Read The Nature of Mathematics Online

76 THE NATURE OF MATHEMATICS 

tangent at the point of a curve, and finding the velocity of 
a particle describing this curve when it gets to that point, 
are identical problems. They are expressed as finding the 
" differential quotient," or the " fluxion " at the point. It is 
now known to be very probable that the abore two methods, 
which are theoretically but not practically the same, were 
discovered independently; Newton discovered his first, and 
Leibniz published his first, in 1684. The finding of the areas 
of curves and of the shapes of the curves which moving particles 
describe under given forces showed themselves, in this calculus, 
as results of the inverse process to that of the direct process 
which serves to find tangents and the law of attraction to 
a given point from the datum of the path described by a 
particle. The direct process is called "differentiation," the 
inverse process " integration." 

Newton's fame is chiefly owing to his application of this 
method to the solution, which, in its broad outlines, he gave, 
of the problem of the motion of the bodies in the solar system, 
which includes his discovery of the law according to which all 
matter gravitates towards is attracted by other matter. 
This was given in his Principia of 1687 ; and for more than 
a century afterwards mathematicians were occupied in extend- 
ing and applying the calculus. 

Then came more modern work, more and more directed 
towards the putting of mathematical methods on a sound logical 
basis, and the separation of mathematical processes from the 
sense-perception of space with which so much in mathematics 
grew and grows up. Thus trigonometry took its place by algebra 
as a study of certain mathematical functions, and it began to 
appear that the true business of geometry is to supply beautiful 
and suggestive pictures of abstract " analytical "or " algebrai- 
cal " or even " arithmetical " as they are called processes of 
mathematics. In the next chapter, we shall be concerned with 
part of the work of logical examination and reconstruction.