METEOROLOGICAL OBSERVATIONS ON BEN NEVIS. 65 



while at Fort William no such lowering of temperature occurs. This is a 

 peculiarity which kites and balloon ascents have recently familiarised us 

 with, and it forms a prime factor in all inquiries into the theory of the 

 cyclone, about which opinion at present is so much divided. 



Heport on the Theory 0/ Point-groups.^ — Part III, 

 Jji/ Frances Hardcastle, Cambridge. 



§ 9. 1818-1857. While Fermat and Descartes, by combining the processes 

 of Algebra and Geometry, were evolving the foundations of that system 

 of co-ordinates which rapidly became the common language of geometers, 

 a contemporary mathematician, Desargues of Lyons (1593-1662), and his 

 pupil Pascal (1623-1662) were occupied with the study of those properties 

 of figures, in space and in the plane, which persist under the operation 

 known as projection. And had it not been for the evil fate which 

 caused the publications of both master and pupil to be lost, and for the 

 oblivion into which even the memory of these writings sank for more than 

 180 years, it is probable that modern synthetic geometry would have been 

 developed from the beginning side by side with analytical geometry, instead 

 of coming into existence, as it did, a whole century and a half later than 

 its rival. The fundamental characteristic of each — that which most dis- 

 tinguished both systems from the geometry of the ancients — is the same, 

 the systematic use of the principle of projection. But it is noteworthy 

 that, although this was present from the beginning in the structure of 

 Cartesian co-ordinates (whereby every point of a curve is projected on to 

 the axis), it was only after the rise of descriptive geometry under Monge 

 (1746-1818) and Carnot (1753-1823) (who explicitly founded it upon 

 projection from ordinary space on to the plane), thatPliicker (1801-1868), 

 by the use of homogeneous co-ordinates, ^ really opened up the projective 

 possibilities inherent in analytical geometry. Throughout the period now 

 to be discussed, the projective standpoint is the one adopted by analytical 

 as well as by synthetic geometers ; the transition to the wider point of 

 view afforded by bi-rational transformation was only effected after the 

 ideas of the theory of functions — at that time still in its infancy — had 

 permeated the whole domain of pure mathematics, and had influenced 

 the theory of higher plane curves to a degree which must have been 

 startling to the mathematicians of the early nineteenth century. 



Among the numerous novel terms introduced by Desargues in his 

 ' Brouillon projet d'une atteinte aux evenements des rencontres d'un cone 

 avec un plan '^ was that of involution,^ and, unlike many of the others, it 

 has survived. Starting from the detinition that six points on a straight 

 line are in involution if certain ratios can be established among the seg- 

 ments formed by them, Desargues proved his famous theorem that a conic 

 and the sides of an inscribed quadrilateral determine six points in invo- 

 lution on any transversal. He did not, however, investigate the still 



' Parts I. and II. appeared in the Brit. Assoc. Beports for 1900, 1902. 



- Moebius's Barycentrischc Calcul was printed in 1827, and was actuallj^ the first 

 publication in which homogeneous co-ordinates were brought forward ; Pliicker's 

 paper in Crelle, vol. v. (1830), gave the first exposition of trilinear co-ordinates. Cf. 

 Clebsch, 'Julius Pliicker zum Gedachtniss,' Ahhandlungen Gotti/iffc/i.xol.xvi. (1872), 

 pp. 1-40. 



^ Discovered in De la Hire's manuscript copy by Chasles in 1845, and printed in 

 Poudra, (Euvres de Desargvos (Paris, 1864), vol. i. pp. 97-230. 



* Poudra, luc. cit., vol. i. p. 101, p. 109, p. 119; also vol. ii. p. 362. 



1903. F 



