AvucustT 8, 1912] 
NATURE 
579 
which is specially suited to his requirements. The 
book also includes the practical work for students 
taking a course in the “Chemistry of Building 
Materials.” 
(6) The “School Chemistry ” of Messrs. Wilson 
and Hedley has been issued as the result of a 
demand for a somewhat shorter course than that 
described in their ‘“‘Elementary Chemistry.” The 
book is characterised by the sound and logical 
method of teaching, largely historical, which is 
happily becoming so common in school-books on 
chemistry. Exception must be taken to the state- 
ments that “the percentage of xenon in air is 
only o'0000026, or 1 part in 38,461,538 parts of 
air’; the two statements are by no means identi- 
cal, and it would be interesting to know what 
weight the authors attach to the eighth significant 
figure in their calculation. 
(7) The French authors remark that ‘“ Books on 
analysis, and especially books on qualitative 
analysis, are very numerous. If one excepts the 
classical treatise of Fresenius, one may say that 
all the others have been written in preparation for 
an examination. This book, in distinction from 
the others, is written for those who wish to learn, 
and not for those who are seeking for diplomas.” 
The product is an interesting volume, in which the 
principles and general methods of analysis, as 
well as the properties of the chief metallic and 
non-metallic radicles and compounds, are 
described. This they hope to supplement later by 
a book on industrial analysis. 
DIFFERENTIAL GEOMETRY. 
Lectures on the Differential Geometry of Curves 
and Surfaces. By Dr. A. R. Forsyth, F.R.S. 
Pp. xxiii+525. (Cambridge: The University 
Press, 1912.) Price 21s. net. 
IFFERENTIAL geometry is a technical and 
rather forbidding term, but the subject is 
of the highest interest, and not to mathematicians 
alone. It includes the whole theory of map- 
drawing; it is required for the problem of soap- 
film surfaces; and if the earth were much different 
from a sphere the theory of geodesics would enter 
into practical questions of navigation and engin- 
eering. 
There are two well-known and_ excellent 
treatises on the subject, by Darboux and Bianchi 
respectively; but hitherto there has been nothing 
corresponding to them in English, so that the 
appearance of the present volume will be wel- 
comed even by those acquainted with its topic, 
and will no doubt lead more Englishmen than be- 
fore to the study of it. 
The general features of Dr. Forsyth’s work are | 
NO. 2232, VOL. 89] 
such as might have been anticipated. As between 
Darboux and Bianchi, it occupies a sort of middle 
position, being less individual and synthetic than 
the one, and less analytical than the other. For 
example, in the chapter on minimal surfaces we 
miss Darboux’s historical notes and correspond- 
ingly progressive treatment; while on the other 
hand we are spared the Riemann-Christoffel sym- 
bols, which play so large a part in Bianchi’s ex- 
position. | Besides Gauss’s fundamental theory, 
it is the Mainardi-Codazzi relations which mainly 
help in developing all the earlier theory of curves 
on surfaces, &c., as here investigated. 
The author’s unrivalled power of dealing with 
complicated analysis is admirably illustrated by 
the section on differential invariants (p. 203-232). 
It would be very difficult indeed to improve upon 
this: it gives a convincing example of the value 
of Lie’s theory of contact-transformations, an 
illustration of Jacobi’s theory of systems of par- 
tial differential equations, a ‘‘ complete”’ set, up 
to a certain stage, of differential invariants, with 
the geometrical interpretation of each, and finally 
sufficient detail to enable a student to work out, 
if he cares to do so, the system of invariants for 
the next stage. 
In trying to estimate the value of a mathe- 
matical treatise, we naturally turn to pages which 
deal with problems still partly, at any rate, un- 
solved, or evidently not reduced to a natural and 
definite conclusion. In the present case we may 
take the theory of geodesics, and the problem of 
deformation of surfaces. With regard to the first, 
there are certain fundamental results due to 
Gauss; the connection of the theory with the strict 
calculus of variation; and the statement of the 
analytical problem in the simplest form  con- 
sistent with present knowledge. On every 
one of these points Dr. Forsyth writes with 
complete mastery, and gives a most valuable set 
of examples. Whether we are likely to get soon 
any substantial contribution to the theory is 
doubtful, but, at any rate, we have here a clear 
account of its present state, and there may at least 
be some more special results awaiting discovery. 
On the problem of the deformation of surfaces 
we have a very interesting chapter (pp. 354—406), 
which, amongst other things, gives the critical 
equation in Darboux’s form, a remarkable 
theorem of Beltrami’s, and a summary of Wein- 
garten’s method. We have also a simple proof 
of the theorem that, in general, a surface cannot 
be deformed while a curve upon it is kept rigid. 
This last is an excellent example of a mathe- 
matical theorem which anybody can understand, 
but which requires very careful discussion to prove 
in a satisfactory way. Everybody can see cases of 
