518 
space ; polyhedra ; the cylinder, cone and sphere, and 
similar solids. At the end of the book will be found 
numerical tables, a biographical table, a table of 
etymologies and an index. The space allotted to the 
different sections is comparable with their relative 
importance, and proper emphasis is laid on funda- 
mental ideas such as congruence, symmetry and 
similarity. 
Another very important feature is that the student is 
consistently stimulated and encouraged to think for 
himself. Marginal queries are frequently inserted, in 
order that he may justify the statements in the text ; and 
some of the proofs are given merely in outline for the 
reader to fill up in detail. On the other hand, figures 
and hints are given with the more difficult exercises. 
The appendix to Book ili., and other paragraphs inserted 
from time to time, ought to be of great help in teaching 
the student how to acquire the difficult art of proceeding 
from the unknown to the known by the method of 
analysis. 
In the theory of parallels, the authors adopt Playfair’s 
axiom; and their treatment of ratio is entirely arith- 
metical. In their opinion the’purely geometric treatment 
is too difficult for the beginner. On this point opinions 
differ, and will probably continue to do so: at the same 
time the arithmetical theory is here given in as nearly 
rigorous a form as the beginner is likely to appreciate. 
Thus it is properly stated as an assumption that a geo- 
metric magnitude may be represented by a number ; and 
the transition from the commensurable to the incom- 
mensurable case is made by the classic process of ex- 
haustion. Of course the strict arithmetical theory is at 
least as hard as the geometrical one, because it involves, 
besides the assumption above stated, either Dedekind’s 
theory of irrational numbers or something equivalent to 
it! But there is something to be said in favour of be- 
ginning with a provisional theory, admittedly imperfect, 
o be made more precise later on. It would be easy to 
add, in a future edition, an appendix giving the strict 
arithmetical and geometrical theories. 
In the discussion of the mensuration of the circle and 
other similar questions, the authors have avoided an 
error into which writers who adopt the arithmetical 
method are very apt to fall. They explicitly state the 
assumption that the circumference of a circle is the limit 
of the perimeter of an inscribed or circumscribed regular 
polygon, and then make use of the proved proposition 
that if, while approaching their respective limits, two 
variables have a constant ratio, their limits have that 
ratio. It is rather curious, by the by, that they omit to 
prove that the volume of a pyramid is the limit of the 
sum of the volumes of the usual set of inscribed prisms. 
In the text, which is beautifully printed by the Athenzeum 
Press, free use is made of abbreviations. The notation 
aé for the rectangle contained by the segments denoted 
by @ and 4 will be. objected to by some people ; but it 
really needs no justification, because the analogy which it 
suggests is too useful to be ignored, and if the student 
1 It may be remarked, in passing, that Euclid’s test of the equality of 
atios really amounts to the establishment of the identity of two 
as Dedekind calls them; for if »#A>—=<mnB according as 
-==<nD, the series of rational numbers 7/# for w hich 7#zA>nB 
lefines a Schnitt, and this is identical with the series for which #zC>2D. 
NO. 1561, VOL, 60] 
NATURE 
[SEPTEMBER 28, 1899 
cannot, after due warning, distinguish @é, the area of a 
rectangle, from a@é, the product of two numbers, it is 
entirely his own fault. 
The figures are very good ; those on solid geometry 
have been very carefully drawn, and are nearly as effective 
as models would be. This isa great help to the beginner : 
he should bear in mind, however, that he must eventually 
be able to use a less pictorial figure, or even construct a 
diagram mentally im cases where an actual figure is too 
complicated to be useful. We should be rather inclined 
to suggest beginning with the more pictorial figures, and 
gradually reducing them to pure diagrams. Between a 
picture and a diagram there is the same sort of difference 
as there is between a photograph of an electrometer and 
a working drawing of the same instrument. 
GaBa MMe 
OUR BOOK SHELF. 
An Elementary Course of Mathematics. By H.S. Hall 
and F. H. Stevens. Pp. ix + 342. (London: Mac- 
millan and Co., Ltd., 1899.) 
IN preparing this book the object kept in mind was, as 
we are told in the preface, to provide in a simple and in- 
expensive volume a short course of arithmetic, algebra and 
Euclid specially adapted to the requirements of students 
who, after leaving school, desire to continue their study 
of elementary mathematics by partly attending evening 
classes and partly working privately at home. 
To attain the end in view, the compilers, in the first 
portion on arithmetic, have restricted themselves to 
simply providing the student with a series of progressive 
exercises arranged to extend over a winter session of 
thirty weeks ; a few additions, exercises with notes and 
hints, conclude this portion. 
Algebra is next dealt with, and no previous knowledge 
is here assumed, so that a progressive but elementary 
course with numerous examples is given, covering the 
usual ground up to quadratic equations. In the last 
section on Euclid only the first book is considered. In 
the case of each proposition a few notes and exercises 
will help the reader to master this book, while ad- 
ditional theorems and a large set of appropriate examples 
are added for further practice. 
For the purpose for which it is intended, this 
elementary course is well adapted. 
Carvell’s Nursery Handbook, with Hints. By J. M. 
Carvell. (London : 
THE contents of this “ Nursery Handbook” are arranged 
under a number of headings; for instance, “ The Nursery,” 
“Sleeping,” ‘“ Clothing,” “Feeding,” &c. But in each 
section the hints given seem to be selected at hap- 
hazard ; small details in some places are noted, while 
many points of importance are omitted. 
In fact, the book seems too disjointed to be of real 
value, and the information too scanty to serve as a 
practical guide. In many instances the directions are 
so short that without amplification they might easily be 
misinterpreted. 
Pp. 70. Barber, 1899.) 
Chats about the Microscope. By Wenry C. Shelley. 
Pp. 101. (London: The Scientific Press, Ltd., 1899.) 
YOUNG naturalists will find in this volume many useful 
hints on the collection and preparation of common 
objects for microscopical study, and will be guided to 
make observations of a number of minute organisms 
easily obtained. 
