## Marin Mersenne Leads an International Effort to Understand Cycloids

## Marin Mersenne Leads an International Effort to Understand Cycloids

# Marin Mersenne Leads an International Effort to Understand Cycloids

*Overview*

French mathematician, theologian, and educator Marin Mersenne (1588-1648) made numerous contributions to mathematics, including the prompting of a greater understanding of cycloids, which in turn directly affected the development of the pendulum clock. By opposing irrational or superstitious interpretations of phenomena, including numbers, Mersenne helped elevate the level of mathematics and mathematical research. Indeed, it was his insistence on empirical evidence, as well as his curiosity about cycloids, that affected Christiaan Huygens's (1629-1695) work, which in turn resulted in Huygens patenting a pendulum clock. Without a doubt, though, Mersenne's great contribution to his time was his indefatigable devotion to collecting, sharing, and distributing scientific and mathematical information among a wide community of correspondents and scholars. In this way, he was a sort of oneman repository or clearinghouse, very much aware of the vital importance of feedback and commentary to scientific progress.

*Background*

Marin Mersenne was the child of a laborer who escaped the poverty of his birth by education—which may have been subsidized by the Jesuits. He studied at the College of Mans, and from 1604 to 1609 studied at Le Fleche, a Jesuit institution, followed by two years of theological studies at the Sorbonne in Paris. In 1611 Mersenne joined the Minim Friars, an order that took its name from its commitment to reducing worldly involvement and focusing on prayer and contemplation, study and reflection. In 1619 Mersenne took up residence in a Minim monastery in Paris, where he lived for the rest of his life.

Minimizing worldly concerns may have been a major concern of Mersenne's religious order, but it hardly kept him from contact with the larger world. His correspondence with the leading scientific and philosophical thinkers of his time quickly became legendary. Mersenne's letters spanned much of the known world, and consisted not only of his own insights, but also of exhortations to the recipients of his letters that they pursue certain fields of inquiry or investigation. Furthermore, Mersenne invariably introduced his own correspondents to others with whom they might pursue independent correspondence. Thus he established a network of communications that reinforced, enriched, and extended the body of knowledge.

Mersenne also devoted much effort to theories of the nature of prime numbers, particularly large prime numbers. Although some of his theories proved to be flawed, today a number of computer programs are devoted to searching for what are still called Mersenne Primes.

Among the many other specific problems that Mersenne pursued was the nature of a particular type of curve known as a cycloid. Cycloids are achieved by tracing the path of a fixed point on the edge of a cylinder rolling smoothly along a straight lane. Cycloids provided a means for calculating the "curve of least time"—the curve by which a fixed point takes the least amount of time to make the transit between two points, A and B. The origins of research into cycloids remain in some dispute. While various sources for its initiation are invoked, there is general agreement that French mathematician Charles Bouvelle (c. 1471-c. 1553) was in 1501 the first to accurately describe the curve's nature, but he did not produce the equations necessary for deriving the specific properties of cycloid curves.

Because those properties promised great practical as well as theoretical and mathematical benefits—particularly in determining the mathematical mechanics of moving objects, as well as in calculating the nature of curves for arches, and the curves described by a pendulum—the search for those properties, including the area of the space contained within the curve, grew heated and acrimonious, with arguments over the accuracy of solutions, and even the authorship of solutions generating anger and accusations.

As a result of these conflicts, and the near-universal attraction of mathematicians to the cycloid question, the cycloid itself came to be known as either the "apple of discord" or the "Helen of Geometers"—a source of irresistible appeal, but over whose appeal battles would be waged.

Galileo (1564-1642) in the late 1500s made an empirical approach to solving the cycloid. As a result of experiments he determined that the area of the cycloid is approximately three times that of the circle that generates the curve, a property he believed accurately would be of use to the building of arches in bridges and other structures. Galileo is also believed to have given the cycloid its name.

Although Galileo's experimental conclusions were useful, the search for a purely mathematical proof continued. It was Marin Mersenne who put into motion the research that resulted in that proof. Along with every other mathematician of the time, Mersenne felt the attraction of the cycloid. His network of correspondents was crucial here, for he used the network to make the cycloidal challenge more widely known. One of the recipients of Mersenne's letters was French mathematician Gilles Personne de Roberval (1602-1675), who used purely mathematical techniques in 1634 to prove that the area of the cycloidal arch is three times the area of the circle. Unfortunately, Roberval neglected to publish his proof, with the result that his achievement was considered plagiarism by those who proved the area later but published sooner.

Other mathematicians applied themselves to determining other properties of the cycloid. Blaise Pascal (1623-1662) had enjoyed a brilliant career as a mathematician, making large contributions to pure mathematics, but had abandoned mathematics in favor of devotion to religious philosophy. Beset by a toothache in 1658, however, Pascal claimed to have experienced a mathematical vision, the result of which was the development of mathematical functions that allowed the determination of the areas of sections of cycloids, the volumes of areas generated by the rolling sphere, as well as the centers of gravity for these sections. According to legend, Pascal derived all of these functions within eight days of his toothache.

What makes the achievement of these mathematicians the more remarkable in regard to cycloidal properties is that these properties and functions were determined without benefit of the most appropriate mathematical tool for tackling problems of motion over space: the calculus. The calculus would have simplified much of the investigation into cycloids, but it had not been invented at the time Galileo, Mersenne, Roberval, Pascal, and others were performing their cycloidal research.

The cycloid's mechanical properties were most determinedly attacked by Christiaan Huygens, who perceived that a pendulum described a cycloidal curve as it moved rhythmically back and forth. Combining his understanding of cycloids with the emergent knowledge of gravity, Huygens found that a body drawn by gravity along a cycloidal arch requires the same amount of time to reach the bottom of the arch.

Huygens's application of his finding to the pendulum, using devices that insured that the pendulum swung in a cycloid, enabled the attachment of a pendulum to the workings of a clock. Huygens built his first pendulum clock in 1657. This represented the largest advance in the clockmaker's art since the development of the waterclock in Greece of the second century b.c.

Huygens would spend much of the rest of his life refining the relationship of the pendulum to the cycloid, moving ever closer to the accurate tracking of the passage of time. As important—and perhaps more so—as the availability of accurate, pendulum-kept time on land, was the applicability of the pendulum clock to keeping accurate time at sea or in transit over large distances, allowing for a far more accurate and efficient means of determining a traveler's precise location.

Above all, it was Huygens's application of his understanding of the cycloid to specific questions of motion, velocity, and the effects of gravity on bodies in motion that extended the realm of the cycloid from pure mathematics and applied mathematics to the world of physics, and the revolution in the understanding of physics that would take place in the century after Huygens's death.

*Impact*

The widespread—if often angry and accusatory—nature of investigations into the properties of cycloids is perhaps the best early example of the growth of *scientific community*. Thanks in no small part to Mersenne's energy as a correspondent and instigator, the full and diverse power and ability of a group of mathematicians and scientists was brought to bear on a single problem. While competition certainly figured in the search for cycloidal solutions, there was also an underlying current of collaboration, however unofficial or even unwanted. This sort of collaboration and review by fellow scientists would become far more common in the century ahead, and would ultimately become the very essence of the worldwide scientific community. Of no small consequence to all of the various approaches to the cycloid was the fact that the knowledge gained, when coupled with Huygens's mechanical insights, established accurate timekeeping as an achievable goal, radically altering the nature of a world whose rhythms had previously been governed by only the loosest sense of time.

**KEITH FERRELL**

*Further Reading*

Dunham, William. *Journey through Genius: The Great Theorems of Mathematics*. New York: John Wiley & Sons, 1990.

Katz, Victor. *A History of Mathematics: An Introduction*. 2nd ed. New York: Addison-Wesley, 1998.

Swetz, Frank J., ed. *From Five Fingers to Infinity: A Journey through the History of Mathematics*. Chicago and Lasalle: Open Court, 1994.