STATEMENT OF LEIBNIZ S PHILOSOPHY 75 



and to substitute for it a more adequate theory of Force 

 and a higher Mathematics. Both the Mathematics and 

 the Physics of the time appeared to Leibniz to be too 

 abstract, and the great object of his speculations was to 

 bring them more into touch with concrete reality. 



The Transition from Synthetic to Analytic Geometry. 



Early in the seventeenth century a considerable ad- 

 vance was made in the science of Mathematics, mainly 

 through the work of Kepler, Cavalieri, and Descartes. 

 The Geometry of the Greeks was synthetic or synoptic. 

 It dealt with ideal figures as discrete wholes, not taking 

 into consideration the possibility of their being analyzed 

 into elements, of which they are combinations or func- 

 tions. Thus the relations of the figures to one another 

 are considered as external. Each is what it is : no one 

 is regarded as having in it the possibility of passing into 

 another. A rectilineal figure is one thing ; a curvilinear 

 figure is another. The barriers between them are re- 

 garded as insurmountable, at least by the methods of 

 exact or demonstrative science. Thus a curve is still 

 a curve, however small may be its curvature. A polygon 

 is still a polygon, however numerous may be its sides. 

 And the kinds of curves are each independent of the 

 others. An ellipse is still an ellipse, however distant 

 one focus may be from the other. 



Kepler's introduction of the notion and the name of 

 infinity into Geometry was the beginning of a great 

 change in mathematical methods. The geometrical 

 figures of the Greeks were all finite, and therefore 

 capable of representation to the eye, or, in other words, 

 capable of being pictured. Every curve must have 

 a definite curvature. Every polygon must Ijave a de- 

 finite number of sides. Kepler, in order to attain to 

 greater exactness in the statement of mathematical 

 relations, suggested that finite (or definite) figures might 



