|An Introduction for
the Student . .
Arguably, the Cycloid Family of
the most distinguished listed of investigators in mathematics.
and Father Mersenne are credited with being the first to name and
its special properties (1599). They were followed by
Fermat, Descartes, Roberval, Wren, Huygens, Desargues, Johann
Leibniz, Newton, Jakob Bernoulli, l'Hôpital and others. One
of the greatest legends in the history of mathematics surrounds
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
calculus, especially the calculus of variations, flourished among these
remarkable men who were determined to understand its many special
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
The Trojan war that followed her capture was one of the fiercest
in ancient times.
At other times,
have called the cycloid an Apple of
Problems related to rotating a
about various lines led mathematicians to problems on surfaces and
of revolution now commonly taught in introductory calculus
These investigations also created opportunities for finding different
for drawing tangents.
History of Pascal's Fascination with the Cycloid . . . .
On November 23, 1654, Blaise Pascal, best know in
Pascal's Triangle, had a deeply moving accident. He barely
death when runaway horses pulling his carriage, bolted off a bridge and
into a stream. Fortunately, the traces to the carriage snapped
Pascal safely on the bridge. Pascal took this as a sign that he
abandon worldly interests, such as mathematics, and devote his talents
to the Christian faith. There followed a number of religious
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
He took this as a sign that he should publish the solutions to the
problems that had distracted him. He worked intensely for eight
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,
autrement Trochoide ou Cycloide, on October 10, 1658. But he
chose to publish this letter under the pseudonym of Amos Dettonville,
anagram on the name Louis de Montalte whom Pascal had made famous
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
in mathematics. The modern student might pause to consider that
Pascal's time, mathematics was possibly viewed by society as almost an
addiction somewhat akin to that of problem gambling and certainly
for the Catholic Church. The age of professional mathematics was
Many university libraries will have a copy of
ŒUVRES Complètes edited by Jacques Chevalier in
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