APRIL 15, 1915] 
NATURE 
V7 1 

surgical science and art, especially those dis- 
coveries which have been made within the last 
half-century. These chapters of history are as 
good as good can be, and here is a book full of 
good reading, written by one of the chief of living 
surgeons. STEPHEN PAGET. 
PURE MATHEMATICS. 
(1) A Treatise on the Analytic Geometry of Three 
Dimensions. Fifth Edition. Vol. xi. By Dr. 
G. Salmon. Pp. xvi+334. (London: Long- 
mans, Green and Co., 1915.) Price 7s. 6d. net. 
(2) A Course of Pure Mathematics. By G. H. 
Hardy. Second Edition. Pp. xit+442. (Cam- 
bridge: At the University Press, 1914.) Price 
12s. net. 
(3) Proceedings of Mathematical 
the London 
Society. Second Series. Volume xii. Pp. 
liii+s500. (London: Francis Hodgson, 1914.) 
nrice 255. 
(1) [TT cannot have been an easy task to pre- 
pare the new edition of Salmon’s classical 
treatise; the result may be considered quite satis- 
factory, although, no doubt, different readers will 
form different opinions about the choice of addi- 
tions that has been made. In this volume, the 
principal ones are as follows :—First of all, a con- 
siderable addition has been made to the section 
on line-geometry. So far as we can judge, this 
has been very well done; it includes Ribaucour’s 
theory of isotropic congruences, a good deal about 
normal congruences, and other interesting matter. 
Next, and partly connected with the foregoing, 
we have an account of curvilinear co-ordinates, 
triply orthogonal systems, and cyclides ; also other 
theorems due to Ribaucour. 
In the part dealing with cubic and quartic 
surfaces, there is a sketch of Segre’s analysis of 
the singularities of cubics; Geiser’s correspond- 
ence of lines on a cubic with bitangents of a plane 
quartic; and some very good articles on cyclides, 
and the special quartics of Steiner and Kummer. 
Towards the end of the book, we have eight pages 
or so on birational transformations, with some 
useful references, and a revised table of singu- 
larities, mainly based on Zeuthen’s memoir of 
1876 (Math. Ann., x.). This last does not seem 
to be quite up to date; for instance, no reference 
has been made to the work of Enriques and 
Castelnuovo, and their discussion of the deficiency 
of a surface, which has yielded a new fundamental 
characteristic, besides that called the deficiency 
in this book. 
Reference is made (Art. 527a) to some recent 
work on the reduction of a cubic to the canonical 
NOwe2ay2, VOL. 05] 

oO 
form 3x; =0, and the proof that this is unique. 
1 
One way of doing this is to obtain the equation 
IIx; = 0, by a combination of invariants and co- 
variants; this the present writer succeeded in 
doing a good while ago, with the help of Salmon’s 
list of invariants, but the result was unpleasantly 
long. 
Perhaps the in the 
revised work is that of the theory of minimal 
surfaces. It would not have taken very many 
pages to give a fair account of this elegant theory, 
now that Lie and Weierstrass have reduced it to 
its simplest form. However, it may fairly be 
said that a student who has this work, and Prof. 
Forsyth’s ‘“ Differential Geometry,” will be able to 
make acquaintance with all the most important 
divisions of the subject, and be able to follow up 
any one in which he is specially interested. 
(2) It is gratifying to see that Mr. Hardy’s 
excellent treatise has so soon reached a second 
edition; and it helps to justify the statement in 
the preface that “it is no longer necessary to 
apologise for treating mathematical analysis as 
a serious subject worthy of study for its own 
sake. The author combines, in a remarkable 
way, strictness of method an agreeable 
style; and his choice of topics seem to us to be 
eminently judicious. The principal additions in 
this re-issue are an account of Dedekind’s theory 
of irrational numbers; a proof of Weierstrass’s 
theorem about points of condensation, of the 
Heine-Borel theorem, and of Heine’s theorem 
about uniform convergence; the notions of “limits 
of indetermination”’ and ‘“‘implicit function” are 
also discussed. To save space, some analytical 
geometry and trigonometry has been deleted. 
The examples are well chosen, and hints towards 
solution are frequently given. This book, and 
Mr. Bromwich’s “Infinite Series,” to which Mr. 
Hardy refers in his preface, ought to do a great 
deal towards making school and college mathe- 
matics more rigorous, without making it repul- 
sive; on the contrary, the apparent paradoxes 
explained, and the latent fallacies exposed, ought 
to provide a certain amount of fun, even for an 
most striking omission 
” 
with 
undergraduate. 
There are one or two very trifling points that 
may be noticed. On page 231, D-1(x~1), when 
x is negative, is defined as log(—x). This is very 
artificial, especially as the figure, page 359, im- 
plies that logx is undefined when x is negative. 
The same complaint of artificiality applies to the 
treatment of differentials (page 280). A rather 
more important point is on the proof (page 313) 
that the sum of a series of positive terms is the 
| same ‘in whatever order the terms are taken.” 
