86 INTRODUCTION 



down with their counting-tables, and (having summoned 

 a friend, if they like) say to one another: Let us 

 calculate.' This sounds like the ungrudging optimism of 

 youth ; but Leibniz was optimist enough t9 cherish the 

 hope of it to his life's end. 



This project of the Logical Calculus or philosophical 

 language connects the mathematics of Leibniz with his 

 theory of knowledge, while the Calculus of Infinitesimals 

 finds immediate application in his revision of Descartes's 

 theories regarding matter and motion. Descartes treated 

 motion and rest synthetically as constant quantitative 

 wholes. Leibniz regards them analytically as consisting 

 of an infinite series of degrees of one constant force. 

 Accordingly Leibniz admits that the Cartesian laws of 

 motion have a certain validity in relation to ' abstract ' 

 motion, but denies that they are adequate to the 'con- 

 crete ' physical phenomena. 



B. MATTER. 



Descartes 's Theory of Matter and Motion. 



As we have already seen, Leibniz's view of matter can 

 be understood only as it appears in contrast with that of 

 Descartes. In accordance with his interpretation of the 

 principle of contradiction, viz. that the essence of a thing 

 consists in that only which is common to all its manifes- 

 tations, or (otherwise expressed) in that only which 

 remains after all varieties or specific determinations have 

 been excluded, Descartes maintained that matter is essen- 

 tially extension. Bodily substance and magnitude or 

 spatial extent are identical. And all the changes in 

 matter or extension are ultimately reducible to motion. 

 Motion is regarded by Descartes as being ' the transference 

 of a portion of matter or a body from the neighbourhood 

 of those bodies which are in direct contact with it, and 

 which we consider as at rest, to the neighbourhood of 

 other bodies or portions of matter V Matter is infinitely 



1 Piincipia, Part ii. 25. Descartes adds : ' By a body, or rather 



