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Apollonius Archimedes Archimedes's axes axis base bisecting Book centre of gravity chap chord circle circumference circumscribed commentary cone conic conic sections conies contained Ctesibius cube curve cylinder diam diameter Diophantus Dioptra distance divided draw drawn earth ellipse equal equation Eratosthenes Eucl Euclid Eutocius follows formula frustum Geminus geometry given ratio gives greater Greek Heron Hipparchus hyperbola inscribed figure intersection lemma length loci mathematics means meet Menelaus method method of exhaustion Metrica moon obtained ordinates Pappus parabola parallel parallelogram perpendicular plane Posidonius problem Proclus produced proof Prop propositions proved Ptolemy radius rectangle regular polygon respectively right angles says segment semicircle side similar solid solution solved sphere spherical spheroid spiral square stades straight line surface tangent Theon Theon of Alexandria theorem tion treatise triangle vertex volume weight whence
Page 230 - Euclidean geometry, and in particular that one which assumes that through a given point only one parallel can be drawn to a given straight line.
Page 332 - Since the volume of a cylinder or a prism is equal to the product of the area of the base and the altitude, we conclude that The volume of a cone or a pyramid is equal to one third the product of the area of the base and the altitude.
Page 21 - I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.
Page 312 - If two triangles have two sides of the one equal to two sides of the other, and also the angles contained by those sides equal, prove that the triangles are congruent.
Page 95 - BOOK II. Proposition 1. If a solid lighter than a fluid be at rest in it, the weight of the solid will be to that of the same volume of the fluid as the immersed portion of the solid is to the whole.
Page 402 - BC) we first find the formula for the area of a triangle in terms of its sides, K=Vs(s — a) (s — b) (s — c).
Page 3 - 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 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.
Page 40 - ... is equal to a triangle with base equal to the circumference and height equal to the radius of the circle, I apprehended that, in like manner, any sphere is equal to a cone with base equal to the surface of the sphere and height equal to the radius*.