496 Anniversary Meeting. [Nov. 30, 



and in it the three principles already alluded to find their most appropriate 

 field of application. The second volume of this treatise is looked forward 

 to with interest, as it will contain a full exposition of the admirable researches 

 on conic sections wherewith Chasles has just crowned his labours. These 

 researches, a brief account of which appeared during the past year in 

 the pages of the ' Comptes Rendus,' have put us in possession of an 

 entirely new method, the nature and utility of which may be rendered 

 intelligible even to those who have not made modern geometry a subject of 

 special study. 



For the determination or construction of the curves usually called 

 conies, and of which the hyperbola, parabola, and ellipse are species, five 

 conditions are requisite and, in general, sufficient. The nature of these 

 five conditions may be such, however, as to admit of their being satisfied 

 by more than one conic. For instance, although one conic only can be 

 described through five given points, there exist two distinct conies, each of 

 which passes through four given points, and touches a given line. Hence 

 arises the important general question, How many conies are there capable 

 of satisfying any five conditions whatever ? By the new method of Chasles 

 we are enabled to answer this question, hitherto a difficult one, with great 

 facility. Starting from the elementary cases where the five conditions are 

 of the simplest possible kind, consisting solely of passages through points 

 and contact with lines, he gradually replaces those conditions by more 

 complex ones, and finally arrives at a simple symmetrical formula 

 which fully answers the above question. Seeing how numerous are the 

 questions in conies which may be ultimately reduced to the one here 

 solved, we may, without exaggeration, assert that in this single formula a 

 great part of the entire theory of conies is virtually condensed. 



The method has been aptly termed by its eminent discoverer a method 

 of geometrical substitution. It involves the consideration of the properties 

 of a system of conies (infinite in number) satisfyingybwr common conditions. 

 Such a system is for the first time defined in a manner closely analogous 

 to that in which curves are distinguished into orders and classes. We 

 merely require to know, first, how many conies of the system pass through 

 an arbitrarily assumed point, and, secondly, how many of them touch any 

 assumed line. These two numbers or characteristics, as they are termed, 

 being once found, all the properties of the system of conies are thereby 

 expressible. For instance, the sum of twice the first characteristic and 

 three times the second gives us the order of the curve upon which the 

 vertices of every conic of the system are situated. 



This new method of characteristics has been already applied to curves 

 of higher orders, as well as to surfaces ; and, considering the magnitude 

 of the new fields of investigation thus opened out, it is probable that, as 

 an instrument of purely geometrical research, the method of Chasles will 

 bear comparison with any other discovery of the century. 



