A History of Greek Mathematics, Volume 2Clarendon Press, 1921 - Mathematics |
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Common terms and phrases
Apollonius Archimedes Archimedes's axes axis base bisected Book centre of gravity chap chord circle circumference circumscribed commentary cone conic contained curve cylinder diameter Diophantus Dioptra draw drawn earth ellipse equal equation equivalent Eratosthenes Eucl Euclid Eutocius follows formula frustum Geminus geometry given in position given ratio gives greater Greek Heron Hipparchus hyperbola inscribed figure intersection lemmas length loci means meet method method of exhaustion Metrica moon obtained ordinates Pappus Pappus's parabola parallel parallelogram perpendicular plane Posidonius problem Proclus produced proof Prop propositions proved Ptolemy radius rectangle regular polygon respectively right angles says sector segment semicircle sides similar solid solution sphere spherical spheroid spiral square stades straight line surface tangent Theon Theon of Alexandria theorem treatise triangle vertex weight whence
Popular passages
Page 47 - Or, to take a case yet stronger, when it is affirmed, that " the area of a circle is equal to that of a triangle having the circumference for its base, and the radius for its altitude...
Page 17 - He performed in fact what is equivalent to integration in finding the area of a parabolic segment, and of a spiral, the surface and volume of a sphere and a segment of a sphere, and the volumes of any segments of the solids of revolution of the second degree.
Page 227 - 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 37 - ... 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 92 - 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 399 - Heron's proof of the formula for the area of a triangle in terms of its sides given on pp.
Page 15 - ... upon a ship of burden with three masts from the king's arsenal which had only been drawn up with great labour by many men, and loading her with many passengers and a full freight, himself the while sitting far off, with no great effort but only holding the end of a compound pulley (iroXvcnraaTos) quietly in his hand and pulling at it, he drew the ship along smoothly and safely as if she were moving through the sea.
Page xii - 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 227 - Playfair's Axiom, though it is actually stated in Proclus on Eucl. I. 31). The proof therefore, apparently ingenious as it is, breaks down. Indeed the method is unsound from the beginning, since (as Saccheri pointed out), before even the definition of parallels by Geminus can be used, it has to be proved that ' the geometrical locus of points equidistant from a straight line is a straight line ', and this cannot be proved without a postulate.
Page 317 - The surface of a sphere is equal to four times the area of a circle...