BREAKING NEWS
Christiaan Huygens

## Summary

Christiaan Huygens
Born14 April 1629
Died8 July 1695 (aged 66)
The Hague, Dutch Republic
NationalityDutch
Alma materUniversity of Leiden
University of Angers
Known for
Scientific career
FieldsNatural Philosophy
Mathematics
Physics
Astronomy
Horology
InstitutionsRoyal Society of London
InfluencesGalileo Galilei
René Descartes
Frans van Schooten
InfluencedGottfried Wilhelm Leibniz
Isaac Newton[2][3]

Christiaan Huygens FRS (/ˈhɡənz/ HY-gənz,[4] also US: /ˈhɔɪɡənz/ HOY-gənz,[5][6] Dutch: [ˈkrɪstijaːn ˈɦœyɣə(n)s] (); Latin: Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is widely regarded as one of the greatest scientists of all time and a major figure in the scientific revolution. In physics, Huygens made groundbreaking contributions in optics and mechanics, while as an astronomer he is chiefly known for his studies of the rings of Saturn and the discovery of its moon Titan. As an inventor, he improved the design of telescopes and invented the pendulum clock, a breakthrough in timekeeping and the most accurate timekeeper for almost 300 years. An exceptionally talented mathematician and physicist, Huygens was the first to idealize a physical problem by a set of parameters then analyse it mathematically,[7] and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon.[8][9] For these reasons, he has been called the first theoretical physicist and one of the founders of modern mathematical physics.[10][11]

In 1659, Huygens derived geometrically the now standard formulae in classical mechanics for the centripetal force and centrifugal force in his work De vi Centrifuga.[12] Huygens also identified the correct laws of elastic collision for the first time in his work De Motu Corporum ex Percussione, published posthumously in 1703. In the field of optics, he is best known for his wave theory of light, which he proposed in 1678 and described in his Traité de la Lumière (1690). His mathematical theory of light was initially rejected in favour of Newton's corpuscular theory of light, until Augustin-Jean Fresnel adopted Huygens's principle to give a complete explanation of the rectilinear propagation and diffraction effects of light in 1821. Today this principle is known as the Huygens–Fresnel principle.

Huygens invented the pendulum clock in 1657, which he patented the same year. His research in horology resulted in an extensive analysis of the pendulum in Horologium Oscillatorium (1673), regarded as one of the most important 17th century works in mechanics. While the first part contains descriptions of clock designs, most of the book is an analysis of pendulum motion and a theory of curves. In 1655, Huygens began grinding lenses with his brother Constantijn to build telescopes for astronomical research. He was the first to identify the rings of Saturn as "a thin, flat ring, nowhere touching, and inclined to the ecliptic," and discovered the first of Saturn's moons, Titan, using a refracting telescope.[13][14] In 1662 Huygens developed what is now called the Huygenian eyepiece, a telescope with two lenses, which diminished the amount of dispersion.

As a mathematician, Huygens developed the theory of evolutes and wrote on games of chance and the problem of points in Van Rekeningh in Spelen van Gluck, which Frans van Schooten translated and published as De Ratiociniis in Ludo Aleae (1657).[15] The use of expectation values by Huygens and others would later inspire Jacob Bernoulli's work on probability theory.[16][17]

## Biography

Portrait of Huygens's father (centre) and his five children (Christiaan at right). Mauritshuis, The Hague.

Christiaan Huygens was born on 14 April 1629 in The Hague, into a rich and influential Dutch family,[18][19] the second son of Constantijn Huygens. Christiaan was named after his paternal grandfather.[20][21] His mother, Suzanna van Baerle, died shortly after giving birth to Huygens's sister.[22] The couple had five children: Constantijn (1628), Christiaan (1629), Lodewijk (1631), Philips (1632) and Suzanna (1637).[23]

Constantijn Huygens was a diplomat and advisor to the House of Orange, in addition to being a poet and a musician. He corresponded widely with intellectuals across Europe; his friends included Galileo Galilei, Marin Mersenne, and René Descartes.[24] Christiaan was educated at home until turning sixteen years old, and from a young age liked to play with miniatures of mills and other machines. His father gave him a liberal education: he studied languages, music, history, geography, mathematics, logic, and rhetoric, but also dancing, fencing and horse riding.[20][23][25]

In 1644 Huygens had as his mathematical tutor Jan Jansz Stampioen, who assigned the 15-year-old a demanding reading list on contemporary science.[26] Descartes was later impressed by his skills in geometry, as did Mersenne, who christened him "the new Archimedes."[8][19][27]

### Student years

At sixteen years of age, Constantijn sent Huygens to study law and mathematics at Leiden University, where he studied from May 1645 to March 1647.[20] Frans van Schooten was an academic at Leiden from 1646, and became a private tutor to Huygens and his elder brother, Constantijn Jr., replacing Stampioen on the advice of Descartes.[28][29] Van Schooten brought his mathematical education up to date, in particular introducing him to the work of Viète, Descartes, and Fermat.[30]

After two years, starting in March 1647, Huygens continued his studies at the newly founded Orange College, in Breda, where his father was a curator. His time in Breda would eventually end when his brother Lodewijk, who was already enrolled, ended up in a duel with another student.[31][32] Constantijn Huygens was closely involved in the new College, which lasted only to 1669; the rector was André Rivet.[33] Christiaan Huygens lived at the home of the jurist Johann Henryk Dauber while attending college, and had mathematics classes with the English lecturer John Pell. He completed his studies in August 1649.[20] He then had a stint as a diplomat on a mission with Henry, Duke of Nassau. It took him to Bentheim, then Flensburg. He took off for Denmark, visited Copenhagen and Helsingør, and hoped to cross the Øresund to visit Descartes in Stockholm. It was not to be.[34]

Although his father Constantijn had wished his son Christiaan to be a diplomat, circumstances kept him from becoming so. The First Stadtholderless Period that began in 1650 meant that the House of Orange was no longer in power, removing Constantijn's influence. Further, he realized that his son had no interest in such a career.[35]

### Early correspondence

A picture of a hanging chain (catenary) in a manuscript of Huygens.

Huygens generally wrote in French or Latin.[36] In 1646, while still a college student at Leiden, he began a correspondence with his father's friend the intelligencer Mersenne, who died quite soon afterwards in 1648.[20] Mersenne wrote to Constantijn on his son's talent for mathematics, and flatteringly compared him to Archimedes on 3 January 1647.[37]

The letters show Huygens's early interest in mathematics. In October 1646 there is the suspension bridge and the demonstration that a hanging chain is not a parabola, as Galileo thought.[38] Huygens would later labelled that curve the catenaria (catenary) in 1690 while corresponding with Gottfried Leibniz.[39]

In the next two years (1647-48), Huygens's letters to Mersenne covered various topics, including the claim by Grégoire de Saint-Vincent of circle quadrature, which Huygens showed to be wrong, the rectification of the ellipse, projectiles, and the vibrating string.[40] Some of Mersenne's concerns at the time, such as the cycloid (he sent Huygens Torricelli's treatise on the curve), the centre of oscillation, and the gravitational constant, were matters Huygens only took seriously towards the end of the 17th century.[41] Mersenne had also written on musical theory. Huygens preferred meantone temperament; he innovated in 31 equal temperament (which was not itself a new idea but known to Francisco de Salinas), using logarithms to investigate it further and show its close relation to the meantone system.[42]

In 1654, Huygens returned to his father's house in The Hague, and was able to devote himself entirely to research.[20] The family had another house, not far away at Hofwijck, and he spent time there during the summer. Despite being very active, his scholarly life did not allow him to escape bouts of depression.[43]

Subsequently, Huygens developed a broad range of correspondents, though picking up the threads after 1648 was hampered by the five-year Fronde in France. Visiting Paris in 1655, Huygens called on Ismael Boulliau to introduce himself, who took him to see Claude Mylon.[44] The Parisian group of savants that had gathered around Mersenne held together into the 1650s, and Mylon, who had assumed the secretarial role, took some trouble from then on to keep Huygens in touch.[45] Through Pierre de Carcavi Huygens corresponded in 1656 with Pierre de Fermat, whom he admired greatly, though this side of idolatry. The experience was bittersweet and somewhat puzzling, since it became clear that Fermat had dropped out of the research mainstream, and his priority claims could probably not be made good in some cases. Besides, Huygens was looking by then to apply mathematics to physics, while Fermat's concerns ran to purer topics.[46]

### Scientific debut

Christiaan Huygens, relief by Jean-Jacques Clérion (c. 1670).

Like some of his contemporaries, Huygens was often slow to commit his results and discoveries to print, preferring to disseminate his work through letters instead.[47] In his early days, his mentor Frans van Schooten provided technical feedback and was cautious for the sake of his reputation.[48]

Between 1651 and 1657, Huygens published a number of works that showed his talent for mathematics and his mastery of both classical and analytical geometry, which allowed him to increase his reach and reputation among mathematicians.[37] Around the same time, Huygens began to question Descartes's laws of collision, which were largely wrong, deriving the correct laws algebraically and by way of geometry.[49] He showed that, for any system of bodies, the centre of gravity of the system remains the same in velocity and direction, which Huygens called the conservation of "quantity of movement". His theory of collisions was the closest anyone has come to the idea of force prior to Newton.[31] These results were known through correspondence and in a short article in Journal des Sçavans, but would remain largely unpublished until after his death with the publication of De Motu Corporum ex Percussione (Concerning the motion of colliding bodies).[49]

In addition to his work on mechanics, he made important scientific discoveries, such as the identification of Saturn's moon Titan in 1655, and the invention of the pendulum clock in 1657, both of which brought him fame across Europe.[20] On 3 May 1661, Huygens observed the planet Mercury transit over the Sun, using the telescope of instrument maker Richard Reeve in London, together with astronomer Thomas Streete and Reeve.[50] Streete then debated the published record of the transit of Hevelius, a controversy mediated by Henry Oldenburg.[51] Huygens passed to Hevelius a manuscript of Jeremiah Horrocks on the transit of Venus, 1639, which thereby was printed for the first time in 1662.[52]

In the same year Huygens, who played the harpsichord, took an interest in Simon Stevin's theories on music; however, he showed very little concern to publish his theories on consonance, some of which were lost for centuries.[53][54] For his contributions to science, the Royal Society of London elected him a Fellow in 1665 when Huygens was but 36 years old.[55]

### France

Huygens, right of centre, from L'établissement de l'Académie des Sciences et fondation de l'observatoire, 1666 by Henri Testelin (c. 1675).

The Montmor Academy was the form the old Mersenne circle took after the mid-1650s.[56] Huygens took part in its debates, and supported its "dissident" faction who favoured experimental demonstration to curtail fruitless discussion, and opposed amateurish attitudes.[57] During 1663 he made what was his third visit to Paris; the Montmor Academy closed down, and Huygens took the chance to advocate a more Baconian program in science. Three years later, in 1666, he moved to Paris on an invitation to fill a position at King Louis XIV's new French Académie des sciences.[58]

While in Paris, Huygens had an important patron and correspondent in Jean-Baptiste Colbert, First Minister to Louis XIV.[59] However, his relationship with the Academy was not always easy, and in 1670 Huygens, seriously ill, chose Francis Vernon to carry out a donation of his papers to the Royal Society in London, should he die.[60] The aftermath of the Franco-Dutch War (1672–78), and particularly England's role in it, may have damaged his relationship with the Royal Society.[61] Robert Hooke, as a representative of the Royal Society, lacked the finesse to handle the situation in 1673.[62]

The physicist and inventor Denis Papin was assistant to Huygens from 1671.[63] One of their projects, which did not bear fruit directly, was the gunpowder engine.[64] Papin moved to England in 1678 to continue work in this area.[65] Also in Paris, Huygens made further astronomical observations using the Observatory recently completed in 1672. He introduced Nicolaas Hartsoeker to French scientists such as Nicolas Malebranche and Giovanni Cassini in 1678.

Huygens met the young diplomat Gottfried Leibniz, visiting Paris in 1672 on a vain mission to meet Arnauld de Pomponne, the French Foreign Minister. At this time Leibniz was working on a calculating machine, and he moved on to London in early 1673 with diplomats from Mainz. From March 1673, Leibniz was tutored in mathematics by Huygens, who taught him analytical geometry.[66] An extensive correspondence ensued, in which Huygens showed at first reluctance to accept the advantages of Leibniz's infinitesimal calculus.[67]

### Final years

Huygens moved back to The Hague in 1681 after suffering another bout of serious depressive illness. In 1684, he published Astroscopia Compendiaria on his new tubeless aerial telescope. He attempted to return to France in 1685 but the revocation of the Edict of Nantes precluded this move. His father died in 1687, and he inherited Hofwijck, which he made his home the following year.[35]

On his third visit to England, Huygens met Isaac Newton in person on 12 June 1689. They spoke about Iceland spar, and subsequently corresponded about resisted motion.[68]

Huygens returned back to mathematical topics in his last years and observed the acoustical phenomenon now known as flanging in 1693.[69] Two years later, on 8 July 1695, Huygens died in The Hague and was buried in an unmarked grave in the Grote Kerk there, as was his father before him.[70]

Huygens never married.[71]

## Mathematics

Huygens first became internationally known for his work in mathematics, publishing a number of important results that drew the attention of many European geometers.[72] Huygens's preferred method in his published works was that of Archimedes, though he made use of the work of Descartes and Fermat in his private notebooks.[20]

Huygens's first publication was in the field of quadrature.

Huygens's first publication was Theoremata de Quadratura Hyperboles, Ellipsis et Circuli (Theorems on the quadrature of the hyperbola, ellipse, and circle), published by the Elzeviers in Leiden in 1651.[47] The first part of the work was in the field of quadrature and contained theorems for computing the areas of hyperbolas and ellipses that paralleled Archimedes's work on conic sections, particularly his Quadrature of the Parabola.[37] Huygens demonstrated that the centre of gravity of a segment of the hyperbola, ellipse, or circle was directly related to the area of that segment. In this way, Huygens was able to extend classical methods to all conic sections, generating new results.[20]

Huygens also included a refutation to Grégoire de Saint-Vincent's claims on circle quadrature, which he had discussed with Mersenne earlier. Quadrature was a live issue in the 1650s and, through Mylon, Huygens intervened in the discussion of the mathematics of Thomas Hobbes. Persisting in trying to explain the errors Hobbes had fallen into, he made an international reputation.[73]

### De Circuli Magnitudine Inventa

Huygens's next publication was De Circuli Magnitudine Inventa (New findings in the measurement of the circle), published in 1654. In this work, Huygens was able to narrow the gap between the circumscribed and inscribed polygons found in Archimedes's Measurement of the Circle, showing that the ratio of the circumference to its diameter must lie in the first third of that interval.[47]

Using a technique equivalent to Richardson extrapolation,[74] Huygens approximated the centre of gravity of a segment of a circle by the centre of the gravity of a segment of a parabola, and thus finding an approximation of the quadrature; with this he was able to refine the inequalities between the area of the circle and those of the inscribed and circumscribed polygons used in the calculations of π.[75] From these theorems, Huygens obtained two set of values, the first between 3.1415926 and 3.1415927, and the second between 3.1415926538 and 3.1415926533.[76]

Huygens also showed that the same approximation with segments of the parabola, in the case of the hyperbola, yields a quick and simple method to calculate logarithms.[77] He appended a collection of solutions to classical problems at the end of the work under the title Illustrium Quorundam Problematum Constructiones (Construction of some illustrious problems).[47]

### De Ratiociniis in Ludo Aleae

Huygens became interested in games of chance after he visited Paris in 1655 and encountered the work of Fermat, Blaise Pascal and Girard Desargues years earlier.[78] He eventually published what was, at the time, the most coherent presentation of a mathematical approach to games of chance in De Ratiociniis in Ludo Aleae (On reasoning in games of chance).[17][79] Frans van Schooten translated the original Dutch manuscript into Latin and published it in his Exercitationum Mathematicarum (1657).[15]

The work contains early game-theoretic ideas and deals in particular with the problem of points.[15] Huygens took from Pascal the concepts of a "fair game" and equitable contract (i.e., equal division when the chances are equal), and extended the argument to set up a non-standard theory of expected values.[80][81]

In 1662 Sir Robert Moray sent Huygens John Graunt's life table, and in time Huygens and his brother Lodewijk dabbled on life expectancy.[47][82]

### Unpublished work

Huygens had earlier completed a manuscript in the manner of Archimedes's On Floating Bodies entitled De Iis quae Liquido Supernatant (About parts floating above water). It was written around 1650 and was made up of three books. Although he sent the completed work to Frans van Schooten for feedback, in the end he chose not to publish it, and at one point suggested it be burned.[37][83]

The work is remarkable for rederiving Archimedes's results for the stability of the sphere and the paraboloid using a method of his own devising, providing original solutions for floating cones, parallelepipeds, and cylinders. Huygens also recognized that for homogeneous solids their specific weight and their aspect ratio are the essentials parameters of hydrostatic stability.[84]

## Natural Philosophy

Hofwijck, Christiaan Huygens's home from 1688.

Huygens was the leading European natural philosopher between Descartes and Newton.[20][85] However, unlike many of his contemporaries, Huygens had no taste for grand theoretical or philosophical systems, and generally avoided dealing with metaphysical issues (if pressed, he adhered to the Cartesian and mechanical philosophy of his time).[10][37] Instead, Huygens excelled in extending the work of his predecessors, such as Galileo, to derive solutions to unsolved physical problems that were amenable to mathematical analysis. In particular, he sought explanations that relied on contact between bodies and avoided action at a distance.[20][86]

In common with Robert Boyle and Jacques Rohault, Huygens advocated an experimentally oriented, corpuscular-mechanical natural philosophy during his Paris years. This approach was sometimes labelled "Baconian," without being inductivist or identifying with the views of Francis Bacon in a simple-minded way.[87]

After his first visit to England in 1661 and attending a meeting at Gresham College where he learned directly about Boyle's air pump experiments, Huygens spent time in late 1661 and early 1662 replicating the work. It proved a long process, brought to the surface an experimental issue ("anomalous suspension") and the theoretical issue of horror vacui, and ended in July 1663 as Huygens became a Fellow of the Royal Society. It has been said that Huygens finally accepted Boyle's view of the void, as against the Cartesian denial of it,[88] and also that the replication of results from Leviathan and the Air Pump trailed off messily.[89]

Newton's influence on John Locke was mediated by Huygens, who assured Locke that Newton's mathematics was sound, leading to Locke's acceptance of a corpuscular-mechanical physics.[90]

### Laws of motion, impact, and gravitation

Depiction from Huygens, Oeuvres Complètes: a boating metaphor underlay the way of thinking about relative motion, and so simplifying the theory of colliding bodies.

The general approach of the mechanical philosophers was to postulate theories of the kind now called "contact action." Huygens adopted this method, but not without seeing its difficulties and failures.[91] Leibniz, his student in Paris, later abandoned the theory.[92] Seeing the universe this way made the theory of collisions central to physics. Matter in motion made up the universe, and only explanations in those terms could be truly intelligible. While he was influenced by the Cartesian approach, he was less doctrinaire.[93] He studied elastic collisions in the 1650s but delayed publication for over a decade.[30]

Huygens concluded quite early that Descartes's laws for the elastic collision of two bodies must be wrong, and he formulated the correct laws.[94] An important step was his recognition of the Galilean invariance of the problems.[95] His views then took many years to be circulated. He passed them on in person to William Brouncker and Christopher Wren in London, in 1661.[96] What Spinoza wrote to Henry Oldenburg about them, in 1666 which was during the Second Anglo-Dutch War, was guarded.[97] Huygens had actually worked them out in a manuscript De Motu Corporum ex Percussione in the period 1652–6. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the Journal des Sçavans in 1669.[30]

Huygens stated what is now known as the second of Newton's laws of motion in a quadratic form.[98] In 1659 he derived the now standard formula for the centripetal force, exerted on an object describing a circular motion, for instance by the string to which it is attached. In modern notation:

${\displaystyle F_{c}={\frac {m\ v^{2}}{r}}}$

with m the mass of the object, v the velocity and r the radius. The publication of the general formula for this force in 1673 was a significant step in studying orbits in astronomy. It enabled the transition from Kepler's third law of planetary motion, to the inverse square law of gravitation.[99] The interpretation of Newton's work on gravitation by Huygens differed, however, from that of Newtonians such as Roger Cotes; he did not insist on the a priori attitude of Descartes, but neither would he accept aspects of gravitational attractions that were not attributable in principle to contact of particles.[100]

The approach used by Huygens also missed some central notions of mathematical physics, which were not lost on others. His work on pendulums came very close to the theory of simple harmonic motion; but the topic was covered fully for the first time by Newton, in Book II of his Principia Mathematica (1687).[101] In 1678 Leibniz picked out of Huygens's work on collisions the idea of conservation law that Huygens had left implicit.[102]

## Horology

Spring-driven pendulum clock, designed by Huygens and built by Salomon Coster (1657),[103] with a copy of the Horologium Oscillatorium (1673),[104] at Museum Boerhaave, Leiden.

Huygens developed the oscillating timekeeping mechanisms that have been used ever since in mechanical watches and clocks, the balance spring and the pendulum, leading to a great increase in timekeeping accuracy. In 1657, inspired by earlier research into pendulums as regulating mechanisms, Huygens invented the pendulum clock, which was a breakthrough in timekeeping and became the most accurate timekeeper for the next 275 years until the 1930s.[105] He contracted the construction of his clock designs to Salomon Coster in The Hague, who built the clock. The pendulum clock was much more accurate than the existing verge and foliot clocks and was immediately popular, quickly spreading over Europe. However, Huygens did not make much money from his invention. Pierre Séguier refused him any French rights, while Simon Douw in Rotterdam and Ahasuerus Fromanteel in London copied his design in 1658.[106] The oldest known Huygens-style pendulum clock is dated 1657 and can be seen at the Museum Boerhaave in Leiden.[107][108][109][110]

Part of the incentive for inventing the pendulum clock was to create an accurate marine chronometer that could be used to find longitude by celestial navigation during sea voyages. However, the clock proved unsuccessful as a marine timekeeper because the rocking motion of the ship disturbed the motion of the pendulum. In 1660, Lodewijk Huygens made a trial on a voyage to Spain, and reported that heavy weather made the clock useless. Alexander Bruce elbowed into the field in 1662, and Huygens called in Sir Robert Moray and the Royal Society to mediate and preserve some of his rights.[111][107] Trials continued into the 1660s, the best news coming from a Royal Navy captain Robert Holmes operating against the Dutch possessions in 1664.[112] Lisa Jardine[113] doubts that Holmes reported the results of the trial accurately, and Samuel Pepys expressed his doubts at the time: The said master [i.e. the captain of Holmes' ship] affirmed, that the vulgar reckoning proved as near as that of the watches, which [the clocks], added he, had varied from one another unequally, sometimes backward, sometimes forward, to 4, 6, 7, 3, 5 minutes; as also that they had been corrected by the usual account.

A trial for the French Academy on an expedition to Cayenne ended badly. Jean Richer suggested correction for the figure of the Earth. By the time of the Dutch East India Company expedition of 1686 to the Cape of Good Hope, Huygens was able to supply the correction retrospectively.[114]

### Pendulums

Diagram showing the evolute of a curve.

In 1673, Huygens published his major work on pendulums and horology entitled Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (The Pendulum Clock: or Geometrical demonstrations concerning the motion of pendula as applied to clocks). It is the first modern work where a physical problem is idealized by a set of parameters then analysed mathematically.[7]

Huygens's motivation came from the observation, made by Mersenne and others, that pendulums are not quite isochronous: their period depends on their width of swing, with wide swings taking slightly longer than narrow swings.[115] He tackled this problem by finding the curve down which a mass will slide under the influence of gravity in the same amount of time, regardless of its starting point; the so-called tautochrone problem. By geometrical methods which anticipated the calculus, Huygens showed it to be a cycloid, rather than the circular arc of a pendulum's bob, and therefore that pendulums needed to move on a cycloid path in order to be isochronous. The mathematics necessary to solve this problem led Huygens to develop his theory of evolutes, which he presented in Part III of his Horologium Oscillatorium.[7][116]

He also solved a problem posed by Mersenne earlier: how to calculate the period of a pendulum made of an arbitrarily-shaped swinging rigid body. This involved discovering the centre of oscillation and its reciprocal relationship with the pivot point. In the same work, he analysed the conical pendulum, consisting of a weight on a cord moving in a circle, using the concept of centrifugal force.

Huygens was the first to derive the formula for the period of an ideal mathematical pendulum (with massless rod or cord and length much longer than its swing), in modern notation:

${\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}$

with T the period, l the length of the pendulum and g the gravitational acceleration. By his study of the oscillation period of compound pendulums Huygens made pivotal contributions to the development of the concept of moment of inertia.[98]

Huygens also observed coupled oscillations: two of his pendulum clocks mounted next to each other on the same support often became synchronized, swinging in opposite directions. He reported the results by letter to the Royal Society, and it is referred to as "an odd kind of sympathy" in the Society's minutes.[117][118] This concept is now known as entrainment.[119]

### Balance spring watch

Drawing of a balance spring invented by Huygens.

Huygens developed a balance spring watch at the same time as, though independently of, Robert Hooke. Controversy over the priority persisted for centuries. In February 2006, a long-lost copy of Hooke's handwritten notes from several decades of Royal Society meetings was discovered in a cupboard in Hampshire, England, presumably tipping the evidence in Hooke's favour.[120][121]

Huygens's design employed a spiral balance spring, but he used this form of spring initially only because the balance in his first watch rotated more than one and a half turns. He later used spiral springs in more conventional watches, made for him by Thuret in Paris from around 1675. Such springs are essential in modern watches with a detached lever escapement because they can be adjusted for isochronism. Watches in Huygens's time, however, employed the very ineffective verge escapement. It interfered with the isochronal properties of any form of balance spring, spiral or otherwise.[122]

In 1675, Huygens patented a pocket watch. The watches, which were made in Paris from c. 1675 following Huygens's design, are notable for lacking a fusee for equalizing the mainspring torque. The implication is that Huygens thought that his spiral spring would isochronize the balance, in the same way that he thought that the cycloid-shaped suspension curbs on his clocks would isochronize the pendulum.[123]

## Optics

Refraction of a plane wave, explained using Huygens's principle as shown in Traité de la Lumière (1690).

In optics Huygens is remembered especially for his wave theory of light, which he first communicated in 1678 to the Académie des sciences in Paris. His theory was published in 1690 under the title Traité de la Lumière[124] (Treatise on light), which contains the first fully mathematized, mechanistic explanation of an unobervable physical phenomenon (i.e., light propagation).[10][125] Huygens refers to Ignace-Gaston Pardies, whose manuscript on optics helped him on his wave theory.[126]

Huygens assumes that the speed of light is finite, as had been shown in an experiment by Ole Christensen Rømer in 1679, but which Huygens is presumed to have already believed.[127] The challenge at the time was to explain geometrical optics, as most physical optics phenomena (such as diffraction) had not been observed or appreciated as issues. Huygens's theory posits light as radiating wavefronts, with the common notion of light rays depicting propagation normal to those wavefronts. Propagation of the wavefronts is then explained as the result of spherical waves being emitted at every point along the wave front (known today as the Huygens–Fresnel principle).[128] It assumed an omnipresent ether, with transmission through perfectly elastic particles, a revision of the view of Descartes. The nature of light was therefore a longitudinal wave.[127]

Huygens had experimented in 1672 with double refraction (birefringence) in the Iceland spar (a calcite), a phenomenon discovered in 1669 by Rasmus Bartholin. At first he could not elucidate what he found.[129] He later explained it using his wavefront theory and concept of evolutes.[125] He also developed ideas on caustics.[130] Huygens's theory of light was not widely accepted, while Newton's rival corpuscular theory of light, as found in his Opticks (1704), gained more support. One strong objection to Huygens's theory was that longitudinal waves have only a single polarization which cannot explain the observed birefringence. However, Thomas Young's interference experiments in 1801, and François Arago's detection of the Poisson spot in 1819, could not be explained through Newton's or any other particle theory, reviving Huygens's ideas and wave models. In 1821, Fresnel was able to explain birefringence as a result of light being not a longitudinal (as had been assumed) but actually a transverse wave.[131] The thus-named Huygens–Fresnel principle was the basis for the advancement of physical optics, explaining all aspects of light propagation until the development of quantum mechanics and the discovery of the photon.[132]

### Lenses

Huygens studied spherical lenses from a theoretical point of view in 1652–3, obtaining results that remained unknown until similar work by Isaac Barrow (1669). His aim was to understand telescopes.[133] Together with his brother Constantijn, Huygens began grinding his own lenses in 1655 in an effort to improve telescopes.[134] He designed in 1662 what is now called the Huygenian eyepiece, with two lenses, as a telescope ocular.[135][129] Lenses were also a common interest through which Huygens could meet socially in the 1660s with Baruch Spinoza, who ground them professionally. They had rather different outlooks on science, Spinoza being the more committed Cartesian, and some of their discussion survives in correspondence.[136] He encountered the work of Antoni van Leeuwenhoek, another lens grinder, in the field of microscopy which interested his father.[137]

Huygens also investigated the use of lenses in projectors. He is credited as the inventor of the magic lantern, described in correspondence of 1659.[138] There are others to whom such a lantern device has been attributed, such as Giambattista della Porta and Cornelis Drebbel, though Huygens's design used lens for better projection (Athanasius Kircher has also been credited for that).[139]

## Astronomy

### Saturn's rings and Titan

Huygens's explanation for the aspects of Saturn, Systema Saturnium (1659).

In 1655, Huygens was the first to propose that the rings of Saturn were "a thin, flat ring, nowhere touching, and inclined to the ecliptic”.[140] Using a refracting telescope with a 43x magnification that he designed himself,[13][14] Huygens also discovered the first of Saturn's moons, Titan.[141] In the same year he observed and sketched the Orion Nebula. His drawing, the first such known of the Orion nebula, was published in Systema Saturnium in 1659. Using his modern telescope, he succeeded in subdividing the nebula into different stars. The brighter interior now bears the name of the Huygenian region in his honour.[142] He also discovered several interstellar nebulae and some double stars.[143]

### Mars and Syrtis Major

In 1659, Huygens was the first to observe a surface feature on another planet, Syrtis Major, a volcanic plain on Mars. He used repeated observations of the movement of this feature over the course of a number of days to estimate the length of day on Mars, which he did quite accurately to 24 1/2 hours. This figure is only a few minutes off of the actual length of the Martian day of 24 hours, 37 minutes.[144]

### Planetarium

At the instigation of Jean-Baptiste Colbert, Huygens undertook the task of constructing a mechanical planetarium that could display all the planets and their moons then known circling around the Sun. Huygens completed his design in 1680 and had his clockmaker Johannes van Ceulen built it the following year. However, Colbert passed away in the interim and Huygens never got to deliver his planetarium to the French Academy of Sciences as the new minister, Fracois-Michel le Tellier, decided not to renew Huygens's contract.[145][146]

In his design, Huygens made an ingenious use of continued fractions to find the best rational approximations by which he could choose the gears with the correct number of teeth. The ratio between two gears determined the orbital periods of two planets. To move the planets around the Sun, Huygens used a clock-mechanism that could go forwards and backwards in time. Huygens claimed his planetarium was more accurate that a similar device constructed by Ole Rømer around the same time, but his planetarium design was not published until after his death in the Opuscula Posthuma (1703).[145]

### Cosmotheoros

Huygens's telescope without tube from Astroscopia Compendiaria tubi optici molimine liberata (1684).

Shortly before his death in 1695, Huygens completed Cosmotheoros. At his direction, it was to be published only posthumously by his brother, which Constantijn Jr. did in 1698.[147] In it he speculated on the existence of extraterrestrial life, on other planets, which he imagined was similar to that on Earth. Such speculations were not uncommon at the time, justified by Copernicanism or the plenitude principle. But Huygens went into greater detail,[148] although without the benefit of understanding Newton's laws of gravitation, or the fact that the atmospheres on other planets are composed of different gases.[149] The work, translated into English in its year of publication and entitled The celestial worlds discover’d, has been seen as being in the fanciful tradition of Francis Godwin, John Wilkins, and Cyrano de Bergerac, and fundamentally Utopian; and also to owe in its concept of planet to cosmography in the sense of Peter Heylin.[150][151]

Huygens wrote that availability of water in liquid form was essential for life and that the properties of water must vary from planet to planet to suit the temperature range. He took his observations of dark and bright spots on the surfaces of Mars and Jupiter to be evidence of water and ice on those planets.[152] He argued that extraterrestrial life is neither confirmed nor denied by the Bible, and questioned why God would create the other planets if they were not to serve a greater purpose than that of being admired from Earth. Huygens postulated that the great distance between the planets signified that God had not intended for beings on one to know about the beings on the others, and had not foreseen how much humans would advance in scientific knowledge.[153]

It was also in this book that Huygens published his method for estimating stellar distances. He made a series of smaller holes in a screen facing the Sun, until he estimated the light was of the same intensity as that of the star Sirius. He then calculated that the angle of this hole was 1/27,664th the diameter of the Sun, and thus it was about 30,000 times as far away, on the (incorrect) assumption that Sirius is as luminous as the Sun. The subject of photometry remained in its infancy until the time of Pierre Bouguer and Johann Heinrich Lambert.[154]

## Legacy

Portrait of Christiaan Huygens by Bernard Vaillant (1686).

During his lifetime, Huygens's influence was immense but began to fade shortly after his death. His skills as a geometer and his mechanical insights elicited the admiration of many of his contemporaries, including Newton, Leibniz, l'Hospital, and the Bernoullis.[47] For his work in physics, Huygens has been considered one of the greatest scientists in history and a prominent figure in the scientific revolution, second only to Newton in both range and the number of results obtained.[155][156]

### Mathematics and physics

In mathematics, Huygens mastered the methods of ancient Greek geometry, particularly the work of Archimedes, and was an adept user of the analytic geometry of Descartes and the infinitesimal techniques of Fermat and others.[83] His mathematical style can be characterized as geometrical infinitesimal analysis of curves and of motion. It drew inspiration and imagery from mechanics, while remaining pure mathematics in form.[72] Huygens brought this type of geometrical analysis to its greatest height but also to its conclusion, as more mathematicians turned away from classical geometry to the calculus for handling infinitesimals, limit processes, and motion.[43]

Huygens was moreover one of the first to fully employ mathematics to answer questions of physics. Often this entailed introducing a simple model for describing a complicated situation, then analyzing it starting from simple arguments to their logical consequences, developing the necessary mathematics along the way. As he wrote at the end of a draft of De vi Centrifuga:[37]

Whatever you will have supposed not impossible either concerning gravity or motion or any other matter, if then you prove something concerning the magnitude of a line, surface, or body, it will be true; as for instance, Archimedes on the quadrature of the parabola, where the tendency of heavy objects has been assumed to act through parallel lines.

Huygens favoured axiomatic presentations of his results, which demanded rigorous methods of geometric demonstration; although he allowed levels of probability in the selection of primary axioms and hypotheses used, the proof of theorems derived from these could never be in doubt.[37] His published works were seen as precise, crystal clear, and elegant, and exerted a big influence in Newton's presentation of his own major work.[157]

Besides the application of mathematics to physics and physics to mathematics, Huygens relied on mathematics as methodology, particularly its predictive power to generate new knowledge about the world.[155] In this respect, he differed from Galileo in that he used mathematics as a method of discovery and analysis, not merely as rhetoric or synthesis, and the cumulative effect of Huygens's highly successful approach created a norm for eighteenth-century scientist such as Johann Bernoulli.[37]

Although never intended for publication, Huygens made use of algebraic expressions to represent physical entities in a handful of his manuscripts on collisions.[49] This would making him one of the first to employ mathematical formulae to describe relationships in physics, as it is done today.[31]

### Later influence

Huygens's very idiosyncratic style and his reluctance in publishing his work greatly diminished his influence in the aftermath of the scientific revolution, as adherents of Leibniz’ calculus and Newton's physics took centre stage.[43][83]

His analysis of curves that satisfy certain physical properties, such as the cycloid, led to later studies of many other such curves like the caustic, the brachistochrone, the sail curve, and the catenary.[27][39] His application of mathematics to physics, such as in his analysis of birefringence, would inspire new developments in mathematical physics and rational mechanics in the next century (albeit in the language of the calculus).[10] Additionally, his pendulum clocks were the first reliable timekeepers fit for scientific use, providing an example for others of work in applied mathematics and mechanical engineering in the years following his death.[7][123]

## Works

Correspondance

Source(s):[20]

• 1650 – De Iis Quae Liquido Supernatant (About the parts above the water, unpublished).[167]
• 1651 – Theoremata de Quadratura Hyperboles, Ellipsis et Circuli, republished in Oeuvres Complètes, Tome XI.[47]
• 1654 – De Circuli Magnitudine Inventa.[37]
• 1655 – Horologium (The clock – short pamphlet on the pendulum clock).[7]
• 1656 – Epistola, qua diluuntur ea quibus 'Εξέτασις [Exetasis] Cyclometriae Gregori à Sto. Vincentio impugnata fuit.[168]
• 1656 – De Saturni Luna Observatio Nova (About the new observation of the moon of Saturn – discovery of Titan).[169]
• 1656 – De Motu Corporum ex Percussione, published posthumously in 1703.[170]
• 1657 – De Ratiociniis in Ludo Aleae (Van reeckening in spelen van geluck, translated into Latin by Frans van Schooten).[15]
• 1659 – Systema Saturnium (System of Saturn).[168]
• 1659 – De vi Centrifuga (Concerning the centrifugal force), published in 1703.[171]
• 1673 – Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (includes his theory of evolutes and designs of pendulum clocks, dedicated to Louis XIV of France).[116]
• 1684 – Astroscopia Compendiaria Tubi Optici Molimine Liberata (Compound telescopes without a tube).[47]
• 1685 – Memoriën aengaende het slijpen van glasen tot verrekijckers (dealing with the grinding of lenses).[10]
• 1686 – Old Dutch: Kort onderwijs aengaende het gebruijck der horologiën tot het vinden der lenghten van Oost en West (instructions on how to use clocks to establish the longitude at sea).[172]
• 1690 – Traité de la Lumière (translated by Silvanus P. Thompson).[8]
• 1690 – Discours de la Cause de la Pesanteur (Discourse about gravity, from 1669?).[47]
• 1691 – Lettre Touchant le Cycle Harmonique (Rotterdam, concerning the 31-tone system).[42]
• 1698 – Cosmotheoros (solar system, cosmology, life in the universe).[153]
• 1703 – Opuscula Posthuma including:[47]
• De Motu Corporum ex Percussione (Concerning the motions of colliding bodies – contains the first correct laws for collision, dating from 1656).
• Descriptio Automati Planetarii (description and design of a planetarium).
• 1724 – Novus Cyclus Harmonicus (a treatise on music, published in Leiden after Huygens's death).[42]
• 1728 – Christiani Hugenii Zuilichemii, dum viveret Zelhemii Toparchae, Opuscula Posthuma ... (pub. 1728) Alternate title: Opera Reliqua, concerning optics and physics[171]
• 1888–1950 – Huygens, Christiaan. Oeuvres complètes. Complete works, 22 volumes. Editors D. Bierens de Haan (tome=deel 1–5), J. Bosscha (6–10), D.J. Korteweg (11–15), A.A. Nijland (15), J.A. Vollgraf (16–22). The Hague:[168]
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).
Tome XX: Musique et mathématique. Musique. Mathématiques de 1666 à 1695 (1940).
Tome XXI: Cosmologie (1944).
Tome XXII: Supplément à la correspondance. Varia. Biographie de Chr. Huygens. Catalogue de la vente des livres de Chr. Huygens (1950).

## Notes

1. ^ The meaning of this painting is explained in Wybe Kuitert "Japanese Robes, Sharawadgi, and the landscape discourse of Sir William Temple and Constantijn Huygens" Garden History, 41, 2: (2013) pp.157-176, Plates II-VI and Garden History, 42, 1: (2014) p.130 ISSN 0307-1243 Online as PDF Archived 9 August 2021 at the Wayback Machine
2. ^ I. Bernard Cohen; George E. Smith (25 April 2002). The Cambridge Companion to Newton. Cambridge University Press. p. 69. ISBN 978-0-521-65696-2. Archived from the original on 16 September 2020. Retrieved 15 May 2013.
3. ^ Niccolò Guicciardini (2009). Isaac Newton on mathematical certainty and method. MIT Press. p. 344. ISBN 978-0-262-01317-8. Archived from the original on 16 September 2020. Retrieved 15 May 2013.
4. ^ "Huygens, Christiaan". Lexico UK Dictionary. Oxford University Press. Retrieved 13 August 2019.
5. ^ "Huygens". Merriam-Webster Dictionary. Retrieved 13 August 2019.
6. ^ "Huygens" Archived 15 March 2021 at the Wayback Machine. Random House Webster's Unabridged Dictionary.
7. Yoder, J. G. (2005), Grattan-Guinness, I.; Cooke, Roger; Corry, Leo; Crépel, Pierre (eds.), "Christiaan Huygens, book on the pendulum clock (1673)", Landmark Writings in Western Mathematics 1640-1940, Amsterdam: Elsevier Science, pp. 33–45, ISBN 978-0-444-50871-3
8. ^ a b c Dijksterhuis, F. J. (2005). Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Kluwer Academic Publishers.
9. ^ Shapiro, A. E. (1989). "Huygens' 'Traité de la Lumière' and Newton's 'Opticks': Pursuing and Eschewing Hypotheses". Notes and Records of the Royal Society of London. 43 (2): 223–247. ISSN 0035-9149.
10. Dijksterhuis, F.J. (2008) Stevin, Huygens and the Dutch republic. Nieuw archief voor wiskunde, 5, pp. 100-107.[1]
11. ^ Andriesse, C.D. (2005) Huygens: The Man Behind the Principle. Cambridge University Press. Cambridge: 6
12. ^ Andriesse, C.D. (2005) Huygens: The Man Behind the Principle. Cambridge University Press. Cambridge: 354
13. ^ a b Louwman, Peter (2004). "Christiaan Huygens and his telescopes". Titan - from Discovery to Encounter. 1278: 103–114. Bibcode:2004ESASP1278..103L. Archived from the original on 2 September 2021. Retrieved 12 April 2021.
14. ^ a b van Helden, Albert (2004). "Huygens, Titan, and Saturn's ring". Titan - from Discovery to Encounter. 1278: 11–29. Bibcode:2004ESASP1278...11V. Archived from the original on 15 April 2019. Retrieved 12 April 2021.
15. ^ a b c d Shafer, G. (2018). Pascal's and Huygens's game-theoretic foundations for probability. [2] Archived 5 December 2020 at the Wayback Machine
16. ^ Schneider, Ivo (1 January 2005). "Jakob Bernoulli, Ars conjectandi (1713)". Landmark Writings in Western Mathematics 1640-1940: 88–104. doi:10.1016/B978-044450871-3/50087-5. ISBN 9780444508713. Archived from the original on 15 April 2021. Retrieved 15 April 2021.
17. ^ a b Schneider, Ivo (2001), Heyde, C. C.; Seneta, E.; Crépel, P.; Fienberg, S. E. (eds.), "Christiaan Huygens", Statisticians of the Centuries, New York, NY: Springer, pp. 23–28, doi:10.1007/978-1-4613-0179-0_5, ISBN 978-1-4613-0179-0, archived from the original on 2 September 2021, retrieved 15 April 2021
18. ^ Stephen J. Edberg (14 December 2012) Christiaan Huygens Archived 2 September 2021 at the Wayback Machine, Encyclopedia of World Biography. 2004. Encyclopedia.com.
19. ^ a b "Christiaan Huygens (1629-1695)". www.saburchill.com. Archived from the original on 13 June 2017. Retrieved 16 February 2013.
20. Henk J. M. Bos (14 December 2012) Huygens, Christiaan (Also Huyghens, Christian) Archived 2 September 2021 at the Wayback Machine, Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
21. ^ R. Dugas and P. Costabel, "Chapter Two, The Birth of a new Science" in The Beginnings of Modern Science, edited by Rene Taton, 1958,1964, Basic Books, Inc.
22. ^ Strategic Affection? Gift Exchange in Seventeenth-Century Holland, by Irma Thoen, p. 127
23. ^ a b "Constantijn Huygens, Lord of Zuilichem (1596–1687), by Adelheid Rech". Archived from the original on 3 July 2017. Retrieved 16 February 2013.
24. ^ The Heirs of Archimedes: Science and the Art of War Through the Age of Enlightenment, by Brett D. Steele, p. 20
25. ^ "entoen.nu: Christiaan Huygens 1629–1695 Science in the Golden Age". Archived from the original on 8 May 2017. Retrieved 16 February 2013.
26. ^ Jozef T. Devreese (31 October 2008). 'Magic Is No Magic': The Wonderful World of Simon Stevin. WIT Press. pp. 275–6. ISBN 978-1-84564-391-1. Archived from the original on 16 June 2016. Retrieved 24 April 2013.
27. ^ a b Yoder, Joella G. (1989). Unrolling Time: Christiaan Huygens and the Mathematization of Nature. Cambridge University Press. pp. 174–175. ISBN 978-0-521-52481-0.
28. ^ H. N. Jahnke (2003). A history of analysis. American Mathematical Soc. p. 47. ISBN 978-0-8218-9050-9. Archived from the original on 30 June 2014. Retrieved 12 May 2013.
29. ^ Margret Schuchard (2007). Bernhard Varenius: (1622–1650). BRILL. p. 112. ISBN 978-90-04-16363-8. Archived from the original on 30 June 2014. Retrieved 12 May 2013.
30. ^ a b c Bunge et al. (2003), Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers, p. 470.
31. ^ a b c Aldersey-Williams, H. (2020). Dutch Light: Christiaan Huygens and the Making of Science in Europe. Pan Macmillan. pp. 122–125. ISBN 978-1-5098-9332-4. Archived from the original on 28 August 2021. Retrieved 28 August 2021.
32. ^ "Christiaan Huygens – A family affair, by Bram Stoffele, pg 80" (PDF). Archived (PDF) from the original on 12 August 2017. Retrieved 16 February 2013.
33. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. pp. 80–. ISBN 978-0-521-85090-2. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
34. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. pp. 85–6. ISBN 978-0-521-85090-2. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
35. ^ a b Bunge et al. (2003), Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers, p. 469.
36. ^ Lynn Thorndike (1 March 2003). History of Magic & Experimental Science 1923. Kessinger Publishing. p. 622. ISBN 978-0-7661-4316-6. Archived from the original on 13 October 2013. Retrieved 11 May 2013.
37. Yoder, J. (1996). "'Following in the footsteps of geometry': The mathematical world of Christiaan Huygens". DBNL. Archived from the original on 12 May 2021. Retrieved 12 May 2021.
38. ^ Leonhard Euler (1 January 1980). Clifford Truesdell (ed.). The Rational Mechanics of Flexible or Elastic Bodies 1638–1788: Introduction to Vol. X and XI. Springer. pp. 44–6. ISBN 978-3-7643-1441-5. Archived from the original on 13 October 2013. Retrieved 10 May 2013.
39. ^ a b Bukowski, J. (2008). "Christiaan Huygens and the Problem of the Hanging Chain". The College Mathematics Journal. 39 (1): 2–11. doi:10.1080/07468342.2008.11922269.
40. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. pp. 78–9. ISBN 978-0-521-85090-2. Archived from the original on 13 October 2013. Retrieved 10 May 2013.
41. ^ Yoder (1989), p. 12.
42. ^ a b c H.F. Cohen (31 May 1984). Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650. Springer. pp. 217–9. ISBN 978-90-277-1637-8. Archived from the original on 13 October 2013. Retrieved 11 May 2013.
43. ^ a b c H. J. M. Bos (1993). Lectures in the History of Mathematics. American Mathematical Soc. pp. 64–. ISBN 978-0-8218-9675-4. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
44. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. p. 134. ISBN 978-0-521-85090-2. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
45. ^ Thomas Hobbes (1997). The Correspondence: 1660–1679. Oxford University Press. p. 868. ISBN 978-0-19-823748-8. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
46. ^ Michael S. Mahoney (1994). The Mathematical Career of Pierre de Fermat: 1601–1665. Princeton University Press. pp. 67–8. ISBN 978-0-691-03666-3. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
47. Howard, N. C. (2003). "Christiaan Huygens: The construction of texts and audiences - ProQuest".
48. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. p. 126. ISBN 978-0-521-85090-2. Archived from the original on 31 December 2013. Retrieved 10 May 2013.
49. ^ a b c Hyslop, S. J. (2014). "Algebraic Collisions". Foundations of Science. 19 (1): 35–51. doi:10.1007/s10699-012-9313-8. Archived from the original on 2 September 2021. Retrieved 28 August 2021.
50. ^ Peter Louwman, Christiaan Huygens and his telescopes, Proceedings of the International Conference, 13 – 17 April 2004, ESTEC, Noordwijk, Netherlands, ESA, sp 1278, Paris 2004
51. ^ Adrian Johns (15 May 2009). The Nature of the Book: Print and Knowledge in the Making. University of Chicago Press. pp. 437–8. ISBN 978-0-226-40123-2. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
52. ^ Venus Seen on the Sun: The First Observation of a Transit of Venus by Jeremiah Horrocks. BRILL. 2 March 2012. p. xix. ISBN 978-90-04-22193-2. Archived from the original on 16 June 2016. Retrieved 23 April 2013.
53. ^ Jozef T. Devreese (2008). 'Magic Is No Magic': The Wonderful World of Simon Stevin. WIT Press. p. 277. ISBN 978-1-84564-391-1. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
54. ^ Fokko Jan Dijksterhuis (1 October 2005). Lenses And Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Springer. p. 98. ISBN 978-1-4020-2698-0. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
55. ^ Gerrit A. Lindeboom (1974). Boerhaave and Great Britain: Three Lectures on Boerhaave with Particular Reference to His Relations with Great Britain. Brill Archive. p. 15. ISBN 978-90-04-03843-1. Archived from the original on 17 June 2016. Retrieved 11 May 2013.
56. ^ David J. Sturdy (1995). Science and Social Status: The Members of the "Académie Des Sciences", 1666–1750. Boydell & Brewer. p. 17. ISBN 978-0-85115-395-7. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
57. ^ The anatomy of a scientific institution: the Paris Academy of Sciences, 1666–1803. University of California Press. 1971. p. 7 note 12. ISBN 978-0-520-01818-1. Archived from the original on 16 June 2016. Retrieved 27 April 2013.
58. ^ David J. Sturdy (1995). Science and Social Status: The Members of the "Académie Des Sciences", 1666–1750. Boydell & Brewer. pp. 71–2. ISBN 978-0-85115-395-7. Archived from the original on 17 June 2016. Retrieved 27 April 2013.
59. ^ Jacob Soll (2009). The information master: Jean-Baptiste Colbert's secret state intelligence system. University of Michigan Press. p. 99. ISBN 978-0-472-11690-4. Archived from the original on 16 June 2016. Retrieved 27 April 2013.
60. ^ A. E. Bell, Christian Huygens (1950), pp. 65–6; archive.org.
61. ^ Jonathan I. Israel (12 October 2006). Enlightenment Contested : Philosophy, Modernity, and the Emancipation of Man 1670–1752: Philosophy, Modernity, and the Emancipation of Man 1670–1752. OUP Oxford. p. 210. ISBN 978-0-19-927922-7. Archived from the original on 17 June 2016. Retrieved 11 May 2013.
62. ^ Lisa Jardine (2003). The Curious Life of Robert Hooke. HarperCollins. pp. 180–3. ISBN 0-00-714944-1.
63. ^ Joseph Needham (1974). Science and Civilisation in China: Military technology : the gunpowder epic. Cambridge University Press. p. 556. ISBN 978-0-521-30358-3. Archived from the original on 16 June 2016. Retrieved 22 April 2013.
64. ^ Joseph Needham (1986). Military Technology: The Gunpowder Epic. Cambridge University Press. p. xxxi. ISBN 978-0-521-30358-3. Archived from the original on 16 June 2016. Retrieved 22 April 2013.
65. ^ Alfred Rupert Hall (1952). Ballistics in the Seventeenth Century: A Study in the Relations of Science and War with Reference Principally to England. CUP Archive. p. 63. GGKEY:UT7XX45BRJX. Archived from the original on 17 June 2016. Retrieved 22 April 2013.
66. ^ Gottfried Wilhelm Freiherr von Leibniz (7 November 1996). Leibniz: New Essays on Human Understanding. Cambridge University Press. p. lxxxiii. ISBN 978-0-521-57660-4. Archived from the original on 16 June 2016. Retrieved 23 April 2013.
67. ^ Marcelo Dascal (2010). The practice of reason. John Benjamins Publishing. p. 45. ISBN 978-90-272-1887-2. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
68. ^ Alfred Rupert Hall (1886). Isaac Newton: Adventurer in thought. Cambridge University Press. p. 232. ISBN 0-521-56669-X.
69. ^ Curtis ROADS (1996). The computer music tutorial. MIT Press. p. 437. ISBN 978-0-262-68082-0. Archived from the original on 17 June 2016. Retrieved 11 May 2013.
70. ^ "GroteKerkDenHaag.nl" (in Dutch). GroteKerkDenHaag.nl. Archived from the original on 20 July 2017. Retrieved 13 June 2010.
71. ^ "never married; from google (christiaan huygens never married) result 1". Archived from the original on 26 November 2016. Retrieved 26 November 2016.
72. ^ a b Bos, H. J. M. (2004). Huygens and mathematics. Titan: From discovery to encounter, pp. 67-80.[3] Archived 31 August 2021 at the Wayback Machine
73. ^ Schoneveld, Cornelis W (1983). Intertraffic of the Mind: Studies in Seventeenth-century Anglo-Dutch Translation with a Checklist of Books Translated from English Into Dutch, 1600–1700. Brill Archive. p. 41. ISBN 978-90-04-06942-8. Archived from the original on 16 June 2016. Retrieved 22 April 2013.
74. ^ Brezinski, C. (2009), "Some pioneers of extrapolation methods", The Birth of Numerical Analysis, World Scientific, pp. 1–22, doi:10.1142/9789812836267_0001, ISBN 978-981-283-625-0
75. ^ Hardingham, C. H. (1932). "On the Area of a Circle". The Mathematical Gazette. 16 (221): 316–319. doi:10.2307/3605535. Archived from the original on 27 August 2021. Retrieved 27 August 2021.
76. ^ Milne, R. M. (1903). "Extension of Huygens' Approximation to a Circular Arc". The Mathematical Gazette. 2 (40): 309–311. doi:10.2307/3605128. ISSN 0025-5572.
77. ^ "Christiaan Huygens | Encyclopedia.com". www.encyclopedia.com. Archived from the original on 26 August 2016. Retrieved 13 March 2021.
78. ^ Thomas Hobbes (1997). The Correspondence: 1660–1679. Oxford University Press. p. 841. ISBN 978-0-19-823748-8. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
79. ^ p963-965, Jan Gullberg, Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 978-0-393-04002-9
80. ^ Vovk, V.; Shafer, G. (2014), "Game-theoretic probability", Introduction to Imprecise Probabilities, John Wiley & Sons, Ltd, pp. 114–134, doi:10.1002/9781118763117.ch6, ISBN 978-1-118-76311-7
81. ^ Schneider, I. (1996). "Christiaan Huygens' non-probabilistic approach to a calculus of games of chance". DBNL.
82. ^ Anders Hald (25 February 2005). A History of Probability and Statistics and Their Applications before 1750. John Wiley & Sons. p. 106. ISBN 978-0-471-72517-6. Archived from the original on 17 June 2016. Retrieved 11 May 2013.
83. ^ a b c Dijksterhuis, E. J. (1953). "Christiaan Huygens; an address delivered at the annual meeting of the Holland Society of Sciences at Haarlem, May 13th, 1950, on the occasion of the completion of Huygens's Collected Works". Centaurus; International Magazine of the History of Science and Medicine. 2 (4): 265–282. doi:10.1111/j.1600-0498.1953.tb00409.x. Archived from the original on 12 August 2021. Retrieved 12 August 2021.
84. ^ Nowacki, H. (2010). "The Heritage of Archimedes in Ship Hydrostatics: 2000 Years from Theories to Applications". The Genius of Archimedes- 23 Centuries of Influence on Mathematics, Science and Engineering: 227–249. doi:10.1007/978-90-481-9091-1_16. Archived from the original on 28 August 2021. Retrieved 28 August 2021.
85. ^ Anders Hald (25 February 2005). A History of Probability and Statistics and Their Applications before 1750. John Wiley & Sons. p. 123. ISBN 978-0-471-72517-6. Archived from the original on 17 June 2016. Retrieved 11 May 2013.
86. ^ William L. Harper (8 December 2011). Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology. Oxford University Press. pp. 206–7. ISBN 978-0-19-957040-9. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
87. ^ R. C. Olby; G. N. Cantor; J. R. R. Christie; M. J. S. Hodge (1 June 2002). Companion to the History of Modern Science. Taylor & Francis. pp. 238–40. ISBN 978-0-415-14578-7. Archived from the original on 17 June 2016. Retrieved 12 May 2013.
88. ^ David B. Wilson (1 January 2009). Seeking nature's logic. Penn State Press. p. 19. ISBN 978-0-271-04616-7. Archived from the original on 16 June 2016. Retrieved 12 May 2013.
89. ^ Stephen Shapin; Simon Schaffer (1989). Leviathan and the Air Pump. Princeton University Press. pp. 235–56. ISBN 0-691-02432-4.
90. ^ Deborah Redman (1997). The Rise of Political Economy As a Science: Methodology and the Classical Economists. MIT Press. p. 62. ISBN 978-0-262-26425-9. Archived from the original on 17 June 2016. Retrieved 12 May 2013.
91. ^ Tian Yu Cao (14 May 1998). Conceptual Developments of 20th Century Field Theories. Cambridge University Press. pp. 25–. ISBN 978-0-521-63420-5. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
92. ^ Garber and Ayers, p. 595.
93. ^ Peter Dear (15 September 2008). The Intelligibility of Nature: How Science Makes Sense of the World. University of Chicago Press. p. 25. ISBN 978-0-226-13950-0. Archived from the original on 16 June 2016. Retrieved 23 April 2013.
94. ^ The Beginnings of Modern Science, edited by Rene Taton, Basic Books, 1958, 1964.
95. ^ Garber and Ayers, pp. 666–7.
96. ^ Garber and Ayers, p. 689.
97. ^ Jonathan I. Israel (8 February 2001). Radical Enlightenment:Philosophy and the Making of Modernity 1650–1750. Oxford University Press. pp. lxii–lxiii. ISBN 978-0-19-162287-8. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
98. ^ a b Ernst Mach, The Science of Mechanics (1919), e.g. pp. 143, 172, 187 <https://archive.org/details/scienceofmechani005860mbp>.
99. ^ J. B. Barbour (1989). Absolute Or Relative Motion?: The discovery of dynamics. CUP Archive. p. 542. ISBN 978-0-521-32467-0. Archived from the original on 5 July 2014. Retrieved 23 April 2013.
100. ^ A.I. Sabra (1981). Theories of light: from Descartes to Newton. CUP Archive. pp. 166–9. ISBN 978-0-521-28436-3. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
101. ^ Richard Allen (1999). David Hartley on human nature. SUNY Press. p. 98. ISBN 978-0-7914-9451-6. Archived from the original on 17 June 2016. Retrieved 12 May 2013.
102. ^ Nicholas Jolley (1995). The Cambridge Companion to Leibniz. Cambridge University Press. p. 279. ISBN 978-0-521-36769-1. Archived from the original on 17 June 2016. Retrieved 12 May 2013.
103. ^ "Boerhaave Museum Top Collection: Hague clock (Pendulum clock) (Room 3/Showcase V20)". Museumboerhaave.nl. Archived from the original on 19 February 2011. Retrieved 13 June 2010.
104. ^ "Boerhaave Museum Top Collection: Horologium oscillatorium, siue, de motu pendulorum ad horologia aptato demonstrationes geometricae (Room 3/Showcase V20)". Museumboerhaave.nl. Archived from the original on 20 February 2011. Retrieved 13 June 2010.
105. ^ Marrison, Warren (1948). "The Evolution of the Quartz Crystal Clock". Bell System Technical Journal. 27 (3): 510–588. doi:10.1002/j.1538-7305.1948.tb01343.x. Archived from the original on 13 May 2007.
106. ^ Epstein/Prak (2010). Guilds, Innovation and the European Economy, 1400–1800. Cambridge University Press. pp. 269–70. ISBN 978-1-139-47107-7. Archived from the original on 16 June 2016. Retrieved 10 May 2013.
107. ^ a b van den Ende, H., Hordijk, B., Kersing, V., & Memel, R. (2018). The invention of the pendulum clock: A collaboration on the real story.
108. ^ van Kersen, Frits & van den Ende, Hans: Oppwindende Klokken – De Gouden Eeuw van het Slingeruurwerk 12 September – 29 November 2004 [Exhibition Catalog Paleis Het Loo]; Apeldoorn: Paleis Het Loo,2004,
109. ^ Hooijmaijers, Hans; Telling time – Devices for time measurement in museum Boerhaave – A Descriptive Catalogue; Leiden: Museum Boerhaave, 2005
110. ^ No Author given; Chistiaan Huygens 1629–1695, Chapter 1: Slingeruurwerken; Leiden: Museum Boerhaave, 1988
111. ^ Howard, N. (2008). "Marketing Longitude: Clocks, Kings, Courtiers, and Christiaan Huygens". Book History. 11: 59–88. ISSN 1098-7371.
112. ^ Michael R. Matthews (2000). Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy. Springer. pp. 137–8. ISBN 978-0-306-45880-4. Archived from the original on 16 June 2016. Retrieved 12 May 2013.
113. ^ Lisa Jardine (1 April 2008). "Chapter 10". Going Dutch: How the English Plundered Holland's Glory. HarperPress. ISBN 978-0007197323.
114. ^ Bunge et al. (2003), Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers, p. 471.
115. ^ Marin Mersenne 1647 Reflectiones Physico-Mathematicae, Paris, Chapter 19, cited in Mahoney, Michael S. (1980). "Christian Huygens: The Measurement of Time and of Longitude at Sea". Studies on Christiaan Huygens. Swets. pp. 234–270. Archived from the original on 4 December 2007. Retrieved 7 October 2010.
116. ^ a b Huygens, Christiaan; Blackwell, R. J. (1986). Christiaan Huygens' the pendulum clock, or, Geometrical demonstrations concerning the motion of pendula as applied to clocks. Ames: Iowa State University Press. ISBN 978-0-8138-0933-5.
117. ^ Thomas Birch, "The History of the Royal Society of London, for Improving of Natural Knowledge, in which the most considerable of those papers...as a supplement to the Philosophical Transactions", vol 2, (1756) p 19.
118. ^ A copy of the letter appears in C. Huygens, in Oeuvres Completes de Christian Huygens, edited by M. Nijhoff (Societe Hollandaise des Sciences, The Hague, The Netherlands, 1893), Vol. 5, p. 246 (in French).
119. ^ Spoor, P. S.; Swift, G. W. (27 July 2000). "The Huygens entrainment phenomenon and thermoacoustic engines". The Journal of the Acoustical Society of America. 108 (2): 588. doi:10.1121/1.429590. ISSN 0001-4966.
120. ^ Nature – International Weekly Journal of Science, number 439, pages 638–639, 9 February 2006
121. ^ Notes and Records of the Royal Society (2006) 60, pages 235–239, 'Report – The Return of the Hooke Folio' by Robyn Adams and Lisa Jardine
122. ^ Whitestone, S. (2012). "Christian Huygens' Lost and Forgotten Pamphlet of his Pendulum Invention". Annals of Science. 69 (1): 91–104. doi:10.1080/00033790.2011.637470. ISSN 0003-3790.
123. ^ a b Marconell, M. H. (1996). Christiaan Huygens: a foreign inventor in the Court of Louis XIV, his role as a forerunner of mechanical engineering (Phd thesis). The Open University. Archived from the original on 30 August 2021. Retrieved 30 August 2021.
124. ^ Christiaan Huygens, Traité de la lumiere... (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1.
125. ^ a b C. Huygens (1690), translated by Silvanus P. Thompson (1912), Treatise on Light, London: Macmillan, 1912; Project Gutenberg edition Archived 20 May 2020 at the Wayback Machine, 2005; Errata Archived 10 June 2017 at the Wayback Machine, 2016.
126. ^ Traité de la lumiere... (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From page 18
127. ^ a b A. Mark Smith (1987). Descartes's Theory of Light and Refraction: A Discourse on Method. American Philosophical Society. p. 70 with note 10. ISBN 978-0-87169-773-8. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
128. ^ Shapiro, A. E. (1973). "Kinematic Optics: A Study of the Wave Theory of Light in the Seventeenth Century". Archive for History of Exact Sciences. 11 (2/3): 134–266. ISSN 0003-9519.
129. ^ a b Bunge et al. (2003), Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers, p. 473.
130. ^ Ivor Grattan-Guinness (11 February 2005). Landmark Writings in Western Mathematics 1640–1940. Elsevier. p. 43. ISBN 978-0-08-045744-4. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
131. ^ Darryl J. Leiter; Sharon Leiter (1 January 2009). A to Z of Physicists. Infobase Publishing. p. 108. ISBN 978-1-4381-0922-0. Archived from the original on 16 June 2016. Retrieved 11 May 2013.
132. ^ Kubbinga, H. (1995). "Christiaan Huygens and the foundations of optics". Pure and Applied Optics: Journal of the European Optical Society Part A. 4: 723–739. doi:10.1088/0963-9659/4/6/004/meta.
133. ^ Bunge et al. (2003), Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers, p. 472.
134. ^ Robert D. Huerta (2005). Vermeer And Plato: Painting The Ideal. Bucknell University Press. p. 101. ISBN 978-0-8387-5606-5. Archived from the original on 17 June 2016. Retrieved 24 April 2013.
135. ^ Randy O. Wayne (28 July 2010). Light and Video Microscopy. Academic Press. p. 72. ISBN 978-0-08-092128-0. Archived from the original on 16 June 2016. Retrieved 24 April 2013.
136. ^ Margaret Gullan-Whur (1998). Within Reason: A Life of Spinoza. Jonathan Cape. pp. 170–1. ISBN 0-224-05046-X.
137. ^ Ivor Grattan-Guinness (11 February 2005). Landmark Writings in Western Mathematics 1640–1940. Elsevier. p. 35. ISBN 978-0-08-045744-4. Archived from the original on 16 June 2016. Retrieved 27 April 2013.
138. ^ Jordan D. Marché (2005). Theaters Of Time And Space: American Planetariums, 1930–1970. Rutgers University Press. p. 11. ISBN 978-0-8135-3576-0. Archived from the original on 17 June 2016. Retrieved 23 April 2013.
139. ^ C. D. Andriesse (25 August 2005). Huygens: The Man Behind the Principle. Cambridge University Press. p. 128. ISBN 978-0-521-85090-2. Archived from the original on 16 June 2016. Retrieved 23 April 2013.
140. ^ Baalke, R. (2011). "Historical Background of Saturn's Rings". solarviews.com. Later, it was determined that Saturn's rings were not solid but made of several smaller bodies. Archived from the original on 11 July 2021.
141. ^ Ron Baalke, Historical Background of Saturn's Rings Archived 21 March 2009 at the Wayback Machine
142. ^ Antony Cooke (1 January 2005). Visual Astronomy Under Dark Skies: A New Approach to Observing Deep Space. Springer. p. 67. ISBN 978-1-84628-149-5. Archived from the original on 17 June 2016. Retrieved 24 April 2013.
143. ^ a b Chapman, A. (1995). "Christiaan huygens (1629–1695): astronomer and mechanician". Endeavour. 19 (4): 140–145. doi:10.1016/0160-9327(95)90076-4. ISSN 0160-9327.
144. ^ "A dark spot on Mars - Syrtis Major". www.marsdaily.com. 3 February 2012. Archived from the original on 21 September 2015. Retrieved 17 May 2016.
145. ^ a b van den Bosch, D. (2018). The application of continued fractions in Christiaan Huygens planetarium.[4] Archived 13 April 2021 at the Wayback Machine
146. ^ "Amin, H. H. N. (2008). Christiaan Huygens' planetarium" (PDF). Archived (PDF) from the original on 14 April 2021. Retrieved 13 April 2021.
147. ^ Aldersey-Williams, Hugh, The Uncertain Heavens Archived 21 October 2020 at the Wayback Machine, Public Domain Review, October 21, 2020
148. ^ Philip C. Almond (27 November 2008). Adam and Eve in Seventeenth-Century Thought. Cambridge University Press. pp. 61–2. ISBN 978-0-521-09084-1. Archived from the original on 1 January 2014. Retrieved 24 April 2013.
149. ^ "Engines of Our Ingenuity 1329: Life In Outer Space – In 1698". www.houstonpublicmedia.org. University of Houston. Archived from the original on 10 April 2017. Retrieved 9 April 2017.
150. ^ Postmus, Bouwe (1987). "Plokhoy's A way pronouned: Mennonite Utopia or Millennium?". In Dominic Baker-Smith; Cedric Charles Barfoot (eds.). Between dream and nature: essays on utopia and dystopia. Amsterdam: Rodopi. pp. 86–8. ISBN 978-90-6203-959-3. Archived from the original on 1 January 2014. Retrieved 24 April 2013.
151. ^ Juliet Cummins; David Burchell (2007). Science, Literature, and Rhetoric in Early Modern England. Ashgate Publishing, Ltd. pp. 194–5. ISBN 978-0-7546-5781-1. Archived from the original on 1 January 2014. Retrieved 24 April 2013.
152. ^ "Johar Huzefa (2009) Nothing But The Facts – Christiaan Huygens". Brighthub.com. 28 September 2009. Archived from the original on 27 November 2020. Retrieved 13 June 2010.
153. ^ a b Jacob, M. (2010). The Scientific Revolution. Boston: Bedford/St. Martin's. pp. 29, 107–114.
154. ^ Russell Mccormmach (2012). Weighing the World: The Reverend John Michell of Thornhill. Springer. pp. 129–31. ISBN 978-94-007-2022-0. Archived from the original on 17 June 2016. Retrieved 12 May 2013.
155. ^ a b Gijsbers, V. (2003). "Christiaan Huygens and the scientific revolution". Proceedings of the International Conference "Titan - from discovery to encounter", 13-17 April 2004, ESTEC, Noordwijk, Netherlands. ESA Publications Division. pp. 171–178. Archived from the original on 12 August 2021. Retrieved 12 August 2021.
156. ^ Simonyi, K. (2012). A Cultural History of Physics. CRC Press. pp. 240–255. ISBN 1568813295.
157. ^ Elzinga, A. (1972). On a research program in early modern physics. Akademiförlaget.
158. Verduin, C.J. Kees (31 March 2009). "Portraits of Christiaan Huygens (1629–1695)". University of Leiden. Archived from the original on 26 August 2017. Retrieved 12 April 2018.
159. ^ Verduin, C.J. (2004). "A portrait of Christiaan Huygens together with Giovanni Domenico Cassini". In Karen, Fletcher (ed.). Titan – from discovery to encounter. 1278. Noordwijk, Netherlands: ESA Publications Division. pp. 157–170. Bibcode:2004ESASP1278..157V. ISBN 92-9092-997-9.
160. ^ Verduin, C. J. K. (2008). A portrait of Christiaan Huygens? Archived 17 March 2016 at the Wayback Machine
161. ^ "Huygen's Probe | Spacecraft". NASA Solar System Exploration. Retrieved 13 October 2021.
162. ^ "(2801) Huygens Asteroid Facts - Universe Guide". www.universeguide.com. Retrieved 13 October 2021.
163. ^ mars.nasa.gov. "Huygens Crater". NASA’s Mars Exploration Program. Retrieved 13 October 2021.
164. ^ "Mons Huygens - The Moon". the-moon.us. Retrieved 13 October 2021.
165. ^ "Huygens Software | Scientific Volume Imaging". svi.nl. Retrieved 13 October 2021.
166. ^ Ziggelaar, A. (1980). "How did the wave theory of light take shape in the mind of Christiaan Huygens?". Annals of Science. 37 (2): 179–187. doi:10.1080/00033798000200181. ISSN 0003-3790.
167. ^ L, H (1907). "Christiaan Huygens, Traité: De iis quae liquido supernatant". Nature. 76 (1972): 381. Bibcode:1907Natur..76..381L. doi:10.1038/076381a0. S2CID 4045325. Archived from the original on 28 July 2020. Retrieved 12 September 2019.
168. ^ a b c Yoder, Joella (17 May 2013). A Catalogue of the Manuscripts of Christiaan Huygens including a concordance with his Oeuvres Complètes. BRILL. ISBN 9789004235656. Archived from the original on 16 March 2020. Retrieved 12 April 2018.
169. ^ Audouin, Dollfus (2004). "Christiaan Huygens as telescope maker and planetary observer". In Karen, Fletcher (ed.). Titan – from discovery to encounter. 1278. Noordwijk, Netherlands: ESA Publications Division. pp. 115–132. Bibcode:2004ESASP1278..115D. ISBN 92-9092-997-9.
170. ^ Huygens, Christiaan (1977). Translated by Blackwell, Richard J. "Christiaan Huygens' The Motion of Colliding Bodies". Isis. 68 (4): 574–597. doi:10.1086/351876. JSTOR 230011. S2CID 144406041.
171. ^ a b Yoeder, Joella (1991). "Christiaan Huygens' Great Treasure" (PDF). Tractrix. 3: 1–13. Archived (PDF) from the original on 13 April 2018. Retrieved 12 April 2018.
172. ^ "Christiaan Huygens, Oeuvres complètes. Tome XXII. Supplément à la correspondance" (in Dutch). Digitale Bibliotheek Voor de Nederlandse Lettern. Archived from the original on 13 April 2018. Retrieved 12 April 2018.