DEMONSTRATION AND NECESSARY TRUTHS. 



i5t 



theory of geometrical reasoning ; the 

 necessity of admitting, among those 

 first principles, 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 pro- 

 positions similar to what are so called. 

 Thus, if instead of the axiom, Mag- 

 nitudes which can be made to coin- 

 cide are equal, we introduce a defini- 

 tion, "Equal magnitudes are those 

 which may be so applied to one 

 another as to coincide ; " the three 

 axioms which follow (Magnitudes 

 Avhich 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 the 

 remainders are equal) may be proved 

 by an imaginary superposition, re- 

 sembling 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 cannot, enclose 

 a space, (or its equivalent, Straight 

 lines which coincide in two points 

 coincide altogether,) and some pro- 

 perty of parallel lines, other than 

 that which constitutes their defini- 

 tion ; one of the most suitable for 

 the purpose being that selected by 

 Professor Play fair : "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 



* We might, it is true, insert this pro- 

 perty 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 

 this we by no means get rid of the as8umx> 

 tion ; we are still obliged to take for granted 

 the geometrical truth that all straight lines 



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 imagi- 

 nary 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 

 approximations 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 with- 

 out qualification or error. The rota- 

 tion of the earth in twenty- four hours, 

 of the same length as in our time, has 

 gone on since the first accurate ob- 

 servations, 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 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 

 order to use them for the further 

 advancement of our knowledge, we 

 must feign that they are exactly true, 

 though 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 ? I answer, they are experi- 

 mental truths ; generalisations from 

 observation. The proposition, Two 

 in the same plane, which have the former 

 of these properties, have also the latter. 

 For if it were possible that they should not, 

 ts at is, if any straight lines in the same 

 plane, other than those which aie parallel 

 according to the definition, had the pro- 

 perty of never meeting although inde- 

 finitely produced, the demonstrations of 

 the subsequent portions of the theory of 

 parallels could not be maintained. 



