in reply to Prof. Young. 25 



to "comparatively few cases." On the contrary, the false 

 cases very seldom occur, and they can generally be corrected 

 by a slight reference to Proposition IV. This is, in fact, the 

 very way in which I dismiss Prof. Young's query respectmg 

 the sum of the geometrical series. I refer to Proposition IV. 

 not to interpret but to correct the resulting expression for the 

 particular case in which the reasoning has failed. Had the 

 expression been a just deduction. Proposition III. would have 

 applied to it, and the sum in that case would have been any- 

 thing. How Professor Young could have misconceived me 

 I am quite at a loss to explain, as I have distinctly pointed 

 out the fault that occurs in deducing it. 



In Prof. Young's first letter, at page 298, he states that 

 "when we are operating with equations of the first degree 

 containing several unknown quantities, the symbol - is, in 

 fact, the very form which the result usually takes when the pro- 

 posed equations involve incompatible conditions." I feel as- 

 sured that Prof. Young, after a little reflection, will not ven- 

 ture again to assert the truth of this hasty and erroneous state- 

 ment. At page 520, however, he has attempted to refute my 

 observation that ^ can never be the symbol of absurdity in 

 the result of an investigation logically conducted ; but it will 

 be seen that my friend's remarks are founded on the same 

 erroneous hypothesis, that the result is justifiable in whatever 

 way it may have been deduced. When it is the result of a 

 logical process it is obvious, since the antecedent equation is 

 satisfied by any value, that all the preceding equations, and 

 consequently the original condition, must likewise be satisfied 

 by any value. The symbol ^ , whenever it is the result of a 

 strict investigation, may therefore possess any value, and can- 

 not possibly be the " symbol of absurdity," however much 

 Prof. Young may be " surprised" at the statement. 



In thus candidly replying to my friend's letter, I am dis- 

 posed to give him every credit for his own opinions. I think, 

 however, that he ought not, for his own sake, to have pro- 

 ceeded to such a length with his remarks, without having, in 

 the first place, made some attempt in support of his objec- 

 tionable premises. Should Prof. Young still entertain the same 

 opinions I shall make no further attempt to change them, 

 though I may be allowed to maintain my own. It will not be 

 necessary to enter into any further details at present, as I have 

 doubtless said quite sufficient for the mathematical readers 

 of the Journal. At least lam well satisfied with having shown, 

 Third Scries. Vol. 9. No. 51. Jtdy 183G. E 



