APRIL 16, 1914] 
NATURE 
52) 
subject for the first time, but to the attention of 
practised workers, who will find both interest and 
advantage from being brought into contact with 
those elemental principles and difficulties, which 
are apt to be lost sight of as they advance along 
their special lines of research. 
PURE MATHEMATICS. 
(1) Plane Geometry. By Prof. W. B. Ford and 
C. Ammerman. Edited by E. R. Hedrick. Pp. 
iIx+213+xxxi. (New York: The Macmillan 
Company; London: Macmillan and Co., Ltd., 
#913.) | Price: 35. 6d. “net. 
(2) Higher Algebra. By Dr. W. P. Milne. Pp. 
x1i+ 586. (London: Edward Arnold, 1913.) 
Price 7s. 6d. ‘net. 
(3) The Twisted Cubic: with Some Account of 
the Metrical Properties of the Cubical Hypevr- 
bola. By P. W. Wood. Pp. x+78. (Cam- 
bridge: University Press, 1913.) Price 2s. 6d. 
net. 
(4) Graphical Methods. By Prof. Carl Runge. 
Pp. viiit+148. (London: Oxford University 
Press; New York: Columbia University Press, 
19r2:), Price 6s. 6d. net. 
(5) Einfiihrung in die Mathematik fiir Biologen 
und Chemiker. By Prof. L. Michaelis. Pp. vii 
+253. (Berlin: Julius Springer, 1912.) Price 
7.80 marks. 
(6) Théorie des Nombres. By E. Cahen. Tome 
Premier. Le Premier Degré. Pp. xii+ 408. 
(Paris: A. Hermann et Fils, 1914.) 
(1) HIS text-book, written by two American 
professors, is arranged on lines similar 
to those of modern English text-books. Fewer 
theorems, more experimental geometry, and 
numerous practical applications are its chief char- 
acteristics. In placing properties of areas after 
those of circles and similar figures, the authors 
have adopted a change of order, which we believe 
will eventually become general. The idea of simi- 
larity is so fundamental and at the same time 
presents comparatively so little difficulty, that it 
is in our view regrettable that in many examina- 
tion syllabuses, and consequently in most school 
courses, it should be postponed to a late period. 
In fact, it is probably true to say that many boys 
leave school without any knowledge of what is 
one of the most valuable, practical, and impor- 
tant branches of geometry. 
(2) There are in this volume distinct merits 
and a certain originality of treatment such as will 
appeal to many teachers of the modern school. 
No attempt has been made to develop the subject 
in a rigorous and logical fashion from the funda- 
mental axioms of number, and we agree with 
NO. 2920," VOL... | 
Dr. Milne’s opinion that students at this period 
of their training are ill-fitted for what is almost 
a philosophical discussion. But at the same time 
it is becoming generally recognised that the harm 
done by inculcating incorrect notions of limits, 
convergence, etc., is so serious and so difficult 
to remedy that many teachers have, with little 
assistance from current text-books, been taking 
their scholarship candidates through a course of 
serious analysis. They will undoubtedly welcome 
the publication of this treatise, which with ad- 
mirable clearness and with an abundance of 
detailed explanations and illustrations, sets out 
the lines upon which accurate investigations of 
the existence of limits and the convergence of 
single and double series must proceed. More 
than half the book is occupied with work of this 
character. 
The scope of treatment is best indicated by an 
enumeration of the headings of the chapters :— 
Rational numbers; irrational numbers; summa- 
tion; binomial theorem; permutations and com- 
binations; exponential and logarithmic series; 
continued fractions; theory of equations; deter- 
minants; miscellaneous theorems, In his selec- 
tion and arrangement of material, the author has, 
therefore, departed considerably from the custom- 
ary plan. What little “Theory of Numbers” 
there is, comes in the first chapter; the usual 
account of probability has been considerably cur- 
tailed; diophantine problems and _ inequalities 
receive very brief treatment; and the chapter on 
permutations follows, instead of precedes, the 
binomial theorem. There is a good collection of 
examples at the end of each chapter; we regard, 
however, as unfortunate the omission of any 
intermediate sets. Clearly it is undesirable for a 
student to read the whole of a chapter before 
doing any examples, and the result of grouping 
them at the end is to saddle the teacher with the 
burden of selection. 
The final collection of four hundred miscellane- 
ous problems and a number of questions of an 
essay type call for special notice. Teachers and 
pupils alike will probably feel the need of an 
index. Dr. Milne has produced an essentially 
scholarly work, and we have no kesitation in 
classing it with those books which are exercising 
a wholesome and valuable influence on mathe- 
matical teaching. 
(3) It is curious that no systematic account 
should have been hitherto published of the pro- 
perties of the space curve of the third order, 
although many mathematicians have given their 
attention to the subject, the fruit of which is to 
be found in a number of isolated memoirs and, 
incidentally, in a few treatises, as, for example, 
