NATURE 
409 
THURSDAY, FEBRUARY 28, 1907. 
SCHOOL MATHEMATICS. 
(1) Trigonometry for Beginners. By J. W. Mercer. 
Pp. xi+351. (Cambridge: University Press, 1906.) 
Price 4s. 
(2) Trigonometry for Beginners. By Rev. J. B. 
Lock and J. M. Child. Pp. viit195. (London: 
Macmillan and Co., Ltd., 1906.) Price 2s. 6d. 
{3) Geometry: an Elementary Treatise on the Theory 
and Practice of Euclid. By S. O. Andrew. Revised 
edition. Pp. xii+218. (London: John Murray, 
1g06.) Price 2s. 
(4) Modern Commercial 
G. H. Douglas. Pp. 
and Co., Ltd., 1906.) 
{s) A New Shilling Arithmetic. By C. Pendlebury, 
assisted by F. E. Robinson. Pp. xii+ 176+ xxxiv. 
(London: George Bell and Sons, 1906.) Price, with 
answers, 1s. 4d. 
(6) Junior Arithmetic Examples. 
chardt. Pp. vilit+171+xl. 
1906.) Price 1s. 6d. 
(7) Clive’s New Shilling Arithmetic. Edited by Dr. 
W. Briggs. Pp. viii+ 160. (London: W. B. Clive, 
University Tutorial Press, Ltd., 1906.) Price 1s. 
{8) Junior Practical Mathematics. By W. J. Stainer. 
Pp. x+350. (London: George Bell and Sons, 1906.) 
Price 3s. 
(9) A Rhythmic Approach to Mathematics. 
L. Somervell, with a preface by Mary Everest 
Boole. Pp. 67. (London: George Philip and Son, 
Ltd., 1906.) Price 2s. 6d. net. 
{1) M® MERCER possesses a fine conception of 
how trigonometry should be presented to 
youths, and his book is admirable and altogether good. 
The development of the subject is very gradual indeed, 
and is constantly enforced by means of concrete ex- 
amples, systematic computations, practical geometry, 
and by judicious graphs. Thus the first ten chapters 
are confined to the development of the simple trigono- 
metrical ratios, that is, to the solution and appli- 
cation of right-angled triangles, under all sorts of 
conditions, such as in problems of elementary sur- 
veying, the resolution of vectors, areas, solutions 
of triangles in general, &c., and later on additional 
illustrations of right-angled triangles occur in the 
chapter on traverse tables. Four-figure mathe- 
matical tables, including logarithms, are in constant 
use, and careful attention is paid to arrangement and 
checking of the numerical work. After this thorough 
grounding, angles. of any magnitude are introduced, 
and triangles are solved by general formule, a useful 
table of log. haversines being here provided. Then 
the radian measure of an angle is explained, and the 
treatment becomes more abstract, the final chapters 
dealing with multiple and submultiple angles, trans- 
formation formulae, identities, equations, and inverse 
notation. Finally, there are two hundred miscel- 
laneous examples grouped in sets of five, and 
collected answers to the very numerous exercises dis- 
NO. 1948, VOL. 75] 
Arithmetic. Part i. By 
163. (London: Macmillan 
Price ts. 6d. 
By W. G. Bor- 
(London: Rivingtons, 
By Edith 
tributed throughout the book. The author is to be 
congratulated on having produced a very notable text- 
book on elementary trigonometry, and one that is 
worthy of adoption in the secondary, technical, and 
public schools throughout the country. 
(2) Messrs. Lock and Child, like the author just 
noticed, start with the very laudable idea of present- 
ing the subject in a more practical and less abstract 
form than is commonly met with, and they are 
successful, though not to the same extent as in the 
previous case. Their development of the subject is 
not so finely graduated; they have not sufficiently 
recognised the fundamental importance of the right- 
angled triangle, and their special five-figure tables, 
without differences, though compact, become tedious 
in use and necessitate undue attention being given to 
the theory of proportional parts, a side issue. The open- 
ing chapters of the book are geometrical, and contain 
descriptions of practical methods of measuring angles, 
both of azimuth and of elevation, a detailed descrip- 
tion of the sextant and theodolite being reserved for 
a later chapter. Suitable exercises and test papers 
are provided at intervals, and the work concludes 
with answers and an index. The book contains many 
good features, and can be recommended especially to 
students preparing for the examination of the Board 
of Education in mathematics, stage 2. 
(3) As the result of increased experience, and also 
to meet the requirements of examiners, the subject- 
matter of Mr. Andrew’s well-known ‘‘ Geometry ’’ has 
been re-arranged and added to, while preserving the 
excellent features of the original work. Thus the 
first seven chapters are mainly experimental, practical, 
and quantitative, and with the ‘intimate first-hand 
knowledge *’ of geometry thus obtained the learner is 
well prepared for the theoretical work of the next 
chapter, in which formal proofs are given, arranged 
in logical sequence. The deductive method is 
employed in subsequent worl, which deals with similar 
figures, the solution of triangles, solid geometry and 
projection, and the mensuration of geometrical solids. 
The plan and scope of the book are excellent, and in 
its revised form the manual will deservedly appeal to 
an increasing number of readers. 
(4) The ‘‘ Commercial Arithmetic”? by Mr. Douglas 
is a very interesting work. Assuming the student to 
possess a knowledge of the fundamental rules of 
arithmetic, the author begins with examples of the 
tabular arrangement of numbers, with checks, 
labour-saving devices, and contracted and approximate 
methods in addition, subtraction, multiplication and 
division, and with the decimalisation of money. In 
the chapter on the calculation of prices we find some 
very neat methods of working. For example, since 
365=240+120+5, we have 365 at, say, 8d. each= 
Sl. +4l.+3s. 4d.=12l. 3s. 4d. In dealing with per- 
centages, commission, discount, and profit and loss, 
examples of commercial book-keeping are given, and 
there is the same regard for special and rapid methods 
of computation. In the calculation of simple interest 
for a specified number of years and days, an ingenious 
method known as the “ third, tenth, tenth ’’ is much 
used. The table given on p. 88 for finding the 
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