May 13, 1915] 

sional space (S,), which is determined by any four | 
of its points which are not coplanar. There is no 
logical reason why we should not assume the 
existence of at least one point which does not 
belong to S,; if we do so, we are able, by similar 
arguments, to show that there will be a space 
Sy, containing Ss, which is determined by any 
five of its points which are independent; that is 
to say, which do not belong to the same three- 
dimensional field. 
The important thing to notice here is that the 
argument is of such a kind that it does not appeal 
to intuition at all. It is true that by drawing 
figures, or,making models, we-can provide images 
which help us to see that for an S,, Ss, or Sy our 
assumptions are not self-contradictory—or rather 
to give us an irrational conviction that things are 
so; for, of course, nothing but a formal proof 
can demonstrate the consistency or inconsistency 
of a set of formal propositions, such as we are 
ultimately dealing with here. 
When the reader has reached this point, he will 
see that we can define and investigate a sequence 
of spaces :— 
Si So, S3, Sho ° 
(each being a field of points) on the assumption 
that when we have reached a space S, there is 
at least one point which does not belong to it. 
A space S, is determined by (n+1) independent 
points; namely, such as do not belong to one and 
the same S,-3. 
_ The author does not go further than the Sy. 
After establishing its (logical) existence, he pro- 
ceeds to discuss perpendicularity and angles; 
symmetry, order, and motion; hyper-surfaces and 
polyhedra; the theory of parallels; and that of 
volume-measurement. Finally, there is a chapter 
on the regular ‘‘ polyhedroids ’’ (hyperpolyhedra) 
in the Sy. 
There are many interesting details and_ illus- 
trations; we may refer to one of these, as it shows 
very well the way in which we are brought to a 
halt in trying to-make actual images of things in 
the S,. If, in an S,, we draw a square, then a 
square on each of its sides, and finally a square 
on one of the outermost sides (so as to make a 
Latin cross) we can see how to fold the squares 
about common edges until they form the faces 
of a cube in our Ss. Suppose, now, that in our 
Ss we draw a cube; then a cube on each of its 
faces; and finally a cube on one of the outermost 
faces of the last. If we could get this solid into 
an Sy, we could “fold” the eight cubes about 
adjacent faces so as to form the boundaries of a 
hypercube. Until we can “see” how to do this, 
we have no proper intuition of an S,, such as we 
have of the lower spaces. G. B. M. 
NO. 2376, VOL. 95] 
NATURE 


283 

THREE BOTANICAL BOOKS. 
(1) Mikrographie des Holses der auf Java vor- 
Vierte Lieferung. By 
Pp. 
kommenden Baumarten. 
Dr. J. W. Moll and H. H. Janssonius. 
336. (Leiden: E. J. Brill, 1914.) 
(2) Practical Field Botany. By A. R. Horwood. 
Pp. xv+193. (London: C.. Griffin and Co., 
IGtds ona») s erice iss. net. 
(3) All About Leaves. By the late F. G. Heath. 
Pp. ix+228. (London: Williams and Norgate, 
1914.) Price 4s. 6d. net. 
(1) [ N this, the fourth instalment of his work 
on the intimate structure of the wood of 
Javan trees, the author deals with the families 
Connaracee, Leguminose, Rosacee, Saxifragez, 
Hamamelidee, and Rhizophoree. The Legumi- 
nose occupy more than half the volume. In this 
family 118 specimens were examined, represent- 
ing forty-nine species and four varieties contained 
in twenty-six genera. Under the heading of each 
family is a list of the literature in chronological 
order, followed by an enumeration of the species 
and varieties examined by the author. A general 
review of the gross and minute anatomy of the 
family as represented by the forms studied is then 
given, and the bearing of the results on the gener- 
ally accepted sub-division of the family is dis- 
cussed. These are, in the main, in conformity 
with the sub-division based on floral structure, 
but it is worthy of note that in Rosacez the differ- 
ences between the groups are greater than is 
usually the case in a single family, and suggest 
the recognition ‘as distinct families of the Amyg- 
dalacee, Chrysobalanacee, and Pomacee. A key 
for determining the species by means of the 
wood-structure is also given. Then follows a 
detailed description of each species or variety; 
first the general topography, which is sometimes 
illustrated by figures, and then an elaborate de- 
scription of the various elements—wood-vessels, 
libriform fibres, wood-parenchyma, medullary 
rays, etc. The account includes measurements 
and the behaviour of the various elements to re- 
agents. It is to be regretted that the author did 
not include photomicrographic reproductions of 
his sections; the few figures which are given are 
poor and show no detail. 
(2) The title of Mr. Horwood’s book is some- 
what misleading. The book contains much 
useful information on the practical study of plants 
in the field, but also much extraneous matter often 
set down in a loose and desultory manner. The 
impression formed by the reader is that the author 
is a man of great industry and some knowledge, 
but does not realise his limitations. The book, if 
judiciously pruned and edited, would make a 
useful little volume. The author’s intention is 
