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