394 Mr. Woolhouse on the Theory of Vanishing Fractions, 
ality, adverse to the results of our ablest modern analysts, and 
directly opposed to well-established truths ;—indeed it would 
almost appear that Professor Young himself had contracted that 
notion. No idea, however, could be imagined more contrary 
to the fact. The new line of theory, adopted and pursued in 
my essay, leads to precisely the conclusions subscribed to by 
all modern analytical writers, and varies only in the substitu- 
tion of strict reasoning in place of the illogical and mysterious 
mode of deduction that has all along rendered this most im- 
portant branch of mathematics a popular paradox. Profes- 
sor Young has quoted two of my most general principles, 
which, with one or two more extracts, will convey a pretty 
correct idea of the particular view I have taken of the subject. 
As these extracts will very much facilitate the present discus- 
sion I shall here annex them. 
I. As a principle, we have no right to reduce a fraction by dividing its 
numerator and denominator by absolute nothing, as the process removes 
from the fractions the indeterminate character which they previously 
possessed, and which they ought to retain. (Gentleman’s Diary, 
Appendix, page 26.) 
IT. If, in the investigation of a geometrical problem, the unknown quan- 
tity is expressed by a fraction which in a particular case becomes a 
vanishing one, the problem in that case will resolve itself into a po- 
rism, and the value of the fraction, or unkuown quantity, will then 
admit of arbitrary assumption; and a similar result will follow in all 
such cases, whatever be the nature of the investigation. (Page 25.) 
IIf. Whenever, in an analytical investigation, the resulting expression 
for a quantity resolves itself into a vanishing fraction, we may observe, 
as a general rule, that either one of the original conditions of the in- 
quiry becomes destroyed, or that two or more of them become depen- 
dent, and, consequently, whichever way it be, that there is at least one 
condition less to fulfill, and that the vanishing fraction is not, restricted 
to any determinate value. (Pages 26, 27.) : 
IV. When a fraction, which in a particular case becomes a vanishing 
one, expresses the value of a quantity which we previously know, from 
the nature of the subject, does not become discontinuous in that case, 
or generally when such a fraction enters in any equation, the other 
terms of which are not discontinuous, the fraction is, under such cir- 
cumstances, necessarily limited to continuous values, and consequently, 
when the terms vanish, it must take the particular value, (described 
in the essay,) or the ordinary result deduced either by the method of 
limits or the usual process of differentiation. (Page 29.) 
The paragraphs IT. and III., which embody the main -prin- 
ciple, are those extracted by Professor Young, who labours 
under a misapprehension if he supposes that I contented my- 
self with testing their accuracy by two particular examples. 
Has Professor Young read the remark on page 26 that im~ 
mediately follows my examples? Speaking of the examples, 
I there add, that “* these are not adduced as curious instances, 
but merely as examples of what always takes place in such 
