82 REPORT — 1902. 



moyen de reduction, de I'etude des lieux plans et solidea.' The terms 

 plane, solid, and linear loci were in general use at that time, with the 

 meanings he attaches to them ; the notion of including them all under 

 one law of formation marks his step towards the true analytical stand- 

 point ; but the end of the paragraph shows the limitation to which his 

 mind was still subject. In fact, the main interest of the new method to 

 Fermat and Descartes lay in its application to the solution of algebraical 

 equations by means of the intersections of geometrical curves ; they did 

 not, apparently, discern that the opposite course would prove the more 

 fruitful, that with a wider knowledge of the theory of algebraical 

 equations future generations would obtain a grasp of the geometry of 

 curves far exceeding that of the ancients. 



The slow emergence of the modern standpoint is shown by the lapse 

 of time — nearly seventy years — between the publication of Descartes' 

 'Geometry ' (1637) and of Newton's 0642-1727) ' Enumeratio linearum 

 tertii ordinis ' (1704). Here we find stated for the first time the defini- 

 tion, now usually adopted, of the order of a curve, viz., the number of 

 points in which it can be cut by a straight line. Descartes had adopted 

 an unfortunate classification,^ not according to degree, but according to 

 'genus' : in his parlance curves of the 2nth and of the (2?i-l)th degrees 

 belong to the nth genus. This had possibly arisen from his investigations 

 into a celebrated problem of Pappus,' or, as Fermat seems to think,^ from 

 an erroneous conclusion respecting the reduction of an equation of degree 

 2n by one degree. In any case it became a source of error ^ and was 

 tacitly abandoned. 



Next in importance to the order of a curve is the number of terms 

 involved in its equation. The statement that the equation of a curve of 

 the nth order contains ^m (w-t-3) coefficients was first made by James 

 Stirling (1692-1770) in his 'Lineas tertii ordinis Newtoniante,' which was 

 published thirteen years after the Enumeratio (i.e., in 1717), and is practi- 

 cally an exposition of and a sequel to Newton's book. Moreover, in the 

 same work -^ Stirling drew attention to the fact that a curve can only pass 

 through hi {11 + ^) points, and that it is determined by this number of 

 points, and thus paved the way for the enunciation, three years later 

 (1720), of the so called Cramer Paradox by his contemporary Maclaurin '' 

 (1698-1746). This young Scotchman was barely twenty-one when his 

 ' Geometrica organica, sive descriptio linearum curvai'um universalis ' was 

 published. Short as it is, 140 quarto pages in all, this treatise at once 

 placed its author in the front rank of geometers, and is justly held to be 

 the foundation of the modern synthetic geometry of higher plane curves,^ 

 in so far as this depends upon theorems dealing with their intersections. 



1 Descanes, QHtirres, edit. Cousin (Paris, 1824), vol. v. p. 338, or Geomeiria, edit. 

 Schooten (Amsterdam, 1659), p. 21. 



2 Cantor, vol. ii. p. 742. 



' Fermat, vol. i. p. 119 (Latin original); vol. iii. p. 110 (French trans.). 



< Cantor, vol. ii. p. 44. " P. 69. 



" For a brief account of the historical oblivion into which the true originator of 

 the paradox had fallen, cf. C. A. Scott, Bull. Am. Math. Soc, vol. iv. (1898), p. 261. 



' Brill and Noether, ' Die Enfcwicklung der Theorie der algebraischen Functionen 

 in iilterer und neuerer Zeit,' Jakresber. d. dcvtsclieii Math. Ver., vol. iii. (1894), p. 129. 

 This valuable report is full of suggestive criticism, most helpful to any student of 

 the papers which it passes in review, but its concise language makes it rather 

 difficult reading. 



