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It has been proved, $ 27, that the cone was the third part of the cylinder, of equal base and altitude; therefore, the hemisphere will be equal to the other two thirds, which make the complement of the cone to the cylinder. If here, again, the circles forming the bases are expressed by the $ 83, of Plane Geometry, and multiplied by the same radius, as altitude, the results of the above will be

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That is, the cone is the one third part, and the hemisphere the two thirds of the cylinder, of equal base and altitude; or the ratio of the cone, hemisphere and cylinder, is 1, 2 and 3, which was to be determined.

Corol. What has been deduced here for the hemisphere, and the cone and cylinder corresponding to it, applies of course equal. ly to the whole spheres as similar figures of equal altitudes, the multiple of which the double of that supposed before ; thence the cone, sphere and cylinder, of equal altitude and upon the same great circle, are also in the ratio of 1, 2, and 3.

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