DEMONSTRATION, AND NECESSARY TRUTHS. I7l 



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 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 maintain them to be hypoth- 

 eses 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 nonentities), it fol- 

 lows that any hypothesis we make respecting an object, to facilitate our 

 study of it, must not involve any thing which is distinctly false, and repug- 

 nant to 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 

 restoring them whenever^ and in as far as, their presence or absence would 

 make any material difference in the truth of our conclusions. Of this na- 

 ture, accordingly, are the first principles involved in the definitions of ge- 

 ometry. That the hypotheses should be of this particular character, is, 

 however, no furtliur necessary, than inasmuch as no others could enable us 

 to deduce concluhious which, with due corrections, would be true of real 

 objects : and in fact, when our aim is only to illustrate truths, and not to 

 investigate them, we are not under any such restriction. We might sup- 

 pose 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 hy- 

 potheses, might form a highly useful intellectual exercise : but as they could 

 only teach us what loould be the properties of objects which do not really 

 exist, they would not constitute 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 conclu- 

 sions will always express, under known liability 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 principles of geom- 

 etry which are involved in the so-called definitions, he has, I conceive, great- 

 ly the advantage of Stewart on another important point in the theory of 

 geometrical reasoning ; the necessity of admitting, among those first prin- 

 ciples, axioms as well as definitions. Some of the axioms of Euclid might, 

 no doubt, be exhibited in the form of definitions, or might be deduced, by 

 reasoning, from propositions similar to what are so called. Thus, if instead 

 of the axiom. Magnitudes which can be made to coincide are equal, we in- 

 troduce a definition, " Equal magnitudes are those which may be so ap- 

 plied to one another as to coincide ;" the three axioms which follow (Mag- 

 nitudes which are equal to the same are equal to one another — If equals 

 are added to equals, the sums are equal — If equals are taken from equals, 



* Mechanical Euclid, pp. 149 et seqq. 



