The Works of Archimedes

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University Press, 1897 - Geometry - 326 pages
Introduction: I. Archimedes. II. Manuscripts and principal editions, order of composition, dialect, lost works. III. Relation of Archimedes to his predecessors. IV. Arithmetic in Archimedes. V. On the problems known as [neuseis] VI. Cubic equations. VII. Anticipations by Archimedes of the integral calculus. VIII. The terminology of Archimedes -- Works: On the sphere and cylinder, books I-II. Measurement of a circle. On conoids and spheroids. On spirals. On the equilibrium of planes, books I-II. The sand-reckoner. Quadrature of the parabola. On floating bodies, books I-II. Book of lemmas. The cattle-problem [including the solution of Wurm's problem by Amthor in Zeitschrift für math. u. phys. [Hist. litt. abth.] v. 25, 1880].

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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 259 - G, it follows that (A -)- B) will remain stationary in the fluid. Therefore the force which causes A by itself to sink must be equal to the upward force exerted by the fluid on B by itself. This latter is equal to the difference between (G -)- H) and G.
Page 255 - 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 257 - Proposition 6 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 246 - 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 232 - ... Aristarchus to the sphere of the fixed stars would contain a number of grains of sand less than 10,000,000 units of the eighth order of numbers [or 10M+? = 10°]. For, by hypothesis, (earth) : (' universe ') = (' universe ') : (sphere of fixed stars). And [p. 227] (diameter of
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".

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