Notices respecting New Books. 301 



explain. Then, again, the substance of his book seems to us of a 

 far inferior quality to Euclid's : this is no more than might be ex- 

 pected; but we will give an instance of what we mean. Euclid's 

 treatment of the Corollaries to the 32nd prop, of Book 1 is not, per- 

 haps, wholly proof against minute criticism ; still if any thing be 

 wanting it could be supplied by a word or two of explanation ; 

 and surely nothing can be plainer or more direct than his method. 

 Mr. Cuthbertson, however, wishes to improve upon it, and he does 

 so as follows : — On p. 39 he gives a Corollary, which is stated 

 thus : — " If A B, B C are two straight lines respectively parallel to 

 D E, E F, then shall the angle A B C be equal to the angle DEE." 

 This is true or not according to the direction in which E ¥ is drawn : 

 e. g. it is true in the case shown in Mr. Cuthbertson's diagram ; 

 but the needful qualification is not given in^the Corollary, nor, so 

 far as we have noticed, anywhere else. On p. 48 this Corollary is 

 used to prove the theorem " if the sides of a polygon be produced 

 in order, the exterior angles shall together be equal to four right 

 angles." The proof consists in taking a point outside the polygon and 

 drawing from it rays parallel to sides respectively. This proof, of 

 course, may be made perfectly sound ; but in the case before us it 

 fails owing to the above-mentioned ambiguity. This is the way in 

 which he treats Euclid's second Corollary, and then he goes on to 

 prove Euclid's first Corollary. The method is in no respect better 

 than Euclid, and the way of stating it inferior to the extent of in- 

 accuracy. 



There is one question of general interest, suggested by a perusal 

 of Mr. Cuthbertson's book, on which we will say a few words, viz. 

 " What are axioms ? " To the mathematician they are merely 

 truths of geometry assumed without proof, as premises needful for 

 proving other truths of geometry. It is usual to answer that 

 axioms are self-evident truths. But, not to say that the question at 

 once arises " Self-evident to whom?'' it is to be observed that the 

 question " How do we come by our knowledge of the axioms of 

 geometry ? " is one with which the mathematician, as such, has 

 nothing to do. There are, of course, two distinct ways of answering 

 this question, and each doubtless capable of numerous modifications. 

 Some hold that the axioms of geometry are what they are in virtue 

 of the conformation of the mind antecedently to all experience of 

 space. Others hold that the axioms are nothing but the expression 

 of our most elementary experiences of space, and that what is 

 called their necessary truth is merely a consequence of the uni- 

 formity of our experiences, joined to the absence of any experience 

 which suggests so much as a type of something inconsistent with 

 them. "We believe this to be a sufficiently correct, though brief, 

 statement of the two rival answers ; and the observation we have 

 to make on them is, that whether either or u either of them be true 

 is a question wholly outside of geometry. 



We may not, perhaps, be justified in doing more than suspecting 

 (but at all events we do very strongly suspect) that the reason of 

 Euclid's 12th axiom being so much objected to is that many matke- 



