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By Larranaga J. C. G.

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In the first case we describe an angle in terms of its measure. In the second case we describe a right angle in terms of the way it is formed. The descriptions are equivalent, but the Greeks used only the latter. With this description the Greeks described a straight (180°) angle as the sum of two right angles. FACT 2: When we cut two parallel lines with a third, transverse line, the interior angles on opposite sides of the transverse line are equal. ) Notice that no measurement is involved. We can Early Greek Geometry 11 a circle and one of its diameters makes it clear that the diameter bisects the circle.

He restored it. In addition to his work on three-dimensional forms, Archimedes studied curves. He wrote an entire treatise entitled On Spirals. Here is how he described the spiral: If a straight line drawn in a plane revolve at a uniform rate about one extremity which remains fixed and return to the position from which it started, and if, at the same time as the line revolves, a point move at a uniform rate along the straight line 32 GEOMETRY beginning from the extremity which remains fixed, the point will describe a spiral in the plane.

It is pure geometrical reasoning, the type of reasoning at which the Greeks excelled. indd 11 3/7/11 11:40 AM 12 GEOMETRY the ­importance of deductive reasoning. Deductive reasoning, the process of reasoning from general principles to specific instances, is the characteristic that makes mathematics special. Mathematics is a deductive discipline. All mathematicians today work by beginning with known principles and then deriving new facts as logical consequences of those principles, but Thales was the first to apply this method rigorously.

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3-Manifolds which are unions of three solid tori by Larranaga J. C. G.

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