70 
Mr Galbraith on« Vanishing Fractions. 
So late as the year ISOS^ in his ‘‘ Principles of Analytical 
Calculation,” Mr Woodhouse, who certainly ranks a»nong the 
first of British mathematicians, though he perhaps may admit 
the accuracy of the conclusions, seems inclined to question the 
legitimacy of the reasoning by which they are obtained. “ The 
value of says he, in the 17th page of the preface, 
oc a 
“ for instance, a) was thought necessarily to equal S a, al- 
though its obvious value is [j, and it was not perceived that, by 
the very process of making the value 2 a, a certain order was 
arbitrarily instituted, and an extension of a rule made This 
opinion has been again combated by Dr Charles Hutton, in the 
last edition (1815) of his Mathematical Dictionary, art. Vanish- 
ing Fractions. A difficulty therefore still seems to attend this 
subject, which we shall endeavour to remove by the following 
artifice. 
Let a be a constant quantity, and ^ a variable, which pos- 
sesses the property of increasing from 0 till it is equal to a. 
Also let m and n be any positive whole numbers greater than 
unity, then may a?, some part of «, be denoted by -, and there- 
m TO m 
fore we have ..d — ^ 
a — X 
a 
na — a 
MI 1 m 1 „TO — 1 
x + ..(A) 
— 1 — 1 ^ ^ V / 
Now, if, for example, mr=:2, and 10, we sliall have, by 
foi-mula (A) ^ 
a 
10 
+ re)*- 
If and nz=:4, we get as before - (4 -j- 1) =— r= 
^1 Again, ifw=:2, and 71=2, we obtain I (2-fl) — 
Q ^ ^ I a. And if m — and n itself be taken, we 
See also page and the Note there from Berkeley’s Analyst. 
