By Thomas L. Heath
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Additional info for The Works of Archimedes: Edited in Modern Notation with Introductory Chapters
Sample text
The next three propositions are proved without fresh recourse to x. 1. Thus the criterion of equimultiples laid down in Def. 5 of Book v. is used to prove [xu. 13] that, if a cylinder be cut by a plane parallel to its bases, the resulting cylinders are to one another as their axes. It is an easy deduction [xu. 14] that cones and cylinders which have equal bases are proportional to their heights, and [xu. 15] that in equal cones and cylinders the bases are reciprocally proportional to the heights, and, conversely, that cones or cylinders having this property are equal.
1. In a paraboloid of revolution any plane section parallel to the axis is a parabola equal to the generating parabola. 2. In a hyperboloid of revolution any plane section parallel to the axis is a hyperbola similar to the generating hyperbola. 3. In a hyperboloid of revolution a plane section through the vertex of the enveloping cone is a hyperbola which is not similar to the generating hyperbola. 4. In any spheroid a plane section parallel to the axis is an ellipse similar to the generating ellipse.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xlix (2) that any cone is one third part of the cylinder which has the same base as the cone and equal height. The other theorems in solid geometry which Archimedes quotes as having been proved by earlier geometers are * : (3) Cones of equal height are in the ratio of their bases, and conversely. (4) If a cylinder be divided by a plane parallel to the base, cylinder is to cylinder as axis to axis. (5) Cones which have the same bases as cylinders and equal height with them are to one another as' the cylinders.