172 KEASONING. 



1 the remainders are equal), may be proved by an imaginary superposition, 

 resembling that by which the fourth proposition of the first book of Euclid 

 is demonstrated. But though these and several others may be struck out 

 of the list of first principles, because, though not requiring demonstration, 

 they are susceptible of it ; there will be found in the list of axioms two or 

 three fundamental truths, not capable of being demonstrated: among which 

 must be reckoned the proposition that two straight lines can not inclose a 

 space (or its equivalent. Straight lines which coincide in two points coin- 

 cide altogether), and some property of parallel lines, other than that which 

 constitutes their definition : one of the most suitable for the purpose being 

 that selected by Professor Playfair : " Two straight lines which intersect 

 each other can not both of them be parallel to a third straight line."* 



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

 mit of being demonstrated, differ from that other class of fundamental 

 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 are only on a par Avith 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 ap- 

 proximations to the truth. Thus in mechanics, the first law of motion (the 

 continuance of a movement once impressed, until stopped or slackened by 

 some resisting force) is true without qualification or error. The rotation 

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

 gone on since the first accui'ate observations, without the increase or dim- 

 inution of one second in all that period. These are inductions which 

 require no fiction to make them be received as accurately true : but along 

 with them there are others, as for instance the propositions respecting the 

 figure of the earth, which are but approximations to the truth ; and in or- 

 der to nse them for the further advancement of our knowledge, Ave must 

 feign that they are exactly true, though they really Avant something of be- 

 ing so. 



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

 what is the evidence on Avhich they rest ? I answer, they are expei'i- 

 mental truths; generalizations from observation. The proposition. Two 

 straight lines can not inclose a space — or, in other Avords, Tavo sti'aight 

 lines Avhich have once met, do not meet again, but continue to diverge — 

 is an induction from the evidence of our senses. 



This oj^inion runs counter to a scientific prejudice of long standing and 

 great strength, and there is probably no proposition enunciated in this 

 work for Avhich a more unfavorable reception is to be expected. It is, 

 however, no ncAV opinion ; and even if it were so, Avould be entitled to be 

 judged, not by its novelty, but by the strength of the arguments by Avhich 

 it can be supported. I consider it very fortunate that so eminent a chani- 



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

 definition so as to require, both 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 tiiis 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. For if it were possible that they should not, that is, 

 if any straight lines in the same plane, other than those which are parallel according to the 

 definition, had the property of never meeting akhough indefinitely produced, the demonstra- 

 tions of the subsequent portions of the theory of parallels could not be maintained. 



