THURSDAY, NOVEMBER 12, 1896. 
GALOISTAN ALGEBRA. 
Lehrbuch der Algebra. Von Heinrich Weber. Erster 
Band. Pp. xvi+ 654. (Braunschweig: Vieweg und 
Sohn, 1895.) 
A “Treatise on Algebra” is rarely found to fulfil the 
4 promise of its title. It is too often a mere collection 
of problems and examples, thrown together without much 
regard to order or method ; such theory as the book con- 
tains is often imperfect, and occasionally even incorrect ; 
and no attempt is made to suggest the idea of an ordered 
system of algebra, which proceeds along natural lines of 
development. 
Prof. Weber's treatise is a work of an entirely different 
stamp. It isdesigned upon a perfectly definite, well-con- 
sidered plan; its foundations are laid with the utmost 
care and precision ; and the reader is carried on from 
stage to stage until he is abreast of some of the most 
interesting, as well as the most recent, of mathematical 
discoveries. The work may be described, in general 
terms, as a treatise on ordinary algebra, with special 
reference to its arithmetical applications ; with the addi- 
tion, subsidiary to the main subject, but very important in 
itself, of the theory of groups. But in order to give any- 
thing like an adequate idea of the author’s scope and 
method, it will be necessary to analyse the different parts 
of his book in some detail. 
The introduction is entirely arithmetical ; at the same 
time it is an indispensable prelude to all that is to follow. 
It contains the elements of the theory of multiplicities, a 
rigorous theory of rational and irrational numbers, and a 
proof of the continuity of real numerical magnitude. 
The demonstrations are mostly inspired by Dedekind ; 
but itis shown that Cantor’s procedure leads to equivalent 
results. It is to be specially observed that the definitions 
of rational fractions, ratios, irrational, negative, and com- 
plex numbers are entirely independent of any hypothesis 
about the existence of divisible concrete quantities ; and 
similarly with regard to the statements and proofs of the 
various propositions. Perhaps in the whole range of 
mathematics no more abstract reasoning can be found 
than that by which the continuity of numerical magni- 
tude has been established ; and it is very instructive 
to compare the vague, illusive glimmering of the truth 
afforded by “intuition” with the precise and logical 
Begriff which has been developed by the persevering 
effort of mathematical speculation. This is one of the 
cases where 
‘obstinate questionings 
Of sense and outward things ” 
have justified their stubbornness by leading to discoveries 
of the highest importance. Arithmetic, and consequently 
the whole of analysis, has now been absolutely and 
finally freed from all necessity of appealing to theories or 
assumptions foreign to its own nature ; and the effect of 
this liberation is already showing itself in many different 
ways. Thus, for example, in the first book of Prof. 
Weber's work will be found (pp. to1-126) a strictly arith- 
metical proof of the fundamental proposition that every 
algebraical equation in one variable with numerical co- 
No. 1411, VOL. 55] 
ee te 
2 
efficients has at least one real or complex numerical root- 
It is true that the proof is accompanied by a geometrical 
figure, but this is merely for the sake of convenience, 
and does not affect the real nature of the demonstration. 
The same thing applies to the proof (pp. 132-6) that the 
roots of an equation are continuous functions of its 
coefficients. 
The first Book is to a great extent preparatory. It 
treats, in order, of rational functions of one or more 
variables, determinants, the existence of roots of alge- 
braic equations, symmetric functions, invariants and 
covariants, and Tschirnhausen’s transformation. The 
portions most worthy of remark are the proof of the 
existence of roots as numerical quantities, already alluded 
to, and the chapter on the Tschirnhausen transformation, 
which gives a very clear account of Hermite’s modified 
form of the process, by means of which the coefficients 
of the transformed equation are expressible as simul- 
taneous invariants of the original quantic /(z) and a certain 
auxiliary quantic T(z). The method is applied to the 
reduction of the general quintic to the normal forms 
which are associated with the names of Bring (or Jerrard) 
and Brioschi. 
Book II. deals with the roots of algebraical equations, 
and comprises six chapters. Of these the first discusses 
the reality of the roots. The solution of quadratic, cubic, 
and biquadratic equations is followed by the proof of an 
important property of the Bezoutiant, namely, that if, by 
a real linear transformation, the Bezoutiant of /(z) is 
expressed as a sum of 7 positive and v negative squares 
which cannot be reduced to a smaller number, then 
7 +yv +1 is the number of distinct roots of f(z)=0, and 
of these 7 — vy + I are real, and the rest imaginary. (It 
is understood, of course, that all the coefficients of /(z) 
are real.) This leads to a digression on Sylvester's 
Law of Inertia of quadratic forms; after which an 
application of the theorem about the Bezoutiant is made 
to the general cubic and biquadratic, and to two special 
quintic equations. 
The Law of Inertia itself is proved in Book I. (p, 
183-4) : it may perhaps be remarked that on p. 184, line 
17, the phrase, ‘‘aus diesen p» Gleichungen” is not very 
clear. What is really meant is the system of p» linear 
equations, by means of which (ex hyfothest) Z,, Z,... 
Z, are expressible in terms of Y,, Yo... Y», Z4,, Z4,.- .Z44. 
The supplementary articles on the law of inertia (pp. 
255-265) explain how the characteristic numbers 7, v for 
a given quadratic form may be deduced from the 
coefficients of the form. 
Chapter viii., which next follows, contains an account 
of Sturm’s Theorem which is very fresh and interesting, 
and includes Hermite’s determinants, which may be used 
instead of the Sturmian functions proper. After this comes 
a sketch of Kronecker’s remarkable theory of character- 
istics, and an application of it to Gauss’s first proof of the 
existence of roots of equations. 
Chapters ix. and x. treat of the separation and ap- 
proximate calculation of roots. On the whole they follow 
the traditional lines ; and we confess that we did not find 
them so interesting as the rest of the work. From the 
author's own point of view, they are, in a sense, super- 
fluous ; and, in fact, no use is made of them subsequently, 
Then from the practical point of view it is hardly satis- 
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