seventeen equal Parts. ^ 173 



wherefore we have this equation, viz. 



Cos 4;^ = cos |-. (2) 



Now, Cos 2cf) = 2 cos ^<{) — I 



Cos 4c^ = 2 (2 cos^ 4) _ 1)1 _ 1 



Cos (J) =2 (2 cos 1- -1)^-1: 



4 

 wherefore, if we put x = cos 4(p = cos j-, and y — cos s we 

 get these equations, viz. 



y = 8x* - Sx^ + I J ^'^'' 



In these equations it is easy to prove that x = cos a = cos 



4na, can never be equal to y = co?. — = cos na: and since the 



cosines of all the multiples of a are only eight in number, it fol- 

 lows that there can be no more than four different values of x 

 and four corresponding values of y, that satisfy the equations. 



In order to illustrate this, let all the values of a and -, that 



4 



have different cosines, he written in two lines, viz. 



<;> : 4a, Sff, 12a, 16a, 20a, 24a, 28a, 32a, 

 5a a 3a la 6a 2a, 

 ~ : a, 2a, 3a, 4a, 5a, 6a, 7a, 8a, 



the arcs placed below in the first line have the same cosines with 

 those above, being found by taking the difference between 17ff, 

 or 34a. Now, if we compare the cosines of the corresponding 

 arcs in the two lines, it will readily appear that we shall have all 

 the different values of x and y, by making (p equal to one or 

 other of the four arcs, 



4a, 8a, 12a, 24a: 

 for when we make (p equal to 



16a, 20a, 28a, 32a, 

 we obtain the same values of x and y as before. 



What has been said relates to the values of <p in multiples of 

 the arc a ; values which are exclusively determined by means of 

 the two equations, 



Cos 4(p = cos — 



Sin 4« = — sin 4*, 



4 



both deduced from equation (I). But U b = — - = 24", and 

 f = iiib, then 



4f — I = « X 360% whence 



