An elusive geometric concept

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!

Written by

Daniel Lignon

PublishedApril 15, 2020UpdatedMay 16, 20261 min read

What is a curve?

Ask the average person this question, and the answer will probably be that it is "a line that turns or bends". And if asked "is a straight line a curve?", the unsurprising answer will be something like "no, because it doesn't turn".

We find much the same kind of definition in Le Robert, the historical dictionary of the French language: *"The word 'curve' describes something whose shape or direction contains no straight or planar part, in general usage and, more specifically, in geometry. It has been used as a noun in its modern sense since the late 17^th century."*

The Larousse French-language dictionary (online), meanwhile, tells us: "A curved line or shape", while the adjective "curved" is defined as "bending without containing any straight-line segment". So it is clear: in everyday language, backed up by dictionary definitions, a curve is a line that turns and contains no straight part. In particular, a straight line cannot be a curve.

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