CHAP, n.] SUPPLEMENT TO CHAP. VTI. 103 



Now we see at once from (4) that the smaller we consider d x to be, the 

 nearer this ratio approaches the limiting value 5x2*. We may suppose 

 d x as small as we please, and then this ratio will differ as little as we 

 please from 5x2*. This value, 5x2*, forms, then, the limit towards which 



d $/ 



the value of the ratio -3 approaches as d x diminishes, but which limit evi- 



w *C 



dently it can never actually reach or exactly equal. Because, in order that 

 this should be the case, d x must be zero. But if d x is zero, that is, if * is 

 not increased, y also is not increased; d y is, therefore, zero, and there is no 

 ratio at all. 



Now, just here comes in what we may regard as the central principle of 

 the calculus. 



If two varying quantities are always equal and always approaching certain 

 limits, then those limits must themselves be equal. 



The principle is too obvious to need demonstration. " Two quantities 

 always equal present but one value, and it seems useless to demonstrate 

 that one variable value cannot tend at the same time towards two constant 

 quantities different from one another. Let us suppose, indeed, that two 

 variables always equal have different limits, A and B ; A being, for ex- 

 ample, the greatest, and surpassing B by a determinate quantity A. 



The first variable having A for a limit will end by remaining constantly 

 comprised between two values, one greater, the other less than A, and hav- 

 ing as little difference from A as you please; let us suppose this difference, 



for instance, less than A . Likewise the second variable will end by re- 



G 



maining at a distance from B less than A. Now it is evident that, then, 



IB 



the two values could no longer be equal, which they ought to be according 

 to the data of the question. These data are then incompatible with the 

 existence of any difference whatever between the limits of the variables. 

 Then these limits are equal." * 



d iJ 

 Now let us apply this principle to equation (4). In this equation -3 is 



a variable always equal to 5 (2 x+d x). But 5 (2 x+d x), as we diminish 



d y 

 d x, approaches constantly the limit 5x2*: and as -= is always equal to 



(L X 



5 (2 x+d *), it also constantly approaches the same limit. These limits, 

 then, are equal, and the limit of -= = 5x2*. 



'' ./' 



Now, if we conceive, and such a conception is certainly possible, d * to 

 be the difference between * and its consecutive or very next value, such that 

 between these two values there is no intermediate value of d x ; then d y 

 will be the difference between two consecutive values of y ; and regarding, 



then, d x and d y in this light, -=-=- will be the limit of the ratio of the in- 



d x 



* The Philosophy of MatliemaUcs. Bledsoe. 



