STATEMENT OF LEIBNIZ S PHILOSOPHY 77 



The Basis of Analytical Geometry. 



This connexion between Algebra and Geometry was 

 definitely established by Descartes in the Analytical 

 Geometry, of which he was the inventor. The basis of 

 the Analytical Geometry is/ the finding of a definite 

 proportion between the space-relations or ratios investi- 

 gated by Geometry and certain numerical ratios. But 

 the space-relations of Geometry are not merely quantita- 

 tive as are the relations of number. To take the simplest 

 of instances, the square upon a line may be represented 

 by the square of a number. But the square of a number 

 n is simply n times n, that is to say, it is the sum of 

 n n's added together. The square of n is a quantity of 

 n's or a simple series of homogeneous units, which may 

 be interchanged within the series without in any way 

 affecting the result. On the other hand, the relation 

 of a geometrical square to the line upon which it is 

 constructed (i.e. to any one of its sides) is not purely 

 quantitative. The square is not a sum of lengths. It is 

 a figure with special characteristics. The line cannot 

 intelligibly be regarded as its unit. It is its side, and 

 as the side of a square it has properties other than those 

 which it would have as a mere line. It is, in fact, 

 part of a unity which is more than merely quantitative. 

 And yet a quantitative ratio can express the relation 

 between the square and its side, in such a way that the 

 properties of the square may be algebraically calculated 

 without direct reference to the geometrical figure. Thus 



plane area. Pascal, however, showed that Cavalieri's method 

 really implied that the infinite series of straight lines is an in- 

 definite ' number of l small ' rectangles, which are so small that 

 the minute triangles between them and the sides of the given 

 triangle may be neglected in the computation. This l indefinite ' 

 of Pascal is the l infinite ' of later mathematicians, and his ( small ' 

 is manifestly their ' infinitely little.' Thus we have here the 

 transition from the ancient to the modern methods. Pascal vindi- 

 cated Cavalieri's method on the ground that it differed only in 

 manner of expression from the method of exhaustions, used in the 

 Greek mathematics. 



