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A Cone is the Third part of a Cylinder of the same Base, and the same height.
If a Cone, and a Cylinder have the Circle A for Base, and the same height; the Cylinder shall be triple to the Cone. For if the Cylinder had a greater Ratio to the Cone, than the Triple; a quan∣tity B less than the Cylinder, would have the same Ratio to the Cone, as
Three to One; and (by the preceding Lemma) there might be inscribed in the Cylinder, a Polygonal Prism greater than the quantity B. Let us suppose that that quantity is that which hath for Base the Polygon CDEFGH. Make also on the same Base, a Pyramid inscri∣bed in the Cone.
Demonstration. The Cylinder, the Cone, the Prism, and the Pyramid, are all of the same height; thence the Prism is triple to the Pyramid (by the 7th.) Now the quantity B is also the triple of the Cone; there is then therefore the same Ratio of the Prism to the Pyramid, as of the quantity B to the Cone; and (by the 14th. of the 5th.) seeing the Prism is greater than the quantity B, the Py∣ramid should be greater than the Cone in which it is inscribed, which cannot be.
But if it were said that the Cone hath a greater Ratio to the Cylinder, than one to three; there might be taken the same method to Demonstrate the con∣trary.