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PROPOSITIONS BOOK XII

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Proposition 1. Similar polygons inscribed in circles are to one another as the squares on their diameters.

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Proposition 2. Circles are to one another as the squares on their diameters.

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Proposition 3. Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

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Proposition 4. If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two equal prisms, then, as the base of the one pyramid is to the base of the other pyramid, so will all the prisms in the one pyramid be to all the prisms, being equal in multitude, in the other pyramid.

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Proposition 5. Pyramids of the same height with triangular bases are to one another as their bases.

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Proposition 6. Pyramids of the same height with polygonal bases are to one another as their bases.

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Proposition 7. Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases.

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Proposition 8. Similar pyramids with triangular bases are in triplicate ratio of their corresponding sides.

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Proposition 9. In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights.

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Proposition 10. Any cone is a third part of the cylinder with the same base and equal height.

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Proposition 11. Cones and cylinders of the same height are to one another as their bases.

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Proposition 12. Similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases.

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Proposition 13. If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis.

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Proposition 14. Cones and cylinders which are on equal bases are to one another as their heights.

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Proposition 15. In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal.

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Proposition 16. Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.

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Proposition 17. Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.

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Proposition 18. Spheres are to one another in the triplicate ratio of their respective diameters.

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 Copyright Applet © 1996/1997 (Juny, 1997) D.E.Joyce Clark University © Drets d´ús cedits 2002/2003 JDL The thirteen books of Euclid's Elements translated from the text of Heiberg with introduction and commentary. Copyright © Thomas Little Heath, 1908