62 



tating his construction, which is described as follows (Cremona Graph. 

 Statics (Beare), p. "ti): 



"The apparatus consists of a perfectly plane circular disc, which may be 

 made of wood ; upon it is pasted a piece of paper ruled in squares. In the 

 center of the disc, which should remain fixed, stands a pin, around which 

 as a spindle another disc of ground glass of equal diameter can turn. Since 

 the glass is transparent, we can with the help of the ruled paper under- 

 neath, immediately draw upon it the circuit corresponding to the given 

 equation. If we now turn the glass plate, the ruled paper assists the eye 

 in finding the circuit which determines a root. A division upon the cir- 

 cumference of the ruled disc enables us by means of the deviation of the 

 first side of the first circuit from the first side of the second, to immediately 

 determine the magnitude of the root. For this purpose the first side of the 

 circuit corresponding to the equation must be directed to the zero point of 

 the graduation." 



Linkages might be found to perform mechanically what must be done 

 by successive approximations in the method above, viz.: to bring the last 

 vertex C/ into co-incidence with D. Kempe has given some linkages for a 

 diflTerent construction. [See Messenger of Mathematics, Vol. 4, 1875, p. 124.] 



III. 



The following constructions are given as illustrations: 



(a.) Roots of 2x'- + 4x -^ 1 = o. [Fig. i.] 



As the co-eflBcients are all real it is preferable, and for real roots neces- 

 sary, to transform the equation by putting x = -^, m = (1, 90°). The 

 equation becomes 2 zH -^ m z + m^ -= O, and A = 2, A B = 4 m, B C == 

 m3 = _ 1. If A' A : () A is a root of this equation then, dividing by m, 

 we find A/ A : m O A «s a root of the original equation. If this is real A' 

 must lie on A B, produced if necessary. Ilemember that A^ is such that 

 O A' A, A^ C B are similar triangles and we see that the angle O A^ C is a 

 right angle when A' lies on A B. Hence the circle on O C as diameter cuts 

 A B in'the sought points A', A'^. From the figure the roots A' A : m O A, 

 A'^ A : m O A are approximately — . 3 and —1.7. 



(b.) Rootsof 2x2 + 2x + 4 = 0. [Fig. ii.] 



Here 0A = 2, AB = 2m, BC = 4m2 ... —4. The circle on O C as di- 

 ameter does not cut A B and the roots are imaginary. Since A^ A, A'' C B 

 are similar, therefore A' is equally distant from A and B, and that distance 

 is mean proportional between O A and C B. A circle with this mean pro- 

 portional as radius and center at A or B will therefore cut the perpendicu- 

 lar erected at the middle point (M) of A B in the sought points A', A'\ 

 The circle with center at M and cutting the circle on C as diameter at 



