84 KEPORT — 1902. 



This fact is of no importance theoretically, and -would be of but slight 

 importance in practice, if the coefficients employed were the most general 

 possible. But when, as in Fermat's example, the coefficients of the 

 highest powers only are unity, the interchange materially complicates 

 the algebra involved in the next stage of division, or in substituting, as 

 he directs, the expression for x obtained from E.i=() in the quadratic 

 equation ; and, presumably for this reason, Fermat left it to the industry 

 of his readei's. Moreover, when the substitution has been effected, a 

 further complication arises from Fermat's method : the irrelevant factor ' 

 which is a necessary consequence of this process is now h\ instead of h\, 

 i.e., (ri? — ctbY instead of unity ; the remaining factor (the resultant) is 

 unchanged, being symmetrical with respect to this interchange of 

 coefficients. 



As regards the general application of his method to a system of n 

 equations among n — \ unknowns, Fermat remarks : 'II est clair que la 

 methode est g^nerale. Si en effet on proposait plus de deux inconnues, 

 la methode, reiterde autant qu'il le faudra, exprimera par exemple la 

 troisifeme en fonction de la premiere et de la seconde, puis la seconde en 

 fonction de la premiere, toujours par le meme moyen.' But we may be 

 permitted to doubt whether the coui'age of most calculators would not 

 fail in undertaking such an elimination without the assistance of a 

 properly devised notation, of which, at this date,^ no trace is found. 



The emancipation of mathematics from the preliminary stage, in which 

 attention is mainly directed to the solution of particular problems, was 

 being effected during Fermat's lifetime — Kepler and Galileo were his 

 seniors by thirty and thirty-seven years respectively, Descartes and 

 Pascal were his contemporaries-^but the chief impulse in this direction 

 was given after his death by Leibnitz (1646-1716) and by Newton. 

 Leibnitz especially saw the cardinal importance of notation. The double 

 suffix notation which he invented {mehrfacher Stellenzeiger) was not 

 mentioned in print until 1700, but had been used by him as early as 

 1678, as is shown by a Latin manuscript note found among his papers 

 after his death.^ The rule here set down for removing the unknowns 

 from any system of linear equations such as, 



\Q + l\x + Uy=0 

 20 + 21a; + 22y=0 

 30 + 31x4- 331/= 



where there is one more equation than there are unknowna, consists in 

 observing the law of combination of the double indicators. A new 

 method for the elimination of the unknowns from two equations of degree 

 higher than the first is then derived from this rule : ' By means of this 

 rule another rule can be found for removing the unknown quantity 

 common to two equations of any degree whatever. Multiply each by an 

 assumed expression of one degree lower, and when these products have 

 been added together, so as to form a single equation, let every term of it 

 be equated to zero ; we thus obtain as many equations as there were 

 coefficients in the assumed expression and one more equation. Hence 



' An ingenious formula for obtaining this factor, in the most general case, is 

 given by Fail de Bruno, Theorie generale de I'elimijiation (Paris, 1859), pp. 47-52. 



* The date of this manuscript is held to be 1638, although it was not printed 

 until 1679, in the Varia Opera, of. Cantor, vol. ii. p. 734. 



' Gerhardt, preface to yol. vii, of Leibnitz's Works (Berlin, Halle, 1849-63), p, 5. 



