ON THE THEORY OF POINT-GROUPS. 87 



fact the only contemporary statement in print on this subject is 

 to be found in a small book on analytical geometry by de Gua de 

 Halves,' published in 1740. He remarks - that to find the necessary 

 condition in order that certain three equations may hold simultaneously : 

 'on pourra se servir des formules que M. Newton a donnees dans son 

 " Arithmetique Universelle," ou, ce qui revient au meme, on pourra encore 

 diviser les deux premieres equations . . . par leur reste, puis les premiers 

 restes par les seconds, et ainsi de restes en restes jusqu'a ce qu'on soit 

 parvenu a en trouver qui ne contiennent plus I'indeterminee x, ces derniers 

 restes, ^tant faits egaux a ze'ro, donneront les equations des conditions.' 



The process of combining two equations by first removing the highest 

 terms, then the lowest, which Maclaurin used in his examples, was sys- 

 tematically described for the first time by Euler (1707-1783) in 1748. Its ex- 

 planation occupies the first half of the chapter ' De intersectione curvarum ' 

 in his ' Introductio in analysin infinitorum.' ^ It also occurs in a memoir 

 presented to the Academy of Sciences in Berlin,'' where it is preceded 

 by the pertinent remark : ' Dans la plupart des cas si Ton se sert des 

 methodes ordinaires d'^liminer, on parviendra a une equation de plus de 

 dimensions que mn.' It is then followed by a discussion of a case 

 in which the number of intersections must fall short of inn : when the 

 equation of a curve of order m, namely, is of the form Vy^ + Qy' + 

 Ei/ -4- S = O, P, Q, R, and S being of dimensions m — 3, m — 2, m — 1, 

 and m respectively in x, since then ' les equations choisies n'expriment 

 pas generalement les courbes des ordres m et n, mais seulement des 

 especes de ces ordres.' This is the first appearance of a class of equa- 

 tions which were very fully discussed later by Bezout. 



Two new methods of elimination were put forward by Euler in these 

 publications ; but neither is rigorously demonstrated, and the unsystematic 

 notation once more proves an obstacle to progress. That of the ' Intro- 

 ductio' is the same as had occurred to Leibnitz, but of which he had 

 written nothing for publication. It consists '' in multiplying each equa- 

 tion by a function of y whose coefficients are undetermined quantities, 

 and then equating to zero the coefficients of the different powers of i/ in 

 the equation formed by subtracting these equations from each other. 

 From this set of linear equations the undetermined coefficients are 

 eliminated and the resultant obtained. The general rule, however, given 

 for the elimination of the undetermined coefficients is very laborious and 

 far inferior to that which Leibnitz bad discovered. This method has 

 taken its place in modern text-books as ' Euler's method,' or, sometimes, 

 as ' Euler'.? second method ' :^ it is really founded on the necessary 

 existence of at least one common root of the two equations if they are 

 to hold simultaneously, and Euler himself explains this in a memoir' 



' Usages de Vanalyse de Descartes jMuir decoucrir sans le secours d^t, Calcul 

 Differeiitiel les Proprietes on Affections ijrincvpales des Lignes Geomctriques de tons 

 Ordres. Cf. Brill and Noether, he. cit., p. 134. 



2 P. 60. 



" See Cantor, vol. iii. p. 576, for an account of this work. 



' ' Demonstration sur le nombre des points ou deux lignes des ordres quel- 

 conques peuvent se couper,' Acad. Berlin, annee 1748, pp. 2.34-248. 



^ See Cantor, vol. iii. p. 577, for detailed description. 



« Encyk. dcr Math. Wis.ien., Leipzig, 1899, Bd. I., p. 246, note 80. 



' 'Nouvelle m6thode d'eliminer les quantit^s inconnues des equations,' Acad. 

 Berlin, ann6e 1764, pp. 91-104. 



