The Works of Archimedes

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University Press, 1897 - Geometry - 326 pages

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Page 121 - Hence all prisms are to one another in the ratio compounded of the ratios of their bases, and of their altitudes. For every prism is equal to a parallelopiped of the same altitude with it, and of an equal base (2.
Page xlviii - Y3 inscribed regular figures of sixteen sides, &r. the preceding process gives the proof that circles are to one another as the squares on their diameters.
Page 253 - Proposition 3. Of solids those which, size for size, are of equal weight with a fluid will, if let down into the fluid, be immersed so that they do not project above the surface but do not sink lower.
Page 255 - If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.
Page 244 - First then I will set out the very first theorem which became known to me by means of mechanics, namely that Any segment of a section of a right-angled cone (ie a parabola) is four-thirds of the triangle which has the same base and equal height, and after this I will give each of the other theorems investigated by the same method. Then, at the end of the book, I will give the geometrical [proofs of the propositions]...
Page clxv - A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.
Page 62 - BA'B' = the cone H'BB'. COR. The segment BAB' is to a cone with the same base and equal height in the ratio of OA' -+ A'M to A'M. Proposition 3. (Problem.) To cut a given sphere by a plane so that the surfaces of the segments may have to one another a given ratio. Suppose the problem solved. Let AA...
Page vii - as it were of set purpose to have covered up the traces of his investigation, as if he had grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results".
Page 219 - Samos brought out a book consisting of some hypotheses, in which the premisses lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the...
Page v - ... gave birth to the calculus of the infinite conceived and brought to perfection successively by Kepler, Cavalieri, Fermat, Leibniz, and Newton.

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