the Differential and Integral Calculus. 31 



to clear up the mystery ascribed to the equations first men- 

 tioned. Similar equations to the sections parallel to the planes 

 zx and z y have been given in Higman's Syllabus, and in 

 other works : perhaps the geometric signification of the equa- 

 tions cited ought not to be overlooked in criticising them. 

 Whether the author, the title of whose book is not mentioned, 



himself clears up the apparent paradox that -~ on the left- 



dz 

 hand side of the equation does not mean the same as — — on 



CI X 



the right side," does not appear. I do not know that I have 

 his book, and therefore shall leave the author to take care of 

 himself. 



Perhaps it is impossible fully to illustrate the subject en- 

 tirely free from paradox : thus the writer of the pamphlet on 

 page 24, says, " the reader must keep in mind, that though x 

 appears in the expression dx, yet dx is entirely independent 

 of x" &c. : if d x have nothing to do with x, the quantity, or 

 whatever x denotes, must have been altogether annihilated, 

 or completely changed, in becoming dx, and in that case x 

 in the latter expression ought to have been some other symbol, 

 to prevent what Berkeley calls njallacia suppositions, or, " a 

 shifting of the hypothesis" 



The writer of the pamphlet in Art. 26, and Professor Miller, 

 (Differential Calculus, Art. 3), state the operations which d x 

 denotes when affixed to a function of x; there is, or at least 

 I fancy there is, a material disparity in their statements ; the 

 reader can if he please turn to the works and judge for him- 

 self; my object for naming the circumstance is, because one 

 of the books was written apparently to recommend the new 

 notation, and the other is the only elementary treatise that I 

 know in which it is used. 



Having cited the above reasons in favour of the d x notation, 

 I will now quote two opinions on the other side of the ques- 

 tion. Mr. W. S. B. Woolhouse, in a very valuable disquisi- 

 tion on the fundamental principles of the Differential and In- 

 tegral Calculus, published in the Appendices to the Gentle- 

 man's Diary for 1835 and 1836, says, " Before closing this 

 paper I cannot refrain from adding a remark on a new plan 

 of differential notation that has lately been introduced, and 

 to a considerable extent adopted at Cambridge. I allude to 



(1- ij ci u 



the substitution of d x y for —+- , d/y for -y-^, and others of a 



similar kind, which possess no recommendation whatever ex- 

 cept it be that of novelty : and I feel convinced that this change 

 is suited only to such persons as are satisfied with mere hocus 



