ISO 



REASONING. 



tested by Dr. Whewell, both in the 

 dissertation appended to his excellent 

 Mechanical Euclid, and in his elabo- 

 rate work on the Philosophy of the 

 Inductive Sciences; in which last he 

 also replies to an article in the Edin- 

 burgh 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 prov- 

 ing 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 defini- 

 tions. This, however, is doing little 

 for Dr. Whewell's purpose ; for it is 

 these very assumptions which are 

 asserted to be 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 con- 

 tained, those by which triangles are 

 bounded, those of which parallelism 

 may be predicated, and the like."* 

 And this is true : but this has never 

 been contradicted. Those who say 

 that the premises of geometry are 

 hypotheses, are not bound to main- 

 tain them to be hypotheses which 

 have no relation whatever to fact. 

 Since an hypothesis framed for the 

 purpose of scientific inquiry must 

 relate to something which has real 

 existence, (for there can be no science 

 respecting non -entities,) it follows 

 that any hypothesis we make respect- 

 ing an object, to facilitate our study 

 of it, must not involve anything which 

 is distinctly false, and repugnant to 



* Mechanical Suclid, pp. 149 ei ttq. 



its real nature : we must not ascribe 

 to the thing any property which it 

 has not ; our liberty extends only to 

 slightly exaggerating some of those 

 which it has, (by assuming it to be 

 completely what it really is very 

 nearly,) and suppressing others, under 

 the indispensable obligation of restor- 

 ing them whenever, and in as far as, 

 their presence or absence would make 

 any material difference in the truth 

 of our conclusions. Of this nature, 

 accordingly, are the first principles in- 

 volved in the definitions of geometry. 

 That the hypotheses should be of this 

 particular character is, however, no 

 further necessary, than inasmuch as 

 no others could enable us to deduce 

 conclusions which, with due correc- 

 tions, would be true of real objects : 

 and in fact, when our aim is only to 

 illustrate truths, and not to investi- 

 gate them, we are not under any 

 such restriction. We might suppose 

 an imaginary animal, and work out 

 by deduction, from the known laws 

 of physiology, its natural history ; or 

 an imaginary commonwealth, and 

 from the elements composing it 

 might argue what would be its fate. 

 And the conclusions which we might 

 thus draw from purely arbitrary 

 hypotheses might form a highly 

 useful intellectual exercise : but as 

 they could only teach us what would 

 be the properties of objects which do 

 not really exist, they would not con- 

 stitute any addition to our knowledge 

 of nature : while, on the contrary, if 

 the hypothesis merely divests a real 

 object of some portion of its properties, 

 without clothing it in false ones, the 

 conclusions will always express, under 

 known lia-bility to correction, actual 

 truth. 



§ 3. But though Dr. Whewell has 

 not shaken Stewart's doctrine as to 

 the hypothetical character of that 

 portion of the first principle* of 

 geometry which are involved in the 

 so-called definitions, he has, I con- 

 ceive, greatly the advantage of Stewart 

 on another important point in the 



