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THE INFINITESIMAL CALCULUS 73 

extending knowledge by using the infinitesimal calculus in all 
branches of pure and applied mathematics, in England com- 
paratively little progress was made. In fact, it was not until 
the beginning of the nineteenth century that there was formed, 
at Cambridge, a Society to introduce and spread the use of 
Leibniz's notation among British mathematicians : to establish, 
as it was said, "the principles of pure d-ism in opposition to 
the dot-age of the university." 

The difficulties met and not satisfactorily solved by Newton, 
Leibniz, or their immediate successors, in the principles of the 
infinitesimal calculus, centre about the conception of a " limit" ; 
and a great part of the meditations of modern mathematicians, 
such as the Frenchman Cauchy (1789-1857), the Norwegian 
Abel (1802-1829), and the German Weierstrass (1815-1897), 
not to speak of many still living, have been devoted to the 
putting of this conception on a sound logical basis. 

We have seen that, if y = a; 2 , -^ = 2a?. What we do in forming 
dx 



dy . , - 

- is to form v L - , which is readily found to be 



, and then consider that, as Aa; approaches more and 
more, the above quotient approaches 2x. We express this by 
saying that the "limit, as h approaches 0," is 2x. We do 
not consider Aa; as being a fixed " infinitesimal " or as an 
absolute zero (which would make the above quotient become 

indeterminate -), nor need we suppose that the quotient reaches 

its limit (the state of Ace being 0). What we need to consider 
is that " Aa; " should represent a variable which can take 
values differing from by as little as we please. That is to 
say, if we choose any number, however small, there is a value 
which Aa; can take, and which differs from by less than that 
number. As before, when we speak of a " variable," we mean 
that we are considering a certain doss. When we speak of a 
" limit," we are considering a certain infinite class. Thus the 
sequence of an infinity of terms 1, f, , , Jg-, and so on, 
whose law of formation is easily seen, has the limit 0. In 
this case is such that any number greater than it is greater 
than some term of the sequence, but itself is not greater