DEMONSTRATION, AND NECESSARY TRUTHS. 3Q3 



clothing it in false ones, the conclusions will always 

 express, under known liability to correction, actual 

 truth. 



$ 3. But although Mr. Whewell has not shaken 

 Stewart's doctrine as to the hypothetical character of 

 that portion of the first principles of geometry which 

 are involved in the so-called definitions, he has, I 

 conceive, greatly the advantage of Stewart on another 

 important point in the 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 

 propositions similar to what are so called. Thus, if 

 instead of the axiom, Magnitudes which can be made 

 to coincide are equal, we introduce a definition, 

 " Equal magnitudes are those which maybe 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, resembling that by which 

 the fourth proposition of the first book of Euclid is 

 demonstrated. But although 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 I agree with 

 Mr. Whewell in placing 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 



