ON THE THEORY OF POINT-GROUPS. 85 



the former rule can be applied. If the two equations from which the 

 common quantities are to be removed are not of the same degree, the 

 coefficients of the higher orders in the equation of lower order are to 

 be taken as zero.' In a letter to I'Hdpital dated April 28, 1693,i 

 Leibnitz refers to this subject, and expresses a desire for tables by which 

 the results of elimination between equations of higher order may be 

 systematically deduced from those of lower order. This is the letter in 

 which he for the first time explained and vindicated the new notation, 

 and it has led most writers '^ to attribute to him the origin of the theory 

 of determinants. However this may be, the Theory of Elimination cer- 

 tainly owes its oi-igin to him, for he was the first to regard it as a matter 

 for separate investigation and to desire its reduction to general laws. 



It would be interesting, were evidence on the subject forthcoming, to 

 think that Leibnitz's new method of elimination had been communicated 

 by him to his friend Tschirnhaus (1651-1708), with whom he was in 

 constant communication on these subjects, and that it is the method 

 alluded to by the latter in his memoir in the ' Acta Eruditorum ' of 1683, 

 where he speaks of well-known rules for obtaining a third equation, in 

 which the unknown is absent, from two given ones containing it. But, 

 on the whole, it is more probable that a method of combination and sub- 

 stitution, which amounts to using the condition for the existence of a 

 greatest common measure was in his mind.^ 



Tables of the nature desired by Leibnitz were published by Newton 

 in his ' Arithmetica Universalis ' (1707) :'' he gives the actual results of 

 the ' extermination ' of a variable from certain typical equations, viz., 

 from two quadratics, from a cubic and a quadratic, from a quartic and a 

 quadratic, and from two cubics, but with no definite account of the steps 

 of his calculation. His method, however, appears to be that of substitu- 

 tion and combination, and is thus essentially different from that of Leibnitz. 

 The arrangement of the terms in the successive results shows an attempt 

 at a systematic derivation of each from the last, but the law is by no means 

 clear. It is noteworthy that in each case the result obtained is given in 

 its simplest form, i.e., the extraneous factor, which is not unity in his 

 notation, has been removed. 



Newton gave no rules applicable to equations of order higher than the 

 fourth, and only considered equations involving one unknown. The more 

 general cases of higher equations and two unknowns were attacked for the 

 first time -^ (unsuccessfully, however) by Maclaurin in the ' Geometrica 

 Organica.' Neglecting for the moment the geometrical application for 

 which he required his result, the interest of Section V. of this treatise 



' Leibnitz, Works, edit. Gerhardt, vol. ii. p. 239. 



- Cf. Gerhardt, GescMchte der Mathematik in Beutschland, 1877, vol. xvii. p. 184, 

 and preface to vol. vii. of Leibnitz's Works, p. 8. Also Brill and Noeiher, loc. cit., 

 p. 126, and Salmon, Higher Algebra, Note on History of Determinants. It has, 

 however, been pointed out by Studnicka (.4. L. Caitchy als forvialer Begriinder der 

 Determinanteii-TJieorie, Prag, 1876) that Cauchy was the first to develop a theory 

 of determinants ; and it seems scientific to distinguish thus between the invention 

 of a new mathematical machine — the suffix notation, which provides a law of 

 formation for the new coefficients — and the separate discussion of its properties. 



' Cantor, vol. iii. p. 109. 



* 1st edit., p. 74. 



* Fermat realised that curves of the mth and rath orders lead, by their intersec- 

 tions, to the solution of an equation of degree mn {cf. Fermat, (Euvres, vol. iii. p. 119, 

 vol. i. p. 130, and Cantor, vol. ii. p. 746), but he gave no attempt at a proof. 



