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XX. Some concluding Itemarks on the Theory of Vanishing 

 Fractions. By J. 11. Young, Esq., Professor of Mathe- 

 matics, Belfast College.* 

 11/1 R. WOOLHOUSE's letter in the preceding Number of 

 -^ -*- this; Magazine (p. 18) renders it befitting that I should 

 offer a few additional obseivations ; although 1 had hoped that 

 the many examples 1 had given of the entire fallacy of the 

 principles in dispute, would have rendered any further recur- 

 rence to the subject on my part quite unnecessary. As far 

 indeed as that letter can be regarded as a " reply" to my ar- 

 guments, I see nothing, I must confess, to justify me in pro- 

 longing the discussion. It clearly, however, establishes the 

 fact that Mr. Woolhouse has all along laboured under a mis- 

 take as to what the processes of mathematicians in the doc- 

 trine of vanishing fractions really are, for he asserts these 

 processes to involve the absurdity of " multiplying and divid- 

 ing by zero", and fincies that, by defending "the usual me- 

 thods," 1 attempt to justify such operations. 



Mr. Woolhouse says that his " especial object in writing 

 the Essay was to show the impropriety of multij)lying and 

 dividing by zero," and he urges my attention to the logical 

 inaccuracy of such processes. Now it appears to me that 

 Mr. Woolhouse would have been as profitably employed in 

 writing to show the impropriety of making two and two equal 

 to five ; for the fallacy which he denounces is never com- 

 mitted ; and if, as indeed his letter shows, he has imagined 

 this fallacy in the reasonings of Lacroix and Bourdon, to 

 which I have already referred him, he has been the subject of 

 a rather singular delusion. With this erroneous impression 

 on his mind, however, it is not diificult to conceive how he 

 has been led to make so many remarkable statements in re- 

 ference to analytical processes and analytical results : he has 

 been combating a doctrine which exists only in his own ima- 

 gination ; and all the '* palpable inconsistencies" which he has 

 discovered belong, not to the true theory which mathemati- 

 cians have laid down, but to the fanciful system which Mr. 

 Woolhouse has set up, unwittingly, in its stead. 



Mr. Woolhouse has evidently adopted the old and exploded 

 notion — indulged, I believe, by no other person living — that 

 the elimination of the vanishing fixctor, — as, for instance, x—a, 



, x^—ar . ...... 



\n such a case as , is a mere artificial contrivance, to 



x — a 



bring out, in the hypothesis of .r = o, one only of the innumer- 

 * Communicated by the Author. 



