# lemniscate of Bernoulli

The lemniscate of Bernoulli, also known simply as a lemniscate, is a curve 'shaped like a figure 8, or a
knot, or the bow of a ribbon' in the words of Jacob Bernoulli in an article published in 1694. Bernoulli named the curve "lemniscate"
after the Greek *lemniskus* for a pendant ribbon (the type fastened
to a victor's garland). It has the Cartesian equation

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

At the time he wrote his article, Bernoulli wasn't aware that the curve he was describing was a special case of a Cassinian oval, which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by Giovanni Fagnano (1715–1797) in 1750; Leonhard Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions.

There is a relationship between the lemniscate and the rectangular hyperbola. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.