**Cartesian equation: **

(*x*^{2} + *y*^{2})^{2} = *a*^{2}(*x*^{2} - *y*^{2})

**Polar equation: **

*r*^{2} = *a*^{2}cos(2*θ*)

**Click below to see one of the Associated curves.**

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In 1694 Jacob Bernoulli published an article in

The general properties of the lemniscate were discovered by Giovanni Fagnano in 1750. Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions.

Inverting the lemniscate in a circle centred at the origin and touching the lemniscate where it crosses the *x*-axis produces a rectangular hyperbola.

The bipolar equation of the lemniscate is *rr*' = *a*^{2}/2.

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JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Lemniscate.html