58 COSMIC PHILOSOPHY. [vt. i. 



truths par excellence, — habitually speaking of them as if they 

 were in some way truer than physical and chemical truths. 

 Bearing in mind what was said a moment ago, it will be 

 sufficient to observe that in mathematics we utter propositions 

 with respect to certain particular relations alone, without 

 regard to other conditions, and hence there is absolutely no 

 room for contingency. Let me conclude this portion of the 

 subject by a citation from Mr. Lewes : — "When we say that 

 twice two is four, or that the internal angles of a triangle 

 are equal to two right angles, we abstract the relations of 

 Number and Form from all other conditions whatever, and 

 our propositions are true, whether the objects counted and 

 measured be hot or cold, large or small, heavy or light, red 

 or blue. Inasmuch as the truths express the abstract rela- 

 tions only, no change in the other conditions can affect these 

 relations ; and truths must always remain undisturbed until 

 a change take place in their terms. Alter the number two, 

 or the figure triangle,by an infinitesimal degree, and the truth 

 is thereby altered. When we say that bodies expand by 

 heat, the proposition is a concrete one, including the variable 

 conditions ; but although these variable conditions prevent 

 our saying that all bodies will under all conditions be always 

 and for evermore expanded by heat, the case is not really 

 distinguished from the former one, since both the Contingent 

 and the Necessary Truth can only be altered by an alteration 

 in the terms. If a body which does not expand by heat 

 (there are such) be brought forward as impugning the truth 

 of our proposition, we at once recognize that this body is 

 under different conditions from those which our proposition 

 included. This is the introduction of a new truth, not a 

 falsification of the old. Our error, if we erred, was in too 

 hastily assuming that all bodies were under the same condi- 

 tions. Hence the correct definition of a Contingent Truth 

 is ' one which generalizes the conditions ' ; while that of a 

 Necessary Truth is ' one which is an unconditional generaii- 



