302 Notices respecting New Books. 



maticians regard the former as the correct answer to the above 

 question. There is not much difficulty in believing that we are 

 born into the world with minds so constituted that as soon as we 

 know the meaning of words we cannot do otherwise than hold that 

 things equal to the same thing are equal to one another ; but no 

 one except a hardened metaphysician could suppose that a belief of 

 the 12th axiom is produced by any thing but an acquaintance with 

 the actual properties of space. Accordingly many wish to substi- 

 tute for it something which is more " self-evident," i. e. something 

 more consonant with their metaphysical views. 



It is not easy to see, on other grounds, what advantage is gained 

 by substituting one axiom for another. No one has any difficulty 

 in understanding what the 12th axiom means, nor in seeing that it 

 is undoubtedly true. If any one will prove the converse of pro- 

 position 27 without assuming more than the first eleven axioms 

 and the first 27 propositions, he will do something worthy of all 

 honour. But when the question is to prove the point by means of 

 a special axiom which differs from Euclid's our interest in the 

 matter is but small, e. g. If any one prefers Playfair's axiom to 

 Euclid's we do not know why he should not ; only we would remark 

 that it is merely a question of preference, that the two axioms are 

 quite coordinate with each other, and that if either is taken for 

 granted the other can be immediately proved. • 



Mr. Cuthbertson, however, takes a different view from this, and 

 he goes to work to improve upon Euclid as follows : — On p. 33 he 

 gives " Deduction Gr," viz. " If points be taken along one of the 

 arms of an angle farther and farther from the vertex, their dis- 

 tances [meaning, as explained, perpendicular distances] from the 

 other arm will at length be greater than any given straight 

 line." It is obvious that this statement as it stands is not true ; 

 however, the needful correction could be supplied without much 

 difficulty; e. g. it would be sufficient for present purposes for it to 

 run, " If points be taken at equal distances &c," and this is appa- 

 rently what is meant. Eurther, the demonstration of the deduction 

 assumes that any angle however small can be multiplied until an 

 angle is obtained greater than a right angle. "We have no objection 

 to this being assumed, only to its being assumed implicitly. In a 

 book which formally specifies the axioms assumed, it ought to 

 have been separately enunciated as an axiom ; and we cannot find 

 that this has been done. On p. 34 Mr. Cuthbertson gives the 

 axiom which he proposes to substitute for Euclid's 12th axiom, viz. 

 " If one straight line be drawn in the same plane as another it 

 cannot first recede from and then approach to the other, neither 

 can it first approach to and then recede from the other on the same 

 side of it." By means of this axiom and deduction Gr, he succeeds 

 in proving Playfair's axiom. In other words (putting accidental 

 defects out of the question), he succeeds in proving one axiom by 

 assuming two. We willingly accord to this the praise of ingenuity ; 

 but we strongly suspect that few besides the author will think it an 

 improvement on Euclid's method. 



