tvith Geometrical Reasoning. 129 



angled triangle is equal to the sum of the squares of the two 

 other sides, " the knowledge that ' the whole is equal to all its 

 parts/ and ' if you take equals from equals the remainders will 

 be equal/ helped him not, I presume, to this demonstration : 

 and a man may, I think, pore long enough on these axioms 

 without ever seeing one jot the more of mathematical truths*/' 

 It is remarkable that Dr. Whewellf, one of the chief opponents 

 of this view, should have cited in his arguments those axioms to 

 which Mr. Locke did not refer, such as the X., XI. or XII. 

 And M. D^Alembert J, whose opinions correspond with those in 

 the Essay on Human Understanding, appears to exclude these 

 three from the list. On this subject Mr. Dugald Stewart is 

 perfectly clear ; he advocates the doctrine that the axioms are 

 not the principles of our reasoning, and he says, " In order to 

 prevent cavil, it may be necessary for me to remark, that when I 

 speak of mathematical axioms I have in view only such as are of 

 the same description with the first nine of those which are pre- 

 fixed to the Elements of Euclid §.^^ This statement he subse- 

 quently reiterates. 



I draw attention to these opinions for the purpose of showing 

 that the substitution of affirmatives for negatives would be of 

 importance in a philosophical point of view, apart from the fact 

 of its being productive of synthetical reasoning, in permitting 

 us to remove from the list of axioms those three which never 

 should have been there, and about which much speculative and 

 unsatisfactory discussion has taken place. 



Having struck out the tenth and twelfth axioms, and, for the 

 present, incorporated the eleventh with the tenth definition, the 

 only substitutions required should be made in the definitions of 

 parallel and straight lines. 



In place of the XXXV. I would propose the following : — 

 Parallel lines are such, that if they meet a third right line, the 

 two interior angles on the same side will be equal to two right 

 angles. 



And instead of the IV. definition : — Uight lines are such that 

 if they coincide in any two points one line must lie entirely in 

 the other. 



It may be urged as an obj ection against the first of these, that 

 it is not as self-evident as the definition it professes to replace ; 

 but it should not be forgotten that Euclid uses two definitions ; 

 and that the one here put forward, by which all the properties of 



* Essay on the Human Understanding, vol. i. p. 305. 



t Philosophy of the Inductive Sciences, 2nd edit. vol. ii. p. 601. 



X Encyclop^die. Discours preliminaire des editeurs, xvi, 



\ Philosophy of the Human Mind, vol. ii. pp. 40 and 527. 



