24 Mr.Woolhouse o?i the Theory of Vanishing Fractions^ 



represents the root of another condition that has been impro- 

 perly derived from it. 



At page 516, Prof. Young states that I have only discussed 

 the converse of the Proposition III. in my former letter. This, 

 however, is not the case, for in that letter I distinctly go into 

 the demonstration of each of the four propositions extracted 

 from my essay. Indeed, immediately after, at page 518, Pro- 

 fessor Young himself admits that I have done so ! ! In allusion 

 to my having rejected as illogical the processes in which mul- 

 tiplications or divisions by zero occur, he there states that " if 

 only results obtained under such restrictions as these are ad- 

 mitted to come under the second and third principles, then the 

 generality of those principles is, of course, at once given up, 

 and my friend and I are thus far agreed." Now these re- 

 strictions are no more than the necessary exclusion of results 

 obtained by imperfect and fallacious reasoning, and cannot 

 lairly be considered as limitations to the generality of the pro- 

 positions. It is now distinctly avowed by Prof. Young that 

 the generality of the propositions objected to is established 

 for all cases in which these restrictions are attended to, or in 

 which the fallacious reasoning does not enter. Proceeding 

 on this admission, it necessarily follows that in every case in 

 which the final result fails to conform with the propositions 

 the fallacious reasoning must have been introduced some- 

 where in the process of solution. In every possible case, there- 

 fore, of nonconformity with the Proposition II. or III. we ob- 

 serve that the fact itself is a sure indication that a multiplica- 

 tion or division by zero has occurred in the investigation, and 

 therefore that the case must be rejected at once as not offering 

 a legitimate result. It would be utterly useless, therefore, to 

 enter into Prof. Young's observations at page 517, on the ex- 

 pression for the radius of curvature*, since neither this nor 

 any other particular example of nonconformity can have the 

 least weight unless my friend can show the operations of mul- 

 tiplying and dividing by zero to be strictly logical. 



Prof. Young is quite out in supposing that the restrictions 

 to correct reasoning will limit the application of the principles 



• I would, however, here remark that the conditions P = 0, Q = 0, 



will not necessarily cause a value of d r to be zero, as Prof. Young alleges. 



. P 

 Such a doctrine would imply that any vanishing fraction -^ would neees- 



sarily attain its greatest or least value when its numerator and denominator 

 both vanish, wliich is not the case. The conditions P =: 0, Q = will 

 not, therefore, determine the points of higher contact referred to by Pro- 

 fessor Young. In his example of the parabola the determination of the 

 true result is purely accidental. 



