210 Mr. Woolhouse's Reply io Prof. Young's 



its value \s, x ■{■ c', is in my opinion totally without meaning. 

 Any person possessing a slight acquaintance with the common 

 definitions of plain algebra needs not to be told that the ex- 

 pression X + a implies an " operation to be performed" just as 



x^ — a^ 

 much as the expression ; and that the expressions are, 



every one of them, " symbolical forms." The fact is that the 



equation, = j; + a, simply indicates a transformation 



of one symbolical form into another, by the operation of divi- 

 sion ; and I have distinctly shown that this operation is not 

 justifiable when x is equal to a, as indeed Professor Young 



tacitly admits when he speaks of the form — ^^^ — as isolated 



or detached from its interpretation. I hold that the questions, 



What is the value of — when x = a? and What is the 



X — a 



g2 q9 



value of ? are not " perfectly distinct" unless the con- 

 dition of continuity be expressly introduced. In the case x = a 

 the operation of division by zero is actually performed in the 

 above equation, and yet Professor Yoiuig pertinaciously asserts, 

 with an undignified attempt at sarcasm, that he has " never 

 seen " any work in which the zero processes occur : the objec- 

 tionable process occurs in every example that he has adduced ; 

 and I am persuaded that he would not, in candour, have insisted 

 on the problem of Clairaut, or indeed on any other example, 

 as an obstacle to my principles, had he attentively considered 

 my general reasoning at the bottom of page 24 of this volume. 

 With respect to this problem I shall refer to Spiller's very 

 neat translation of Lacroix's Algebra, as the more likely to be 

 in the hands of the English student, and just observe that the 

 operations are strict as far as the quadratic root at the top 

 of page 167, but that the succeeding operation in which the 

 numerator and denominator of the second of the two fractions 

 contained in the surd are multiplied by 1 — ?w, is unwarrant- 

 able in the case m = 1. If the surd be expanded as it is im- 

 mediately and legitimately presented by the quadratic, and the 

 value ?« = 1 substituted, the operation will lead to the proper 



results, viz. — , oo . This will sufficiently account for my 



having passed over " in silence so decided an argument" as 

 wholly inapplicable to the case. 



Professor Young refers particularly to the examples in 

 Bourdon's Algebra, see page 118, fourth edition, where 

 Bourdon deduces the symbols of his results from the general 



