Christiaan Huygens’s The Pendulum Clock explains how his accurate pendulum clock was first built in 1656, the mathematics behind its accuracy, the important properties of pendula, centrifugal force, and the acceleration of gravity.
When Christiaan Huygens was only seventeen years old, he developed a proof showing that the distance a freely falling body covered increased with the square of the time it has fallen. His proud father passed this on to his friend, Marin Mersenne,
Huygens worked on these problems over the years, and he reported his results in his most important book, Horologium oscillatorium sive de mortu pendulorum ad horologia aptato demonstrationes geometricae (1673; The Pendulum Clock
During these investigations, Huygens also related the speed with which a suspended object swings to the tension in the cord holding it and to the centrifugal force. He arrived at the correct result that the centrifugal force is proportional to the square of the velocity and inversely proportional to the radius of the circular path.
One of the great technological problems of Huygens’s day was how to determine the position of a ship exactly when it was far out at sea. Good star maps already existed, so that in principle, a navigator could measure how far above the horizon a few bright stars were, and then he could calculate where on Earth one had to be to have that view of the sky. Of course, the stars wheel across the sky each night, so that the navigator also needed to know at what time the measurements were made. Unfortunately, mechanical clocks were accurate only to within about 15 minutes per day, which could produce navigational errors of 250 miles that were clearly unacceptable.
Mechanical clocks driven by a spring tended to run fast or slow depending upon how tightly the spring was wound. Huygens was a keen follower of Galileo
Huygens continued to analyze and refine his clocks, finally publishing his results in The Pendulum Clock in 1673. He had discovered that the period of a pendulum depended upon the square root of the pendulum’s length; that is, if a pendulum was twice as long as another pendulum, its period was the square root of two (1.41) times as long.
A simple pendulum consists of a light string or rod extending downward from a pivot and having a weight, called a bob, on the lower end. Huygens found that a lens-shaped bob had less air resistance and made the clock more accurate. If the mass of the string or rod cannot be ignored compared with the mass of the bob, and especially if mass is attached to the pendulum above the bob, the pendulum is no longer “simple.” Such a pendulum, whose mass is not concentrated in a small bob, is called a “compound” or a “physical” pendulum. Huygens showed that a compound pendulum acted like a simple pendulum with its mass concentrated at a point called the “center of oscillation.” This allowed Huygens to predict the effect of adding small masses above the bob to adjust the pendulum’s period. Analyzing the physical pendulum also required Huygens to develop the concept of “rotational moment of inertia,” the effect a mass distribution has on the ease with which it can be made to rotate about an axis.
Huygens discovered that the period of a pendulum is constant only if the amplitude of its swing is small, a condition that could not be maintained on a rocking ship at sea. He used clever mathematical analysis to show that the pendulum would be isochronous; that is, its period would remain constant regardless of the swing’s amplitude if the path of the bob were a cycloid. A cycloid is the shape traced out by a point fixed on the rim of a wheel as the wheel rolls along the ground. A cycloid can be pictured as a wire that is first bent into the shape of a semicircle, and then the ends are pulled a bit farther apart, making the curve flatter.
Huygens made the bob follow a cycloidal path by suspending the pendulum between a pair of guide plates shaped like a portion of a cycloid and by using a ribbon for the upper part of the pendulum that was between the guide plates. In action, near the end of the pendulum’s swing, the ribbon swung up against a guide plate and matched its contour. This shortened the pendulum’s length and made the period nearly independent of the amplitude of the swing.
He made his ships’ clocks in pairs so that if one stopped or needed repair, the other would keep on running. In practice, he hung them side by side from a wooden beam, and he was astounded to find that regardless of how they were started, after about thirty minutes the pendula were exactly 180 degrees out of phase. (When one pendulum was at the extreme right end of its swing, the other was at the extreme left end of its swing.) He correctly concluded that otherwise-imperceptible vibrations were traveling along the support beam from one clock to the other, a condition known as “weak coupling between the pendula.” Huygens believed that this effect would help keep his clocks accurate, and, in fact, they were accurate to within about 10 seconds per day.
Christiaan Huygens’s expression for centrifugal force, when combined with Johannes Kepler’s
Probably more than any other three scientists, Galileo, Huygens, and Newton diverted science into the channel in which it now flows. All three were adept with their hands; they ground lenses, built models, and made the equipment they needed for experiments. They all were excellent observers and meticulous experimenters. Their work represents an increase in both the use and sophistication of mathematics. Galileo established mathematics as essential to understanding nature, Huygens used more subtle and abstract mathematics to arrive at his results, and Newton went beyond the boundaries of geometry, algebra, and trigonometry to invent calculus. Without Huygens, Newton might not have achieved so much.
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Andriesse, Cornelis D. Titan kan niet slapen: Een biografie van Christiaan Huygens. Translated by Sally Miedema as Titan: A Biography of Christiaan Huygens. Utrecht, the Netherlands: University of Utrecht, 2003. A fine, recent biography. Huygens discovered Saturn’s moon Titan and was a “titan” among the scientists of his day. -
Bennett, Matthew, et al. “Huygens’s Clocks.” Proceedings of the Royal Society of London 458 (2002): 563-579. An excellent description of clock pairs built by Huygens for navigation at sea, and of his discovery of their natural synchronization. Only the well-prepared should brave the mathematical proofs of section 4. -
Huygens, Christiaan. Christiaan Huygens’ “The Pendulum Clock: Or, Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks.” Translated with notes by Richard J. Blackwell, and introduced by H. J. M. Bos. Ames: Iowa State University Press, 1986. The first English translation of Huygens’s major work on the pendulum clock. -
Klarreich, Erica G. “Huygens’s Clocks Revisited.” American Scientist 90 (2002): 322-323. A popular-level summary of the Matthew Bennett et al. article. -
Struik, Dirk J. The Land of Stevin and Huygens: A Sketch of Science and Technology in the Dutch Republic During the Golden Century. Boston: Kluwer, 1981. A short, illustrated work that centers on Huygens as the major claim to fame of the Netherlands for the seventeenth century scientific revolution. -
Yoder, Joella G. Unrolling Time: Huygens and the Mathematization of Nature. New York: Cambridge University Press, 2004. A 252-page account of the interrelationship between mathematics and physics in the work of the Dutch mathematician, physicist, and astronomer. Excellent at putting concepts in their historical context.
Pierre de Fermat; Galileo; Francesco Maria Grimaldi; Robert Hooke; Christiaan Huygens; Johannes Kepler; Marin Mersenne; Sir Isaac Newton; Blaise Pascal.