150 REASONING 



on which the reasonings of the science are founded, do not, any more 

 than in other sciences, exactly coiTespond with the fact ; t>ut we srip- 

 pose that they do so, for the sake of tracing the consequences which 

 follow fiom the supposition. The opinion of Dugald Stewart respect- 

 ing the foundations of geometry, is, I conceive, substantially correct ; 

 that it is built upon 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 con- 

 clusions as certain as those of geometiy, that is, as strictly in accord- 

 ance with the hypotheses, and as in'esistibly compelling assent 07i 

 condition that those hypotheses are true. 



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

 necessary truths, the necessity consists in reality only in this, that they 

 necessarily 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 necessarily following from some 

 assumption, which, by the conditions of the inquiry, is not to be ques- 

 tioned. In this relation, of course, the derivative truths of every de- 

 ductive 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 svipposed certain for the purposes 

 of the particular science. And therefore the conclusions of all deduc- 

 tive sciences were said by the ancients to be necessary propositions. 

 We have obsei-^'ed already that to be predicated necessarily was char- 

 acteristic of the predicable Propiium, 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- . 

 deavored to enforce, has been contested by a living philosopher, Mr. 

 Whewell, both in the dissertation appended to his excellent Mechani- 

 cal Enclid, and in his more recent 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 gi'eat scientific emi- 

 nence), in which Stewart'^ opinion was defended against his foirner 

 strictures. Mr. Whewell's mode of refuting Stewart is to prove against 

 him (as has also been done in this work), that the premisses of geom- 

 etry are not definitions, but assumptions of the real existence of things 

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

 Mr. Whewell's purpose, for it is these very assixmptions which we say 

 are hypotheses, and which he, if he denies that geometry is founded on 

 hypotheses, must show to be absolute truths. All he does, however, 

 is to observe, that they at any rate are not arbitrary hypotheses ; that 

 we should not be at liberty to substitute other hypotheses for them; 

 that not only " a definition, to be admissible, must necessarily refer to 

 and agree with some conception which we can distinctly frame in our 

 thoughts," but that the straight lines, for instance, which we define, 

 must be "those by which angles are contained, those by which trian- 

 gles are bounded, those of which jiarallelism may be predicated, and 

 the like."* And this is true; but this has never been contradicted. 

 * Whewell's Mechanical Euclid, p. 149, et segq. 



