theorem : 



" The square of the sine and the square of the sine of the 

 complement is equal to the square of the radius " ; i.e. simply 



sin 2 6 + cos 2 <9 = 1. 



Starting, then, with the sine of twelve degrees and continually 



finding the sines of half-arcs, we obtain the following series 



of tables : 



I 



Beginning with any of the complementary angles in table I and 



