242 Mr. 0. V. Boys on a Machine 



I am not aware of any other machines for solving equations 

 of a high order, except Mr. Cunynghame's, which will be 

 described this afternoon. Machines for solving simultaneous 

 equations also exist. 



Like Mr. Hinton, I made use of the equality of moments 

 by arranging a series of levers to operate upon one another. 

 Thus, let the levers be called successively 1, 2, 3, 4, &c; then 

 1 is on a stationary axis ; it has at unit distance from its axis 

 and on either side of it pivots, from each of which hangs a 

 pan or hook labelled —a and +a. If, then, a weight of a 

 units is put upon either of these pans, the moment of the force 

 upon the beam will be —or +a units. 



Let a second beam 2 be connected with 1 by a sliding joint 

 which is permanently at unit distance from the axis of 2 ; 

 let the joint also carry a scale-pan, and let there be another 

 scale-pan at unit distance on the other side. These are labelled 

 + b and — b. A w eight of b units hung on either of these 

 pans will produce a turning moment on b of + or — b units, and 

 a turning moment on a of + or — bx units, where x + 1 units is 

 the distance between the axes of 1 and 2. Such a pair of 

 beams of course will solve a simple equation whether the root 

 be positive, negative, whole, or fractional; for as the second 

 beam is made to traverse, it must pass some position where 

 a + bx=0, and on the two sides of this position the moments 

 on 1 w T ill be opposite in sign . Therefore by shifting the beam 

 till the arm changes position, the root can be exhibited by a 

 pointer on a scale. 



Similarly, if a third beam be mounted opposite 1 with a 

 fixed axis, and if it be provided with pans as before, labelled 

 — c and +c, and if this be connected with 2, as 2 is with 1, 

 then on pushing 2 along between these two beams a weight 

 of c units on one of the c pans will exert on 2 a moment of 

 + ex, and on 1 of + ex 2 units. If, then, a quadratic equation 

 of the form a + bx + ex 2 = has to be solved, weights of a, b, 

 and c units are placed on the proper pans and the beam 2 slid 

 along until the beam 1 shifts its position. The two places at 

 which this occurs are the two real roots of the quadratic. 

 When the quadratic has no real roots, the machine can be still 

 employed to find the impossible roots, as will be explained later. 



A fourth beam placed opposite to 2 and connected with 3, 

 as 2 is with 1, will in the same way make the machine capable 

 of finding the real roots of a cubic equation, and so by adding 

 more beams to the stationary and movable sets, an equation 

 of any degree can in theory be solved. Of course, as the 

 number of joints increases, the friction and elasticity of the 

 working parts increase enormously, and so with the best 



