Huygens (incorrectly Huyghens), Christian, a Dutch natural philosopher, born at the Hague, April 14, 1629, died there, July 8, 1695. He was the second son of Constantine Huygens, secretary and counsellor of the stadtholders Frederick Henry, William II., and William III. His father taught him the rudiments of education and the elements of mechanics. At the age of 15 he became the pupil of Stampion, and at 16 he was sent to Leyden to study law with Vinnius, who dedicated to him his first' commentary on the Institutes of Justinian. He there also pursued mathematical studies, and afterward at Breda in the university, which was under the direction of his father. In 1650, after a journey to Denmark with Henry, count of Nassau, he began those mathematical and physical researches which afterward made him famous. In 1651 he published at Leyden his first work, on the quadrature of the hyperbola, the ellipse, and the circle, and in 1654 a paper entitled De Circuli Magnitudine inventa nova. In 1655 Huygens went for the first time to France, and received the degree of doctor of laws from the faculty of the academy of Angers. On his return to Holland he turned his attention to the construction of telescopes, in connection with his elder brother Constantine. With one of these instruments, having a focal length of 10 ft., and more powerful than any ever before made, he discovered the first (now called the fourth) satellite of Saturn, and published the discovery at the Plague in 1656. During the next year he wrote a paper on the calculus of probabilities.

Pascal and Fermat had already written upon the subject, but the treatise of Huygens was more profound, and 50 years afterward James Bernoulli employed it as an introduction to his Art Conjectandi. It was also translated into Latin by his former tutor Schooten under the title De Ratiociniis in Ludo Aleae, by which it is also known in 's Gravesande's edition of Huygens's works. Schooten published it in his Exercitationes Mathematical, to demonstrate, as he says, the utility of algebra. About this time Huygens sent a paper to Wallis on the area of the cis-soid, and to Pascal a calculation for hyperbolic conoids, and spheroids in general, and on the quadrature of a portion of a cycloid, in which papers he employed methods having the highest characteristics of original thought. But his attention was not wholly devoted to merely theoretical mathematics, for about this time he introduced one of the most practical and important of all inventions. Galileo had observed the isochronism of small vibrations of the pendulum, and had employed it as a measurer of time, but his method required an assistant to count the oscillations, and was of course far from being exact.

To keep the pendulum in motion and cause it to register its successive vibrations was one of the problems which Huygens attempted, and which he succeeded in solving by the invention of the pendulum clock, a description of which, under the title of Horologium, he dedicated to the states general of Holland in 1658. (See Clocks and Watches.) In 1659 he constructed a telescope of 22 ft. focal length, in which he used a combination of two eye pieces, and again examined Saturn, making the discovery of the ring of the planet. The singular appearance which it sometimes presents of being accompanied by two luminous bodies, one on either side, had been observed by Galileo, but his telescope had not sufficient power to permit him to discover its cause. Huygens's instrument enabled him to make out that the phenomenon in question, which at regular times appeared and disappeared, was produced by the oblique position of the ring with regard to the earth and to the sun. From an analysis of the phenomenon he predicted the disappearance of the ring in 1671, and the prediction was verified.

He published an account of these observations at the Hague in 1659, in a volume also containing an account of several other discoveries, such as that of the great nebula in the sword of Orion, the bands upon the disks of Jupiter and Mars, and the fact that the fixed stars have no sensible magnitude. It was also accompanied by a description of a method for measuring the diameter of the planets. The micrometer used by him has been superseded by others, but it served the purpose of making correct measurements. In 1660 he visited France and England, and soon after published his celebrated theorems on the laws of the impact of bodies, in which most of the principles of the laws of motion are established. In 1665, at the invitation of Colbert, he went to France and became a member of the academy of sciences, then recently formed. Apartments were assigned to him in the royal library, and he resided in Paris for the greater part of the next 15 years, during which time he presented many papers to the academy, some of which still remain unpublished in its archives.

In 1670 he visited Holland to restore his health, which had become impaired by his great labors; and on his return to Paris in the following year he completed his great work Horologium Oscillatorium (fol., Paris, 1673). To this book are appended 13 theorems on centrifugal force, which will be noted further on. About this time he invented the spiral spring which is applied to the balance wheel of watches, a description of which was published in the journal of the academy of sciences in 1675. The invention was claimed by Hooke of England and Hautefeuille of France, but the evidence that it is the invention of Huy-gens is too strong to be any longer questioned. It is said that the first watch provided with a hair spring was made by Thuret under Huy-gens's direction, and was sent to England. In 1675 he again went to Holland for the benefit of his health, and in 1676 he read before the academy of sciences his famous treatise on light, and also a treatise on the cause of gravity, in which he attempts to account for the force by supposing that ethereal matter revolves about the earth with a velocity greater than that of the planet, and compares it to the force which causes bodies a little heavier than water, and lying lightly upon the smooth bottom of a cylindrical vessel containing water, to move toward the centre when the circular motion of the vessel by which its fluid contents have been caused to revolve is arrested.

In 1681 he returned to his native country, and immediately began the construction of an automatic planitarium to represent the true motion of the bodies of the solar system. This invention led to the important discovery of continued fractions, which he found it necessary to employ in order to establish the relation between the number of teeth contained in two wheels which play into one another. After this he resumed for several years, in conjunction with his brother Constantine, the construction of telescopes. He made two objectives, one of 170 and another of 210 ft. focal length, which he presented to the royal society of London. Asa telescope of such dimensions would be difficult to manage, Huy-gens proposed to dispense with the tube and place the object glass in an elevated position so that it could be adjusted to any angle, and then to place the eye piece at the focus. This arrangement continued to be used until the introduction of reflecting telescopes. While Huygens was absorbed in these occupations a great revolution was going on in the mathematical world.

Leibnitz had invented the differential calculus, which he published in 1684, and had proposed as a test to the followers of the old methods the problem of finding the curve of equable approach, or that which a suspended body must follow in order to approach or recede from equal heights in equal times. Huygens accomplished the solution by the old methods, but he was the only one who succeeded. Soon after this Newton published his Principia, and Huygens, with a desire of becoming acquainted with the author, visited England for the third time, and on his return published his treatise on light under the title Traite de la lumiere, ou sont expliquees les causes de ce qui lui arrive dans la reflexion, dans la refraction et particulierement dans Vetrange refraction du cristal d'lslande (Ley-den, 1690). Soon after this he investigated the properties of the catenary curve, a problem which had just been proposed by James Bernoulli, who had become proficient in the methods of the differential calculus; but Huygens solved the question by the old methods, which was considered a wonderful achievement.

He nevertheless found the task so difficult that his opposition to the differential calculus was shaken, and he entered at once into correspondence with Leibnitz. He had previously, whenever meeting with difficulties, attributed them to himself and not to defects in the methods. After examining the differential calculus he admitted its superiority, immediately commenced its use, and soon gave a wider development to the invention than it had yet attained. At his death he left his manuscripts to the library of Leyden, intrusting their publication to two of his pupils, Voider and Fullen. - Huygens was never married, and aside from his scientific pursuits his life was not eventful. He had a fine personal appearance, and his character was eminently noble. Newton spoke of him as the summus Hugenius, and considered his style as an author more classic than that of any other mathematician of that time. He was affable and kind, and was easily accessible to young students, whom he was always delighted to assist in their investigations. His labors were immense, and the practical value of their results is inestimable.

His discovery of the laws of the double refraction of light in Iceland spar, and of polarization, perhaps as much as any other cause, led to the reexamination of the undulatory theory, and, with the necessary adaptations, to its employment to account for all the phenomena of radiation of both heat and light. In accordance with this theory the most important researches in modern physics have been made, as those upon the diather-manous properties of bodies, and upon the absorption of radiant heat by gases and vapors, by which great light has been thrown on the science of meteorology. Besides his invention of the pendulum clock and of the balance wheel to the watch, the first chronometers taken aboard ships were made under his direction, and he was far in advance of all others of his day in astronomical observations. His discovery of the isochronism of the cycloid was one of the most important in mathematics; and not inferior to it is the invention of the involution and evolution of curves, and the establishment of the proposition that the cycloid is its own evolute. He also, in his Horologium Oscillatorium, gives a method for finding the centre of oscillation, which was the first successful solution of a dynamical problem in which connected material points are supposed to act on one another.

The difficulty of this subject is shown by the fact that Newton fell into an error in regard to it in attempting to solve the problem of the precession of the equinoxes. The question of the centre of oscillation had been proposed by Mersenne in 1646, and although some cases had been solved on the principle of the centre of percussion, it was beyond the reach of any methods then known. Huygens was only a boy of 17 when the question was proposed, and could then see no principle by which it could be solved; but when he published his Horologium Oscillato-rium in 1673, the principles which he assumed led to correct results in all cases. The two first theorems appended to that work state: 1, that if two equal bodies move in unequal circles in equal times, the centrifugal forces will be proportional to the diameters of the circles; and 2, that if the velocities are equal, the centrifugal forces will be in the inverse ratio of the diameters. To arrive at these conclusions required the application of the second law of motion (i. e., that the motion which a force gives to a body is compounded with the motion which it previosly had) to the limiting elements of the curve, in the manner in which Newton afterward demonstrated the theorems of Huygens in his Principia. Huygens's own demonstrations of these theorems were found after his death among his papers.

In his treatise on the impact of bodies (De Motu Corporum ex Percussione), Huygens must have assumed the third law of motion, which Newton afterward expressed by saying that "action and reaction are equal and opposite," by which we understand that the quantity of motion in the impact of bodies remains unchanged, one of the first grand principles in the doctrine of the conservation of force. His works were edited by 's Gravesande under the titles of Opera varia (2 vols. 4to in 1, Leyden, 1724) and Opera Reliqua(2 vols. 4to, Amsterdam, 1728).