ON THE THEORY OF POINT-GROUPS. SI 



Jie^^ort on the Tlteory of Polnt-growps. — Part II. 

 Bij FR.\^x'ES Hardc.\stlk, Camhrhhjr. 



CONTEKTS. 



I'AGr: 



§ 0. The Title of the Report 81 



§ 6. General Historical Introduction 81 



§ 7. The Theory of Elimination, from Leibnitz to Cramer, 1603-1 7 JO . . 83 

 § 8. Memoirs on the Intersections of Plane Curves, from Maclaurin to Lami', 



1720-1818 00 



§ 5. The Title op the Report. 



The first instalment of this report is printed in the 'British Associa- 

 tion Report' for 1900 under the title 'A Report on the Present State of 

 the Theory of Point-groups.' In view, however, of the space it has been 

 found necessary to assign to the historical development of the subject, 

 this title has been changed by the omission of the words limiting its scope 

 to contemporary times. The sections of which the present instalment 

 consists are numbered consecutively with Part I. After a short general 

 introduction (§ 6) and a section on the Tlieory of Elimination (§ 7) the 

 first period of the historical outline given in § 2 is expanded in § 8. 



§ G. General Historical Introduction. 



In the middle of the seventeenth century two men's names stand out 

 prominently in the history of pure mathematics. Descartes (159G-1G50) 

 and Fermat (lGOl-1665), both Frenchmen, were born within five years of 

 each other, and although the exclusive epithet of Cartesian has been 

 bestowed by posterity upon the technical device which each independently 

 invented for the treatment of geometrical problems, it is doubtful whether 

 Format's ideas were not of Avider significance,^ and in the investigations 

 which boar most on our purpose he certainly showed the greater insight, 

 notwithstanding certain unfortunate deviations from fact in his criticism 

 of his rival. ^ 



The paper containing Fermat's exposition of the method of coordinates ^ 

 begins by a detailed investigation of the equations of a straight line and 

 of each of the conic sections in turn, but carries this idea no further. In 

 his prefatory words, however, we note his realisation of tlie possibility of 

 a more general application of the method : ' Toutes les fois que dans une 

 Equation finale on trouve deux quantity's inconnues, on a un lieu, 

 I'extremite de I'une d'elles decrivant une ligne droite on courbe. 

 Toutes les fois que I'extremite de la quantite inconnue qui decrit le lieu 

 suit une ligne droite ou circulaire, le lieu est dit ;;^a?i ; si elle decrit une 

 parabole, une hyperbole ou une ellipse, le lieu est dit solide ; pour d'autres 

 courbes on I'appelle lieu de ligne. Nous n'ajouterons rien sur ce dernier 

 cas, car la connaissance des lieux de ligne se deduit tres facilement, au 



' Cf Cantor, Gcxcliirhtc dcr ISIathematili, vol. ii. p. 7-15. This learned and 

 withal interesting history mppiies much information concerning works published 

 before the year 1758, with which it closes. 



- Fermat, (Eiivres (edit. Tannery and Henrj-, Paris, 1891), vol. i. p. 121, editors' 

 note. Cf. also Cantor, vol. ii. p. 744. 



' 'Introduction aux lieux plans et solides,' loc. c't., vol. i. pp. 91-110 (Latin 

 original) ; vol. iii. pp. 85-101 (French trans.). 



1902. a 



