DEMONSTRATION, AND NECESSARY TRUTHS. 257 



principles, axioms as well as definitions. Some of the axioms 

 of Euclid might, no doubt, be exhibited in the form of defini- 

 tions, 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 applied to one another as to coincide ;" the three axioms 

 which follow (Magnitudes 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 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 suscep- 

 tible 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 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 con- 

 stitutes their definition : one of the most suitable for the pur- 

 pose 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 fundamental principles which are involved in the 



* 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 indefi- 

 nitely 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 indefi- 

 nitely produced, the demonstrations of the subsequent portions of the theory of 

 parallels could not be maintained. 



VOL. I. 17 



