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THE INFINITESIMAL CALCULUS 71 

d~s 

say a, then - = a . dt : and, integrating both sides : 
at 

ds 

=fa.dt = afdt = at + b, 

CLt 

where 6 is a new constant. Integrating again, we have : 

at 2 
s=aft.dt + bfdt = + bt + c; 

J 

which is a more general form of Galileo's result. 

Thus, the infinitesimal calculus brought about a great ad- 
vance in our powers of describing nature. And this advance 
was mainly due to Leibniz's notation : Leibniz himself attri- 
buted all of his mathematical discoveries to his improvements 
in notation. Those who know something of Leibniz's work 
know how conscious he was of the suggestive and economical 
value of a good notation. And the fact that we still use and 
appreciate Leibniz's "/" and "d" even though our views as 
to the principles of the calculus are very different from those of 
Leibniz and his school, is perhaps the best testimony to the 
importance of this question of notation. This fact that Leibniz's 
notations have become permanent is the great reason why I 
have dealt with his work before the analogous and prior work 
of Newton. 



Isaac Newton (1642-1727) undoubtedly arrived at the 
principles and practice of a method equivalent to the infini- 
tesimal calculus much earlier than Leibniz, and, like Roberval, 
his conceptions were obtained from the dynamics of Galileo. 
He considered curves to be described by moving points. If 
we conceive a moving point as describing a curve, and the 
curve referred to co-ordinate axes, then the velocity of the 
moving point can be decomposed into two others parallel to 
the axes of x and y respectively ; these velocities are called 
the " fluxions " of x and y, and the velocity of the point is the 
fluxion of the arc. Reciprocally the arc is the " fluent " of 
the velocity with which it is described, From the given equa- 
tion of the curve we may seek to determine the relations 
between the fluxions and this is equivalent to Leibniz's 
problem of differentiation ; and reciprocally we may seek the 
relations between the co-ordinates when we know that between