CHAP. XIII.] CLARKE AND SPINOZA. 193 



5th. It has not had a cause from within (because no part is 

 necessary, and if no part is necessary, the whole cannot be ne- 

 cessary). 



Omitting, merely for brevity, the subsidiary proofs contained 

 in the parentheses of the fourth and fifth premiss, we may repre- 

 sent the premises as follows : 



Let x = Something has always existed. 



y There has existed some one unchangeable and in- 

 dependent being. 

 z = There has existed a succession of changeable and 



dependent beings. 



p = That series has had a cause from without. 

 q = That series has had a cause from within. 

 Then we have the following system of equations, viz. : 

 1st. x = 1 ; 



2nd. x"= v (y (1 - z) + z (1 - y)} ; 

 3rd. z = v{p(l-q) + (\-p)q}', 

 4th. p = ; 

 5th. q = : 



which, on the separate elimination of the indefinite symbols v, 

 gives 



1-# = 0; (1) 



s{y* + (l-y)(l-*)}-0; (2) 



*(j*+-0->>(l-*))-0; (3) 



P = 0; (4) 



q = 0. (5) 



The elimination from the above system of re, p, q, and y, con- 

 ducts to the equation 



z = 0. 



And the elimination of #, p, q, and z, conducts in a similar man- 

 ner to the equation 



jr-i; 



Of which equations the respective interpretations are : 



1st. The whole of existing things has not been comprehended 



in a succession of changeable and dependent beings. 



2nd. There has existed some one unchangeable and independent 



being. 



