220 
Mr. Knight on the construction 
Prop. II. 
3. To express a number (x) by the product of a series of fractions 
converging continually towards unity. 
Let n, n't n ”, &c. be numbers much less than x\ in the 
equation x—x, change x, in the second member, into x-\- n, 
and multiply by such a factor as will restore the equality ; 
there arises x= (a? -j- n ) x If» in the second member 
of this equation, we change x into x -f* n ' , in the last factor 
, and multiply by such a fractional factor as shall again 
restore the equality, we have 
, . x x + n' x(x + n+n') 
(* + ») X X 
If here, in like manner, we change x into x-\ -n" in the last 
factor, and restore the equality as before, by annexing a new 
factor, then 
x= (#-} -n) 
x + n' (,r + ?g 1 ') (x + n + n 1 -f w") 
* x + n + n X (x + n+n”) (x + n' + n") X 
x ( x+n+n')(x + n + n ") (x+n 1 +n'') 
(x + «) (j; + «') (x + n") {x+n+n' + n") 
and the same process may be repeated as long as we think it 
necessary. Now it is plain that the last annexed factor, as 
we continue these operations, must always approach nearer 
to unity than that which was the last -before ; thus, n being 
very small compared with x, — • does not much differ from 
unity, and when x-\-n' is put for x in this fraction (n J being 
also very 'small compared with x) its value will be nearly 
the same as before : of course the annexed factor 
(x+n) (x+n) 
will differ very little from unity : and it will differ from it 
