76 INTRODUCTION 



be regarded as consisting of an infinite (or indefinite) 

 number of elements. Thus he considered a circle to be 

 composed of an infinite number of triangles, having 

 their common vertex at the centre and forming the 

 circumference by their bases 1 . Such an analytic con- 

 ception of the figure is, of course, not capable of being 

 pictured. But it at once suggests the possibility of 

 representing the figure, not by a rough drawing or 

 image, but by an infinite numerical series the terms 

 of which are so related to one another that their sum is 

 finite. Accordingly, in thus considering the finite as 

 made up of an infinite number of elements, we have 

 promise of a connexion between Geometry and Algebra, 

 of such a kind that geometrical relations may be sym- 

 bolized algebraically and the knowledge of them may be 

 extended and generalized by calculation. Such a con- 

 nexion would mean the reduction of the discontinuous 

 concepts of Synthetic Geometry to the comparative 

 continuity of Algebraic Concepts or Numbers. It would 

 thus lessen the abstractness of Geometry, and make it 

 more adequate to the continuity of nature, or, looking 

 at the same thing from the dpposite point of view, it 

 would enable the continuous system of space-relations 

 to be more completely brought within the range of 

 mathematical demonstration. For instance, problems 

 w^hich the Greeks had to solve by the indirect and 

 unsuggestive method of reductio ad absurdum would now 

 be capable of a direct demonstrative solution, and there 

 would arise many new problems which the old methods 

 could not touch. / 



1 In a similar way Cavalieri afterwards suggested that the area 

 of a triangle might be conceived as made up of an infinite number 

 of straight lines, each parallel with the base. The lengths of these 

 lines he regarded as forming an infinite series in arithmetical pro- 

 gression, of which the first term is zero. The sum of this series is 

 equal to half the product of the last term (i.e. the length of the 

 base of the triangle) and the number of terms (i.e. the altitude of 

 the triangle). As against this it was pointed out that, since a line 

 has no breadth, no number of straight lines can ever make up a 



