The essential cycloid
equations:
Many famous mathematicians
have investigated the cycloid.

Def:
The cycloid is the locus of a point on the circumference of a circle
where
a circle of radius a rolls
along a fixed straight line.

With
national rivalries and individual competition among the most
distinguished
of mathematicians, the cycloid has been called the
"Helen
of Geometers"

the most beautiful curve in the world.
No
topic in mathematics has a more outstanding list of
investigators.
Indeed, those who have published on the cycloid and its related curves
constitute a virtual "Who's Who" of mathematics.
Galileo
(1599) apparently named the curve and attempted to find its area by
weighing
various pieces of metal slices representing the rolling disc. His
student Torricelli, as well as Fermat, Roberval, and Descartes all
published
articles on finding its exact area. Roberval and Sir Christopher
Wren, the great British architect, succeeded in calculating the length
of the arc. In 1658 Pascal offered a prize for the solution to a
number of problems of "la Roulette." Wallis entered the
competition,
but apparently Pascal never awarded the prize.
The
mechanically minded were also fascinated. Gear teeth were
proposed
by Desargues (1630s) for a cycloid as it rolled along its fixed
straight
line. The first pendulum clock, invented by Huygens, contained a
device for making the pendulum "isochronous"  equal in length of time
 by using the cycloidal arc and the evolute of the cycloid as a guide.
Knowing
the brachistochrone was related to the cycloid, James (Johann)
Bernoulli
(1698) challenged others to investigate its properties. Leibniz,
Newton, Jakob Bernoulli and L'Hospital all accepted this challenge by
publishing
solutions.
It
is fun to guess how Galileo, Torricelli, Fermat, Roberval, Descartes,
Wren,
Pascal, Wallis, Desargues, Huygens, two Bernoullis, Leibniz, Newton and
L'Hospital would have reacted if they had been able to see how easily
contemporary
students can display and animate "their" cycloid.
