160 
NAT ORE 
PARRIL bo, BOL! 
in Grace and Young’s ‘Algebra of Invariants,” 
where the invariants and covariants of a binary 
cubic are interpreted in terms of the geometry 
of the skew cubic. This is now remedied by Mr. 
Wood’s tract, which discusses the subject 
ab initio. The first section deals with the projec- 
tive properties of the curve, developed analytically 
with the use of homogeneous coordinates, and 
the second is specially concerned with the cubical 
hyperbola (the case in which the curve has three 
real and distinct points at infinity), and discusses 
the properties of asymptotes, diameters, vertices, 
centre, axis, inscribed and circumscribing 
quadrics, the rectangular cubical hyperbola, etc. 
No one who is interested in geometry can fail 
to appreciate the collection of properties which 
Mr. Wood has made, and many will, no doubt, be 
encouraged by the way in which the initial stages 
have been simplified, to pursue the subject 
beyond the limits which space has here rendered 
necessary. 
(4) The purpose of this book is to supply the 
theoretical basis upon which graphical methods 
rest, and to discuss in general terms the manner 
in which applications may be made to shorten 
the labour involved in the heavy computation with 
which the physicist and engineer are so often 
faced. In many cases special graphical methods 
have been invented to cope with a particular kind 
of problem, and in view of the fact that there is 
little inter-communication between those working 
in different spheres, the opportunity of making 
use in one department of a device that has been 
of value in another is often missed, on account 
of the failure to recognise the generality of the 
principle which has been employed. The subject- 
matter is divided into three sections: the first 
deals with graphical arithmetic, the evaluation of 
integral functions, and the treatment of complex 
quantities; the second with the representation of 
functions of one or more variables, the principle 
of the slide rule, and the idea of conformal repre- 
sentation; and the last with the calculus and the 
solution of differential equations. 
(5) This course of pure mathematics is most 
distinctly a lower limit of the equipment every 
scientific student should possess. The ideas of 
the calculus are at last beginning to find their 
way into the ordinary curriculum, more rapidly 
on the Continent than in England, and the time 
cannot be far distant when it will be impossible 
for boys who specialise in chemistry or physics to 
leave school ignorant of infinitesimal methods. 
The book deals with revision of arithmetic, 
algebra, geometry, and trigonometry ; graphs of 
functions; differential and integral calculus, with 
special reference to expansion in series; and dif- 
NO. 2320, VOL.' 93] 
ferential equations. 
make rather dull reading, little indication being 
gently the original 
The sections of the calculus 
given of the nature of the applications that it 
permits. The systematic treatment of what may 
be called the grammar of the subject should, 
however, enable the reader to acquire some degree 
of facility in performing ordinary operations. 
(6) What is the criterion that distinguishes the 
theory of numbers from other branches of ana- 
lysis? To this question, M. Cahen makes the 
following reply :—-‘‘The Theory of Numbers is a 
science in which division is possible only in special 
cases, whereas elsewhere division is impossible 
only in special cases.” And this statement gives 
in brief the limits he has set himself in this 
treatise. The first eight chapters deal with addi- 
tion, multiplication, subtraction, and divisions of 
integers, H.C.F. and L.C.M., and fractions; the 
next four with systems of diophantine equations 
of the first degree; then follow chapters on linear 
substitution and groups, linear and_ bilinear 
forms, congruences, matrices, prime numbers. 
The treatment is thorough in character, and 
the work is set out so clearly that no student, 
however small his previous knowledge may be of 
the theory of arithmetic, should find it difficult 
to follow the argument; and if he reads through 
this volume carefully and tests his progress by 
working out some of the examples provided, he 
should obtain a firm grasp of this important 
modern subject. Text-books such as these form 
an admirable preparation for the student who 
wishes to make a more specialised study of the 
subject, for, by giving him a sound groundwork, 
they make it possible for him to consult intelli- 
memoirs which mark the 
growth of the theory, and which no text-book, 
however comprehensive, can in reality replace. 
OUR BOOKSHELF, 
Dental Diseases in Relation to Public Health. By 
Dr. J. Sim Wallace. Pp. viii+g90: (London: 
Office of The Dental Record, 1914.) Price 3s- 
net. 
Tuis book consists of three chapters. They are 
addresses given by the author in ‘‘response to 
requests.”’ Chap. i. sets forth in detail the pre- 
valence of dental diseases, the serious effects they 
exercise on general health—especially during 
childhood—and the methods by which such 
diseases may be prevented. There are, however, 
within its pages statements based on loose figures, 
which are ¢alculated to mar the effect the writer 
has in view by causing an impression of exaggera- 
tion of unverified inference. On the strength of 
the statement—itself too wide a generalisation— 
that 75 per cent. of the total population have 
irregularities of the teeth, we have presented to 
us the wild statement that “the number of teeth 
