of the Binomial Theorem. 307 
x-\ -a 
x-\~a 
x*-J -ax 
ax-\-a* 
x'-\-v.ax-{-a*z=.x-{-a\ 
x-\ -a 
x 3 -j-2ax B -j- a* x 
ax'-\-2CL x-\ -a 3 
x 3 -\-^ax^-\-^a i x-\-a 3 =x-{-a\ 3 
x-j-a 
x'-^sax 3 -^^ x*-\ -a 3 x 
ax 3 -\-§a* ^*+3^ 3 x-\- a* 
a: 4 4-4j ax 3 -\-6a* ,z l -f 4# 5 x-±a*—x-\-at, 
&c. 
In the same manner the value of x — a\ n may be obtained ; 
and its only difference from the value of x + a\ n will consist 
in having the negative sign prefixed to such terms as have an 
odd power of a. And as the powers of any other quantity, 
either simple or compound, may be obtained gradually by 
multiplying the last found power by the root, in order to find 
the next higher power, it is manifest that the principles of 
multiplication are the most simple and evident, to which we 
can resort, for the demonstration of the binomial theorem,. 
These principles, therefore, will be used throughout the whole 
of the following investigations on the subject, and by them 
every case of the theorem will be established. 
