90 REPORT— 1902. 



the first line of ' facteurs-premiers ' by reason of their known values. 

 And since each ' facteur- second' can be ultimately reduced, by means of 

 the multiplication theorem, to the sum of products of those in the first 

 line — the total sum of the significant figures in every such product being 

 equal to that in the original bracket — it follows that this sum will be, 

 as required, the power of y. The characteristic property of the multi- 

 plication theorem, although only stated with regard to the roots of the 

 equation, justifies the conclusion ' that Cramer had conceived the idea of 

 ' weight ' as applied to the expression in terms of the coefficients of a 

 symmetric function of the roots, for the sum of the significant figures in 

 the bracket of a ' facteur-second ' (which primarily denotes the degree in 

 the roots) is shown in the proof we have sketched to persist throughout 

 the process of tracing it back to the first line and to be thus equal to 

 the sum of the suffixes in the equivalent expression in terms of the co- 

 efficients. 



§ 8; Memoirs on the Intersections of Plane Curves, from Maclaurin 



TO Lame, 1720-1818. 



The progress of analytical geometry as a whole during this period of 

 about a hundred years is mainly along the lines which Newton had begun 

 to explore. The classic treatises of Euler and Cramer fall within its 

 limits, but, original and fruitful as these works were, they formed rather 

 the climax of the purely Cartesian epoch than the beginning of the modern 

 era. And it is noteworthy, as corroboration of this fact, that both were 

 published before the year 1758, which saw the foundation of the Turin 

 Academy under Lagrange, the ' father of modern mathematics.' The 

 classification of curves according to the degree of their equations, the 

 investigation of all possible types of the same order, of the infinite 

 branches and of the singularities of a given curve, such were the chief 

 problems which occupied these geometers. The notation employed is 

 invariably the Cartesian equation written at full length ; the point was 

 thus the primary element in the plane, and the properties of curves 

 were studied as belonging to configurations of points. The number of 

 intersections of a pair of curves and the possibility of passing a curve of 

 given order through a certain number of given points are questions which 

 really belong to a domain beyond the vision of the mathematicians of the 

 eighteenth century — a domain in Avhich a curve itself is the primary 

 element and systems of curves the subject-matter of investigation. They 

 arose, however, naturally enough, the moment the significance of the 

 coefficients in the equation of a curve was recognised, and were only 

 partially answered by Maclaurin, Euler, and Cramer before Lame, in 1818, 

 took a more important step in their elucidation than he himself probably 

 realised. 



The enunciation of the paradox connected with these questions is due 

 to Maclaurin.- It forms the second corollary to the Lemma of Section V. in 

 his 'Geometrica Organica,' and may be rendered as follows : 'A line of 

 order n can cut another of the same order in 'n? points. Hence two lines 



' Brill andNoether, Jtf.7;ra&er. d. deutschen Math. Frr.,vol. iii. (1894), p. 1.S7. Cf. 

 Cantor, vol. iii. p. .588. The. reference given by the former to §§ 9, 10 of Cramer's 

 Appendix II. is somewhat misleading, as these only deal with the degree in the 

 roots ; but if the whole proof is studied their conclusion is fully established. 



- See § 6. 



