﻿Solution of Equations. 899 



equation to be solved, in order to apply these Lazytongs 

 methods, though, of course, every linkage equation implies 

 one of the more ordinary sort in terms o£ powers of an 

 unknown quantity, and the problem might have been stated 

 in that form. 



In the so-called problem of u doubling the cube " what is 

 meant is, that a cube of edge a being given, the edge b of a 

 cube of double the volume of the first shall be found. 



a 3 

 Here — = J, and b is required. As a is less than b 



b s 



a a 



write r = cos 0, whence - 7 -> = cos 3 = A. 

 b ' ¥ 2 



The value of cos 3 in cosines of multiples of is 

 cos 30 4- 3 cos 



which, therefore, is to be equated to J by the terms of the 

 problem, and. leads at once to the form adapted to linkage 

 treatment : 



= 2-3 cos 6— cos 30, 



from which cos or j can be found. R 2 is in this case 



zero, and the linkage will accordingly require the treatment 

 indicated in fig. 3. 



The insertion of two mean proportionals between a the 

 smaller and b the larger of two straight lines is one very 

 nearly the same as the last. If # 1? x 2 are the two magnitudes 



required, and — = cos 0. then — = cos 2 0, and T = cos 3 0. 

 x x x» b 



Therefore 



cos 30 + 3 cos 6 



= 4? — 3 cos 0— cos 30, 



b 



indicating the linkage required, one of the first sort. If, 



x* 



however, we make - = tail 0, — = tan 2 0, r = tan 3 Q, we 



Xt J Xo ' b 



are no longer under the necessity of taking . as less than 



3 N 2 



