614 
NATURE 
[Oct. 28, 1886 
their hours of reflection they know to be wrong. And 
since action takes the line of least resistance at the 
moment of temptation, there follows remorse in the hour 
of bitter remembrance. In the moment of trial, knowledge 
of right is not absent but submerged. The development 
of ethical doctrine as it passed through the hands of Plato 
and Aristotle, of Zeno and Epicurus, is treated with marked 
clearness and ability. 
In the chapter on Christianity and mediaeval ethics the 
main characteristics of Christian morality occupy a con- 
siderable space, the only writer whose doctrines are 
expounded at any considerable length being Thomas 
Aquinas. . 
The last chapter deals with modern ethics, chiefly 
English. The author, however, tells us, in his preface, 
that he has not attempted to deal with contemporary 
modes of ethical thought—with which he has been 
engaged controversially—except in a very brief and 
summary way. The motive is admirable ; but the fact is 
to be regretted. As a “manual for students” the book 
would have been more complete had contemporary ethics 
formed the subject of a concluding chapter. There are 
many who will take up this book as a summary of the 
subject as a whole from the hands of one of its masters, 
and who will be disappointed to find so meagre an account 
of modern transcendental ethics and of the moral theory 
as “sanctioned” by evolution. The writer who has 
treated the ethics of twenty centuries with such marked 
impartiality could safely have been trusted to preserve a 
due “ objectivity ” of treatment in dealing with the modes 
of ethical thought in his own time. 
All genuine students of human thought and endeavour 
will thank Prof. Sidgwick for having presented them with 
this altered and enlarged edition, in a handy form, of his 
article in the “ Encyclopedia Britannica.” 
@ ire. 
PROFESSOR CHRYSTAL’S “ALGEBRA” 
Algebra: an Elementary Text-Book for the Higher 
Classes of Secondary Schools and for Colleges. By G. 
Chrystal, M.A. Part I. (Edinburgh: Adam and 
Charles Black, 1886.) 
ss HERE are few things where the want of an en- 
lightened scientific public strikes an expert more 
than the matter of scientific text-books.” “ For our teach- 
ing of algebra, I am afraid, we can claim neither the 
sanction of antiquity nor the light of modern times. 
Whether we look at the elementary, or at what is called 
the higher teaching of this subject, the result is unsatis- 
factory. .. . In the higher teaching, which interests me 
most, I have to complain of the utter neglect of the all- 
important notion of algebraic form.” “The logic of the 
subject, which, both educationally and scientifically speak- 
ing, is the most important part of it, is wholly neglected. 
The whole training consists in example-grinding. What 
should have been merely the help to attain the end has 
become the end itself. The result is that algebra, as we 
teach it, is neither an art norascience, but an ill-digested 
farrago of rules, whose object is the solution of examin- 
ation problems.” “The problem for the writer of a 
text-book has come now, in fact, to be this—to write a 
looking through it, can mark a single -passage which the 
candidate for a minimum pass can safely omit. ... When 
our system sets such mean ends before the teacher, and 
encourages such unworthy conceptions of education, is it 
to be wondered at that the cry arises that pupils degene- 
rate beneath even the contemptible standards of our ex- 
aminations.... The cure for all this evil is simply to 
give effect to a higher ideal of education in general, and 
of scientific education in particular. Science cannot live 
among the people, and scientific education cannot be 
more than a wordy rehearsal of dead text-books, unless 
we have living contact with the working minds of living 
men.” 
Such being some of the author’s weighty utterances in 
his famous British Association address to Section A 
(NATURE, vol. xxxii. pp. 446-449), it was with much 
interest we read the announcement that he was writing 
a treatise on algebra, and it is with much pleasure we 
have perused this first instalment of 542 pages. This is 
no ordinary treatise: school text-books abound, and 
more are on the way. This bears traces everywhere of a 
master’s genius; those are but clever arrangements of 
well-known materials. 
This is an elementary volume because “it begins at 
the beginning of the subject” ; it is not written, however, 
for babes. It will have been noticed how the address 
' quoted above insisted upon the “all-important notion of 
algebraic form”: at the commencement Prof. Chrystal 
lays down generally the three fundamental laws, and 
thence proceeds deductively. This he does because this 
idea of algebraic form is “ the foundation of all the modern 
developments of algebra, and the secret of analyticat 
geometry, the most beautiful of all its applications.” The 
following abstract of the interesting preface will best indi- 
cate the writer’s aim. Outside algebra proper the reader 
is expected to be familiar with the definition of the trigo- 
nometrical functions, and to have a knowledge of their 
fundamental addition-theorem. The first object is to 
“develop algebra as a science, and thereby to in- 
crease its usefulness as an educational discipline.” 
Sources of information are indicated, and a most 
admirable feature is the introduction of numerous his- 
torical notes. With regard to some of the early chapters, 
which are specially hard reading for junior students, Prof. 
Chrystal writes that they were “written as a suggestion 
to the teacher how to connect the general laws of algebra 
with the former experience of the pupil. In writing 
this chapter I had to remember that I was en- 
gaged in writing, not a book on the philosophical 
nature of the first principles of algebra, but the first 
chapter of a book on their consequences.” 
The subject is broken up into twenty-two chapters, and, 
as the arrangement—“ the result of some ten years’ expe- 
rience as a University teacher ”—deviates somewhat frony 
ordinary usage, we give the headings :—(1) Fundamental 
laws and processes (association, commutation, and distri- 
bution, with historical note) ; (2) laws of indices, theory of 
degree ; (3) theory of quotients, first principles of theory 
of numbers; (4) distribution of products (= and I,) 
principle of substitution, homogeneity, symmetry, prin- 
ciple of indeterminate coefficients ; (5) transformation 
of the quotient of two integral functions ; (6) G.C.M. 
book so neatly trimmed and compacted that no coach, on \ and L.C.M.; (7) factorisation of integral functions ; 
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