126 
NATURE 
. 
[OcTOBER 19, 1916 
intimate as that of the soul to the body; we can- 
not get between them even in thought, but the 
difference is one of kind and not of degree.” There 
is much in the volume about the wonders of the 
inorganic domain, especially under the eyes of 
modern chemists and physicists, but the refrain 
is always what Tyndall called “the mystery and 
the miracle of vitality.” Thus, to mention half of 
the fascinating studies, we have discussions of 
“The Breath of Life,” “The Living Wave,” “The 
Baffling Problem,” “Scientific Vitalism,” and 
“The Vital Order.” 
It is not easy to describe the life of the feaihive 
without the postulate of psychical organisation, 
what Maeterlinck called the Spirit of the Hive; 
so to Burroughs it appears necessary to recognise 
a more than physico-chemical unity of the organ- 
ism, in which the cells are the bees, and thus he 
speaks of the Spirit of the Body. But this vitality 
is potential in all matter, though it finds oppor- 
tunity to manifest itself with emphasis in proto- 
plasm. Vitality begins in the inmost sanctuary of 
the molecules, “but whether as the result of their 
peculiar and very complex compounding or as the 
cause of the compounding—how are we ever to 
know?” The striking essay entitled ‘A Bird of 
Passage” develops the idea that life plays a very 
small part in the total scheme of things, “the 
great cosmic machine would go on just as well 
without it.” Yet it is only in the highest expres- 
sions of life that the total scheme of things 
acquires any meaning at all. And the author 
ends with the thought, which he knows to be 
beyond science, that there is a kind of universal 
mind pervading not only living matter, but the stuff 
of which the whole world has been spun. As the 
reader is warned in the preface, there is consider- 
able reiteration in the course of the essays, but 
with a writer like Burroughs the impression left 
is that of music with a recurrent theme. 
[2 AL 
DIOPHANTINE ANALYSIS. 
Mathematical Monographs. No. 16, Diophantine 
Analysis. By R. D. Carmichael. Pp. vi+118. 
(New York: Jj. W iley and Sons, Inc. ; London : 
Chapman and Hall, Ltd., 1915.) Price 5s. 6d. 
net. 
HE remarkable thing about Diophantine ana- 
lysis is that, although it is quite respectably 
old, it is still in that stage where the amateur is 
on an equal footing with the professional. If it 
be true, as we are inclined to think, that Fermat’s 
last theorem admits of a Diophantine proof, this is 
as likely to be discovered by a schoolboy as by a 
professor steeped in all the lore of modern 
analysis. 
Prof. Carmichael’s book is welcome because it 
gives, either in the text or in the examples, a great 
deal of the actual results hitherto obtained ; and the 
author has done something towards sorting out 
these results and adumbrating a real theory. In 
NO. 2451, VOL. 98] 
this respect chap ii. (on multiplicative domains) is. 
the most valuable. Chap. v. gives a brief but 
up-to-date account of what is known about 
Fermat’s last theorem. Important sections are - 
those which treat of Fermat’s methods of 
“descent” and of “double equations”; these, at 
any rate, are definite processes capable of exten- 
sion to various cases. 
The weak point of the book, in our opinion, is. 
that the author never looks at a problem from a 
geometrical point of view. Of course, in the last 
resort, geometry is irrelevant; but in research it 
is very valuable. For instance, let F(x, y, s) be 
a homogeneous cubic; then from the theory of 
curves we can conclude that if F=o has an in- 
tegral solution (xj, y,, 3;), it has a sequence 
(Xn) Vn) Sn) Of integral solutions, which in most 
cases corresponds to a compact set of points on 
the curve F=o. The proof of this is most easily 
obtained from elliptic functions; there ought to be 
a purely Diophantine proof, but the difficulty is. 
that we have to estimate the “nearness ” of a solu- 
tion (x’, y’, 3’) to a solution (x, y, s), and 
(x2, Yo, 2) in the sequence is not generally 
“near” to (x,, 3, 3,) in the geometrical sense. 
Again, if we have a unicursal surface, such as. 
that given by the parametric equations, ; 
= eam 
NER? 
_aAp+1) 
) pa Au—1) 
At+p 
A+p 
whence x*/a?+y?/b®—2?/c?=1, this suggests cor- 
responding Diophantine theorems. Then, too, we 
have to consider solutions which, though not in- 
tegral in the ordinary sense, are integral in certain 
algebraic fields; for instance, if 2p=—1+1/3, 
then (25—6p, 1—9p, 8+30p) is a solution of 
x°+7y — z8=o0, which is integral. in the field (p), 
although it is not so in the field (1). In the latter 
field we have the solution (1, 1, 2); the reader is 
left to discover whether there are any other ordi- 
nary integral solutions, and if so, how many. 
There are numerous exercises in the book which 
ought to stimulate the reader; some of them are 
practically suggestions for research. As a rule, 
it is unfair to expect a mathematical writer to give 
exact references to the sources of his examples; 
but in this case we wish Prof. Carmichael had 
been a little more definite, because in this subject 
even a short note on a very special problem may 
possibly contain the germ of an important dis- 
covery. As an instance of what we mean, Eisen- 
stein’s proof of the irreducibility of (1—x?)/(r1—x), © 
when Pp is prime, is based on a theorem of his 
which must surely admit of some generalisation. 
To find whether any given polynomial is irre- 
ducible or not-is practically such a laborious task 
(though theoretically possible) that special theo- 
rems like Eisenstein’s are always welcome. 
We hope that this book will have a wide circula- 
tion among mathematicians of all ages and capaci- 
ties; it is rather a disgrace to the moderns that in 
this field they have added so little to the work of 
that great triumvirate, Diophantus, Fermat, and 
Euler. G) Bain 
