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ARCHIMEDES 
first of the three propositions, proved to be to the area of 
the auxiliary circle as the minor axis to the major; equilateral 
polygons of 4 n sides are inscribed in the circle and compared 
with corresponding polygons inscribed in the ellipse, which are 
determined by the intersections with the ellipse of the double 
ordinates passing through the angular points of the polygons 
inscribed in the circle, and the method of exhaustion is then 
applied in the usual way. Props. 7, 8 show how, given an ellipse 
with centre C and a straight line CO in a plane perpendicular to 
that of the ellipse and passing through an axis of it, (1) in the 
case where OC is perpendicular to that axis, (2) in the case 
where it is not, we can find an (in general oblique) circular 
cone with vertex 0 such that the given ellipse is a section of it, 
or, in other words, how we can find the circular sections of the 
cone with vertex 0 which passes through the circumference of 
the ellipse; similarly Prop. 9 shows how to find the circular 
sections of a cylinder with GO as axis and with surface passing 
through the circumference of an ellipse with centre C, where 
CO is in the plane through an axis of the ellipse and perpen 
dicular to its plane, but is not itself perpendicular to that 
axis. Props. 11-18 give simple properties of the conoids and 
spheroids, easily derivable from the properties of the respective 
conics; they explain the nature and relation of the sections 
made by planes cutting the solids respectively in different ways 
(planes through the axis, parallel to the axis, through the centre 
or the vertex of the enveloping cone, perpendicular to the axis, 
or cutting it obliquely, respectively), with especial reference to 
the elliptical sections of each solid, the similarity of parallel 
elliptical sections, &c. Then with Prop. 19 the real business 
of the treatise begins, namely the investigation of the Volume 
of segments (right or oblique) of the two conoids and the 
spheroids respectively. 
The method is, in all cases, to circumscribe and inscribe to 
the segment solid figures made up of cylinders or ‘ frusta of 
cylinders ’, which can be made to differ as little as we please 
from one another, so that the circumscribed and inscribed 
figures are, as it were, compressed together and into coincidence 
with the segment which is intermediate between them. 
In each diagram the plane of the paper is a plane through 
the axis of the conoid or spheroid at right angles to the plane