Marcu 25, 1897] 
NATURE 
483 
series of integers comprised in that form. 
notion may be extended to the case when a and § are 
any two algebraical integers belonging to the same 
P| 
Now this | 
‘corpus ; and the extension may be made in two ways. | 
In Dedekind’s theory x and y, as before, stand for | 
rational integers, and our attention is directed not so 
much to the form +a+ _y8 as to the series of numbers | 
it represents. This series is called an ideal, and denoted 
by [a, 2]; so far as aand 8 are concerned, it is found to 
possess properties precisely analogous to those of the | 
greatest common measure. Kronecker, on the other 
hand, keeps the form a+ 8 explicitly, using -r, y as 
mere symbols, or umbree; and the highest common 
‘divisor of a and P is defined as follows. The norm of 
xa + y8 is a rational homogeneous form in +, y which 
is the product of a rational integer and a primitive form 
F ; by the highest common divisor of a and 8 we mean 
(va + y8)/F. This definition has, of course, to be 
subsequently justified. 
In a certain sense, then, the difference between the 
two methods is merely one of symbolic ; but as in other 
similar cases (e.g. the methods of Cartesian and of 
homogeneous coordinates), it sometimes happens that 
propositions which are easily proved by the one are 
difficult for the other, and versa. Kronecker’s 
theory was not worked out in detail in his famous 
“Festschrift”; Prof. Weber has now made it easily 
intelligible by adopting it, with some modification, as 
the basis of his exposition. Simplicity is gained by 
‘omitting primitive forms, such as F above, in the 
expression for divisors; and by means of a few new 
terms, such as “functional,” the discussion is made at 
once concise and clear. 
It should be added that the reader will find in this 
book not only a thorough account of the elements of 
the subject from Kronecker’s point of view, but a guide 
to its most recent developments. Thus, for instance, it 
contains Minkowski’s theorem on the minimum values of 
quadratic forms, with important applications to minimum 
representatives of ideal classes; and a summary of 
Hensel’s very important investigations, by which it 
becomes possible to give an exf/ic’t representation of 
the prime ideals (or functionals) which belong to a given 
corpus. 
Chapter xx., on quadratic corpora, shows the relation 
of Gauss’s theory of quadratic forms to the general 
theory. Chapters xxi.-xxiv. are devoted mainly to the 
proof of Kronecker’s theorem that all Abelian numerical 
corpora are cyclotomic ; in other words, that the roots 
of a// Abelian equations with rational integral coefficients 
are rational functions of roots of unity. The proof of 
ice 
| 
this involves a long series of propositions, many of which | 
are extremely valuable in themselves ; we may instance 
the determination of the number of classes belonging to 
a given corpus, and the corollary that in every algebraical 
corpus there are an infinite number of prime ideals of 
the first degree. Perhaps the proof of Kronecker’s 
theorem may some day be attained by a less laborious 
route ; meanwhile it is a remarkable example of those 
arithmetical truths which are easily stated and easily 
understood, but, as yet, require for their demonstration 
an elaborate mathematical apparatus. 
Prof. Weber's concluding chapter (xxv.), on transcen- 
NO. 1430, VOL. 55] 
dental numbers, contains a proof of the transcendence 
of e and 7, and forms an elegant coronis for a work 
which is so important and so original that it is, to a 
great extent, above the range of ordinary criticism. As 
an introduction to, and exposition of, the theory of 
rational algebra and its arithmetical applications, it is 
simply invaluable. A student of real capacity, familiar 
with the technique of elementary algebra, may, by read- 
ing this work, together with Dedekind’s wonderful tracts 
(“ Ueber Stetigkeit,” &c., and “Was sind u. was sollen 
die Zahlen?”) and the last two editions of Dirichlet’s 
| “Zahlentheorie,” equip himself for exploration in that 
strange unearthly region of arithmetic which attracts 
some sedentary spirits in much the same way as Arctic 
travel charms a Franklin or a Nansen. And even 
though he may not be one of the few who make dis- 
coveries of real importance, he will at least be able to 
appreciate intelligently the work that has been done, 
and the progress that has been made in developing the 
most abstract part of the only science that deserves to 
be called exact. 
Gratitude has been defined by somie practical cynic as 
the expectation of benefits to come: we must plead guilty 
to some such feeling on reading Prof. Weber’s promise 
of a sequel, which is to deal with applications of the 
theory of algebraical numbers to the theory of elliptic 
functions ; an application already partially carried out 
in his “ Elliptische Functionen und algebraische Zahlen.” 
And we cannot help remembering that, in conjunction 
with Prof. Dedekind, Prof. Weber has laid the found- 
ations of a thoroughly arithmetical treatment of algebraic 
functions of one variable, in which alone (in our opinion) 
will be found a complete justification of the results to 
which Riemann was led by his geometrical method. Is 
it too much to hope that Prof. Weber may sometime be 
willing to develop these principles into a treatise on 
algebraical and Abelian functions ? G. B. M. 
THE WORSHIP OF TREES. 
The Sacred Tree; or, the Tree in Religion and Myth. 
By Mrs. J. H. Philpot. Pp. xvi + 179. (London: 
Macmillan and Co., Ltd., 1897.) 
HE further we are able to penetrate the mists which 
hang over the early history of mankind, the 
more sure we become that the primeval ancestors of 
our race regarded certain trees with veneration and 
awe; and it seems quite possible that in the earliest 
times the tree was a symbol of a supernatural and 
almighty power, which we might describe by the word 
“sod.” We shall not attempt to express in years the 
amount of the time which must have passed since tree 
worship began ; but it will be sufficient, in the course of 
this short notice, to give a few proofs of its existence in 
the times which antedate the literature and history of all 
countries except those of Egypt and Southern Babylonia. 
The study of the tree in its relation to religion 
and myth has occupied the minds of some of. our 
best anthropologists, and though we are inclined to 
think that presently certain people will find the tree in 
every ancient piece of work and symbol—just as some 
investigators find the Christian cross everywhere, and 
others find the lotus in every ornament—still there is no 
