[ 241 ] 



XXXI. On a Machine for Solving Equations. 

 By C. V. Boys*. 



[Plate III.] 



AT a late meeting of the Physical Society Mr. C.H. Hinton 

 showed some apparatus that he had made to explain the 

 nature of equations. One of the instruments for representing 

 the case of a simple equation was manageable enough; but 

 the second, which dealt with the quadratic, could hardly be 

 employed to solve a quadratic, as the root was not capable of 

 continuous variation. It could, therefore, only after con- 

 siderable trouble in trials and errors had been taken, find 

 those real roots which consisted of whole numbers between 

 the limits of the machine, say + and —10. 



At the time that Mr. Hinton was trying to devise a 

 quadratic machine (with the object not so much of solving 

 quadratic equations, as of assisting boys to understand the 

 method employed, now about ten years ago), he showed me 

 what he had done; and it then occurred to me thai: continuous 

 variation might be employed in a machine so that fractional 

 roots could be found, and that the degree need not be limited 

 to a quadratic, but that a cubic or one of any power of the 

 form a + bx + ex 1 + dx z + &c. = could be solved, I then half 

 completed the machine I have now the honour of exhibiting 

 before the Physical Society, but laid it on one side till the 

 present time. 



I find that Mr. A. B. Kempe has invented and described! a 

 machine with the same object. He replaces the expression x 

 by cos 0, and then by a known process changes the equation 

 in which the terms are powers of cosines of 6 into one in 

 which the terms are cosines of multiples of 0. The equation 

 is then in a state to be dealt with by his machine. A series of 

 levers jointed on one another, the first hinging on the inter- 

 section of a pair of rectangular axes, have their lengths 

 variable, so that each one in order can be set so as to be equal 

 to the coefficient of the corresponding term; moreover, by 

 mechanism each one makes with the last the same angle that 

 the first makes with the axis of x. Thus, on rotating the first, 

 the end of the last describes a curved line cutting at one or 

 more points the axis of y\ every angle 6 of the first which 

 causes this intersection is noted, and the value being inserted 

 in the original equation gives at once one of the real roots. 



* Communicated by the Physical Society : read December 12, 1885. 

 t ' Messenger of Mathematics,' 1873, vol. ii. p. 51. 



