section on 
APRIL 25, 1907 | NATURE 603 
reproduction which takes in the | SOME RECENT MATHEMATICAL WORKS. 
mechanism of pollination, asexual reproduction by | Space and Geometry. By Dr. Ernst Mach. Trans- 
means of bulbils, experiments on regeneration, on the lated by Thos. J. McCormack. Pp. 148. (London: 
behaviour of potato tubers, and on grafting. The 
instructions are well arranged, and they form, with 
accessory explanations, a fairly continuous whole. 
A useful appendix is added, in which the needful out- 
fit in apparatus and reagents is given, together with 
hints on laboratory methods. The book is intended 
partly for the “cultivated layman” and partly for 
the students of the Gymnasium and Realschule. It 
will, however, prove useful to the teachers in English 
universities, as well as to others who have discovered 
the wisdom of making even advanced students per- 
form for themselves elementary experiments. 
We are inclined to think that the cultivated lay- 
man will be frightened by the first twenty pages of 
the book, which contain a large number of rough 
gualitative estimations of the chemical compounds 
occurring in plants. This is excellent for the labor- 
atory, but is hardly readable by one who does not 
repeat the experiments—and we cannot imagine the 
cultivated layman working his way through them. 
This, however, is not the fault of the authors, and 
it is only fair to say that the book in general is far 
from being unreadable. 
In a future edition the authors would be well 
advised to give scientific names, if only for the sake 
of foreign readers, who cannot be supposed to 
know what plants are meant by Sommerwurz or 
Mauerpfeffer. In some few cases the instructions 
want a little re-editing. Thus, in exp. 123, p. 82, 
the student is directed to compare the assimilation 
of a withered leaf with that of a fresh one, but he 
is not told that the absence of assimilation in the 
withered leaf is due to the closure of its stomata. 
The experiment is, in fact, incomplete; what is miss- 
ing is a repetition of Stahl’s proof that the leaves 
of certain plants the stomata of which do not close 
on withering are capable of assimilating in that 
condition. At p. 45 the treatment of the function of 
the stoma in gaseous interchange is not all that 
could be wished. The reader will have a singular 
view of Brown and Escombe’s researches if his know- 
ledge is confined to what he can learn in the present 
volumie. 
The experiments (p. 52) on the effect of freezing 
leaves would be more instructive if the ice-injection 
of the intercellular spaces were studied on a hardy 
plant such as ivy. In the second experiment, on 
p. 78, a Tropzolum leaf is recommended for use in 
experiments on the passage of air through vegetable 
membranes. But this is hardly allowable, since the 
leaf in question is well supplied with stomata on 
both surfaces. 
_In spite of a few oversights in its pages, we do 
not hesitate to recommend the work of the brothers 
Linsbauer to our readers. The methods prescribed 
are simple and trustworthy, and the book has a merit 
which is rare in text-books, namely, that it is 
obviously written with sincere interest in the problems 
set before the learner. 18g 1D): 
NO. 1956, VOL. 75] 
Kegan Paul and Co., 1906.) Price 5s. net. 
Irrational Numbers and their Representation by 
Sequences and Series. By Dr. Henry Parker 
Manning. Pp. vit123. (New York: J. Wiley and 
Sons; London: Chapman and Hall, Ltd., 1906.) 
| Auslese aus meiner Unterrichts- und Vorlesungs- 
praxis. By Dr. Hermann Schubert. Vol. iii. 
Pp. 250. (Leipzig: G. J. Géschen, 1906.) 
Lecons de Géométrie supérieure. By M. E. Vessiot. 
Pp. viii+322. (Lyons: Delaroche et Schneider ; 
Paris: A. Hermann, n.d.) Price 12 frances. 
La Géométrie analytique génerale. By H. Laurent. 
Pp. viit+is1. (Paris: A. Hermann, 1906.) Price 
6 francs. 
N. H. Abel: sa Vie et son Giuvre. By Ch. Lucas 
de Pesloiian. Pp. xiii+169; with portrait. (Paris: 
Gauthier-Villars, 1906.) Price 5 francs. 
Theory of the Algebraic Functions of a Complex 
Variable. By Dr. John Charles Fields. Pp. vii+ 
186. (Berlin: Mayer and Miiller, 1906.) 
Recherches sur lV’Elasticité. By P. Duhem. Pp. 218. 
(Paris: Gauthier-Villars, 1906.) 
F reform of mathematical teaching is to mean 
anything real, it is necessary that the teacher 
should possess a much more extended survey of his 
subject than is conveyed in the ordinary English text- 
book. There could be no more suitable book for 
giving the elementary or secondary teacher some 
intelligent ideas about geometry than Dr. Mach’s 
series of essavs. In them the subject is treated in 
its physiological, its psychological, and its physical 
aspects. 
The first essay thus deals with the relation of the 
spatial concept to the senses. In the second we have 
an attempt to trace the natural development of geo- 
metry from psychological causes, while the last essay 
discusses the subject from the point of view of physical 
inquiry. Incidentally, a number of illustrations are 
introduced, some of which are admirably adapted for 
teaching purposes. There could not be a better object- 
lesson in the elementary properties of Euclidean space 
than the indefinitely extended pavement formed of 
equal and similar triangles discussed on p. 59. From 
it can be read off all the principal properties of 
parallels and parallelograms, the relation between the 
three angles of a triangle, and also the main proper- 
ties of similar triangles the sides of which are com- 
mensurable. 
Dr. Manning’s book on irrational numbers con- 
tains a presentation in a simple form of another 
field of mathematical inquiry, such as is also 
eminently_ suited for placing in the hands of the 
ordinary schoolmaster. We have decided that the 
geometry of proportion shall be taught to schoolboys 
without reference to irrational quantities, but we have 
not yet eliminated a spirit of reckless extravagance 
in the quite unnecessary use of infinite series, often 
with total disregard for their convergency. In Dr. 
Manning’s treatment an irrational number is defined 
