304 REASONING. 



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 indemon- 

 strable as those which admit 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 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 approxima- 

 tions 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 a particle of qualification or 



* "We might, it is true, insert this property into the definition of 

 parallel lines, framing the definition so as to require, loth 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. For if it were 

 possible that they should not, that is, if any straight lines other 

 than those which are parallel according to the definition, had the 

 property of never meeting although indefinitely produced, the 

 demonstrations of the subsequent portions of the theory of parallels 

 could not be maintained. 



