180 THE ELEMENTS DOWN TO PLATO’S TIME 
near to the base], what must we think of the surfaces forming 
the sections 1 Are they equal or unequal ? For, if they are 
unequal, they will make the cone irregular as having many 
indentations, like steps, and unevennesses; but, if they are 
equal, the sections will be equal, and the cone will appear to 
have the property of the cylinder and to be made up of equal, 
not unequal, circles, which is very absurd.’ 
The phrase ‘ made up of equal . . . circles ’ shows that 
Democritus already had the idea of a solid being the sum of 
an infinite number of parallel planes, or indefinitely thin 
laminae, indefinitely near together: a most important an 
ticipation of the same thought which led to such fruitful 
results in Archimedes. This idea may be at the root of the 
argument by which Democritus satisfied himself of the truth 
of the two propositions attributed to him by Archimedes, 
namely that a cone is one third part of the cylinder, and 
a pyramid one third of the prism, which has the same base 
and equal height. For it seems probable that Democritus 
would notice that, if two pyramids having the same height 
and equal triangular bases are respectively cut by planes 
parallel to the base and dividing the heights in the same 
ratio, the corresponding sections of the two pyramids are 
equal, whence he would infer that the pyramids are equal as 
being the sum of the same infinite number of equal plane 
sections or indefinitely thin laminae. (This would be a par 
ticular anticipation of Cavalieri’s proposition that the areal or 
solid content of two figures is equal if two sections of them 
taken at the same height, whatever the height may be, always 
give equal straight lines or equal surfaces respectively.) And 
Democritus would of course see that the three pyramids into 
which a prism on the same base and of equal height with the 
original pyramid is divided (as in Eucl. XII. 7) satisfy this 
test of equality, so that the pyramid would be one third part' 
of the prism. The extension to a pyramid with a polygonal 
base would be easy. And Democritus may have stated the 
proposition for the cone (of course without an absolute proof) 
as a natural inference from the result of increasing indefinitely 
the number of sides in a regular polygon forming the base of 
a pyramid. 
Tannery notes the interesting fact that the order in the list