Dn. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 171 



light were inherent and not borrowed. But we cannot conceive one of the parallelograms on the 

 same base and between the same parallels larger than the other ; for we find that, if we attempt to 

 do this, when we separate the parallelograms into parts, we have to conceive one triangle larger than 

 another, both having all their parts equal ; which we cannot conceive at all, if we conceive tlie 

 triangles distinctly. We make this impossibility more clear by conceiving the triangles to be placed 

 so that two sides of the one coincide with two sides of the other ; and it is then seen, that in 

 order to conceive the triangles unequal, we must conceive the two bases which have the same 

 extremities both ways, to be different lines, though both straight lines. This it is impossible to 

 conceive : we assent to the impossibility as an axiom, when it is expressed by saying, that two 

 straight lines cannot inclose a space; and thus we cannot distinctly conceive the contrary of the pro- 

 position just mentioned respecting parallelograms. 



4. But it is necessary, in applying this distinction, to bear in mind the terms of it ; — that we 

 cannot distitictty conceive the contrary of a necessary truth. For in a certain loose, indistinct way, 

 persons conceive the contrary of necessary geometrical truths, when they erroneously conceive false 

 propositions to be true. Thus, Hobbes erroneously held tiiat he had discovered a means of geome- 

 trically doubling the cube, as it is called, that is, finding two mean proportionals between two given 

 lines; a problem which cannot be solved by plane geometry. Hobbes not only proposed a construction 

 for this purpose, but obstinately maintained that it was right, when it had been proved to be 

 wrong. But then, the discussion showed how indistinct the geometrical conceptions of Hobbes 

 were; for when his critics had proved that one of the lines in his diagram would not meet the other 

 in the point which his reasoning supposed, but in another point near to it ; he maintained, in replv, 

 that one of these points was large enough to include the other, so that they might be considered as 

 the same point. Such a mode of conceiving the opposite of a geometrical truth, forms no excep- 

 tion to the assertion, that this opposite cannot be distinctly conceived. 



5. In like manner, the indistinct conceptions of children and of rude savages do not invalidate 

 the distinction of necessary and expei-iential truths. Children and savages make mistakes even with 

 regard to numbers ; and might easily happen to assert that 27 and 38 are equal to 63 or 6-1. 

 But such mistakes cannot make such arithmetical truths cease to be necessary truths. When 

 any person conceives these numbers and their addition distinctly, by resolving them into parts, or in 

 any other way, he sees that their sum is necessarily 65. If, on the ground of the possibility of 

 children and savages conceiving something different, it be held that this is not a necessary truth, it 

 must be held on the same ground, that it is not a necessary truth that 7 and 4 are equal to 11 ; for 

 children and savages might be found so unfamiliar with numbers as not to reject the assertion that 

 7 and 4 are 10, or even that 4 and 3 are 6, or 8. But 1 suppose that no persons would on such 

 grounds hold that these arithmetical truths are truths known only by experience. 



6. Necessary truths are established, as has already been said, by demonstration, proceeding from 

 definitions and axioms, according to exact and rigorous inferences of reason. Truths of experience 

 are collected from what we see, also according to inferences of reason, but proceeding in a less exact 

 and rigorous mode of proof. The former depend upon the relations of the ideas which we have 

 in our minds : the latter depend upon the appearances or phenomena, which present themselves to 

 our senses. Necessary truths are formed from our thoughts, the elements of the world within us ; 

 experiential truths are collected from things, the elements of the world without us. The truths of 

 experience, as they appear to us in the external world, we call Facts ; and when we are able to find 

 among our ideas a train which will conform themselves to the apparent facts, we call this a Theory. 



7. This distinction and opposition, thus expressed in various forms ; as Necessary ami 

 Experiential Truth, Ideas and Senses, Thoughts and Things, Theory and Fact, may be termed 

 the FundamPMtnl Antlthenin of I'liiloxop/11/ ; for almost all the discussions of philosophers liave 

 been employed in asserting or denying, explaining or obscuring tliis antithesis. It may be ex- 



Vol.. VIII. Part II. Z 



