of Mr. Cayley and Mr. Jerrard. 461 



tone of his remarks we are led to infer that he is opposing a 

 conclusion of mine with respect to a and a 5 . But what does he 

 say of [e\) ? Not one word. The very existence of that equation 

 is ignored by him throughout. I can only marvel at what I have 

 not inaptly, I think, termed the curious irrelevancy of his objection. 



In his paper for October he does indeed hazard the assertion 

 that "notwithstanding the equation (e\) or anything else what- 

 ever," Lagrange's theory is applicuble to the case of u = a, t 

 v = a b ; but a moment's reflection on what has been demon- 

 strated in art. 2 will convince us of the character of Mr. Cayley's 

 assertion. 



4. In recklessness of assertion, however, Mr. Cayley is far out- 

 done by Mr. Cockle. What, for instance, shall I say of the con- 

 cluding statement of the latter mathematician in his separate 

 paper for October ? — " In employing Lagrange's theory we may 

 confine our attention to (E,), and need not concern ourselves 

 about (e^) in any way." What assertions are there involved in 

 his alleged proof of this strange proposition * ? It is amazing 

 to see how completely the human intellect may be overmastered 

 by prejudice. 



Lesson* 



In rescuing with such pains and difficulty from the class of im- 

 possible problems the noted problem of equations, a lesson (which 

 I wish not to put away from me) presses itself upon my mind. 



If in a question in mathematics which comes wholly within 

 the province of our reason, we are, in fixing the limits of possi- 

 bility, thus liable to fall into an error, glaring when pointed out, 

 and yet so wide-spread as almost to overbear opposition, what 

 infant's weakness is it, in " The mysterious scheme of things in 

 the midst of which we find ourselves f/ 3 that stretches beyond 

 us on every side in boundless immensity, to take as sure standards 

 of impossibility the things which are impossible with men. 



The lesson is a humbling one ; but it is not less instructive on 

 that account. 



November 1862. 



* Figure to yourself a geometrician following an analogous method to 

 that pursued by Mr. Cockle, and disputing the validity of the proof of the 

 sixth proposition of the first hook of Euclid's ' Elements ' on such grounds 

 as these : — 



Denoting by Q a triangle, the angles at the base of which are equal, 

 then, if the sides be not equal, we can cut off towards the base of the 

 triangle a second triangle denoted by e, which shall be equal to the first or 

 given one ; but we must not draw any inference from a comparison of (* 

 and t : as we may confine our attention to (5, and need not concern ourselves 

 about t in any way. This will perhaps enable you to realize the curious 

 nature of the statement mentioned in the text. 



t See Butler's ' Analogy/ part 1, conclusion. 



