June 20. £912] 
NATURE 
oo 
fieant chlorophyll granules, wherein takes place the 
wondrous process of the transformation of the 
sun’s rays into chemical energy, source of all the 
manifestations of life on our planet’; the flowers 
in illustration of “the wonderful ties which bind 
together the two kingdoms of Nature” and the 
forest as the sublimest picture of struggle and 
elimination and survival. In one respect we con- 
fess our disappointment that Prof. Timiriazeff 
should have seen fit to lend the weight of his 
authority to the side of the mechanistic biologists, 
and should have thought it necessary to refer to 
“ Neovitalism” as a “morbid outgrowth.” 
MODERN MATHEMATICS FOR TEACHERS. 
Monographs on Topics of Modern Mathematics 
Relevant to the Elementary Field. Edited by 
J. W. A. Young. Pp. viii+416. (London: 
Longmans, Green and Co., 1911.) Price 
ios. 6d. net. 
ECENT work on the first principles of 
mathematics has been so far-reaching and 
revolutionary that most, if not all, of those 
acquainted with the results are anxious to bring 
them to bear upon general education. For this 
there are two main reasons: in the first case, it 
cannot be right to go on pretending to teach a 
subject in a strictly logical way when all sorts 
of assumptions, many of them wrong, are being 
tacitly made; and secondly (this is still more im- 
portant), the philosophical side of the new theories 
is bound, sooner or later, to have a profound effect 
upon educated thought. It is, for instance, a 
great achievement that mathematicians have now 
got definite concepts of three distinct “infinite 
numbers,” as contrasted with the vague “~~ ” of 
former times; that they have proved the possi- 
bility of three distinct geometries, in two of which 
the axiom of parallels does not hold good; and 
that there is some prospect of bringing the theories 
of electricity and gravitation under one compre- 
hensive hypothesis—it may be by a restatement 
of the laws of motion, or even by the assumption 
of a sort of four-dimensional space. 
The trouble is that the treatises which deal with 
these matters scientifically are full of strange 
symbols and elaborate detail, so that it is hope- 
less to expect an average mathematical teacher to 
study them. Prof. Young and his colleagues have 
therefore done a real service by providing for 
secondary-school teachers chapters on nine im- 
portant topics, partly but not wholly demonstra- 
tive, and mainly designed to give them a reason- | 
ably sufficient account of what has been done, so 
The most important chapters are undoubtedly 
i.tii., which treat of the foundations of geometry, 
modern pure geometry, and non-Euclidean geo- 
metry. They ought to make plain the character 
of a complete system of axioms and postulates, 
the notions of order, congruence, segment, and so 
on; the principles of duality and projectivity, and 
the properties of the elementary figures; and 
finally the justification of introducing the two non- 
Euclidean geometries. The treatment of the last- 
named is (quite rightly, we think) in great measure 
analytical ; the fact is that our false intuition of 
space is so ingrained that few of us will give it 
up until we are faced by a consistent set of funda- 
mental formule. 
Chapter iv., on the foundations of algebra, is 
perhaps the most rigorous of the nine. A set of 
twenty-seven postulates is drawn up, suited for 
ordinary complex algebra, and it is shown that 
they are sufficient, consistent, and independent. 
An appendix contains a note on Dedekind’s theory 
of cuts and Cantor’s method of sequences. What 
seems to us a defect in this chapter is that it 
assumes ordinary complex algebra as the most 
comprehensive one, and this is remarkable as 
coming from a compatriot of the Peirces. Surely 
in a chapter on general algebra some reference 
should have been given to quaternions, and to 
| those systems where ab=o does not require that 
either a or b should be o. And we do not agree 
without reservation to the remark on p. 200, 
“both the arithmetical and the geometrical sys- 
tems are equally entitled to stand as representa- 
tives of the type of algebra in question.” To 
justify this, we must at least assume the Dedekind- 
Cantor postulate. 
Chapter vi., on the function-concept and ele- 
mentary notions of the calculus, does not go very 
far, and may, perhaps, be taken to give more 
value to the “graph” than it really does; but it 
is lucid and interesting, and, at any rate, gives 
Dirichlet’s definition of a one-valued function, and 
refers to Weierstrass’s proof that a continuous 
function need not have a differential coefficient. 
Curiously enough, in dealing with maxima and 
minima the author omits to notice the case when 
dy/dx is infinite, and does not bring out the real 
point that dy/dx must change sign. There is 
some rather vague talk about different ranges of 
the independent variable; for analytical purposes 
the range must be some definite one-dimensional 
arithmetical field, and clearness would be gained 
by saying that, for strict differentiation, it must 
be a segment of the arithmetical continuum. For 
instance, let f(x)=1 when x is rational, and 
that in the light of their new knowledge they may | /(x)=o0 when x is not: this is a definite one-valued 
modify their teaching. 
NO. 2225, voL. 89] 
function, 
and if the field of x is the field of 
