86 REPORT — 1902. 



centres on the attempt to eliminate one unknown between two equa- 

 tions in two unknowns and to determine the dimension of the remaining 

 unknown in the result. In the Lemma with which this section opens 

 one equation only is assumed to be of an arbitrary number of dimensions, 

 n, in X and y, while the second is a quadratic and a cubic equation 

 respectively. Maclaurin substitutes for all powers of the variable y 

 equal to and greater than 2 and 3 respectively in the equation of 

 degree n, and thus obtains equations in 2/ lower by one degree than 

 the lowest of the original pair of equations. In the simpler case (that 

 of the quadratic and the equation of degree n) this gives a value 

 for 1/, which simply requires substitution in the quadratic in order to 

 lead to the required equation in x — of degree 2n — and does not intro- 

 duce any extraneous factor. But when the original system consists 

 of a cubic and an equation of degree n, leading to a system of a cubic and 

 a quadratic in )j whose coefficients are functions of x, the process of 

 eliminating y — whether by direct G.C.M. method or by Maclaurin's 

 shortened process of combining the equations, first removing the highest 

 term and then the lowest term — necessarily introduces an extraneous 

 factor which involves x. To find this factor, even with an unsystematic 

 notation, is easy enough in this simple case, and Maclaurin shows, quite cor- 

 rectly, that the irreducible resultant is then of degree 3n ; but he saw clearly 

 that all his knowledge of Newton's ' method of divisors ' would not avail 

 him in the completely general case of equations of degrees m and n. He 

 therefore relegates this question to Corollary I., which, freely translated, 

 runs : ' Hence the intersections of lines of order m and n are seen to be 

 vm in number. We have, it is true, hitherto searched in vain for a 

 universal proof of this fact by reason of the difficulty of finding divisors 

 in harder equations.' That he did not employ a better notation is all 

 the more remarkable when we consider that in a chapter of his ' Algebra ' 

 (planned about the year 1729,' but published, after his death, in 1748) 

 he ' exterminates ' the unknowns from systems of two and three linear 

 equations, and gives a rule for four equations which shows his clear 

 appreciation of the symmetry of the result : ' If 4 equations are given, 

 involving 4 unknown Quantities, their Values may be found much after 

 the same Manner Vjy taking all the Products that can be made of 4 

 opposite Coefficients (i.e., belonging to different equations and to dfferent 

 variables) and always prefixing contrary signs to those that involve 

 the Product of two opposite Coefficients.' This is exactly the idea 

 involved in the modern solution by means of determinants. And con- 

 sidering that Leibnitz had insisted on the value of his double suffix 

 notation in the paper in the 'Acta Eruditorum ' for 1700,^ written in 

 reply to a tract published in London, Maclaurin almost seems to have 

 gone purposely out of his way to avoid its use,^ and, in so doing, perhaps 

 lost a chance of overcoming some of the obstacles to his method of elimina- 

 tion. Even the best possible notation, however, would have been com- 

 paratively valueless without the realisation of the intimate theoretical 

 connection between the process of finding the G.C.M. and the problem of 

 elimination, and for this the time was not ripe. The method, or its 

 equivalent, was used in practice, but was not explicitly explained. In 



' Cantor, vol. iii. p. 568. 



- Leibnitz, Worl/s, vol. v. pp. 340-349. 



' Cantor, vol, iii. p. 670. 



