The works of Archimedes

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University Press, 1897 - Geometry - 326 pages
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Page 222 - ... of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface. Now it is easy to see that this is impossible; for, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean...
Page xlix - Eudoxus on solids which are held to be most irrefragably established, namely, that any pyramid is one third part of the prism which has the same base with the pyramid and equal height, and that any cone is one third part of the cylinder which has the same base with the cone and equal height. For, though these properties also were naturally inherent in the figures all along, yet they were in fact unknown to all the many able geometers who lived before Eudoxus, and had not been observed by any one.
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 259 - H-)+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 [Prop. 6]. Hence A is depressed by a force equal to H, ie its weight in the fluid is H, or the difference between (G + B) and G.
Page clxxii - If a straight line of which one extremity remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line revolves, a point move at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.
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 221 - 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 257 - 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.

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