Christiaan Huygens, Lord of Zeelhem, was a Dutch scientist born on April 14, 1629, and died on July 8, 1695. He was a mathematician, physicist, engineer, astronomer, and inventor who played an important role in the Scientific Revolution. In physics, Huygens made important contributions to the study of light and motion. As an astronomer, he studied Saturn’s rings and discovered its largest moon, Titan. As an engineer, he improved telescope designs and invented the pendulum clock, which was the most accurate timekeeper for nearly 300 years. Huygens was also a skilled mathematician who wrote one of the first modern books that used math to explain physical problems. His work on light included the first mathematical explanation of how light behaves, even though it was not widely accepted at first.

In 1656, Huygens wrote a book called De Motu Corporum ex Percussione, which explained the correct laws of how objects move when they collide. This book was published after his death in 1703. In 1659, he used geometry to develop a formula for centrifugal force, a concept in physics, a decade before Isaac Newton. Huygens is best known for his wave theory of light, which he described in his book Traité de la Lumière in 1690. At first, his theory was not accepted, but later, Augustin-Jean Fresnel used Huygens’s ideas to explain how light travels in straight lines and bends around objects. This combined idea is now called the Huygens–Fresnel principle.

Huygens invented the pendulum clock in 1657. It was made in Paris by Isaac II Thuret. His research on clocks led to a detailed study of pendulums in a book called Horologium Oscillatorium, published in 1673. This book is considered one of the most important works on mechanics from the 17th century. It includes descriptions of clock designs and explains how pendulums move and how curves can be studied. In 1655, Huygens and his brother Constantijn began making lenses for refracting telescopes. He discovered Titan and was the first to explain Saturn’s unusual appearance as being caused by a ring that does not touch Saturn. In 1662, he created a telescope design with two lenses, now called the Huygenian eyepiece, which reduces light distortion.

As a mathematician, Huygens studied shapes called evolutes and wrote about probability in a book called Van Rekeningh in Spelen van Gluck. This book was translated and published in 1657 as De Ratiociniis in Ludo Aleae by Frans van Schooten. His work on probability influenced later scientists, including Jacob Bernoulli, who studied probability theory.

Biography

Christiaan Huygens was born on April 14, 1629, in The Hague, Netherlands, to a wealthy and important Dutch family. He was the second son of Constantijn Huygens and named after his grandfather. His mother, Suzanna van Baerle, died shortly after giving birth to his sister. The Huygens family had five children: Constantijn (born 1628), Christiaan (1629), Lodewijk (1631), Philips (1632), and Suzanna (1637).

Constantijn Huygens was a government advisor, poet, and musician. He worked closely with leaders in the Netherlands and corresponded with famous scientists and thinkers across Europe, including Galileo Galilei, Marin Mersenne, and René Descartes. Christiaan received his early education at home until he was 16. He enjoyed building small models of machines and mills. His father taught him many subjects, including languages, music, history, geography, mathematics, logic, and rhetoric, as well as dancing, fencing, and horseback riding.

In 1644, Christiaan studied mathematics with Jan Jansz Stampioen, who gave him a challenging list of books on science. Later, Descartes and Mersenne praised his skills in geometry, with Mersenne calling him the "new Archimedes."

At 16, Constantijn sent Christiaan to Leiden University to study law and mathematics. He studied there from May 1645 to March 1647. Frans van Schooten Jr., a professor, became Christiaan’s tutor, helping him learn the latest ideas in mathematics from thinkers like Viète, Descartes, and Fermat.

After two years, Christiaan continued his studies at Orange College in Breda, where his father was a manager. The college operated until 1669, and its leader was André Rivet. Christiaan lived with a lawyer named Johann Henryk Dauber while attending college and studied mathematics with an English teacher named John Pell. His time in Breda ended around the time his brother Lodewijk had a duel with another student. Christiaan left Breda in August 1649 and worked briefly as a diplomat with Henry, Duke of Nassau. He traveled to Germany and Denmark but was unable to visit Descartes in Sweden, as Descartes had died before he could do so.

Although his father wanted him to be a diplomat, Christiaan had no interest in that career. The political changes in the Netherlands after 1650 reduced his father’s influence, and he realized his son preferred science and mathematics.

Christiaan wrote mostly in French or Latin. In 1646, while still a student, he began writing letters to his father’s friend, Marin Mersenne, who died in 1648. Mersenne had praised Christiaan’s math skills and compared him to the ancient scientist Archimedes.

In his letters, Christiaan discussed scientific ideas. In October 1646, he explained that a hanging chain forms a curve different from a parabola, which Galileo had thought it was. He later named this curve the "catenary" in 1690. Between 1647 and 1648, he wrote about topics like the law of free fall, the shape of an ellipse, and the motion of a vibrating string.

In 1654, Christiaan returned to his father’s home in The Hague and focused fully on research. He spent summers at a nearby house called Hofwijck. Despite his achievements, he sometimes struggled with sadness.

Christiaan corresponded with many scientists, though communication became harder after 1648 due to political unrest in France. In 1655, he met scientists in Paris, including Ismael Boulliau and Claude Mylon. Through another scientist, Pierre de Carcavi, he wrote to Pierre de Fermat, whom he admired. However, Fermat had stopped working on major research topics, which made their exchange difficult.

Christiaan often shared his ideas through letters instead of publishing them immediately. His mentor, Frans van Schooten, helped him refine his work.

Between 1651 and 1657, Christiaan published several mathematical works that showed his skill in geometry and increased his reputation. He questioned Descartes’s incorrect laws about collisions and developed the correct laws using algebra and geometry. He discovered that the center of gravity in a system of moving objects remains constant, which he called the conservation of "quantity of movement." His ideas influenced later scientists but were not widely known until 1703.

Christiaan also made important scientific discoveries. He identified Saturn’s moon Titan in 1655, invented the pendulum clock in 1657, and explained Saturn’s rings in 1659. These achievements made him famous across Europe. In 1661, he observed Mercury passing in front of the Sun with a telescope and debated the accuracy of another scientist’s records. He also shared a manuscript about the 1639 Venus transit with another scientist.

In 1661, Sir Robert Moray gave Christiaan a table of life expectancy data, which he and his brother Lodewijk used to study how long people lived. Christiaan created the first graph showing a continuous distribution of death rates and used it to solve problems about shared life insurance. He also played the harpsichord, an early keyboard instrument.

Mathematics

Christiaan Huygens first gained international recognition for his work in mathematics. He published several important results that caught the attention of many European mathematicians. In his published works, Huygens preferred the methods used by Archimedes. However, he also used techniques from Descartes's analytic geometry and Fermat's methods for very small measurements in his private notes.

Huygens's first published work was Theoremata de Quadratura Hyperboles, Ellipsis et Circuli (Theorems on the quadrature of the hyperbola, ellipse, and circle), released in Leiden in 1651. The first part of the work included theorems for calculating the areas of hyperbolas, ellipses, and circles. These theorems were similar to Archimedes's work on conic sections, especially his Quadrature of the Parabola. The second part of the work addressed claims by Grégoire de Saint-Vincent about the quadrature of a circle, which Huygens had previously discussed with Mersenne.

Huygens showed that the center of gravity of a segment of a hyperbola, ellipse, or circle is directly connected to the area of that segment. He then explained how triangles inside conic sections relate to the center of gravity of those sections. By applying these theorems to all conic sections, Huygens expanded classical methods to create new mathematical results.

Quadrature and rectification were important topics in the 1650s. Through Mylon, Huygens participated in a debate involving Thomas Hobbes. He continued to highlight his mathematical achievements, which helped him gain a reputation worldwide.

Huygens's next work was De Circuli Magnitudine Inventa (New findings on the magnitude of the circle), published in 1654. In this work, he improved upon Archimedes's Measurement of the Circle by narrowing the range between the values of circumscribed and inscribed polygons. He showed that the ratio of a circle's circumference to its diameter, or pi (π), must fall within the first third of that range.

Using a method similar to Richardson extrapolation, Huygens simplified the inequalities in Archimedes's approach. By applying the center of gravity of a parabolic segment, he approximated the center of gravity of a circular segment, leading to a faster and more accurate calculation of the circle's area. From these results, Huygens calculated two sets of values for π: the first between 3.1415926 and 3.1415927, and the second between 3.1415926533 and 3.1415926538.

Huygens also demonstrated that using parabolic segments could quickly calculate logarithms for hyperbolas. At the end of his work, he included solutions to classical problems under the title Illustrium Quorundam Problematum Constructiones (Construction of some illustrious problems).

Huygens became interested in games of chance after visiting Paris in 1655. He studied the work of Fermat, Blaise Pascal, and Girard Desargues. He later published De Ratiociniis in Ludo Aleae (On reasoning in games of chance), which was the most organized explanation of using mathematics to analyze games of chance at the time. Frans van Schooten translated the original Dutch text into Latin and published it in 1657.

This work introduced early ideas from game theory, focusing on the problem of points. Huygens adopted Pascal's ideas about "fair games" and equal division when chances are equal. He expanded these ideas to develop a theory of expected values. His ability to apply algebra to probability showed how combining Euclidean geometry with symbolic reasoning could solve complex problems.

Huygens included five difficult problems at the end of his book. These problems became a standard test for those who wanted to demonstrate their mathematical skills in games of chance for the next sixty years. Notable people who studied these problems included Abraham de Moivre, Jacob Bernoulli, Johannes Hudde, Baruch Spinoza, and Leibniz.

Earlier, Huygens completed a manuscript titled De Iis quae Liquido Supernatant (About parts floating above liquids), written around 1650. It consisted of three books. Although he sent the work to Frans van Schooten for feedback, he ultimately decided not to publish it. Some of its results were not rediscovered until the 1700s and 1800s.

In this manuscript, Huygens reworked Archimedes's solutions for the stability of a sphere and a paraboloid using Torricelli's principle, which states that objects move only if their center of gravity descends. He then proved a general rule that, for a floating object in balance, the distance between its center of gravity and its submerged part is at its smallest. Using this rule, Huygens solved original problems about the stability of floating cones, parallelepipeds, and cylinders, sometimes considering full rotations. His method was similar to the principle of virtual work. Huygens was also the first to recognize that, for these solid objects, their specific weight and shape are the key factors in determining their stability in water.

Natural philosophy

Christiaan Huygens was one of the most important European scientists between René Descartes and Isaac Newton. Unlike many of his peers, Huygens did not focus on creating large, complex theories or discussing philosophical ideas. Instead, he worked to solve difficult physics problems that could be studied using math. He especially focused on how objects interact when they touch each other, rather than using ideas about forces acting from a distance.

During his time in Paris, Huygens supported a scientific approach that relied on experiments and the idea that the natural world could be explained through mechanical processes. In 1661, he visited England and learned about Robert Boyle’s experiments with an air pump at Gresham College. Later, Huygens redesigned Boyle’s experiments to test new ideas. This process took many years and led to important discoveries, including Huygens accepting Boyle’s belief that a vacuum (empty space) could exist, even though some scientists, like Descartes, disagreed.

Huygens helped John Locke understand Isaac Newton’s work by confirming that Newton’s math was correct. This encouraged Locke to support a type of physics that used tiny particles to explain natural events.

Many scientists of the time, including Huygens, believed that forces only acted when objects touched each other. This idea was central to physics, as it made the study of collisions important. Huygens agreed with some of Descartes’s ideas but was not strict about following them. He studied how objects bounce off each other in the 1650s but did not publish his findings for over a decade.

Huygens discovered that Descartes’s rules for collisions were incorrect and created the correct laws, including the idea that the total motion of objects remains the same during collisions. He also recognized that the laws of motion are the same for all observers moving at a constant speed. Huygens wrote about these ideas in a manuscript called De Motu Corporum ex Percussione between 1652 and 1656. He shared his results with scientists in London in 1661, and later announced them to the Royal Society in 1668. His work was published in 1669.

In 1659, Huygens calculated the constant for gravitational acceleration and wrote a version of Newton’s second law of motion. He also developed a formula for centrifugal force, the force that pushes objects outward when they move in a circle. This formula is still used today and is written as:

$$ F = m omega^2 r $$

where m is the object’s mass, ω is its angular speed, and r is the radius of the circle. Huygens collected his findings in a work titled De vi Centrifuga, which was not published until 1703. His work helped create a general understanding of force before Newton’s Principia Mathematica was written in 1687.

Huygens’s ideas about centrifugal force were important for studying the motion of planets. They helped scientists move from Kepler’s laws of planetary motion to Newton’s law of gravity. However, Huygens did not believe that gravity acted from a distance, unlike some scientists who followed Newton.

Huygens’s work on pendulums brought him close to understanding simple harmonic motion, a concept fully explained later by Newton. In 1657, he invented the pendulum clock, which greatly improved timekeeping. His clock was much more accurate than earlier designs, losing only about 15 seconds per day compared to 15 minutes for older clocks. However, Huygens did not earn much money from his invention because others copied his design.

Huygens wanted to create a reliable pendulum clock for use on ships to help sailors find their location at sea. But the movement of the ship disturbed the pendulum, making it unreliable. Trials with the clock on voyages showed mixed results. Eventually, scientists like Jean Richer corrected for the shape of Earth, which helped improve the accuracy of pendulum clocks for navigation.

In 1673, Huygens published a major work called Horologium Oscillatorium, which explained how pendulums work using math. He studied why pendulums swing at different speeds depending on how far they move and discovered that a pendulum must follow a specific curved path, called a cycloid, to swing at a constant rate. This problem, called the tautochrone, led Huygens to develop new mathematical tools, such as the theory of evolutes.

Huygens’s work laid the foundation for modern physics and engineering, influencing later scientists like Newton and Leibniz. His inventions and discoveries helped shape the scientific understanding of motion, gravity, and timekeeping.

Legacy

Christiaan Huygens was called the first theoretical physicist and a founder of modern mathematical physics. His influence was strong during his lifetime but became less known after he died. His skills in geometry and mechanical inventions impressed many of his peers, including Isaac Newton, Gottfried Wilhelm Leibniz, Guillaume de l'Hôpital, and the Bernoullis. For his work in physics, Huygens is considered one of the greatest scientists of the Scientific Revolution, second only to Newton in the depth of his ideas and the number of discoveries he made. He also helped create systems for scientific research in Europe, making him an important figure in the development of modern science.

In mathematics, Huygens studied ancient Greek geometry, especially the work of Archimedes, and used the methods of analytic geometry and infinitesimal techniques developed by Descartes and Fermat. His mathematical approach focused on analyzing curves and motion using geometry. He drew ideas from mechanics but kept his work purely mathematical. Huygens was one of the last to use this type of analysis, as more mathematicians later turned to calculus for studying infinitesimals, limits, and motion.

Huygens used mathematics to solve physics problems. He often created simple models to explain complex situations, then analyzed them step by step, developing the necessary math as needed. He preferred to present his results using clear, logical proofs based on geometric principles. While he allowed some uncertainty in the starting assumptions, the proofs themselves were always certain. His writing style influenced how Newton presented his own major works.

Huygens used mathematics not only to study physics and vice versa but also as a method to discover new knowledge about the world. Unlike Galileo, who mainly used math for explanation or synthesis, Huygens consistently used it to develop theories about natural phenomena. He insisted that physical concepts must align closely with mathematical models. This approach set a standard for later scientists like Johann Bernoulli, Jean le Rond d'Alembert, and Charles-Augustin de Coulomb.

Although Huygens never intended to publish some of his work, he used algebra to describe physical relationships in a few manuscripts about collisions. This made him one of the first to use mathematical formulas to explain physics, as is done today. He also came close to the modern idea of a limit while studying optics, though he never applied it outside of that field.

Huygens’s reputation as Europe’s greatest scientist was surpassed by Newton by the end of the 17th century. However, as Hugh Aldersey-Williams noted, Huygens’s achievements in some areas were greater than Newton’s. Although his writings anticipated the format of modern scientific articles, his preference for classical methods and reluctance to publish limited his influence after the Scientific Revolution, as followers of Newton and Leibniz gained more prominence.

Huygens studied curves with special physical properties, such as the cycloid, which led to later research on other curves like the caustic, brachistochrone, sail curve, and catenary. His use of mathematics in physics, including studies of impact and birefringence, inspired future developments in mathematical physics and mechanics. He also created the first reliable timekeeping devices—the pendulum and balance spring—which were used in scientific measurements for the first time. His work laid the foundation for combining applied mathematics with mechanical engineering in later centuries.

During his lifetime, Huygens and his father commissioned several portraits, including:

• 1639 – A painting of Constantijn Huygens with his five children by Adriaen Hanneman, Mauritshuis, The Hague
• 1671 – A portrait by Caspar Netscher, Museum Boerhaave, Leiden (loaned from Haags Historisch Museum)
• c. 1675 – A depiction of Huygens in Établissement de l'Académie des Sciences et fondation de l'observatoire, 1666 by Henri Testelin, Musée National du Château et des Trianons de Versailles, Versailles
• 1679 – A medallion portrait in relief by Jean-Jacques Clérion
• 1686 – A pastel portrait by Bernard Vaillant, Museum Hofwijck, Voorburg
• 1684–1687 – Engravings by G. Edelinck based on a painting by Caspar Netscher
• 1688 – A portrait by Pierre Bourguignon, Royal Netherlands Academy of Arts and Sciences, Amsterdam

The European Space Agency’s probe on the Cassini spacecraft, which landed on Titan, Saturn’s largest moon, in 2005, was named after Huygens.

Monuments to Christiaan Huygens can be found in cities across the Netherlands, including Rotterdam, Delft, and Leiden.

Works

• 1650 – De Iis Quae Liquido Supernatant (About parts floating above liquids), unpublished.
• 1651 – Theoremata de Quadratura Hyperboles, Ellipsis et Circuli, published again in Oeuvres Complètes, Tome XI.
• 1651 – Epistola, qua diluuntur ea quibus 'Εξέτασις [Exetasis] Cyclometriae Gregori à Sto. Vincentio impugnata fuit, additional information.
• 1654 – De Circuli Magnitudine Inventa.
• 1654 – Illustrium Quorundam Problematum Constructiones, additional information.
• 1655 – Horologium (The clock), a short pamphlet explaining the pendulum clock.
• 1656 – De Saturni Luna Observatio Nova (About the new observation of the moon of Saturn), describes the discovery of Titan.
• 1656 – De Motu Corporum ex Percussione, published after Huygens’s death in 1703.
• 1657 – De Ratiociniis in Ludo Aleae (Van reeckening in spelen van geluck), translated into Latin by Frans van Schooten.
• 1659 – Systema Saturnium (System of Saturn).
• 1659 – De vi Centrifuga (Concerning the centrifugal force), published after Huygens’s death in 1703.
• 1673 – Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae, includes a theory of evolutes and designs of pendulum clocks, dedicated to Louis XIV of France.
• 1684 – Astroscopia Compendiaria Tubi Optici Molimine Liberata (Compound telescopes without a tube).
• 1685 – Memoriën aengaende het slijpen van glasen tot verrekijckers, discusses the grinding of lenses.
• 1686 – Kort onderwijs aengaende het gebruijck der horologiën tot het vinden der lenghten van Oost en West (in Old Dutch), instructions on using clocks to determine longitude at sea.
• 1690 – Traité de la Lumière, discusses the nature of light propagation.
• 1690 – Discours de la Cause de la Pesanteur (Discourse about gravity), additional information.
• 1691 – Lettre Touchant le Cycle Harmonique, a short tract about the 31-tone system.
• 1698 – Cosmotheoros, discusses the solar system, cosmology, and extraterrestrial life.
• 1703 – Opuscula Posthuma, includes: De Motu Corporum ex Percussione (Concerning the motions of colliding bodies), contains the first correct laws for collision, dating from 1656; Descriptio Automati Planetarii, provides a description and design of a planetarium.
• 1724 – Novus Cyclus Harmonicus, a treatise on music published in Leiden after Huygens’s death.
• 1728 – Christiani Hugenii Zuilichemii, dum viveret Zelhemii Toparchae, Opuscula Posthuma (alternate title: Opera Reliqua), includes works in optics and physics.
• 1888–1950 – Huygens, Christiaan. Oeuvres complètes. Complete works, 22 volumes. Editors: D. Bierens de Haan (1–5), J. Bosscha (6–10), D.J. Korteweg (11–15), A.A. Nijland (15), J.A. Vollgraf (16–22). The Hague: Tome I: Correspondance 1638–1656 (1888). Tome II: Correspondance 1657–1659 (1889). Tome III: Correspondance 1660–1661 (1890). Tome IV: Correspondance 1662–1663 (1891). Tome V: Correspondance 1664–1665 (1893). Tome VI: Correspondance 1666–1669 (1895). Tome VII: Correspondance 1670–1675 (1897). Tome VIII: Correspondance 1676–1684 (1899). Tome IX: Correspondance 1685–1690 (1901). Tome X: Correspondance 1691–1695 (1905). Tome XI: Travaux mathématiques 1645–1651 (1908). Tome XII: Travaux mathématiques pures 1652–1656 (1910). Tome XIII, Fasc. I: Dioptrique 1653, 1666 (1916). Tome XIII, Fasc. II: Dioptrique 1685–1692 (1916). Tome XIV: Calcul des probabilités. Travaux de mathématiques pures 1655–1666 (1920). Tome XV: Observations astronomiques. Système de Saturne. Travaux astronomiques 1658–1666 (1925). Tome XVI: Mécanique jusqu’à 1666. Percussion. Question de l'existence et de la perceptibilité du mouvement absolu. Force centrifuge (1929). Tome XVII: L’horloge à pendule de 1651 à 1666. Travaux divers de physique, de mécanique et de technique de 1650 à 1666. Traité des couronnes et des parhélies (1662 ou 1663) (1932). Tome XVIII: L'horloge à pendule ou à balancier de 1666 à 1695. Anecdota (1934). Tome XIX: Mécanique théorique et physique de 1666 à 1695. Huygens à l'Académie royale des sciences (1937