604 
for guidance in the task which in his future career he 
may have to face, of collecting workmen in inaccessible 
districts, of housing them, and of looking after them 
generally. 
The work contains 281 illustrations, some being 
drawings to scale and others reproductions of photo- 
graphs, showing in a striking manner the operations 
of mining and quarrying. All are excellent, and, like 
the letterpress, are exceptionally well printed. A very 
full and accurate index greatly adds to the value of 
the book. IB. HB: 
SOME BOOKS ON QUATERNIONS. 
Etude sur les Quantilés mathématiques. Grandeurs 
dirigées, Quaternions. By Prof. Claro Cornelio 
Dassen, D. es Sc. Pp. vit133. (Paris: A. Her- 
mann, 1903.) Price 5 francs. 
Introduction to Quaternions. By the late Profs. Philip 
Kelland, F.R.S., and P. G. Tait. Prepared by 
C. G. Knott, D.Sc. Pp. viit+208. (London: 
Macmillan and Co., Ltd., 1904.) Price 7s. 6d. 
Bibliography of Quaternions and Allied Systems of 
Mathematics. Drawn up for the International 
Association for Promoting the Study of Quaternions, 
&c. By Alexander Macfarlane, D.Sc., IE ID, 
F.R.S.E., General Secretary of the Association. 
Pp. 86. (Dublin: University Press, 1904.) 
iis may perhaps be rather an exaggerated statement, 
but it is none the less to a great extent true, that 
mathematicians tend to divide themselves into two 
classes, quaternionists and non-quaternionists, and that 
these two classes frequently become involved in 
polemical controversies. But at the present time the 
notion of vector quantities is of frequent occurrence in 
physics, and it is important that everyone should have 
an opportunity of understanding the laws and nature 
of vector operations. It is not unfrequently stated that 
forces are vectors, because they are directed quantities, 
and therefore they are compounded by the parallel- 
ogram law. But the moment of inertia of a body 
about a line is also a directed quantity in the sense 
that its magnitude depends on the direction of the 
line, although moments of inertia, as every mathe- 
matician knows, are not compounded by the parallel- 
ogram law. Clearly dogmatic statements about 
vectors are dangerous for teaching purposes. 
Dr. Dassen’s book deals with a much wider field 
than the study of quaternions proper. The first part 
treats of ‘‘ the concept of quantity,’’ i.e. the funda- 
mental laws of algebra. The second part is divided 
into three chapters, which treat of directed quantities 
in space of one, two and three dimensions respectively, 
so that it is only in this last chapter that the particular 
system of algebra associated with the name of Sir 
W. R. Hamilton is discussed. 
Prof. Knott’s edition of Kelland and Tait’s book is 
a typical English text-book of a kind such as writers 
of other nationalities rarely produce. By this we imply 
that it is full of worked illustrative examples, and at 
the end of each chapter is a copious collection of ex- 
amples for exercise. The first five chapters, extend- 
NO. 1800, VOL. 69] 
NATURE 
[APRIL 28, 1904 
ing up to p. 89, would form an excellent course for a 
student whose time is short. 
The chapter on dynamical applications is an im- 
portant innovation which might well be extended in 
the interest of science students by the addition of more 
elementary examples. Chapter vi., dealing as it does 
with ‘‘cones and their sections,’ possesses little 
interest for the modern student. It would have been 
more useful thirty years ago, when it was the fashion 
not to teach the calculus until the student had learnt 
by heart a series of propositions for the parabola, a 
second series for the ellipse, and a third series (scarcely 
ever properly understood) for the hyperbola. Nowa- 
days more suitable illustrations may be obtained from 
physical problems without wasting time over these 
elaborate discussions about conics. On the other hand, 
chapter vii., dealing with central surfaces of the 
second order, is less detailed in its treatment, and 
consequently likely to be more helpful. 
Dr. Macfarlane is to be congratulated on the very 
complete list of papers which he has succeeded in- 
drawing up for the association of which he is secre- 
tary. These papers, we notice, cover different systems 
of multiple algebras, discussions on the geometric re- 
presentation of complex magnitude, and nearly every 
book, pamphlet or paper on mathematics or physics 
in which quaternions or allied systems of mathematics 
play any part whatever, such, for example, as Clerk 
Maxwell’s ‘‘ Matter and Motion’? or Minchin’s 
‘“Statics.”’ The only improvement that can be 
suggested is a classification of the papers by which 
those merely containing references to quaternions 
should be separated from those which treat of a sub- 
stantial portion of the general theory of algebras of 
vector quantities. 
OUR BOOK SHELF. 
A Manual of Zoology. By Richard Hertwig. Trans- 
lated and edited by J. S. Kingsley. Pp. xi+704. 
(London: G. Bell and Sons, 1903.) Price 12s. 6d. 
net. 
Tue English student of zoology has now the choice of 
a large number of excellent text-books, varying in 
style, in thoroughness, in wealth of illustration, but 
sufficient for the purposes of his education up to the 
time when his more advanced studies require him to 
turn his attention to the writings of foreign authors 
in their original language. To add to the list, the 
translation of a foreign text-book requires some special 
justification. It must provide something, some special 
treatment of a subject or some philosophical specula- 
tion which has not been provided by those who, with 
a knowledge of the needs of our own students, have 
written text-books for their guidance. The great 
reputation of Prof. Hertwig would lead us to expect a 
text-book from his pen that would justify its translation 
into the language of any country where zoology is 
seriously studied, and in some respects we are not 
disappointed. i 
The book opens with a definition of general terms 
and a short history of the development of the science 
from the time of Aristotle to the present day, which 
are admirably concise and clear. The pages dealing 
with general morphology and physiology are also 
decidedly excellent, better, perhaps, than are the corre- 
sponding parts of any other modern text-book. In 
the portion dealing with special zoology, however, 
