58 Notices respecting New Books. 



theorem. The latter is merely a logical rule, and relates to the 

 form of the reasoning. The following remark will put in a clear 

 light the impropriety of classing (3) and (7) together under the 

 same head. If it is right to rank as seventh amongst the axioms 

 of geometry the rule that " a conditional proposition may be con- 

 verted by negation," then the rule that " a universal affirmative 

 proposition can only be converted by limitation " ought to be ad- 

 mitted into the list ; and in fact it would be hard to say what rule 

 of deductive logic could well be omitted*. 



It is one thing to make here and there a remark on the form of the 

 reasoning employed in proving this or that proposition, and another 

 to erect a rule of logic into an axiom of geometry : the latter con- 

 fuses things that differ ; the propriety of making the former is a 

 matter of opinion. Considering, however, that elementary geometry 

 should be taught to rather young boys, we are strongly inclined to 

 think it best to keep the rules of logic out of sight, or at least that 

 they should be noticed, if at all, in the oral instruction of the 

 teacher, and should not form a substantial part of the text-book. 

 It must be remembered that when we are concerned with given 

 matter we reason correctly, or can be shown to have reasoned in- 

 correctly without explicit reference to the rides of logic. In a 

 book of geometry, therefore, logical rules are extraneous matter, 

 and as such are nearly certain to be treated inadequately, if admitted 

 at all. Mr. Wilson illustrates this remark by devoting a page and 

 a half to the conversion of propositions (pp. 3, 4) ; but he notices 

 only conditional propositions, and- his reason for doing so is that 

 " if A is B then C will be D is the type of a theorem," although 

 his own enunciations often take a different form (e. g. Theorems L, 

 VI., X., &c). 



Mr. Wilson has four enunciations under the head of Geometrical 

 Axioms. Two of them run thus : — " (2) Two straight lines which 

 have two points in common lie wholly in the same straight line. 

 (3) A finite straight line has one and only one point of bisection " 

 (p. 9). The former of these is a true axiom ; the latter admits of 

 proof, and is therefore a theorem assumed to be true merely as a 

 matter of convenience. Suppose Euclid had assumed the first eight 

 and had begun his reasoning with the ninth proposition ; his subse- 

 quent reasoning would have been unaffected ; but surely it would 

 have been an abuse of language if he had classed together as axioms 

 " Two straight lines cannot inclose a space," and " The angles at the 

 base of an isosceles triangle are equal." Yet this is just the sort 

 of thing which Mr. Wilson has done. 



(2) On coming to the definitions, it may be well to notice that in 

 definition we have to do with nothing but the meaning of words. 

 It follows that there must be some words which are incapable of 

 definition, and whose meanings must be arrived at by a direct ap- 

 peal to the intuitions of sense — to experience, in fact. There may 

 be good reasons for saying that a line is " length without breadth ; " 

 but it is hardly a definition, as the words contained in it (" length " 

 * Mr. Wilson's sixth axiom also is purely logical. 



