Bibliography. 219 
Its chief excellence consists in the clearness and precision of the steps 
by which he advances. He combines clearness with sufficient length, so 
that his steps are abundantly evident, at the same time that there is diffi- 
culty enough to render them interesting to the more intelligent student. 
He does not pretend to originality, but his style and method are pe- 
culiar to himself, and his demonstrations are either altogether new, or 
happy modifications of those of other writers. With regard to the 
binomial theorem he says in his preface, ‘‘ a rigid demonstration of it, 
at once simple and elementary, has been much sought for by mathe- 
maticians. The one here given depends upon a principle which is the 
foundation of the differential calculus, and is in fact little else than a 
translation, of the very simple demonstration afforded by that science, 
into the elementary language of algebra.” We have examined the 
demonstration with some care. The principle is similar to that used 
by Bourdon in his later editions, (we have not seen his earlier,) and by 
other algebraists. Prof. Davies, in his translation of Bourdon, gives 
Euler’s method, which though ingenious seems by no means so clear 
or so elegant, and certainly more abstruse and less direct. Euler 
first deduces the binomial formula in the case when the exponent is a 
positive integer, and then proves that the same formula exposes the 
expansion of binomials affected with negative and fractional exponents. 
~ Professor Chauvenet has made the demonstration at once elegant and 
direct, by first proving the fundamental principle somewhat in the form 
of a lemma. This principle is found in the peculiar nature of the 
vr —y” 
quotient of “ , whatever be the nature of the exponent m. He 
‘first beautifully shows in a few lines that this quotient is always exact, 
and the series limited, when m is positive and integral; and then in 
an equally striking manner demonstrates that when xy, although 
: 0 : AP AY i 
the quotient then becomes 0° (the expression for an infinitesimal quan- 
tity,) that still it is for all values, if the exponent is equal to mx”~1, the 
well known form of the differential co-efficient of #”. When these 
principles are established, the demonstration becomes at once direct and 
clear, and as elementary as the student can desire. He has inter- 
spersed through the work numerous and appropriate examples; into 
this chapter in particular he has introduced a beautiful collection, 
many of them original, others from French works, which illustrate fully 
the application of the binomial theorem. 
The chapter on the “nature and use of logarithms” is extremely 
happy, well calculated to interest the student and to place the subject 
before him in an entirely new light. We would call the attention of 
teachers of mathematics particularly to it. ‘The mode of deriving the 
