An Introduction for
the Student . .
. .
Arguably, the Cycloid Family of
curves has
the most distinguished listed of investigators in mathematics.
Galileo
and Father Mersenne are credited with being the first to name and
discuss
its special properties (1599). They were followed by
Torricelli,
Fermat, Descartes, Roberval, Wren, Huygens, Desargues, Johann
Bernoulli,
Leibniz, Newton, Jakob Bernoulli, l'Hôpital and others. One
of the greatest legends in the history of mathematics surrounds
Pascal's
publication of solutions to various cycloid problems.
One might assert that a
fascination with the
motion of the cycloidal curves led a century of civilization's greatest
mathematicians into modern mathematics. Certainly, the birth of
the
calculus, especially the calculus of variations, flourished among these
remarkable men who were determined to understand its many special
qualities.
Because of the frequency of
disputes in the
17th century, the cycloid became known as the "Helen
of Geometers." The name is appropriately based on
Greek mythology. Helen was the most beautiful woman in the
world.
The Trojan war that followed her capture was one of the fiercest
conflicts
in ancient times.
At other times,
mathematicians
have called the cycloid an Apple of
Discord.
Problems related to rotating a
cycloidal arch
about various lines led mathematicians to problems on surfaces and
volumes
of revolution now commonly taught in introductory calculus
courses.
These investigations also created opportunities for finding different
methods
for drawing tangents.

A Brief
History of Pascal's Fascination with the Cycloid . . . .
On November 23, 1654, Blaise Pascal, best know in
mathematics for
Pascal's Triangle, had a deeply moving accident. He barely
escaped
death when runaway horses pulling his carriage, bolted off a bridge and
into a stream. Fortunately, the traces to the carriage snapped
leaving
Pascal safely on the bridge. Pascal took this as a sign that he
should
abandon worldly interests, such as mathematics, and devote his talents
to the Christian faith. There followed a number of religious
experiences
that deeply influenced his writings. In particular, his Provincial
Letters and Pensées de M. Pascal sur la Religion
brought him considerable fame. The readership has been estimated
in the millions.
However, Pascal returned to mathematics for one brief
period of months.
Though always frail of health, he found he could not sleep because of a
bad toothache. To forget about the intense pain, he made himself
focus on the cycloid. Much to his amazement, the pain
disappeared.
He took this as a sign that he should publish the solutions to the
cycloid
problems that had distracted him. He worked intensely for eight
days.
His solutions included both area and volume at various intervals in the
cycle of the revolving curve. He researched the work of others on
the cycloid and published Histoire de la Roulette,
appelé
autrement Trochoide ou Cycloide, on October 10, 1658. But he
chose to publish this letter under the pseudonym of Amos Dettonville,
an
anagram on the name Louis de Montalte whom Pascal had made famous
through
his Provincial Letters.
Historians speculate that Pascal may have wished to
avoid the criticism
that he had lapsed from grace and once again reverted to his worldly
interest
in mathematics. The modern student might pause to consider that
in
Pascal's time, mathematics was possibly viewed by society as almost an
addiction somewhat akin to that of problem gambling and certainly
trouble
for the Catholic Church. The age of professional mathematics was
only emerging.
Many university libraries will have a copy of
Pascal
ŒUVRES Complètes edited by Jacques Chevalier in
1954.
We will include a selection of these images for your pleasure.
