into seventeen equal Parts. 175 



m—n = ~— — 2 {m + ??)- + 4 



m + n = ^-^'^ - 2(7»-«)4 + 4 ; 



and, by adding and subtracting these, 



8m = m* + 6ra^ «^ + «4 _ igwz* _ 16«i + 32 

 — 2ra = Tre' n + 7W«' — Smn. 

 From the second of these equations we get, 



m ' 



4 39 



w+ = 64 H + 4m — I6m^ + m* ; 



and these values being substituted in the first equation, we sh;dl 

 obtain after reducing, 



«i6 — Sm-* + 4m5 + 8m' — 1 = 0. 

 Divide all the terms of this equation by ra', then 



w' - -V -8w + - + 4 = 0: 



and if we now put z = m— — , and substitute, we shall obtain, 



x' — 5« + 4 = 0, 

 or, (z - 1) (z^ + z - 4) = 0. 



From the factor s; — 1 = 0, we get 2; = 1, and hence 



m = 1 



m 



n' = S — — —rnP- 



m 



m+n m—n 



These formulae determine two values of each of the quantities 

 fw, «, X, 7/ ; and the values of x and y so obtained, are the co- 

 sines which, as has been shown above, belong to the quindeca- 

 gon. 



For the polygon of seventeen sides, we have 

 z' + z — 4 = 



1 

 w — — = z 



1)1 



n^ = 8 — — - ?«' 



VI 



~ \ ' "^ ~ 4 * 



Now, if we put /J = Z_i^L_^ p' — _ J the two values 



of 



