Quadratrix of Hippias

Cartesian equation:
y = x cot(πx/2a)
Polar equation:
r = 2/(πsin(θ))


Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point


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The quadratrix was discovered by Hippias of Elis in 430 BC. It may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts.

Later it was studied by Dinostratus in 350 BC who used the curve to square the circle.

Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. Plato describes him as a vain man being both arrogant and boastful. He had a wide but superficial knowledge. His only contribution to mathematics seems to be the quadratrix.

Other Web sites:

Xah Lee

M D Huberty


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JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Quadratrix.html