26 
NATURE 
[ NoveMBER 12, 1896 
factory to fillup three pages with an account of Daniel 
Bernoulli’s method of approximation, and omit all men- 
tion of Horner's algorithm. Still these chapters are re- 
deemed from commonplace by a very elementary proof of 
Newton’s Rule (first demonstrated by Sylvester), a geo- 
metrical excursus, after Klein, and some very curious 
theorems of Laguerre’s relating to equations with no 
imaginary roots. 
Chapter xi. discusses continued fractions, arithmetical 
equivalence, and the theory of the reduction of quadratic 
irrational numbers. A quadratic irrational (a, 6, G is 
defined to mean 
Vi iyi aS /D sr b_ 2a 
2 JD -6 
where 
D= 6 + 4ac, 
and a, 4,c are ordinary integers. Thus @ is a definite 
root of the equation 
cw =at bw; 
so that the author makes two alterations in the traditional 
notation of Gauss and Dirichlet. One of these, the 
substitution of 4 for 24, needs no justification, and has, 
indeed, become almost inevitable ; the reason for the 
change of sign in cis less obvious, especially as on p. 390 
the typical quadratic equation is written 
A+B + Ca2=0 
with 
B? - 4AC =D. 
It is true that, in consequence of this additional modi- 
fication, there is a trifling gain of typographical elegance ; 
but this seems to be outweighed by other disadvantages. 
This chapter concludes with the approximate calcula- 
tion of roots by means of continued fractions, a brief dis- 
cussion of rational roots, and a very meagre treatment of 
what is really a fundamental problem, namely the resolu- 
tion of a polynomial with integral coefficients into its 
irreducible factors. It is true that a method is given 
which is theoretically sufficient, but this is quite useless in 
practice ; while, on the other hand, the purely tentative 
method illustrated by an example can only be made con- 
clusive by special artifices. Some account ought, we 
think, to have been given of Kronecker’s algorithm 
(Crelle, vol. 92), which, although tedious, is really 
practicable, and has the advantage of giving a definite 
answer after a finite number of trials which may be 
estimated beforehand. 
Chapter xii. contains the elementary theory of the roots 
of unity, primitive roots to a modulus, indices, quadratic 
residues, and the law of quadratic reciprocity : the proof 
of this last is the trigonometrical one of Eisenstein. 
We now come to Book III., on “ Algebraical Quanti- 
ties,” and here the essential and characteristic part of the 
work may be said to begin. The key-note is struck at the 
commencement of chapter xili. by the definition of a 
numerical corpus (Zah/korper). The notion of a corpus, 
which is of the most fundamental character, is due to 
Dedekind, and is as follows. Let us take a finite 
or infinite system of elements a, B, y, &c., concerning 
which nothing is assumed except that they can enter into 
rational combination according to the rules of ordinary 
algebra ; then the totality of all rational functions of a, f, 
NO. [41 1, VOL.'55| 
y, &c., except those which involve division by zero, con- 
stitutes a corpus, denoted by 
Aa, Boy... -), 
The simplest corpus is that of all rational numbers. 
This is contained in every other corpus ; for if be any 
element of the corpus, then by definition the corpus con- 
tains w @, that is, unity ; and from this all other rational 
numbers may be derived by rational operations only. 
If the elements of a corpus are all numbers, it is called 
a numerical corpus; but the elements may be _ inde- 
pendent variables, or even variables subject to algebraical 
conditions. 
If Q(a, 8, y...) is any corpus, and x any quantity not 
contained in it, the corpus Q(x, a, 8, y...) is said to be 
derived from Q(a, 2, y .. .) by the adjunction of x. 
If z is an undetermined variable, the polynomial 
Je) = a2” + asm. t+ am 
is said to be a function in 9 when all the coefficients a,, 
. . a, belong to a. 
When Q@ is given, we may, if we like, regard all the 
quantities belonging to it as rational: for this reason 
Kronecker calls a corpus a domain of rationality. 
A function in @ is reducible (in @) if it can be resolved 
into the product of two functions in®, A function which 
is irreducible in @ may be reducible in a corpus derived 
from © by adjunction. 
Let /(z) be an irreducible function in Q, of the 7th 
degree in z ; then the equation 
7 (s) =0 
is assumed to have 7 conjugate roots 2,, 22...2,. If & 
is a numerical corpus, the roots have actual numerical 
values ; but this is really immaterial so far as the general 
algebraic theory of the corpus is concerned. 
By the separate adjunction of the roots, we obtain the 
conjugate corpora Q(2,), Q(z,) ... Q(z,), each of which is 
called an algebraical corpus of the zth degree. These 
conjugate corpora are not necessarily all different ; they 
may, in fact, be all identical, and the corpus is then 
called a Galoisian or normal corpus. In this case all the 
roots 2; are expressible as rational functions of any one 
of them, and this leads to the definition of a normal (or 
Galoisian) equation. 
In the chapter we are now considering, the author 
proves the important theorem that the simultaneous ad- 
junction of several algebraic quantities is equivalent to 
the adjunction of one only, provided that it be ap- 
propriately chosen; develops the distinction between 
primitive and imprimitive corpora ; defines the Galoisian 
resolvent of an equation; and shows that the corpus 
Q(z, 2... 2,), derived from # conjugate corpora, is 
normal. This last proposition is the master-key 
to the whole Galoisian theory. After this we have a 
discussion of the substitutions of a normal corpus ; it is 
important to observe that their number is equal to its 
degree. Then comes a brief digression on the elements 
of the theory of permutation-groups ; and this leads to 
the very important conclusion that the group of p sub- 
stitutions of a normal corpus which contains 7 conjugate 
algebraical corpora is isomorphic with a certain group 
of permutations of 7 things. This may be regarded as 
a vroup of permutations of the #7 conjugate roots, and 
Qy - 
