164 
better méthods of feeding.” The importance of 
sufficient food is emphasised, but the author might 
have dealt even more fully with this point. As the 
dairy industry advances, cowkeepers increase their 
rations, until, in districts where the industry is very 
highly developed, as in many parts of the home 
counties, they tend to give too much food for 
profitable milk production. 
It has been shown by experiment that an in- 
crease in the quantity of the ration leads to an 
increased production of milk, but the return 
obtained per unit of food becomes less and less, 
and finally is not worth as much as the food costs, 
The great difficulty about dairy farming in the 
States seems to be the labour supply. ‘The 
special objections raised to the labour on the 
dairy farm are the long hours, the steady, regular 
work, and the nature of the work.” The same 
diificulty is felt in British agriculture, and to meet 
it machinery is being invented. Strenuous efforts 
are being made to perfect the cow-milking 
machine, which will considerably ease matters. 
E. J. Russet. 
ALGEBRAIC NUMBERS. 
The Elements of the Theory of Algebraic Num- 
bers. By Prof. L. W. Reid. With an introduc- 
tion by Prof. D. Hilbert. Pp. xix+454. (New 
York: The Macmillan Co. ; London : Macmillan 
and Co., Ltd., 1910.) Price 15s. net. 
T is almost a misfortune that Gauss and Smith 
were such consummate masters of mathe- 
matical style. Nearly everyone who writes on an 
arithmetical topic is tempted to imitate them, at 
least in their brevity and severe deductive method, 
and the result is that many are frightened away | 
from a delightful study because of its dry. and, so 
to speak, inaccessible aspect. 
Prof. Reid’s book ought to do much to remove 
this misapprehension ; he has assumed no previous 
knowledge of the subject beyond elementary arith- | 
metic, and he has been careful to give a very large 
number of special examples, as well as discussions 
of special fields. Thus the reader is able to see, 
much more than is usually possible, the really 
inductive character of arithmetical research, and 
is provided with material on which he may make 
experiments of his own. : 
As an introduction to the general theory of 
algebraic numbers, the work follows mainly the 
treatment of Hilbert and Dedekind; the principal 
difference from the latter being that, for instance 
in a quadratic field, an ideal (a, 8) is (ultimately) 
defined as the aggregate of integers \a+ 8, where 
A, # are any two integers in the field, and a, B 
are given integers therein. This saves a good deal 
NATURE 
[APRIL 18, 1912 
kind’s own definition of an ideal is adopted, and 
detracts little, if anything, from the naturalness 
of the sequence of theorems. 
After four chapters dealing with the ordinary 
rational theory (including the law of quadratic 
reciprocity), we have four others, each devoted to 
a special quadratic field, namely, those derived 
from / —1, /—3, V2, “—5 respectively. In 
the last of these it is made perfectly clear how 
the law of resolution into prime factors appears to 
break down, and how it is restored by the intro- 
duction of ideals. Moreover, examples are given 
to show the distinction, in this field, between prin- 
cipal and non-principal ideals. 
The next four chapters give general theorems 
on algebraic numbers, a discussion of the general 
quadratic field, its discriminant and ideals, and 
the theory of congruences with respect to ideal 
moduli in such a field. 
Finally, there are two chapters, of a rather more 
advanced kind, on the units of the general quad- 
ratic field, and on the number of its ideal classes. 
In the latter use is made of Minkowski’s remark- 
able theorem that every ideal class contains am 
element whose norm does not exceed |/d|, 
where d is the discriminant of the field. Prof. 
Reid has elsewhere published a list of classes of 
cubic fields calculated on the same principle; but 
the work is tentative and laborious, and it is still 
a desideratum, even for cubic fields, to determine, 
by some simple method, the fundamental units. 
and representatives of each ideal class. Fortun- 
ately, however, in working with an ideal, any one 
of its forms will do in using it as a modulus, find- 
ing its prime factors, and so on: just as (6, 9) or 
(6, 9, 27) define 3 (as a greatest common measure) 
just as well as 3 itself for purposes of this kind. 
It is very gratifying to see that the higher 
arithmetic is attracting more and more attention, 
and it is certain that books like Prof. Reid’s will 
greatly help to popularise “the queen of the 
sciences,’’ as Gauss so affectionately called it. 
GuBaMe 
A HISTORY OF EUROPEAN CULTIVATED 
PLANTS AND DOMESTIC ANIMALS. 
Kulturpflanzen und Haustiere in ihrem Uebergang 
aus Asien nach Griechenland und Italien sowie 
in das iibrige Europa. Historisch-linguistische 
Skizzen von Victor Hehn. Achte Auflage. Neu 
herausgegeben von O. Schrader. Mit botani- 
schen Beitragen von A. Engler und F. Pax. Pp. 
Xxvili+665. (Berlin: Gebriider Borntraeger, 
1gi1.) Price 17 marks. 
ICTOR HEHN’S book, the result of years of 
V labour, first appeared in 1870. A second 
of rather delicate reasoning, necessary if Dede- | edition was called for in 1874, to which the essay 
NO. 2216, VOL. 89] 
Bt So A ttm - 
