92 HEPORT— 1902. 



iudeterminees s'il y a parmi les equations deux ou plusieurs, qui sont deja 

 dans les autres, et qui par consequent ne contribuent rien a la determination 

 renferme'es des inconnues.' ^ This indicates more insight into the possi- 

 bilities in the nature of a system of equations than any passage in Cramer ; 

 but when he proceeds to make use of this conclusion he fails to bring out 

 the complete resolution of the paradox. He first considers the cases of 

 a straight line and of a conic and shows how they may remain indeter- 

 minate, even after assignation of two and five points respectively on 

 them. These cases present no difficulty, but are not really relevant to 

 the paradox, since in their case v'^ is less than \v{y-\-'i) : they arise when 

 tlie two assigned points coincide, and when of the five points four lie on 

 a straight line. The cubic, which is the next curve he considers, and to 

 which the paradox is applicable, is confessedly beyond his grasp : ' II est 

 cependant fort difficile de definir ces cas gendralement, comme j'ai fait pour 

 les lignes du second ordre, puisque le calcul, a cause du grand nombre des 

 points et des coefficients, deviendrait trop complique. Neanmoins il 

 n'est pas difficile de decouvrir plusieurs cas particuliers, ou ce defaut dans 

 la determination a lieu ; desquels on ne conclura pas difficilement, que 

 le nombre de tels cas pent etre infiniment grand, ce qui suflit pour mon 

 dessein.' ^ The particular cases alluded to are, fii'st, when the nine points 

 are so situated that the cubic through them must degenerate into a conic 

 and a straight line : this is analogous to the case of the conic ; 

 and, next, when they are arranged in the shape of a square, so that the 

 equation of a cubic through them is my{y'^-a'^)=::nx{x'^-d-), ' oii le rapport 

 des coefficients m et n pent etre quelconque, de sorte qu'une infinite de 

 lignes du troisieme ordre pent etre indiquee, qui passent toutea par ces 

 points donne's.' ^ 



In view of the fact that Euler brings forward this example definitely 

 as a special case, and that he does not even hint at a similar one in the 

 discussion of quartics and quintics with which his paper closes, it is hardly 

 likely that he recognised the univei'sal application of the principle which 

 underlies it ; in fact, he actually deduces the equations of the two 

 independent component cubics (each consisting of three straight lines) 

 from the original equations by writing ??i- = 0, ?^=0, in succession. The 

 principle itself was not formulated until more than sixty years had 

 passed, by Lame' (1795-1870), so slowly do ideas grow. And, precisely as 

 the theory of curves was originally explored in the interests of algebra 

 before the reverse process set in, so here, the intersections of curves 

 were studied with a view to the determination of the curves on which 

 they lie long before the properties of the systems of curves themselves were 

 investigated. 



In the preface to his ' Examen des differentes m^thodes employees 

 pour resoudre les problemes de geom^trie,' ■* Lame acknowledges that the 

 general reflections on pure mathematics which occupy a quarter of his 

 book were suggested to him by the problems which it contains ; this fact, 

 characteristic of the practical engineer, does not detract from the theoretical 

 importance of the general statement here made for the first time,'' viz., 

 that an equation whose left-hand side is formed by the additive combina- 

 tion of the left-hand sides of two or three given equations of the same 

 degree, each multiplied by an undetermined quantity, is the adequate repre- 



' Loc. cit., p. 227. = Loc. cit., p. 231. ' Zoc. cit., p. 231. 



' Paris, 1818, pp. 124, * Zoc. cit., p. 28. 



