ON THE THEORY OF POINT-GROUFS. 91 



o£ oi'der n may sometimes pass through the same it- points ; and thus 

 given points whose number is ^n{n + S) do not suffice to determine a line 

 of order Jt in such a way that the curve which can be drawn through them 

 is unique ; yet, in truth, since the coefficients in the general equations of 

 a line of order w are ^n{n + 3) in, number, it is clear that, if more points 

 are given, it is perhaps not possible to draw a line of order n through 

 them, and the problem may be impossible. So nine points do not as fully 

 determine a line of order three as five do a line of order two ; yet ten are 

 too many for the determination of a line of order three.' Shortly 

 expressed, this amounts to the enigmatic statement that yi{n + 3) points 

 both do and do not determine a curve of order n uniquely. Maclaurin 

 makes no attempt at an explanation, and thirty years elapsed before the 

 matter was again put forward in print. It then appeared simultaneously 

 in two publications — in Cramer's ' Analyse ' and in a memoir by Euler 

 entitled 'Sur une contradiction apparente dans la doctrine des lignes 

 courbes.' ^ 



Cramer had spent part of a two years' leave of absence from his 

 Geneva professorship in England, and it is perhaps due to this that he 

 alone of foreign mathematicians gives the reference to Maclaurin's 

 corollary in a footnote to his own statement of the paradox : ^ ' Une 

 contradiction apparente ... est celle-ci. Puisqu'une ligne de I'ordre vi 

 ne pent rencontrer une ligne de I'ordre n qu'en mn points, une ligne de 

 I'ordre v ne rencontrera une autre ligne de meme ordre qu'en v"^ points. 

 Si done v^ est egal ou plus grand que le nombre ^v{v + o), qui est celui des 

 points qui determinent une ligne de I'ordre v, oi\ pourra faire passer plus 

 d'une ligne de I'ordre v par iiV{v + 3) points, ce qui semble contraire 

 a I'article § 38.' The explanation he ofiers is as follows : ' Cette contradic- 

 tion se leve par la remarque qui termine § 38, c'est qu'encore qu'on ait 

 autant d'equations qu'il en faut, generalement parlant, pour determiner 

 tous les coefficients de I'equation prise pour i-epresenter la courbe qui doit 

 passer par un certain nombre de points donnes, il peut pourtant arriver 

 que ces coefficients restent indetermines. Alors I'equation prise reste 

 indetermine et repre'sente une infinite de courbes du meme ordi'e.' His 

 use of the plural, ' ces coefficients,' is worth noting ; in § 38 it is even more 

 emphatic : ' quelques-uns de ces coefficients ' ; he probably did not recog- 

 nise that a single infinity only of curves is involved.^ 



Singularly enough, Cramer, in enunciating the paradox, makes no 

 mention of Euler, whose paper had been communicated to the Berlin 

 Academy two years previously and printed in the year of publication of 

 the ' Analyse.' The same volume of the ' Memoires de I'Academie de 

 Berlin ' contains an historical paper by Cramer himself, and it is rather 

 improbable that he had no knowledge of Euler's work.^ Euler's paper is 

 written with characteristic naivete, almost suggesting that he is writing 

 down his own train of thought step by step, in meditating over the knotty 

 point. By means of the simplest examples he illustrates the cause of 

 a possible indetermination in one or more of n unknown quantities, con- 

 nected by n equations, summing up thus : ' Une des quantites inconnues 

 restera indeterminee si une des equations proposees est renfermee dans 

 les autres. De plus, deux ou plusieurs quantities inconnues resteront 



' Acad. Berlin, annce 1748, pp. 219-233. - Analyse, pp. 78, 70. 



" Cf. Scott, Bull. Am. Math. Soc, vol. iv. (1898), p. 262. 



* Cf. Cantor, vol. iii. p. 797, where doubt is thrown on Cramer's statement that 

 his Analyse was written quite independently of Euler's Introductio 



