1865.] President's Address. 495 



The term anharmonic ratio, now universally employed, is due to 

 Chaslcs ; the ratio itself, however, appears to have been known to Pappus, 

 the eminent Alexandrian geometer of the fourth century. Chasles, indeed, 

 has shown that this ratio probably constituted an essential feature of those 

 three famous books on Porisms, which Euclid is known to have written, 

 but of whose nature vague indications merely have been transmitted to 

 us in the mathematical collections of Pappus. Robert Simson of Glasgow, 

 the well-known translator of Euclid's ' Elements,' was the first who satis- 

 factorily solved the enigma concerning the real nature of Porisms, and 

 he also succeeded in partially restoring the three lost books. Chasles, 

 however, was the first to restore them completely ; and this he has done in 

 a work* which is admitted to be a valuable addition to the history of 

 geometrical science, as well as a model of ingenious and philosophical 

 divination. 



Chasles has contributed to the advancement of pure geometry, not only 

 by means of the three complete works already alluded to, but also through 

 the publication of numerous smaller memoirs. Of these the following, 

 by no means the only important ones, demand a passing reference. 



The papers on " Stereographic Projections " converted a method ori- 

 ginally devised for the construction of maps into a powerful instrument 

 of geometrical transformation. Two able memoirs on " Cones of the 

 Second Order" and on "Spherical Conies," thanks to the translation, 

 published in 1841, by Dr. Graves of Trinity College, Dublin, had a direct 

 influence on pure geometry in our own country. A paper " On the Corre- 

 spondence between Variable Objects " furnished us with a principle of the 

 greatest utility in all higher geometrical investigations. In several other 

 memoirs the method of generating curves of higher orders by means of 

 homographic pencils of curves of inferior orders is perfected, and new 

 properties are thereby deduced of plane curves of the third and fourth 

 orders. The theory of non-plane curves, especially those of the third and 

 fourth orders, had its origin, for the most part, in Chasles's memoirs ; and 

 the modern science of kinematics is indebted to him for two valuable papers 

 on the finite and infinitesimal displacements of a Solid Body. The pro- 

 blem of the attraction of Ellipsoids, rendered celebrated by the investiga- 

 tions of Newton, Maclaurin, Ivory, Legendre, Lagrange, and Laplace, re- 

 ceived from Chaslcs its first complete synthetical solution. In this problem, 

 too, originated the conception of confocal surfaces of the second order, the 

 theory of which he has since greatly perfected. 



The first volume of Chasles's fourth work (a Treatise on Conic Sections f) 

 appeared during the present year: it is a sequel to his 'Higher Geometry ;' 



* Les trois livres dc Porismcs d'Euclide, retablie ponr la premiere fois, d'apres la notice 

 et Ics lemraes dc Pappus, ct conformement au sentiment de R. Simson sur la forme des 

 cnonces de ces propositions. Paris, 1860. 



t Traite dcs Sections Coniqucs, faisaiit suite au traito dc Geometric Snpcricurc. 

 Paris, 1865. 



