494 Anniversary Meeting. [Nov. 30, 



origin of truths, brings us into actual contact with each individual truth, 

 and at the same time reveals to us the mysterious chain by which all are 

 connected." 



The elaborate work here quoted* is unique of its kind ; it is our highest 

 authority on all matters connected with the history of geometry, of which 

 science it carefully traces the development from the time of Thales and Py- 

 thagoras, down to the earlier part of the present century. Although pro- 

 fessing to be an aperru merely, it nevertheless represents a vast amount of 

 historical research, and is moreover enriched by copious notes containing 

 the results of important original investigations. 



In the year 1846 the foundation of a chair of modern geometry was de- 

 cided upon by the Faculty of Sciences at Paris, and Chasles was at once 

 chosen to supply a demand which his own researches had in a great measure 

 created. Thus commenced that personal influence on the younger geo- 

 meters of his country which still continues, and is traceable in all their 

 productions. Another result of this appointment, by which geometers of 

 all nations have greatly profited, was the publication, in 1852, of his 'Trea- 

 tise on the Higher Geometry '(, a work in which the three fundamental 

 principles of pure geometry are, for the first time, fully and systematically 

 expounded. These principles embrace the modern theories of anharmonic 

 ratios, of homographic divisions and pencils, and of geometric involution. 

 An anharmonic ratio is in reality a ratio of two ratios, the latter having 

 reference to two pairs of segments determined by any four points of 

 a line. On one peculiar property of this ratio that of its remaining 

 unaltered by projection all modern geometry may be said to be 

 founded. Homographic divisions consist of two rows of points, in the 

 same straight line or in different ones, which so correspond that the 

 anharmonic ratio of any four points of one row is equal to that of 

 the corresponding points of the other row. Finally, two homographic rows, 

 in the same straight line, are said to form an* involution when to any point 

 whatever of that line one and the same point corresponds, no matter to 

 which of the two rows the first point may be conceived to belong. Usually 

 there are two points in such an involution, each of which coincides with its 

 own corresponding point ; by a mere accident of position, however, the 

 actual existence of these double points may be destroyed, whilst all other 

 properties of the involution remain intact. In this contingency originated 

 a mode of speech of the greatest utility in geometry. The double points 

 are said to be real in the one case, and imaginary in the other. For the 

 undisguised and philosophic introduction of imaginary points and lines 

 into pure geometry we are chiefly indebted to Chasles. 



* Aperfu historique sur 1'origine et le developpement des mcthodes en Geometric, 

 particulierement de celles qui se rapportent a la Geometric Moderne ; stiivi d'un Memoirc 

 de Geometric sur deux principes generaux de la science, la dualite et riiomograpliie. 

 Bruxelles, 1837. German translation by Dr. SShncke; Halle, 1837. 



t Traite de Geometric Superieurc. Paris, 1852. 



