112 A SHORT HISTORY OF SCIENCE 



and proceeded to investigate the variations which an algebraic ex- 

 pression undergoes when one of its variables assumes a continuous 

 succession of values. Hankel. 



In one of the most brilliant passages of his Aperqu historique 

 Chasles remarks that, while Archimedes and Apollonius were the 

 most able geometricians of the old world, their works are distinguished 

 by a contrast which runs through the whole subsequent history of 

 geometry. Archimedes, in attacking the problem of the quadrature 

 of curvilinear areas, established the principles of the geometry which 

 rests on measurements; this naturally gave rise to the infinitesimal 

 calculus, and in fact the method of exhaustions as used by Archimedes 

 does not differ in principle from the method of limits as used by 

 Newton. Apollonius, on the other hand, in investigating the proper- 

 ties of conic sections by means of transversals involving the ratio of 

 rectilineal distances and of perspective, laid the foundations of the 

 geometry of form and position. Ball. 



The works of Archimedes and Apollonius marked the most 

 brilliant epoch of ancient geometry. They may be regarded, more- 

 over, as the origin and foundation of two questions which have occu- 

 pied geometers at all periods. The greater part of their works 

 are connected with these and are divided by them into two classes, 

 so that they seem to share between them the domain of geometry. 



The first of these two great questions is the quadrature of curvi- 

 linear figures, which gave birth to the calculus of the infinite, con- 

 ceived and brought to perfection successively by Kepler, Cavalieri, 

 Fermat, Leibnitz and Newton. 



The second is the theory of conic sections, for which were in- 

 vented first the geometrical analysis of the ancients, afterwards 

 the methods of perspective and of transversals. This was the pre- 

 lude to the theory of geometrical curves of all degrees, and to that 

 considerable portion of geometry which considers, in the general 

 properties of extension, only the forms and situations of figures, 

 and uses only the intersection of lines or surfaces and the ratios of 

 rectilineal distances. 



These two great divisions of geometry, which have each its pe- 

 culiar character, may be designated by the names of Geometry of 

 Measurements and Geometry of Forms and Situations, or Geometry 

 of Archimedes and Geometry of Apollonius. Chasles (Gow). 



