296 Prof. Young’s Observations upon 
ing statements and positions, in reference to this important in- 
quiry, of a very peculiar kind, and which appear to me to be not - 
only opposed to well-established truths, but calculated, under 
the protection of his name, to retard—what I am sure that 
gentleman is most anxious to promote—the spread of pure 
scientific truth. It is from the same anxiety for truth that 
I here venture, very briefly, to examine the more prominent 
of the positions adverted to, and this I do with the same sen- 
timents of respect and regard which I have long entertained for 
his talents and friendship. I cannot, perhaps, in strictness, 
say that my own defence requires that I should reply to the 
animadversions which Mr. Woolhouse has made upon the 
views which, in conimon with so many others, [ entertain on 
the subject of vanishing fractions, although I think I am pri- 
vileged to show that my friend has not supplied the place of 
these views, which he has very unsparingly censured, by others 
that will bear the test of careful examination. With many of 
the observations in Mr. Woolhouse’s Essay I am disposed en- 
tirely to agree, as being in strict accordance with the usual 
notions of this doctrine; but the new theory, for which the 
Essay is chiefly remarkable, seems to me to have been much 
too hastily framed; it is embodied in the very general propo- 
sitions which follow : 
I. “ If, in any investigation of a geometrical problem, the 
unknown quantity is expressed by a fraction which, in a parti- 
cular case, becomes a vanishing one, the problem in that case 
will resolve itself into a porism, and the value of the fraction, 
or unknown quantity, will then admit of arbitrary assumption; 
and a similar result will follow in all such cases, whatever be 
the nature of the investigation.” 
II. ‘* Whenever, in an analytical investigation, the resulting 
expression for a quantity resolves itself into a vanishing frac- 
tion, we may observe, as a general rule, that either one of the 
original conditions of the inquiry becomes destroyed, or that 
two or more of them become dependent, 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.” (Gentleman’s Diary, 1836, pp. 25, 26.) 
That these propositions are fallacious, Mr. Woolhouse would, 
I think, have soon seen, if he had attempted their demonstra- 
tion, instead of contenting himself with testing their accuracy 
by two particular examples, which, as far as they go, seem 
indeed, at first sight, to corroborate their truth, although upon 
examination such will not be found to be the case; and if we 
were to interpret every vanishing fraction agreeably to this 
theory, we should frequently be involved in the most pal- 
pable errors. Indeed it is remarkable that my friend did not 
