504 
One new subject introduced is the symmetry 
of systems, which is really a summary of the work 
of P. Curie, and English readers will be thankful 
for having so easy a means of acquiring a know- 
ledge of this important subject. 
The book does not pretend to be an encyclo- 
pedia of physics, but it treats of the whole sub- 
ject so as to bring students up to a standard 
when they can feel confident in taking up research 
on some definite subject. In other words, it meets 
the requirements of the standard of the Honours 
B.Sc. Examination. The whole book is clearly 
written, and teachers will have no hesitation in 
leaving students alone with it. 
Another excellent feature is the treatment 
of a gravitational field first, and then later 
an electrostatic field where it is only necessary 
to give analogies with the former case. Here, 
generally, the Cartesian notation is used, but the 
Vector notation is explained without much use 
being made of it. 
One of the most striking drawbacks of the book 
is the lack of an index. (ieaRe 
LOGIC, TEACHING AND PRACTICE IN 
MATHEMATICS. 
(1) The Algebra of Logic. By Louis Couturat. 
Authorised English Translation by Lydia G. 
Robinson. With a Preface by P. E. B. Jour- 
dain. Pp. xiv+98. (London and Chicago: 
The Open Court Publishing Company, 1914.) 
Price..3s. 6d. net: 
(2) An Algebra for Preparatory Schools. 
Trevor Dennis. Pp. viii+155. 
University Press, 1913.) Price 2s. 
(3) Test Papers in Elementary 
C..V. Durell. Pp. viii+ 233. ~ (London: Mae- 
millan and Co., Ltd., 1914.) Price 3s. 6d. 
(4) Practical Mathematics for Technical Students. 
Panticlc, By PsS)-Usherwood andy@.) yee 
Trimble. Pp. 370. (London: Macmillan and 
Cor atd. voi.) SPriceigsaiod: 
(5) 4 Text-book on Spherical Trigonometry. By 
Prof. R. E. Moritz. Pp. vi+67. (New York: 
John Wiley and Sons, Inc. ; London: Chapman 
and Hall, Ltd., 1913.) Price 4s. 6d. net. 
(6) Plane and Spherical Trigonometry (with Five- 
Places Tables)... «By~Prof. “Ro Ey Montz ep: 
xvi+357+67+96. (New York: John Wiley 
and Sons, Inc.; London: Chapman and Hall, 
Ets, 1913.) Price wos. xod..net: 
(1) HIS is an excellent translation of M. 
Couturat’s well-known “Algébre de la 
Logique.” The conciseness and modernity of 
M. Couturat’s book is very apparent when we 
NOLP2233, VOL.) oi 
By 
(Cambridge 
Algebra. By 
NATURE 
[JuLY 16, 1914 
compare it with the first two volumes of Schréder’s 
bulky, prolix, and somewhat antiquated work. 
The book under review is, as is remarked in the 
preface (p. viil.; cf. pp. 92-93), an exposition of 
the beautiful and simple calculus of symbolic 
logic, regarded as a branch of universal algebra. 
Leibnitz distinguished the two most important 
aspects of a symbolic language designed for pur- 
poses of reasoning, under the names calculus ratio- 
cinator and characteristica universalis : the latter 
indicates broadly the route taken by Frege, Peano, 
Russell, and Whitehead; the former the route 
taken by Boole, Jevons, Venn, Schréder, and 
others, and is that described in the present work. 
Between these two routes, the “logic of rela- 
tions” lies, and this is not dealt with here; but 
we are given a complete presentation of the im- 
portant modern work of Whitehead (1898, 1901), 
Johnson (1901), Poretzsky (1899—1904), and 
Huntington (1904) on the logic of classes and 
propositions. Miss Robinson has added several 
valuable notes to her translation. The volume is 
neat and handy, and is an important addition to 
our English mathematical literature. 
(2) “The development of the subject is based 
on psychological rather than logical principles.” 
Here we have a sensible admission of the relation 
in which a text-book should stand to a scientific 
treatise, which the text-books of the past helped 
to obscure. The book consists of a series of 
graduated papers which exactly follow the lines 
of the syllabus issued by the curriculum committee 
of the Headmasters’ Conference, and. approved 
by the general committee of the Mathematical 
Association for all boys, except mathematical 
specialists, in public schools. There is no “book- 
work,” and the subject is developed wholly by- 
means of examples; but some attempt has been 
made to give the pupil an impression of the exis- 
tence of foundations of the subject and some sense 
of their nature. The very first question of the 
book is rather characteristic: ‘““A boy has 5d. and 
is given 3d. How much has he now? How 
much has he if he had 5d. and receives x pence? 
.”’, and soon. The last question of this paper 
is: ‘Make up more questions like these and give 
the answers.” This is an excellent way of 
teaching, and there is a freshness about the 
book. 
(3) This is a collection of papers designed 
primarily for out-of-school work, and _ conse- 
quently includes only few graphical questions. 
The papers follow the traditional course of ele- 
mentary algebra, along the thorny path of quad- 
ratics, logarithms, the progressions, combinations, 
and easy probability, up to the giddy heights of — 
