IVO REASONING. 



pies of geometry, thus appears to be fictitious. The assertions on which 

 the reasonings of the science are founded, do not, any more than in other 

 sciences, exactly correspond with the fact ; but we suppose that they do 

 so, for tlie sake of tracing the consequences which follow from the suppo- 

 sition. The opinion of Dugald Stewart respecting the foundations of ge- 

 ometry, 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 sti'ictly in accordance with the hypotheses, and as irresistibly compel- 

 ling assent, o?i condition that those hypotheses are true.* 



When, therefore, it is affirmed that the conclusions of geometry are nec- 

 essary truths, the necessity consists in reality only in this, that they cor- 

 rectly 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 in- 

 vestigation, is that of legitimately following from some assumption, which, 

 by the conditions of the inquiry, is not to be questioned. In this relation, 

 of course, the derivative truths of every deductive science must stand to 

 the inductions, or assumptions, on which the science is founded, and which, 

 whether true or untrue, certain or doubtful in themselves, are always sup- 

 posed certain for the purposes of the particular science. And therefore 

 the conclusions of all deductive sciences were said by the ancients to be 

 necessary propositions. We have observed already that to be predicated 

 necessarily was characteristic of the predicable Proprium, and that a pro- 

 prium was any property of a thing which could be deduced from its es- 

 sence, that is, from the properties included in its definition. 



§ 2. The important doctrine of Dugald Stev/art, which I have endeav- 

 ored to enforce, has been contested by Dr. Whewell, both in the disserta- 

 tion appended to his excellent Mechanical Euclid^ and in his elaborate 

 work on the Philosophy of the Inductive Sciences; in which last he also 

 replies to an article in the Edinburgh Review (ascribed to a writer of 

 great scientific eminence), in which Stewart's opinion was defended against 

 his former strictures. The supposed refutation of Stewart consists in 

 proving against him (as has also been done in this work) that the premises 

 of geometry are not definitions, but assumptions of the real existence of 

 things corresponding to those definitions. This, however, is doing little for 

 Dr. Whewell's purpose ; for it is these very assumptions which are as- 

 serted to be hypotheses, and which he, if he denies that geometry is founded 



* It is justly remarked by Professor Bain (Logic, ii., 134) that the word Hypothesis is here 

 used in a somewhat peculiar sense. An hypothesis, in science, usually means a supposition 

 not proved to be true, but surmised to be so, because if true it would account for certain 

 known facts ; and the final result of the speculation may be to prove its truth. The hypothe- 

 ses spoken of in the text are of a different character ; they are known not to be literally true, 

 while as much of them as is true is not hypothetical, but certain. The two cases, however, 

 resemble in the circumstance that in both we reason, not from a truth, but from an assump- 

 tion, and the truth therefore of the conclusions is conditional, not categorical. This suffices 

 to justify, in 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 assumption 

 that what is very nearly true is exactly so. This unreal exactitude might be called a fiction, 

 as properly as an hypothesis ; but that appellation, still more than 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 experience. 



