A History of Greek Mathematics, Volume 2

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Clarendon Press, 1921 - Mathematics - 1032 pages

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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 228 - If a straight line is perpendicular to each of two straight lines, at their point of intersection, it is perpendicular to the plane in which the two lines lie.
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*.
Page 206 - Hence it is plain that triangles on the same or equal bases, and between the same parallels, are equal, seeing (by cor.

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