30 
ARCHIMEDES 
the whole segment of the parabola acting at H balances the 
triangle CFA placed where it is. 
But the centre of gravity of the triangle CFA is at W, 
where GW — 2 WK [and the whole triangle may be taken as 
acting at W\ 
Therefore (segment ABC) : AGFA = WK : KH 
1:3, 
so that 
(segment ABC) = CFA 
= 4AABG. 
Q. E. D. 
It will be observed that Archimedes takes the segment and 
the triangle to be made up of parallel lines indefinitely close 
together. In reality they are made up of indefinitely narrow 
strips, but the width (dx, as we might say) being the same 
for the elements of the triangle and segment respectively, 
divides out. And of course the weight of each element in 
both is proportional to the area. Archimedes also, without 
mentioning moments, in effect assumes that the sum of the 
moments of each particle of a figure, acting where it is, is 
equal to the moment of the whole figure applied as one mass 
at its centre of gravity. 
We will now take the case of any segment of a spheroid 
of revolution, because that will cover several propositions of 
Archimedes as particular cases. 
The ellipse with axes AA', BB' is a section made by the 
plane of the paper in a spheroid with axis A A '. It is required 
to find the volume of any right segment ADC of the spheroid 
in terms of the right cone with the same base and height. 
Let DC be the diameter of the circular base of the segment. 
Join AB, AB', and produce them to meet the tangent at A' to 
the ellipse in K, K', and DC produced in E, F. 
Conceive a cylinder described with axis A A' and base the 
circle on KK' as diameter, and cones described with i(} as 
axis and bases the circles on EF, DC as diameters. 
Let N be any point on AG, and let MOPQNQ'P'O'M' be 
drawn through N parallel to BB' or DC as shown in the 
figure. * 
Produce A'A to H so that HA — A A'. 
Now 
It is now 
By the pro 
therefore 
whence 
Add NQ 2 t 
Therefore, 
But MiY 2 , 
circles with A