368 Analyses of Books. [May, 
Because all fractions, whether mixed, compound, or continued, 
may be reduced to simple fractions, having their numerators and 
denominators whole numbers, and the theorem has been proved 
to be universally true for such fractions; and because every 
irrational number may be either accurately or so nearly expressed 
by a fraction that the difference shall be less than any assignable 
quantity, it follows that the theorem is true for all numbers 
rational or irrational. 
To extend this theorem to imaginary exponents, we must 
observe, that as the form of an irrational exponent is not 
changed by making it imaginary, so neither is the form of any 
coefficient which is a function of this exponent ; consequently 
the theorem is likewise true for imaginary powers, and is, there- 
fore, universally true. f 
A demonstration of the binomial theorem might easily have 
been given for fractional powers, by pursuing the same route 
that I have for the demonstration of whole numbers ; namely, 
by extracting the successive roots, and observing the law which 
connects the quotients of the coefficients of the succeeding by 
those of the preceding terms. I have, however, chosen the 
present method, because it is more simple and natural, and 
because it exhibits a connective dependance between the proofs 
of whole and fractional numbers that was supposed not to exist, 
and displays the resources of the elementary branches of a 
science, which has itself, for this purpose in its full extent, often 
been thought to be not sufficiently general. 
Pe as pee BR OS eS 
ArTic.Le VI. 
ANALYSES OF Books. 
Recherches sur Uidentité des Forces Chemiques et Electriques. 
Par M. HH. C. Uirsted, Professeur a ? Université Royale de 
Copenhagne, et Membre de la Societé Royale des Sciences de la 
méme Ville, &c. Traduit de ? Allemand par M. Marcel de 
Serres, Ex-Inspecteur des Arts et Manufactures, et Professeur 
de la Faculté des Sciences @ ?Université Imperiale; de la 
Socteté Philomatique de Paris, &c. Paris, 1813. 
+ 
In the fifth volume of the Annals of Philosophy, p. 5 (Jan. 
1815), 1 gave some account of this work, mentioning at the 
same time that I had not seen the book itself, but derived my 
information from the German journals, and from an outline given 
by Von Mons in his translation of Sir H. Davy’s elementary 
work on chemistry. Some time after this notice of mine appeared, 
I received a letter from Professor CErsted informing me that the 
account of his book in the German journals was far from aceu- 
