62 

NATURE 
[Marcu 18, 1915 

a weak lens, but the lamp is pushed up within 
the focus of the first lens, and it is very rarely 
that an operator changes his condenser when he 
changes his objective, as laid down in § 89. 
The only other criticism is that in projecting 
the rings and brushes in convergent polarised 
light, the objective does not focus the back lens 
of the second convergent system on the screen, 
but its back focal plane; this mistake is, how- 
ever, generally made in the optical text-books. 
It should be added that there is a useful biblio- 
graphy and a copious index. 

MATHEMATICAL 
(1) Elements of Geometry. By S. Barnard and 
J. M. Child. Parts ivi. Pp. ix+465. 
(London: Macmillan and Co., Ltd., 1914.) 
Price 4s. 6d. 
WORKS. 
(2) A Foundational Study in the Pedagogy of 
Arithmetic. By Dr. H. B. Howell. xi+ 328. 
(New York: The Macmillan Co.; London: 
Macmillan and Co., Ltd., 1914.) Price 5s. 6d. 
net. 
(3) Constructive Text-book of Practical Mathe- 
matics. By H. W. Marsh. Volume iit. Tech- 
nical Geometry. Pp. xiv+244. (New York: 
J. Wiley and Sons, Inc.; London: Chapman 
and Hall, Ltd., 1914.) Price 5s. 6d. net. 
(4) Algebraic Invariants. By Prof. L. E. Dick- 
son. Pp. x+100. {New York: J. Wiley and 
Sons, Inc.; London, Chapman and Hall, Ltd., 
1914.) Price 5s. 6d. net. 
(5) Plane Trigonometry and Tables. By G. Went- 
worth and D. E. Smith. Pp. v+188. Trigo- 
nometric and Logarithmic Tables. By G. 
Wentworth and D. E. Smith. Pp. v+104. 
(London: Ginn and Co., 1914.) Price 5s. 
(1) HIS is not a modified edition of the 
authors’ well-known text-book entitled 
“A New Geometry for Schools.” In form, 
grouping, method, and to some extent, subject- 
matter, it differs widely from their former work. 
The experience gained from eleven years’ teach- 
ing on modern lines has led them to certain 
conclusions which they embody in the present 
volume. To some extent, they revert to older 
methods, as, for example, in the treatment of 
parallels, tangents, the early theorems of the 
second book of Euclid, and the grouping of cog- 
nate theorems. In other respects they adopt a 
more radical position by introducing  trigono- 
metrical ratios and sections on solid geometry 
(mainly numerical) at an early stage. They have 
compressed an immense amount of matter into 
a compact form, and the array of numerical and 
theoretical: exercises is almost alarming in its 
extent.) : { 
NO. 2368, VOL. 95] 

(2) The science of pedagogy is arousing in- 
creasing interest in this country, although, 
curiously enough, less among those engaged in 
teaching at our public schools than at primary 
and other secondary schools. But the literature 
of the subject has received comparatively few 
British contributions. The object of the author 
is, in his own words, to summarise the anthro- 
pological, experimental, pedagogical, psycho- 
genetic and phylogenetic resources now available, 
which bear on the part arithmetic should play in 
a scientific curriculum. His investigation is four- 
fold: (1) genetic, (2) psychological, (3) statistical, 
(4) didactical. Not the least interesting and 
valuable part of the book is the account of experi- 
ments, arranged and conducted by the author on 
the determination of ability in number appre- 
hension and fundamental processes. 
(3) It is strange to pick up a text-book on 
geometry and fail to find a single diagram before 
page 150. The explanation lies in the fact that 
the author believes that each student should, by 
his own researches, write his own text-book. 
‘That such a mode of procedure as is indicated by 
the plan and structure of this book has been in 
operation for nearly twenty years is conclusive 
evidence of the enthusiastic personality of the 
author : that many teachers, however, could adopt 
his method seems to us highly improbable. His 
purpose is to enable each pupil to build up for 
himself a logical geometrical structure, by the aid 
of appropriate suggestions made at the right time. ' 
The form these are to take and the quality of 
the work expected of the pupil is set out in great 
detail. We doubt, however, whether the author 
will make many converts. 
(4) This monograph provides a simple and 
admirable introduction to the theory of invariants 
of algebraic forms. It is divided into three parts. 
The first deals with linear transformation from 
the point of view of (1) change of axes of reference, 
(2) projective geometry. Jacobians and Hessians 
are discussed, the latter in connection with the 
solution of the cubic and the points of inflexion 
of a cubic curve. The second treats of algebraic 
properties, such as weight, annihilators, recipro- 
city, differential operators, ete. And the con- 
cluding section introduces the symbolic notation of 
Aronhold and Clebsch. Some carefully selected 
sets of examples are provided. 
(5) The form and plan of this text-book indicate 
that the changes in the teaching of elementary 
trigonometry have taken place along similar lines 
on each side of the Atlantic. The practical aspect 
of the subject is now regarded as requiring chief 
emphasis in the early stages and the abstract 
theory is postponed. Considerable space is de- 
