F989 5. 
LXXI. ‘On the Theory of Vanishing Fractions, in reply to 
the Observations of Professor Young. By W.S. B. Woot- 
HOUSE.* 
[|X my short but comprehensive essay on the principles of 
* the differential and integral calculus, printed in the Ap- 
pendix to the Gentleman’s Diary for the years 1835-36, my 
expressed object was to remove, as far as was practicable, the 
perplexing difficulties usually experienced by those students 
who very naturally desire to bring the subject under the gui- 
dance and dominion of their reasoning faculties. The theory 
of vanishing fractions is well known to be the chief source of 
these difficulties ; and it so happens, unfortunately for begin- 
ners, that writers have hitherto paid little or no attention to 
the strict interpretation that ought to be given, in the general 
sense, to a fraction when the values of its numerator and de- 
nominator have both absolutely disappeared or become equal 
to zero. This part of the subject has been diligently exa- 
mined in the second part of my essay, where it is shown that 
a fraction in such a state may consistently possess any value 
whatever, if it be not limited by a special condition, but that 
one particular value only will fulfill the law of continuity 
assumed by the successive values immediately before or after 
the disappearances take place. It appears, however, that my 
explanations, there given, are insufficient to satisfy the scruples 
of Prof. Young, who has, in opposition to the principle I have 
adopted, entered into a very general statement of his own views 
at page 295 of the last Number of the Philosophical Magazine. 
The opinions of one so deservedly eminent as my esteemed 
friend are entitled to high consideration, and 1 am duly sen- 
sible of the respectful and condescending manner in which his 
observations are expressed. In a mathematical discussion of 
this kind, however, it would have been more desirable had 
Prof. Young attached less weight to his supposed evidence of 
authority, and applied himself more closely to the demonstra- 
tion of his statements, nearly all of which are at direct variance 
with my judgement, and therefore, to me, far from being satis- 
factory. I here propose a brief and explicit examination of 
the most important points that Professor Young has advanced, 
and, I hope, with the same earnest anxiety for the spread of 
scientific truth that he has been pleased to ascribe to me. 
To any person unacquainted with the inquiry, Professor 
Young’s assertions would seem to imply that my views were of a 
strange and revolutionary description; that they were, in re- 
* Communicated by the Author, 
Third Series. Vol. 8. No. 48. May 1836. 2 S 
