ON THE THEORY OF POINT-GROUPS. 83 



§ 7. The Theory op Elimination, from Leibnitz to Cramer, 



1693-1750. 



We must go back fifty years before the birth of analytical geometry 

 to find the first appearance of a problem which is fundamental to the 

 theory of elimination in one of its aspects, the problem, namely, of 

 obtaining the greatest common measure of two algebraical expressions. 

 In 1585 a Belgian mathematician, Simon Stevin (1548-1620), published an 

 algebra in. which the first successful attempt at a solution was made. He 

 divided one expression by the other, and this again by the remainder, until 

 no remainder is left ; the last divisor is then the greatest common 

 measure ; the fractions are left as they appear in the course of the work. 

 Vieta (1540-1603), the greatest algebraist of the sixteenth century, wrote 

 a treatise on algebraical equations in 1591 (published, after his death, in 

 1615), but did not investigate this question. In fact it does not seem 

 to be mentioned again in pi'int for more than a century, when Rolle 

 (1652-1719) gave it a place in his 'Traite d'Alg^bre,' published in 1690. 

 By this time abbreviations of the work by means of multiplication and 

 division were probably in use, since we find them freely employed by 

 Reyneau (1656-1728) in his ' Analyse demontr^e ' of 1708. 



In the meanwhile Fermat was attacking the problem of rationalising 

 an equation which Vieta had left in a very unsatisfactory condition, and 

 the method he adopted, although stated as a series of proportion sums in 

 the manner of his time, amounts precisely to the elimination of an 

 unknown from two equations by using the condition that their left-hand 

 sides should have a greatest common measure. The example he gives ^ 

 consists of a cubic and a quadratic equation, each containing one variable 

 which is to be eliminated. Calling these equations P=0, Q=0, respec- 

 tively, we may write P=9'o Q + Rj, and Rj is then the first remainder 

 after dividing P by Q. Fermat's process, however, reverses the usual 

 arrangement of terms in P and Q, writing them in ascending not in 

 descending order, for he arranges his proportions so that the antecedents 

 contain the variable while the consequents are free from it ; thus : 



aQO^ + a^x- + arX : «3=6Qa;- + 6ia; : b^ 



is derived from 'P=aQX^ + a^x--{-a.x + a^=:0 



Q=baX^ + biX + b.y=0 



where in his notation 



«(,:=?, n,=0, a.2=0, rt3=::^ — a^ 

 bQ-=l, b^=d, b2=n^ — cib 



and thus his expression for R,, after dividing out by the factor x, diflffers 

 from the usual one by the interchange of «o ^^^ ^3) "i ^^^ ";> ^o ^^^ 



b , and of - for x, being in modern notation 



* Fermat, OHwt'es {edit. Tannery and Henrj', Paris, 1891) •Nouvean traitement 

 en analytique des inconnues secondes et d'ordre superieur,' vol. i. pp. 181-188 (Latin 

 original); voi. iii. pp. 157-163 (French trans.V 



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