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52 THE NATURE OF MATHEMATICS 

with the Grecian spirit. The method of discovery seemed 
much more important than correctness of demonstration. About 
the same time as the invention of analytical geometry by 
Descartes came the invention of a method for finding the 
areas of surfaces, the positions of the centres of gravity of 
variously shaped surfaces, and so on. In a book published 
in 1635, and in certain later works, Bonaventura Cavalieri 
(1598-1647) gave his "method of indivisibles," in which the 
cruder ideas of his predecessors, notably of Kepler (1571-1630) 
were developed. According to Cavalieri, a line is made up of 
an infinite number of points, each without magnitude, a surface 
of an infinite number of lines, each without breadth, and a 
volume of an infinite number of surfaces, each without thick- 
ness. The use of this idea may be illustrated by a simple 
example. Suppose it is required to find the area of a right- 
angled triangle. Let the base be made up of n points (or 
indivisibles), and similarly let the side not perpendicular to the 
base be made of na points, then the ordinates at the successive 
points of the base will contain a, 2a . . ., na points. There- 
fore the number of points in the area is a + 2a + . . . -f na ; 
the sum of which is ^(n 2 a+na). Since n is very large, we 
may neglect %na, for it is inconsiderable compared with |n 2 a. 
Hence the area is composed of a number |(no)n of points, 
and thus the area is measured in square units by multiplying 
half the linear measure of the altitude by that of the base. 
The conclusion, we know from other facts, is exactly true. 

Cavalieri found by this method many areas and volumes and 
the centres of gravity of many curvilinear figures. It is to 
be noticed that both Cavalieri and his successors held quite 
clearly that such a supposition that lines were composed of 
points was literally absurd, but could be used as a basis for a 
direct and concise method of abbreviation which replaced with 
advantage the indirect, tedious, and rigorous methods of the 
ancient Greeks. The logical difficulties in the principles of 
this and allied methods were strongly felt and commented on 
by philosophers sometimes with intelligence ; felt and boldly 
overcome by mathematicians in their strong and not unreason- 
able faith ; and only satisfactorily solved by mathematicians 
not the philosophers in comparatively modern times. 

The method of indivisibles whose use will be shown in