493 
1879 .] Notices of Books . 
In fa (ft, throughout his book, the author is far more inclined to 
attack Euclid’s Rivals than to defend Euclid himself, so that a 
one-sided view of the question is shown, and Euclid’s faults are 
kept steadily in the back-ground. Euclid and Minos then discuss 
the method of procedure in examining Modern Rivals, and a few 
alterations proposed by them. 
In the Second Adi Minos is discovered still sleeping, and to 
him enters a shade which is addressed as “ Niemand," with 
copies of the Modern Rivals to be discussed. These are divided 
into two classes — those which rejedl Euclid’s treatment of paral- 
lels, including Legendre, Cooley, Cuthbertson, Wilson, Pierce, 
and Willock; and those which adopt Euclid’s treatment of paral- 
lels — these are Chauvenet, Loomis, Morell, Reynolds, Wright, 
the Syllabus of the Association for the Improvement of Geome- 
trical Teaching, and Wilson’s Manual founded on that Syllabus. 
In the Fourth Adi the shade of Euclid again converses with 
Minos, and, with the exception of some one or two slight addi- 
tions and alterations, such as a proof that all right angles are 
equal, it is decided that Euclid’s Manual ought to be left un- 
touched. 
The chief points of attack on Euclid’s Modern Rivals are 
Mr. Wilson’s two works — “Elementary Geometry” and the 
“ Manual founded on the Association Syllabus.” The author 
makes comparatively short work of Legendre’s book as unsuited 
to beginners, though doubtless valuable to advanced students ; 
and of Cooley’s, in which a certain theorem breaks down through 
a faulty definition of parallel lines. But to Mr. Wilson’s two 
Manuals he devotes nearly a third of his volume. Much of the 
criticism on these, however, is mere cavilling ; for instance, at 
page 177, Minos says, speaking of Wilson’s “ Syllabus ” Manual : 
— “ At p. 57 I see an ‘ Exercise * (No. 5). 4 Show that the angles 
of an equiangular triangle are equal to two -thirds of a right 
angle.' In this attempt I feel sure I should fail. In early life I 
was taught to believe them equal to two right angles — an anti- 
quated prejudice no doubt ; but it is difficult to eradicate these 
childish instindls.” This is mere straw-splitting ; strictest accu- 
racy would of course require the insertion of “ each ” before 
“ equal,” but if the sum of the interior angles had been intended 
to be understood “ together ” before “ equal ” would have been 
absolutely necessary. The very next paragraph, containing an 
accusation of the fallacy Petitio Principii, is another instance of 
cavilling criticism ; for to take a line greater than half another, 
surely we need only take it greater than the whole, and that in- 
volves no knowledge of the bisecting point of the line. 
At page 160 there is a criticism on the definition of a right 
angle as given by the Association for the Improvement of Geo- 
metrical Teaching in their Syllabus. This is* — “ When one 
straight line stands upon another straight line, and makes the 
adjacent angles equal, each of the angles is called aright angle.” 
