30 JOURNAL OF THE 



Therefore by the definition of differential above, du ^area 

 APQB = ydx. This is readily proved by the method of 

 limits thus: Note that, 



VAA- < A?^ < {y + A>') A-i-; 

 hence dividing- by J.r and observing that as J.i" diminishes 



An 

 indefinitely, ( y -f aj') and hence — (which is still nearer/) 



Ar 

 approach y indefinitely in value, 

 A// dii 

 . ' . lim. — = — = J/ . . . . (ii). 

 A.r dx 



Similarly we can prove, by either method, that if V 

 represents the volume generated by CDPA revolving about 

 <?X, that ^V is represented by the volume generated by 

 APQB about OX . • . ^V = '/ dx. 



Referring to eq. (ii) and fig. 2 it is seen that if for any curve u = area 

 CDPA can be expressed as a function of Ji% or m =y" (a-), then by the 

 usual notation, 



Au 



lim. =/i (x); 



Al- 

 and this equals jj/ by (11). 



On calling za an infinitesimal that tends towards zero in the same time 

 as AX, we can write, 



Ati = (/' (.r) -f w) AX, 

 since on dividing by ax and taking the limit, we are conducted to the 

 preceding equation. But by the Leibnitz method the term [wax) is thrown 

 away, on differentiating tc =y (x), when ax and au are infinitely small, 

 so that Au =/'^j-Ar = ^ax. 



Another error is made, however, by regarding the area au = APSB as 

 equal to APQB, which, combined with the preceding result, gives cor- 

 rectly, area APQB=jj/ax. As Leibnitz regarded du and dx, as identical 

 with AM and A'", when the latter were infinitely small, the}^ should replace 

 the latter in the above equations to express them b}' his notation. 



Thus truth is again evolved from error, and it can be similarly shown 

 in other cases, though a general demonstration seems difficult, if not im- 

 possible. 



The "Method of Indivisables " by which "lines were considered as 

 composed of points, surfaces as composed of lines and volumes as com- 



