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History of Mathematics

  Blaise Pascal and the Cycloid

"Apple of Discord"

The graphics in this deposit were created using Maple software.

The Cycloid

To see continuous animation and extensive web "deposits" on the cycloid family of curves:

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.

Burton, David M.,  The History of Mathematics, McGraw Hill, 5th ed., 2003, pp. 422-425

Eves, Howard,  An Introduction to the History of Mathematics, Saunders College Publishing, 6th ed., 1990,  pp. 331-332.

Shikin, Eugene V.,  Handbook and Atlas of Curves, CRC Press, 1995.

Yates, R. C.,  Curves and their Properties, NCTM, 1952.  Also in A Handbook on Curves and their Properties, various publishers including the NCTM.

Weisstein, Eric. W.,  CRC Concise Encyclopedia of MATHEMATICS, Chapman & Hall/ CRC, 2nd ed., 2003.

For Mathematica® code that will create the cycloid:
     Gray, A.,  MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®,   2nd. ed., CRC Press, 1998.