Gl 



are to try in the equation for x. To find the result of the trial, construct 

 the triangle A^ B' B similar to O A' A, and then the triangle B' C^ C, also 

 similar to O A' A. We have O A = a, A' A = a r, and hence A' B = A^ A 

 + AB = ar+b; also by similar triangles, B^ B = r A^ B --= a r- -{- b r, and 

 hence B^ C = B^ B + B C = a r- + b r + c. Again by similar triangles, 

 C^ C = r (a r2 + b r + c) = a r3 ^ b r^ + c r and hence C D = C C + C D 

 = a r^ + b r^ + c r + d, the remainder sought ; moreover, the co-eflacients 

 of the quotient are represented by O A, A/ B, B^ C. The problem is to so 

 choose the first point A' that the last vertex C of the series of similar tri- 

 angles O A' A, A^ B' B, B' C C, shall coincide with D : then A' A : O A is 

 a root of the given equation. With the ability to construct a series of sim- 

 ilar triangles with ease, a position for A^ may be approximated to without 

 much difficulty. Observe that O A^, A^ B^, W C^ are equi-mulliples of 

 O A A' B, B^ C. This follows from the similar triangles A' A, A^ B^ B, 

 B' C C, which give O A^ : O A = A' B' : A^ B = B^ C : B' C both as to 

 tensor and angle parts. Hence the circuit O A^ B' C represents the quo- 

 tient on the new scale in which A/ instead of O A represents the first 

 co-eflBcient a. 



If the co-eflScients of the given equation are all real numbers and only 

 the real roots are sought, the method fails, since A^ must be taken on A 

 produced giving no triangle A' A. In such a case, put x = -^— where m 

 is a given versor, say (1, 60°), or (1, 90°); the equation becomes ; 



a z^ + m b z- + m- c z + m^ d ^ o. 

 The figure O, A, B, C, D that represents the co-efficients of this equation 

 will have equal angles at A, B, C, viz.: the supplement of the angle of m 

 (since a, b, c, d are real numbers, their angles are O or 180°). We are to 

 seek for roots of this equation whose angles are, angle of m or angle of m— 

 180°. (Since z = mx, therefore angle z = angle m ^ angle x.) Thus A' 

 must be taken on A B produced ; and since the angles at A, B, C, are 

 equal, it follows that the similar triangles required will have their vertices 

 B', C^ on B C, C D, produced, so that the construction of these triangles is 

 simplified, e. g., to find B^ draw from A^ a line making with O A^ an angle 

 equal to the angle A; that line meets B C in B'. The broken line O A^B' C 

 has its angles A^, B^ equal to the angles A, B, and its vertices A', B', C in 

 the sides A B, B C, C D; trials of this construction must be made until C 

 co-incides with D, when A^ A : m O A is the real root of the equation in x. 



Taking m=(l, 90°), this is Lill's construction for the real roots of an 

 equation with real co-eflScients. Lill has devised an instrument for facili- 



