### Overview

In the period between the death of Galileo (1564-1642) and the rise to fame of Isaac Newton (1642-1727), Christiaan Huygens (1629-1695) stood alone as the world's greatest scientific intellect. His treatment of impact, centripetal force, and the pendulum helped clarify the ideas of mass, weight, momentum, and force, thus making it possible for dynamics and astronomy to advance beyond mere geometrical description, while his wave theory of light helped initiate modern physical optics. Beyond such specifics, Huygens exercised a profound influence on the progress of science through his use of quantitative methods.

### Background

The scientific achievements of Huygens were realized under the aegis of a methodology that successfully combined empiricism and rationalism. The empiricist tradition, which found its canonical formulation in Francis Bacon's (1561-1626) *Novum Organum*, was primarily concerned with building knowledge of the world through direct observation and experimentation. The rationalist tradition, whose foremost exponent was René Descartes (1596-1650), eschewed perceptual knowledge as fallible, preferring instead to focus on the certainty attainable through *a priori* reasoning.

Descartes sought to place knowledge of the world on a secure foundation by organizing it into an axiomatic structure similar to Euclid's geometry. A few self-evident truths were proposed as axioms and used to deduce the body of existing empirical knowledge. This rational reconstruction from first principles was meant to undermine the metaphysical speculations of scholasticism and replace the architectonic of Aristotelian physics with a completely new Cartesian physics and cosmology.

A central feature of Cartesian scientific methodology was its reliance on mechanistic explanation. Accordingly, Descartes maintained that the analysis of any natural phenomenon must proceed by considering the motions and direct-contact interactions of matter's various particles. Descartes believed he had metaphysically demonstrated the truth of this principle and the other structural elements of his system, but he conceded that the details of his explanations might be modified by future mathematical and experimental advances.

Huygens accepted the need for mechanistic explanations but was dissatisfied with Descartes' limited application of mathematics to physical phenomena. Though the Cartesian program embodied the ideal of mathematization in its general structure, Descartes produced little in the way of detailed mathematical analysis of physical phenomena. This was disconcerting to Huygens, who felt the subtleties of physical phenomena could only be captured by combining mathematical laws with mechanistic explanation. He also challenged the Cartesian devaluation of experimental knowledge. Though aware of the inadequacies of naive empiricism, Huygens realized, as Galileo had before him, that the mathematical analysis of natural phenomena depends critically on the careful definition and quantification of relevant physical conditions. In this regard, experimental work plays an essential regulative role in the process of scientific discovery.

Critical of the inadequacies inherent to empiricism and rationalism, Huygens combined the best features of each to craft his own research methodology. His strong penchant for mathematization and mechanistic explanation, tempered by a deep understanding of the importance of experimental data, produced stunning scientific successes.

### Impact

One of the first problems Huygens addressed was Cartesian impact theory. Descartes believed that once the clockwork mechanism of the universe was set in motion the universe would run indefinitely, requiring no divine intervention; to suppose otherwise implied God was imperfect. Consequently, Descartes maintained that the amount of motion initially imparted to the different parts of the universe must be conserved. He defined a body's motion as the product of its mass and speed. Though motion can be transferred between bodies through collisions, he claimed the total quantity of motion must remain constant. Unfortunately for Descartes, his conservation law disagreed with experiments.

In 1652 Huygens applied himself to the problem and showed that Descartes's principle holds only if speed is taken as a directed quantity--velocity. Huygens collected his results in 1656 in *De motu corporum ex percussione*. When the Royal Society began focusing on the same problem in 1666, John Wallis (1616-1703) and Christopher Wren (1632-1723) were asked to examine the problem anew, and Huygens was solicited for a report of his discovery. The results of all three scientists, obtained independently and published together in the *Philosophical Transactions* (1669), established the law of conservation of momentum.

Additionally, Huygens showed that for elastic collisions the product of the mass times the square of the velocity--called *vis viva* (living force) during the seventeenth century--is conserved. The debate over the nature of *vis viva* was one of the main threads leading to the development of the concept of energy and the conservation principle thereof.

Huygens's mathematical analysis of physical problems had immediate application in observational astronomy. Aided by his theoretical researches in optics, Huygens and his brother Constantijn developed lens-polishing techniques that reduced spherical aberration. They incorporated these lenses and other improvements into their telescopes. With their first instrument, Huygens discovered Saturn's satellite Titan and fixed the planet's period of revolution at 16 days (1655). The following year he provided a correct description of Saturn's ring. He later made the first observation of Martian surface markings and determined that planet's rotational period (1659). Huygens also invented a two-lens eyepiece and an improved micrometer.

Huygens rendered astronomy a greater service in 1657 with his invention of the pendulum clock. There had been many attempts to produce accurate pendulum clocks, and Huygens's first device was a combination of existing elements. This explains why his priority over the invention was challenged by other scientists. Nevertheless, Huygens's design was original in its application of a freely suspended pendulum whose motion was transmitted to the clockwork by means of a fork and handle. He also introduced an endless chain that allowed the clock to be wound without disturbing its progress. More importantly, he realized the pendulum is not quite tautochronous--that its period depends on the amplitude of swing. Huygens solved this problem by devising, through trial and error, fulcrum attachments that altered the arc of the pendulum bob so the period was independent of amplitude. He described his device in *Horologium* (1658).

The appearance of Huygens's clock inaugurated the era of accurate time keeping and revolutionized the art of exact astronomical measurements. Many towns in Holland quickly built tower clocks, and Jean Piccard (1620-1682) later instituted a program of regular horological measurements at the Paris Observatory.

Dissatisfied with the qualitative nature of this work, Huygens undertook a systematic mathematical analysis of the simple pendulum (1659). He quickly derived the relationship between pendulum length and period of oscillation for small amplitudes. In considering the general case, he discovered that the period of a pendulum will not be completely tautochronous unless the arc of swing is a cycloid. Huygens developed the theory of evolutes to mathematically demonstrate the correspondence between his empirically constructed fulcrum plates and those necessary to force the pendulum bob to describe a cycloidal path. By 1669 his study of the center of oscillation had yielded a general rule for determining the length of a simple pendulum equivalent to a compound pendulum. All of these results were published in *Horologium Oscillatorium* (1673), which Florian Cajori (1859-1930) ranked as the greatest work of science next to Newton's *Principia*.

Huygens's mechanistic tendencies are most evident in his studies of gravity and light. His 1659 gravitational researches presupposed and built upon Descartes's vortex theory--gravity is caused by particles of subtle matter swirling with great speed around Earth. Huygens maintained that vortex particles have a tendency (conatus) to move away from Earth's center. In realizing their conatus, vortex particles exert a force on ordinary particles of matter through direct contact, which brings about in the latter a conatus to move toward Earth's center. Thus, the centrifugal force of vortex particles produces a centripetal force in ordinary matter. Fleeing vortex particles are continually replaced, thus maintaining a constant gravitational force.

Next, Huygens established the law of centrifugal force for uniform circular motion as well as the similarity of the centrifugal and the gravitational conatus. He also distinguished between *quantitas materiae* and weight, the former being proportional to the space occupied by ordinary matter, while the latter was treated as a gravitational effect proportional to *quantitas materiae*. This is likely the earliest insight into the distinction between mass and weight. Though Huygens rejected Newton's theory of universal gravitation because it required action-at-a-distance, his own mechanistic account failed to explain satisfactorily how subtle vortical-matter transferred centripetal conatus to ordinary matter.

Huygens's application of mechanistic principles to optical phenomena culminated in his wave theory of light, published in 1690 under the title *Traité de la lumière*. He conceived of light as a disturbance propagated by mechanical means at a finite speed through a subtle medium of closely packed elastic particles. According to Huygens's principle, a vibrating particle transfers its motion to those touching it in the direction of motion. Each particle so disturbed becomes the source of a hemispherical wave-front. Where many such fronts overlap there is light.

Considerable research on the wave nature of light had been done before Huygens, but the significance of his work lies in its systematic treatment that allowed the theory to be fruitfully applied and developed. He mathematically demonstrated the rectilinear propagation of light, deduced the laws of reflection and refraction, and accounted for double refraction in the mineral known as Iceland Spar. However, he was not able to explain polarization.

Newton's corpuscular theory of light dominated eighteenth-century optical thinking, but it was eclipsed by Huygens's wave theory in the early nineteenth century. Though the two views were later synthesized in the quantum theory of light during the early years of the twentieth century, Huygens's principle remains the basis of modern physical optics.

Huygens' work fell into relative oblivion shortly after his death. Nevertheless, his achievements remain an enduring testament to the explanatory power of quantitative methods in the analysis of physical phenomena.