DEMONSTRATION AND NECESSARY TRUTHS. 



149 



properties and of the irregularities, 

 and to reason as if these did not 

 exist : accordingly, we formally an- 

 nounce in the definitions, that we in- 

 tend to proceed on this plan. But it 

 is an error to suppose, because we re- 

 solve to confine our attention to a 

 certain number of the properties of 

 an object, that we therefore conceive, 

 or have an idea of, the object denuded 

 of its other properties. We are think- 

 ing, all the time, of precisely such 

 objects as we have seen and touched, 

 and with all the properties which 

 naturally belong to them ; but, for 

 scientific convenience, we feign them 

 to be divested of all properties, except 

 those which are material to our pur- 

 pose, and in regard to which we de- 

 sign to consider them. 



The peculiar accuracy, supposed to 

 be characteristic of the first principles 

 of geometry thus appears to be ficti- 

 tious. The assertions on which the 

 reasonings of the science are founded 

 do not, anymore than in other sciences, 

 exactly correspond with the fact, but 

 we suppose that they do so for the 

 sake of tracing the consequences 

 which follow from the supposition. 

 The opinion of Dugald Steward re- 

 specting the foundations of geometry, 

 is, I conceive, substantially correct ; 

 that it is built on hypotheses ; that 

 it owes to this alone the peculiar 

 certainty supposed to distinguish it ; 

 and that in any science whatever, by 

 reasoning from a set of hypotheses, 

 we may obtain a body of conclusions 

 as certain as those of geometry, that 

 is, as strictly in accordance with the 

 hypotheses, and as irresistibly com- 

 pelling assent, on condition that those 

 hypotheses are true.* 



* It is justly remarked by Professor Bain 

 (Logic, ii. 134) that the word Hypothesis is 

 used here in a somewhat peculiar sense. 

 An hypothesis, in science, usually means 

 a supposition not proved to be true, btit 

 surmised to be so because if true it would 

 account for certain known fact.5 ; and the 

 final result of the speculation may be to 

 prove its truth. The hypotheses .sjjoken 

 of in the text are of a different character ; 

 they arc known not to be literally true, 

 while as much of them as is true is not 



When, therefore, it is affirmed that 

 the conclusions of geometry are neces- 

 sary truths, the necessity consists in 

 reality only in this, that they correctly 

 follow from the suppositions from 

 which they are deduced. Those 

 suppositions are so far from being 

 necessary, that they are not even 

 true ; they purposely depart, more or 

 less widely, from the truth. The 

 only sense in which necessity can be 

 ascribed to the conclusions of any 

 scientific investigation, is that of 

 legitimately following from some as- 

 sumption, which, by the conditions 

 of the inquiry, is not to be questioned. 

 In this relation, of course, the deriva- 

 tive truths of every deductive science 

 must stand to the inductions, or as- 

 sumptions, on which the science is 

 founded, and which, whether true or 

 untrue, certain or doubtful in them- 

 selves, are always supposed certain 

 for the purposes of the particular 

 science. And therefore the conclu- 

 sions of all deductive sciences were 

 said by the ancients to be necessary 

 propositions. We have observed al- 

 ready that to be predicated necessarily 

 was characteristic of the predicable 

 Proprium, and that a proprium was 

 any property of a thing which could 

 be deduced from its essence, that is, 

 from the properties included in its 

 definition. 



§ 2. The important doctrine of 

 Dugald Stewart, which I have en- 

 deavoured to enforce, has been con- 

 hypothetical, but certain. The two cases, 

 however, resemble in the circumstance 

 that in both we reason, not from a truth, 

 but from an assumption, and the truth 

 therefore of the conclu.sions is conditional, 

 not categorical. This suffices to justify, iu 

 point of logical propriety, Stewart's use of 

 the term. It is of course needful to bear 

 in mind that the hypothetical element in 

 the definitions of geometry is the assump- 

 tion tliat what is very nearly true is exactly 

 so. Thi.s unreal exactiivide might be called 

 a fiction, as properly as an hypothesis ; but 

 that appellation, still more tlmn the other, 

 would fail to point out the close relation 

 which exists between the fictitious point 

 or line and the points and lines of which 

 we have exjierience. 



