Greek geometers were the model for mathematicians when Descartes revolutionized geometry by introducing coordinates, making it possible to approach geometry through algebraic methods. Cartesian geometry was born.
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To locate points in a plane, Cartesian coordinates are generally used. Descartes is credited with inventing this method, so much so that it bears his name. Wrongly?
Almost two thousand years of geometry! That is how long it took, with Descartes, to develop the first tools for studying those mathematical objects we all think we know: curves. How far we have come since the tentative explorations of ancient Greek scholars!
Starting with two distinct points in the plane and performing only finitely many ruler-and-compass constructions, can we obtain any point in the plane? If not, which points can we obtain in this way?
From yesterday's "touching line" to today's tangent, from "vanishing quantities" to asymptotes, the story has been a long mathematical epic. Here are a few glorious episodes from this geometric quest to "approximate curves with straight lines."
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