414 SCIENCE. 
An Elementary Course in the Integral Calculus. 
By DANIEL ALEXANDER MurRAy, Ph.D., 
Instructor in Mathematics in Cornell Univer- 
sity. New York, Cincinnati, Chicago, The 
American Book Company. 1898. Pp.xiv + 
288. Price. 
The late Judge Cooley, it is said, made it a 
point to advise his students never to buy a law 
book not containing a suitable index. If Pro- 
fessor Lambert’s book were provided with this 
common convenience and with a table of an- 
swers and if the pages were less crowded and 
the margins not so narrow, the size of the vol- 
ume, which contains fourteen chapters includ- 
ing three dealing with differential equations, 
would more nearly agree with its scope. Even 
then, however, the book would belong, where 
the author doubtless intended it should belong, 
to the class of text-books which in order to dis- 
tinguish them from their more comprehensive 
and cumbrous rivals, are sometimes described 
as thin. It is hardly to be imagined that, 
among those extra-scientific features of a book 
that may properly be considered in determining 
its acceptability as a text-book for classes, mere 
thinness could count for much. Certainly a 
slight difference of length, breadth, thickness or 
weight could not be decisive. Perhaps the most 
competent teachers are apt to prefer the thin 
text-book as less likely to dishearter and over- 
whelm the beginnner by multiplicity and as 
leaving more room for personal view-point and 
individuality ; but then the least competent, 
too, are, for obvious reasons, prone to the like 
preference. In this case the thin book will 
prove friendly to sciolism rather than to knowl- 
edge, as the student will hardly escape the im- 
pression that the science is as thin as the book. 
The English is in general clear, precise and 
correct. The style is, however, uniformly dry, 
the reader being soberly conducted through the 
‘enchanted realm of open mystery’ with 
scarcely a change of mood or variation of pulse. 
Books for boys, however logical and scientific, 
one could wish might be more vital and vitaliz- 
ing, more human, more cheerful and sympa. 
thetic, addressed not so exclusively to the 
analytic and formularizing powers, but to the 
appreciation also, to the faculties of estrange- 
ment and curiosity, of wonder and admiration, 
[N. 8. Von. X. No. 247. 
looking not less towards knowledge but more 
towards culture. But even if we may not 
rightfully expect inspiration, we may, at all 
events, demand direction, orientation, judicious 
accentuation. The author does occasionally 
remark that some notion is fundamental, but in 
general the student is left to his own resources 
for discriminating the more from the less im- 
portant matters. Cardinal theorems, at least, 
might have received the common emphasis of 
italics. 
The first hundred pages are concerned with 
algebraic functions, which, by the way, again 
receive the old definition. Transcendental 
functions follow. Integration, in which con- 
siderable use is made of trigonometric substitu- 
tions, is throughout treated simultaneously 
with differentiation. By this arrangement, it 
became possible to introduce at an early stage 
a goodly variety of physical, geometric and en- 
gineering problems which serve to illustrate 
the practical utility of the calculus. In addi- 
tion, by way of encouraging practice, numerous 
exercises, invariably called problems, have been 
inserted. 
The author desires ‘by a logical presenta- 
tion of principles to inspire confidence in 
the methods of infinitesimal analysis.’ It is 
really not at all necessary. College atmosphere 
is saturated with belief in the validity and 
power of this subtile analysis. The faith is 
acquired by a kind of ‘cerebral suction.’ The 
average student has too much of this ‘ confi- 
dence’ even before he begins the study, and as 
a rule too much also at the end. A genuine 
intellectual conviction, thougo it may not fol- 
low doubt, certainly can not precede it ; and the 
rigorist’s first object would seem to be not to 
inspire, nor to preserve, but rather to mitigate 
the student’s unearned confidence. To beget a 
wholesome skepticism is an indispensable pre- 
liminary, but this is as little undertaken in this 
book as in the majority of its competitors. 
It should by no means be inferred that the 
book is devoid of modern elements. The no- 
tions of absolute and uniform convergence, for 
example, are introduced, and the conditions 
for term-by-term integration and differentiation 
of power-series are considered ; but the work is 
not preéminently ‘logical.’ The infinitesimal 
