152 REASONING. 



osition that two straight lines cannot inclose a space, (or its equivalent, 

 Straight lines which coincide in two points coincide altogether,) and 

 some property of parallel lines, other than that which constitutes their 

 definition : the most suitable, perhaps, being that selected by Professor 

 Playfair : " Two straight lines which intersect each other cannot both 

 of them be parallel to a third straight line."* 



The axioms, as well those which are indemonstrable as those which 

 admit of being demonstrated, differ from that other class of funda- 

 mental principles which are involved in the definitions, in this, that 

 they are true without any mixture of hypothesis. That things which 

 are equal to the same thing are equal to one another, is as true of the 

 lines and figures in nature, as it would be of the imaginary ones 

 assumed in the definitions. In this respect, however, mathematics 

 is only on a par with most other sciences. In almost all sciences 

 there are some general propositions which are exactly true, while the 

 greater part are only more or less distant approximations to the truth. 

 Thus in mechanics, the first law of motion, (the continuance of a move- 

 ment once impressed, until stopped or slackened by some resisting force,) 

 is true without a particle of qualification or error ; it is not affected by 

 the frictions, rigidities, and miscellaneous disturbing causes, which 

 qualify, for example, the theories of the lever and of the pulley. The 

 rotation of the earth in twenty-four hours, of the same length as in our 

 time, has gone on since the first accurate observations, without the 

 increase or diminution of one second in all that period. These are 

 inductions which require no fiction to make them be received as accu- 

 rately true : but along with them there are others, as for instance the 

 propositions respecting the figure of the earth, which are but approxi- 

 mations to the truth ; and in order to use them for the further advance- 

 ment of our knowledge, we must feign that ihey are exactly true, 

 although they really want something of being so. 



§ 4. It remains to inquire, what is the ground of our belief in axioms 

 — what is the evidence on which they rest 1 I answer, they are ex- 

 perimental truths ; generalizations from observation. The proposition, 

 Two straight lines cannot inclose a space — or in other words. Two 

 sti'aight lines which have once met, do not meet again, but continue to 

 diverge — is an induction from the evidence of our senses. 



This opinion runs counter to a philosophic prejudice of long stand- 

 ing and great strength, and there is probably no one proposition enun- 

 ciated in this work for which a more unfavorable reception is to be ex- 

 pected. It is, however, no new opinion ; and even if it were so, would 

 be entitled to be judged, not by its novelty, but by the strength of the 

 arguments by which it can be supported. I consider it very fortunate 

 that so eminent a champion of the contrary opinion as Mr. Whewcll, 

 has recently found occasion for a most elaborate treatment of the whole 

 theory of axioms, in attempting to construct the philosophy of the 



* We might, it is true, insert this property into the definition of parallel lines, framing- the 

 definition so as to require, bo/h that when produced indefinitely they shall never meet, and 

 also that any straight line which intersects one of them shall, if prolonged, meet the other. 

 But by doing this we by no means get rid of the assumption ; we are still obliged to take 

 for granted the geometrical truth, that all straight lines in the same plane, which have the 

 former of these properties, have also the latter. P'or if it were possible that they should 

 not, that is, if any straight lines other than those which are parallel according fo the defini- 

 tion, had the property of never meeting although indefinitely produced, the demonstrationa 

 of the subsequent portions of the theory of parallels could not be nijdntained. 



