Editing Christiaan Huygens

{{Short description|Dutch mathematician and physicist (1629–1695)}}
{{for|the ocean liner|MS Christiaan Huygens}}
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{{Infobox scientist
| name              = Christiaan Huygens
| honorific_suffix  = {{postnominals|FRS|size=100%}}
| image             = Christiaan Huygens-painting.jpeg
| caption           = Huygens by [[Caspar Netscher]] (1671), [[Museum Boerhaave]], [[Leiden]]<ref>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 [https://www.researchgate.net/publication/313059385_Japanese_robes_sharawadgi_and_the_landscape_discourse_of_Sir_William_Temple_and_Constantijn_Huygens Online as PDF] {{Webarchive|url=https://web.archive.org/web/20210809203016/https://www.researchgate.net/publication/313059385_Japanese_robes_sharawadgi_and_the_landscape_discourse_of_Sir_William_Temple_and_Constantijn_Huygens |date=9 August 2021 }}</ref>
| birth_date        = {{Birth date|df=yes|1629|04|14}}
| birth_place       = [[The Hague]], [[Dutch Republic]]
| death_date        = {{Death date and age|df=yes|1695|07|08|1629|04|14}}
| death_place       = The Hague, Dutch Republic
| field             = {{ubl|[[Mathematics]]|[[Physics]]|[[Astronomy]]|[[Mechanics]]|[[Horology]]}}
| work_institution  = {{ubl|[[Royal Society of London]]|[[French Academy of Sciences]]}}
| alma_mater        = {{ubl|[[Leiden University|University of Leiden]]|[[University of Angers]]}}
| academic_advisors = [[Frans van Schooten]]
| known_for         = {{collapsible list|[[Aerial telescope]]|[[Balance spring]]|[[Birefringence]]|[[Centrifugal force]]|[[Centripetal force]]|[[Collision]] formulae|[[Titan (moon)|Discovery of Titan]]|Explanation of [[Saturn's rings]]|[[Evolute]]|[[Gambler's ruin]]|[[Gunpowder engine|Huygens's engine]]|[[Huygenian eyepiece]]|[[Huygens–Fresnel principle]]|[[Lemniscate of Gerono|Huygens's lemniscate]]|[[Huygens–Steiner theorem]]|[[Septimal tritone|Huygens's tritone]]|[[Involute]]|[[Injection locking]]|[[Repetition pitch]]|[[Magic lantern]]|[[Pendulum clock]]|[[Problem of points]]|[[Tautochrone curve]]|[[Tractrix]]|[[Wave|Wave theory]]|[[31 equal temperament]] musical tuning}}
| signature         = File:Huygens black & white signature.jpg
}}
{{Classical mechanics}}
'''Christiaan Huygens''', [[Halen|Lord of Zeelhem]], {{postnominals|FRS}} ({{IPAc-en|ˈ|h|aɪ|ɡ|ən|z}} {{respell|HY|gənz}},<ref>{{Cite encyclopedia |url=http://www.lexico.com/definition/Huygens,+Christiaan |archive-url=https://web.archive.org/web/20200318044013/https://www.lexico.com/definition/huygens,_christiaan |url-status=dead |archive-date=2020-03-18 |title=Huygens, Christiaan |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}}</ref> {{IPAc-en|USalso|ˈ|h|ɔɪ|ɡ|ən|z}} {{respell|HOY|gənz}};<ref>{{Cite Merriam-Webster|Huygens|access-date=13 August 2019}}</ref> {{IPA|nl|ˈkrɪstijaːn ˈɦœyɣə(n)s|lang|ChristianHuygensPronunciation.ogg}}; also spelled '''Huyghens'''; {{langx|la|Hugenius}}; 14 April 1629 – 8 July 1695) was a Dutch [[mathematician]], [[physicist]], [[engineer]], [[astronomer]], and [[inventor]] who is regarded as a key figure in the [[Scientific Revolution]].<ref name=":24" /><ref name=":14" /> In physics, Huygens made seminal contributions to [[optics]] and [[mechanics]], while as an astronomer he studied the [[rings of Saturn]] and discovered its largest moon, [[Titan (moon)|Titan]]. As an engineer and inventor, he improved the design of telescopes and invented the [[pendulum clock]], the most accurate timekeeper for almost 300 years. A talented mathematician and physicist, his works contain the first idealization of a physical problem by a set of [[Mathematical model|mathematical]] [[parameter]]s, and the first mathematical and mechanistic explanation of an [[unobservable]] physical phenomenon.<ref name=":7">{{Citation |last=Yoder |first=J. G. |author-link=Joella Yoder |title=Christiaan Huygens, book on the pendulum clock (1673) |date=2005 |url=https://www.sciencedirect.com/science/article/pii/B978044450871350084X |work=Landmark Writings in Western Mathematics 1640–1940 |pages=33–45 |editor-last=Grattan-Guinness |editor-first=I. |place=Amsterdam |publisher=Elsevier Science |language=en |isbn=978-0-444-50871-3 |editor2-last=Cooke |editor2-first=Roger |editor3-last=Corry |editor3-first=Leo |editor4-last=Crépel |editor4-first=Pierre}}</ref><ref name=":6">Dijksterhuis, F.J. (2008) Stevin, Huygens and the Dutch republic. ''Nieuw archief voor wiskunde'', ''5'', pp. 100–107.[https://research.utwente.nl/files/6673130/Dijksterhuis_naw5-2008-09-2-100.pdf]</ref>

Huygens first identified the correct laws of [[elastic collision]] in his work ''De Motu Corporum ex Percussione'', completed in 1656 but published posthumously in 1703.<ref name=":28">[[Gabbey, Alan]] (1980). Huygens and mechanics. In H.J.M. Bos, M.J.S. Rudwick, H.A.M. Snelders, & R.P.W. Visser (Eds.), ''Studies on Christiaan Huygens'' (pp. 166-199). Swets & Zeitlinger B.V.</ref> In 1659, Huygens derived geometrically the formula in [[classical mechanics]] for the [[centrifugal force]] in his work ''De vi Centrifuga'', a decade before [[Isaac Newton]].<ref>Andriesse, C.D. (2005) ''Huygens: The Man Behind the Principle''. Cambridge University Press. Cambridge: 354</ref> In optics, he is best known for his [[wave theory of light]], which he described in his ''[[Treatise on Light|Traité de la Lumière]]'' (1690). His theory of light was initially rejected in favour of Newton's [[corpuscular theory of light]], until [[Augustin-Jean Fresnel]] adapted 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 [[Horology|horological research]] resulted in an extensive analysis of the [[pendulum]] in ''[[Horologium Oscillatorium]]'' (1673), regarded as one of the most important 17th century works on mechanics.<ref name=":7" /> While it contains descriptions of clock designs, most of the book is an analysis of pendular motion and a theory of [[curve]]s. In 1655, Huygens began grinding lenses with his brother Constantijn to build [[refracting telescope]]s. He discovered Saturn's biggest moon, Titan, and was the first to explain Saturn's strange appearance as due to "a thin, flat ring, nowhere touching, and inclined to the ecliptic."<ref name=":3">{{Cite journal|last=van Helden|first=Albert|date=2004|title=Huygens, Titan, and Saturn's ring|journal=Titan – from Discovery to Encounter|url=http://adsabs.harvard.edu/abs/2004ESASP1278...11V|volume=1278|pages=11–29|bibcode=2004ESASP1278...11V|access-date=12 April 2021|archive-date=15 April 2019|archive-url=https://web.archive.org/web/20190415083622/http://adsabs.harvard.edu/abs/2004ESASP1278...11V|url-status=live}}</ref> In 1662, he developed what is now called the [[Huygenian eyepiece]], a telescope with two lenses to diminish the amount of [[Dispersion (optics)|dispersion]].<ref name=":2">{{Cite journal|last=Louwman|first=Peter|date=2004|title=Christiaan Huygens and his telescopes|journal=Titan – from Discovery to Encounter|url=http://adsabs.harvard.edu/abs/2004ESASP1278..103L|volume=1278|pages=103–114|bibcode=2004ESASP1278..103L|access-date=12 April 2021|archive-date=2 September 2021|archive-url=https://web.archive.org/web/20210902210758/https://ui.adsabs.harvard.edu/abs/2004ESASP1278..103L/abstract|url-status=live}}</ref>

As a mathematician, Huygens developed the [[Evolute|theory of evolutes]] and wrote on [[Game of chance|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).<ref name=":25" /> The use of [[expectation value|expected value]]s by Huygens and others would later inspire [[Jacob Bernoulli|Jacob Bernoulli's]] work on [[Ars Conjectandi|probability theory]].<ref>{{Cite journal|date=2005-01-01|title=Jakob Bernoulli, Ars conjectandi (1713)|url=https://www.sciencedirect.com/science/article/pii/B9780444508713500875|journal=Landmark Writings in Western Mathematics 1640–1940|language=en|pages=88–104|doi=10.1016/B978-044450871-3/50087-5|last1=Schneider|first1=Ivo|isbn=9780444508713|access-date=15 April 2021|archive-date=15 April 2021|archive-url=https://web.archive.org/web/20210415033934/https://www.sciencedirect.com/science/article/pii/B9780444508713500875|url-status=live}}</ref><ref name=":5">{{Cite journal|last=Shafer|first=G.|date=2018|title=Marie-France Bru and Bernard Bru on Dice Games and Contracts|journal=Statistical Science|volume=33|issue=2|pages=277–284|doi=10.1214/17-STS639|issn=0883-4237|doi-access=free}}</ref>

==Biography==
[[File:Adriaen Hanneman - Constantijn Huygens and his-five-children.png|thumb|upright=.9|[[Constantijn Huygens|Constantijn]] surrounded by his five children (Christiaan, top right). [[Mauritshuis]], [[The Hague]].]]
Christiaan Huygens was born into a rich and influential Dutch family in [[The Hague]] on 14 April 1629, the second son of [[Constantijn Huygens]].<ref>[[Stephen J. Edberg]] (14 December 2012) [http://www.encyclopedia.com/doc/1G2-3404703173.html Christiaan Huygens] {{Webarchive|url=https://web.archive.org/web/20210902210759/https://www.encyclopedia.com/people/science-and-technology/physics-biographies/christiaan-huygens|date=2 September 2021}}, ''Encyclopedia of World Biography''. 2004. Encyclopedia.com.</ref><ref name="opendoor">{{cite web |title=Christiaan Huygens (1629–1695) |url=http://www.saburchill.com/HOS/astronomy/016.html |url-status=live |archive-url=https://web.archive.org/web/20170613164310/http://saburchill.com/HOS/astronomy/016.html |archive-date=13 June 2017 |access-date=16 February 2013 |website=www.saburchill.com}}</ref> Christiaan was named after his paternal grandfather.<ref name="completedictionary">[[Henk J. M. Bos]] (14 December 2012) [http://www.encyclopedia.com/doc/1G2-2830902105.html Huygens, Christiaan (Also Huyghens, Christian)] {{Webarchive|url=https://web.archive.org/web/20210902210759/https://www.encyclopedia.com/people/science-and-technology/physics-biographies/christiaan-huygens |date=2 September 2021 }}, ''Complete Dictionary of Scientific Biography''. 2008. Encyclopedia.com.</ref><ref>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.</ref> His mother, [[Suzanna van Baerle]], died shortly after giving birth to Huygens's sister.<ref>''Strategic Affection? Gift Exchange in Seventeenth-Century Holland'', by Irma Thoen, p. 127</ref> The couple had five children: [[Constantijn Huygens Jr.|Constantijn]] (1628), Christiaan (1629), [[Lodewijck Huygens|Lodewijk]] (1631), Philips (1632) and Suzanna (1637).<ref name="father">{{Cite web |url=http://www.essentialvermeer.com/history/huygens.html |title=Constantijn Huygens, Lord of Zuilichem (1596–1687), by Adelheid Rech |access-date=16 February 2013 |archive-date=3 July 2017 |archive-url=https://web.archive.org/web/20170703211137/http://www.essentialvermeer.com/history/huygens.html |url-status=live }}</ref>

[[Constantijn Huygens]] was a diplomat and advisor to the [[House of Orange-Nassau|House of Orange]], in addition to being a poet and a musician. He [[Republic of Letters|corresponded widely]] with intellectuals across Europe, including [[Galileo Galilei]], [[Marin Mersenne]], and [[René Descartes]].<ref>''The Heirs of Archimedes: Science and the Art of War Through the Age of Enlightenment'', by Brett D. Steele, p. 20</ref> Christiaan was educated at home until the age of sixteen, and from a young age liked to play with miniatures of [[mill (grinding)|mills]] and other machines. He received a [[liberal education]] from his father, studying languages, [[music]], [[history]], [[geography]], [[mathematics]], [[logic]], and [[rhetoric]], alongside [[dancing]], [[fencing]] and [[horse riding]].<ref name="completedictionary" /><ref name="father" />

In 1644, Huygens had as his mathematical tutor [[Jan Jansz de Jonge Stampioen|Jan Jansz Stampioen]], who assigned the 15-year-old a demanding reading list on contemporary science.<ref>{{cite book|author=Jozef T. Devreese|title='Magic Is No Magic': The Wonderful World of Simon Stevin|url=https://books.google.com/books?id=f59h2ooQGmcC&pg=PA275|access-date=24 April 2013|date=31 October 2008|publisher=WIT Press|isbn=978-1-84564-391-1|pages=275–6|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616194009/https://books.google.com/books?id=f59h2ooQGmcC&pg=PA275|url-status=live}}</ref> Descartes was later impressed by his skills in geometry, as was Mersenne, who christened him the "new [[Archimedes]]."<ref name=":0">{{Cite book |last=Dijksterhuis |first=F. J. |title=Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. |publisher=Kluwer Academic Publishers |year=2005}}</ref><ref name="opendoor"/><ref name=":8">{{Cite book|last=Yoder|first=Joella G.|title=Unrolling Time: Christiaan Huygens and the Mathematization of Nature|publisher=Cambridge University Press|year=1989|isbn=978-0-521-52481-0|pages=174–175}}</ref>

===Student years===
At sixteen years of age, Constantijn sent Huygens to study law and mathematics at [[Leiden University]], where he enrolled from May 1645 to March 1647.<ref name="completedictionary"/> [[Frans van Schooten|Frans van Schooten Jr.]], professor at Leiden's Engineering School, became private tutor to Huygens and his elder brother, Constantijn Jr., replacing Stampioen on the advice of Descartes.<ref>{{cite book|author=H. N. Jahnke|title=A history of analysis|url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PA47|access-date=12 May 2013|date=2003|publisher=American Mathematical Soc.|isbn=978-0-8218-9050-9|page=47|archive-date=30 June 2014|archive-url=https://web.archive.org/web/20140630095949/http://books.google.com/books?id=CVRZEXFVsZkC&pg=PA47|url-status=live}}</ref><ref>{{cite book|author=Margret Schuchard|title=Bernhard Varenius: (1622–1650)|url=https://books.google.com/books?id=dmArFPaY5ZgC&pg=PA112|access-date=12 May 2013|date=2007|publisher=BRILL|isbn=978-90-04-16363-8|page=112|archive-date=30 June 2014|archive-url=https://web.archive.org/web/20140630095832/http://books.google.com/books?id=dmArFPaY5ZgC&pg=PA112|url-status=live}}</ref> Van Schooten brought Huygens's mathematical education up to date, particularly on the work of [[François Viète|Viète]], Descartes, and [[Pierre de Fermat|Fermat]].<ref name=":32">{{Cite journal |last=Paoloni |first=B. |date=2022 |title=L'art de l'analyse de Christiaan Huygens de l'Algebra à la Geometria |url=https://brill.com/view/journals/rds/143/3-4/article-p423_6.xml |journal=Revue de Synthèse |volume=143 |issue=3–4 |pages=423–455 |doi=10.1163/19552343-14234034 |s2cid=254908971 |issn=1955-2343}}</ref>

After two years, starting in March 1647, Huygens continued his studies at the newly founded [[Orange College of Breda|Orange College]], in [[Breda]], where his father was a [[curator]]. Constantijn Huygens was closely involved in the new College, which lasted only to 1669; the rector was [[André Rivet]].<ref>{{cite book|author=Andriesse|first=C. D.|url=https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA80|title=Huygens: The Man Behind the Principle|publisher=Cambridge University Press|year=2005|isbn=978-0-521-85090-2|pages=80–82|access-date=|archive-url=https://web.archive.org/web/20160617112715/https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA80|archive-date=17 June 2016|url-status=live}}</ref> 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 (mathematician)|John Pell]]. His time in Breda ended around the time when his brother Lodewijk, who was enrolled at the school, duelled with another student.<ref name=":14" /><ref>{{Cite web |title=Christiaan Huygens – A family affair, by Bram Stoffele, pg 80. |url=http://www.proevenvanvroeger.nl/eindopdrachten/huygens/huygensfamily.pdf |url-status=live |archive-url=https://web.archive.org/web/20170812225640/http://proevenvanvroeger.nl/eindopdrachten/huygens/huygensfamily.pdf |archive-date=12 August 2017 |access-date=16 February 2013}}</ref> Huygens left Breda after completing his studies in August 1649 and had a stint as a diplomat on a mission with [[Henry, Duke of Nassau]].<ref name="completedictionary"/> After stays at [[County of Bentheim (district)|Bentheim]] and [[Flensburg]] in Germany, he visited [[Copenhagen]] and [[Helsingør]] in Denmark. Huygens hoped to cross the [[Øresund]] to see Descartes in [[Stockholm]] but was prevented due to Descartes' death in the interim.<ref name=":14" /><ref>{{Cite web |last=Stoffele |first=B. |date=2006 |title=Christiaan Huygens – A family affair - Proeven van Vroeger |url=https://www.yumpu.com/en/document/view/15920743/christiaan-huygens-a-family-affair-proeven-van-vroeger |access-date=2023-11-27 |website=Utrecht University |language=en}}</ref>

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.<ref name="Dictionary, p. 469">Bunge et al. (2003), ''Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers,'' p. 469.</ref>

===Early correspondence===
[[File:KettingHyugens.jpg|thumb|upright=.9|Picture of a hanging chain ([[catenary]]) in a manuscript of Huygens]]
Huygens generally wrote in French or Latin.<ref>{{cite book|author=Lynn Thorndike|title=History of Magic & Experimental Science 1923|url=https://books.google.com/books?id=Sr923sVWH_QC&pg=PA622|access-date=11 May 2013|date=1 March 2003|publisher=Kessinger Publishing|isbn=978-0-7661-4316-6|page=622|archive-date=13 October 2013|archive-url=https://web.archive.org/web/20131013023137/http://books.google.com/books?id=Sr923sVWH_QC&pg=PA622|url-status=live}}</ref> In 1646, while still a college student at Leiden, he began a correspondence with his father's friend, [[Marin Mersenne]], who died soon afterwards in 1648.<ref name="completedictionary"/> Mersenne wrote to Constantijn on his son's talent for mathematics, and flatteringly compared him to Archimedes on 3 January 1647.<ref name=":10" />

The letters show Huygens's early interest in mathematics. In October 1646 there is the [[suspension bridge]] and the demonstration that a [[Catenary|hanging chain]] is not a [[parabola]], as Galileo thought.<ref>{{cite book|author=Leonhard Euler|editor=[[Clifford Truesdell]]|title=The Rational Mechanics of Flexible or Elastic Bodies 1638–1788: Introduction to Vol. X and XI|url=https://books.google.com/books?id=gxrzm6y10EwC&pg=PA44|access-date=10 May 2013|date=1 January 1980|publisher=Springer|isbn=978-3-7643-1441-5|pages=44–6|author-link=Leonhard Euler|archive-date=13 October 2013|archive-url=https://web.archive.org/web/20131013023135/http://books.google.com/books?id=gxrzm6y10EwC&pg=PA44|url-status=live}}</ref> Huygens would later label that curve the ''catenaria'' ([[catenary]]) in 1690 while corresponding with [[Gottfried Wilhelm Leibniz|Gottfried Leibniz]].<ref name=":16">{{Cite journal|last=Bukowski|first=J.|date=2008|title=Christiaan Huygens and the Problem of the Hanging Chain|url=https://doi.org/10.1080/07468342.2008.11922269|journal=The College Mathematics Journal|volume=39|issue=1|pages=2–11|doi=10.1080/07468342.2008.11922269|s2cid=118886615}}</ref>

In the next two years (1647–48), Huygens's letters to Mersenne covered various topics, including a mathematical proof of the [[Free fall|law of free fall]], the claim by [[Grégoire de Saint-Vincent]] of [[squaring the circle|circle quadrature]], which Huygens showed to be wrong, the rectification of the ellipse, projectiles, and the [[vibrating string]].<ref>{{cite book|author=Andriesse|first=C. D.|url=https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA78|title=Huygens: The Man Behind the Principle|publisher=Cambridge University Press|year=2005|isbn=978-0-521-85090-2|pages=78–79|archive-url=https://web.archive.org/web/20131013023116/http://books.google.com/books?id=6FTqA9fwxFMC&pg=PA78|archive-date=13 October 2013|url-status=live}}</ref> Some of Mersenne's concerns at the time, such as the [[cycloid]] (he sent Huygens [[Evangelista Torricelli|Torricelli]]'s treatise on the curve), the [[Center of percussion|centre of oscillation]], and the [[gravitational constant]], were matters Huygens only took seriously later in the 17th century.<ref name=":7" /> 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.<ref name="Cohen1984">{{cite book|author=H.F. Cohen|title=Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650|url=https://books.google.com/books?id=itKDhDRaik8C&pg=PA217|access-date=11 May 2013|date=31 May 1984|publisher=Springer|isbn=978-90-277-1637-8|pages=217–9|archive-date=13 October 2013|archive-url=https://web.archive.org/web/20131013023051/http://books.google.com/books?id=itKDhDRaik8C&pg=PA217|url-status=live}}</ref>

In 1654, Huygens returned to his father's house in The Hague and was able to devote himself entirely to research.<ref name="completedictionary" /> 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.<ref name=":9">{{cite book|author=H. J. M. Bos|url=https://books.google.com/books?id=lSGvPI6LHvwC&pg=PA64|title=Lectures in the History of Mathematics|publisher=American Mathematical Soc.|year=1993|isbn=978-0-8218-9675-4|pages=64–65|archive-url=https://web.archive.org/web/20160616195750/https://books.google.com/books?id=lSGvPI6LHvwC&pg=PA64|archive-date=16 June 2016|url-status=live}}</ref>

Subsequently, Huygens developed a broad range of correspondents, though with some difficulty after 1648 due to 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]].<ref>{{cite book|author=C. D. Andriesse|title=Huygens: The Man Behind the Principle|url=https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA134|access-date=10 May 2013|date=25 August 2005|publisher=Cambridge University Press|isbn=978-0-521-85090-2|page=134|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616202240/https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA134|url-status=live}}</ref> 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 to keep Huygens in touch.<ref>{{cite book|author=Thomas Hobbes|title=The Correspondence: 1660–1679|url=https://books.google.com/books?id=GYF_mBtgIVwC&pg=PA868|access-date=10 May 2013|date=1997|publisher=Oxford University Press|isbn=978-0-19-823748-8|page=868|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616202651/https://books.google.com/books?id=GYF_mBtgIVwC&pg=PA868|url-status=live}}</ref> Through [[Pierre de Carcavi]] Huygens corresponded in 1656 with Pierre de Fermat, whom he admired greatly. 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 [[Applied mathematics|apply mathematics]] to physics, while Fermat's concerns ran to purer topics.<ref>{{cite book|author=Michael S. Mahoney|title=The Mathematical Career of Pierre de Fermat: 1601–1665|url=https://books.google.com/books?id=My19IcewAnoC&pg=PA67|access-date=10 May 2013|date=1994|publisher=Princeton University Press|isbn=978-0-691-03666-3|pages=67–8|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616213214/https://books.google.com/books?id=My19IcewAnoC&pg=PA67|url-status=live}}</ref>

===Scientific debut===
[[File:Christiaan Huygens by Jaques Clerion.jpg|thumb|upright=.9|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.<ref name=":12" /> In his early days, his mentor Frans van Schooten provided technical feedback and was cautious for the sake of his reputation.<ref>{{cite book|author=C. D. Andriesse|title=Huygens: The Man Behind the Principle|url=https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA126|access-date=10 May 2013|date=25 August 2005|publisher=Cambridge University Press|isbn=978-0-521-85090-2|page=126|archive-date=31 December 2013|archive-url=https://web.archive.org/web/20131231214115/http://books.google.com/books?id=6FTqA9fwxFMC&pg=PA126|url-status=live}}</ref>

Between 1651 and 1657, Huygens published a number of works that showed his talent for mathematics and his mastery of [[Euclidean geometry|classical]] and [[Analytic geometry|analytical geometry]], increasing his reach and reputation among mathematicians.<ref name=":10" /> Around the same time, Huygens began to question Descartes's laws of [[collision]], which were largely wrong, deriving the correct laws algebraically and later by way of geometry.<ref name=":13">{{Cite journal|last=Hyslop|first=S. J.|date=2014|title=Algebraic Collisions|url=https://doi.org/10.1007/s10699-012-9313-8|journal=Foundations of Science|language=en|volume=19|issue=1|pages=35–51|doi=10.1007/s10699-012-9313-8|s2cid=124709121|access-date=28 August 2021|archive-date=2 September 2021|archive-url=https://web.archive.org/web/20210902210801/https://link.springer.com/article/10.1007%2Fs10699-012-9313-8|url-status=live}}</ref> He showed that, for any system of bodies, the [[Center of mass|centre of gravity]] of the system remains the same in velocity and direction, which Huygens called the [[Conservation law|conservation of "quantity of movement"]]. While others at the time were studying impact, Huygens's theory of collisions was more general.<ref name=":14">{{Cite book |last=Aldersey-Williams |first=H. |url=https://books.google.com/books?id=7n7VDwAAQBAJ&q=In+the+case+of+two+bodies+which+meet%2C+the+quantity+obtained+by+taking+the+sum+of+their+masses+multiplied+by+the+squares+of+their+velocities+will+be+found+to+beequal+before+and+after+the+collision.%E2%80%99&pg=PP86 |title=Dutch Light: Christiaan Huygens and the Making of Science in Europe |date=2020 |publisher=Pan Macmillan |isbn=978-1-5098-9332-4 |pages= |language=en |access-date=28 August 2021 |archive-url=https://web.archive.org/web/20210828235517/https://books.google.com/books?id=7n7VDwAAQBAJ&pg=PP86&lpg=PP86&dq=In+the+case+of+two+bodies+which+meet,+the+quantity+obtained+by+taking+the+sum+of+their+masses+multiplied+by+the+squares+of+their+velocities+will+be+found+to+beequal+before+and+after+the+collision.%E2%80%99&source=bl&ots=98NF_u7tbn&sig=ACfU3U08HzNIaxw5RlsDedPFq0KsaPbisw&hl=en&sa=X&ved=2ahUKEwi-ioTf8NTyAhU3RjABHQnyAXcQ6AF6BAgCEAM#v=onepage&q=In%20the%20case%20of%20two%20bodies%20which%20meet,%20the%20quantity%20obtained%20by%20taking%20the%20sum%20of%20their%20masses%20multiplied%20by%20the%20squares%20of%20their%20velocities%20will%20be%20found%20to%20beequal%20before%20and%20after%20the%20collision.%E2%80%99&f=false |archive-date=28 August 2021 |url-status=live}}</ref> These results became the main reference point and the focus for further debates through correspondence and in a short article in ''[[Journal des sçavans|Journal des Sçavans]]'' but would remain unknown to a larger audience until the publication of ''De Motu Corporum ex Percussione'' (''Concerning the motion of colliding bodies'') in 1703.<ref>{{Cite book |last=Meli |first=Domenico Bertoloni |url=https://books.google.com/books?id=I6QreZN02joC |title=Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century |publisher=JHU Press |year=2006 |isbn=978-0-8018-8426-9 |pages=227–240 |language=en}}</ref><ref name=":13" />

In addition to his mathematical and mechanical works, Huygens made important scientific discoveries: he was the first to identify [[Titan (moon)|Titan]] as one of [[Saturn|Saturn's]] moons in 1655, invented the pendulum clock in 1657, and explained Saturn's strange appearance as due to a [[Rings of Saturn|ring]] in 1659; all these discoveries brought him fame across Europe.<ref name="completedictionary" /> On 3 May 1661, Huygens, together with astronomer [[Thomas Streete]] and Richard Reeve, observed the planet [[Mercury (planet)|Mercury]] transit over the Sun using Reeve's telescope in London.<ref>Peter Louwman, Christiaan Huygens and his telescopes, Proceedings of the International Conference, 13 – 17 April 2004, ESTEC, Noordwijk, Netherlands, ESA, sp 1278, Paris 2004</ref> Streete then debated the published record of [[Hevelius]], a controversy mediated by [[Henry Oldenburg]].<ref>{{cite book|author=Adrian Johns|title=The Nature of the Book: Print and Knowledge in the Making|url=https://books.google.com/books?id=ERpBdEUdhz8C&pg=PA437|access-date=23 April 2013|date=15 May 2009|publisher=University of Chicago Press|isbn=978-0-226-40123-2|pages=437–8|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617101026/https://books.google.com/books?id=ERpBdEUdhz8C&pg=PA437|url-status=live}}</ref> Huygens passed to Hevelius a manuscript of [[Jeremiah Horrocks]] on the [[transit of Venus, 1639|transit of Venus in 1639]], printed for the first time in 1662.<ref>{{cite book|title=Venus Seen on the Sun: The First Observation of a Transit of Venus by Jeremiah Horrocks|url=https://books.google.com/books?id=KlA6UCyOboUC&pg=PR19|access-date=23 April 2013|date=2 March 2012|publisher=BRILL|isbn=978-90-04-22193-2|page=xix|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616202523/https://books.google.com/books?id=KlA6UCyOboUC&pg=PR19|url-status=live}}</ref>

In that same year, [[Sir Robert Moray]] sent Huygens [[John Graunt]]'s [[life table]], and shortly after Huygens and his brother Lodewijk dabbled on [[life expectancy]].<ref name=":12" /><ref>{{cite book |author=Anders Hald |url=https://books.google.com/books?id=pOQy6-qnVx8C&pg=PA106 |title=A History of Probability and Statistics and Their Applications before 1750 |date=25 February 2005 |publisher=John Wiley & Sons |isbn=978-0-471-72517-6 |page=106 |access-date=11 May 2013 |archive-url=https://web.archive.org/web/20160617035707/https://books.google.com/books?id=pOQy6-qnVx8C&pg=PA106 |archive-date=17 June 2016 |url-status=live}}</ref> Huygens eventually created the first graph of a continuous distribution function under the assumption of a uniform [[Mortality rate|death rate]], and used it to solve problems in [[Annuity|joint annuities]].<ref>Hacking, I. (2006). ''The emergence of probability'' (p. 135). Cambridge University Press.</ref> Contemporaneously, Huygens, who played the [[harpsichord]], took an interest in [[Simon Stevin|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.<ref>{{cite book|author=Jozef T. Devreese|title='Magic Is No Magic': The Wonderful World of Simon Stevin|url=https://books.google.com/books?id=f59h2ooQGmcC&pg=PA277|access-date=11 May 2013|date=2008|publisher=WIT Press|isbn=978-1-84564-391-1|page=277|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616213542/https://books.google.com/books?id=f59h2ooQGmcC&pg=PA277|url-status=live}}</ref><ref>{{cite book|author=Fokko Jan Dijksterhuis|title=Lenses And Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century|url=https://books.google.com/books?id=KDBXCvx0-0oC&pg=PA98|access-date=11 May 2013|date=1 October 2005|publisher=Springer|isbn=978-1-4020-2698-0|page=98|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616221604/https://books.google.com/books?id=KDBXCvx0-0oC&pg=PA98|url-status=live}}</ref> For his contributions to science, the [[Royal Society]] of London elected Huygens a Fellow in 1663, making him its first foreign member when he was just 34 years old.<ref>{{Cite journal |last=Kemeny |first=Maximilian Alexander |date=2016-03-31 |title="A Certain Correspondence": The Unification of Motion from Galileo to Huygens |url=https://ses.library.usyd.edu.au/handle/2123/15733 |journal=The University of Sydney |language=en |pages=80}}</ref><ref>{{cite book|author=Gerrit A. Lindeboom|title=Boerhaave and Great Britain: Three Lectures on Boerhaave with Particular Reference to His Relations with Great Britain|url=https://books.google.com/books?id=yOIUAAAAIAAJ&pg=PP15|access-date=11 May 2013|date=1974|publisher=Brill Archive|isbn=978-90-04-03843-1|page=15|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113611/https://books.google.com/books?id=yOIUAAAAIAAJ&pg=PP15|url-status=live}}</ref>

===France===
[[File:Christiaan-huygens2.jpg|thumb|upright=.9|Huygens, right of centre, from ''{{lang|fr|L'établissement de l'Académie des Sciences et fondation de l'observatoire}}, 1666'' by [[Henri Testelin]] (c. 1675)]]
The [[Montmor Academy]], started in the mid-1650s, was the form the old Mersenne circle took after his death.<ref>{{cite book|author=David J. Sturdy|title=Science and Social Status: The Members of the "Académie Des Sciences", 1666–1750|url=https://books.google.com/books?id=xLsNxkRXiNAC&pg=PA17|access-date=11 May 2013|date=1995|publisher=Boydell & Brewer|isbn=978-0-85115-395-7|page=17|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616211016/https://books.google.com/books?id=xLsNxkRXiNAC&pg=PA17|url-status=live}}</ref> Huygens took part in its debates and supported those favouring experimental demonstration as a check on amateurish attitudes.<ref>{{cite book|title=The anatomy of a scientific institution: the Paris Academy of Sciences, 1666–1803|url=https://books.google.com/books?id=_G-MCYFN7R4C&pg=PA7|access-date=27 April 2013|date=1971|publisher=University of California Press|isbn=978-0-520-01818-1|page=7 note 12|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616201537/https://books.google.com/books?id=_G-MCYFN7R4C&pg=PA7|url-status=live}}</ref> He visited Paris a third time in 1663; when the Montmor Academy closed down the next year, Huygens advocated for a more [[Baconian method|Baconian]] program in science. Two years later, in 1666, he moved to Paris on an invitation to fill a leadership position at [[Louis XIV|King Louis XIV]]'s new French [[French Academy of Sciences|Académie des sciences]].<ref>{{cite book|author=David J. Sturdy|title=Science and Social Status: The Members of the "Académie Des Sciences", 1666–1750|url=https://books.google.com/books?id=xLsNxkRXiNAC&pg=PA71|access-date=27 April 2013|date=1995|publisher=Boydell & Brewer|isbn=978-0-85115-395-7|pages=71–2|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617112244/https://books.google.com/books?id=xLsNxkRXiNAC&pg=PA71|url-status=live}}</ref>

While at the Académie in Paris, Huygens had an important patron and correspondent in [[Jean-Baptiste Colbert]], First Minister to Louis XIV.<ref>{{cite book|author=Jacob Soll|title=The information master: Jean-Baptiste Colbert's secret state intelligence system|url=https://books.google.com/books?id=vVjru_uV_hoC&pg=PA99|access-date=27 April 2013|date=2009|publisher=University of Michigan Press|isbn=978-0-472-11690-4|page=99|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616220714/https://books.google.com/books?id=vVjru_uV_hoC&pg=PA99|url-status=live}}</ref> His relationship with the French Académie 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.<ref>A. E.  Bell, ''Christian Huygens'' (1950), pp. 65–6; [https://archive.org/stream/christianhuygens029504mbp#page/n79/mode/2up archive.org.]</ref> However, the aftermath of the [[Franco-Dutch War]] (1672–78), and particularly England's role in it, may have damaged his later relationship with the Royal Society.<ref>{{cite book|author=Jonathan I. Israel|title=Enlightenment Contested : Philosophy, Modernity, and the Emancipation of Man 1670–1752: Philosophy, Modernity, and the Emancipation of Man 1670–1752|url=https://books.google.com/books?id=7qAeKpIIxCsC&pg=PA210|access-date=11 May 2013|date=12 October 2006|publisher=OUP Oxford|isbn=978-0-19-927922-7|page=210|author-link=Jonathan I. Israel|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113007/https://books.google.com/books?id=7qAeKpIIxCsC&pg=PA210|url-status=live}}</ref> [[Robert Hooke]], as a Royal Society representative, lacked the finesse to handle the situation in 1673.<ref>{{cite book |author=Lisa Jardine |title=The Curious Life of Robert Hooke |date=2003 |publisher=HarperCollins |isbn=0-00-714944-1 |pages=180–3|author-link=Lisa Jardine }}</ref>

The physicist and inventor [[Denis Papin]] was an assistant to Huygens from 1671.<ref>{{cite book|author=Joseph Needham|title=Science and Civilisation in China: Military technology : the gunpowder epic|url=https://books.google.com/books?id=BZxSnd2Xyb0C&pg=PA556|access-date=22 April 2013|date=1974|publisher=Cambridge University Press|isbn=978-0-521-30358-3|page=556|author-link=Joseph Needham|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616221807/https://books.google.com/books?id=BZxSnd2Xyb0C&pg=PA556|url-status=live}}</ref> One of their projects, which did not bear fruit directly, was the [[gunpowder engine]], a precursor of the [[internal combustion engine]] that used [[gunpowder]] as its fuel.<ref>{{Cite book |url=https://books.google.com/books?id=6zYTD5kUyZkC&pg=PA2 |title=Internal Combustion Engines |date=1988 |publisher=Academic Press |isbn=978-0-12-059790-1 |editor-last=Arcoumanis |editor-first=Constantine |series=Combustion Treatise |location= |pages=2}}</ref><ref>{{cite book|author=Joseph Needham|title=Military Technology: The Gunpowder Epic|url=https://books.google.com/books?id=hNcZJ35dIyUC&pg=PR31|access-date=22 April 2013|date=1986|publisher=Cambridge University Press|isbn=978-0-521-30358-3|page=xxxi|author-link=Joseph Needham|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616211010/https://books.google.com/books?id=hNcZJ35dIyUC&pg=PR31|url-status=live}}</ref> Huygens made further astronomical observations at the Académie using the [[Paris Observatory|observatory]] recently completed in 1672. He introduced [[Nicolaas Hartsoeker]] to French scientists such as [[Nicolas Malebranche]] and [[Giovanni Cassini]] in 1678.<ref name=":14" /><ref>{{Cite journal|last=Abou-Nemeh|first=S. C.|date=2013|title=The Natural Philosopher and the Microscope: Nicolas Hartsoeker Unravels Nature's "Admirable Œconomy"|url=https://doi.org/10.1177/007327531305100101|journal=History of Science|language=en|volume=51|issue=1|pages=1–32|doi=10.1177/007327531305100101|s2cid=141248558|issn=0073-2753}}</ref>

The young diplomat Leibniz met Huygens while visiting Paris in 1672 on a vain mission to meet the French Foreign Minister [[Arnauld de Pomponne]]. Leibniz was working on a [[Stepped reckoner|calculating machine]] at the time and, after a short visit to London in early 1673, he was tutored in mathematics by Huygens until 1676.<ref name="Leibniz1996">{{cite book|author=Gottfried Wilhelm Freiherr von Leibniz|title=Leibniz: New Essays on Human Understanding|url=https://books.google.com/books?id=vD6nSUSbL7IC&pg=RA1-PR82|access-date=23 April 2013|date=7 November 1996|publisher=Cambridge University Press|isbn=978-0-521-57660-4|page=lxxxiii|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616191719/https://books.google.com/books?id=vD6nSUSbL7IC&pg=RA1-PR82|url-status=live}}</ref> An extensive correspondence ensued over the years, in which Huygens showed at first reluctance to accept the advantages of Leibniz's [[infinitesimal calculus]].<ref>{{cite book|author=Marcelo Dascal|title=The practice of reason|url=https://books.google.com/books?id=4XhKkK9Ms70C&pg=PA45|access-date=23 April 2013|date=2010|publisher=John Benjamins Publishing|isbn=978-90-272-1887-2|page=45|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617035625/https://books.google.com/books?id=4XhKkK9Ms70C&pg=PA45|url-status=live}}</ref>

===Final years===
[[File:Hofwijck (3).JPG|thumb|222px|[[Hofwijck]], Huygens's summer home; now a museum]]
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.<ref name="Dictionary, p. 469"/>

On his third visit to England, Huygens met Newton in person on 12 June 1689. They spoke about [[Iceland spar]], and subsequently corresponded about resisted motion.<ref>{{cite book |author=Alfred Rupert Hall |title=Isaac Newton: Adventurer in thought |date=1996 |publisher=Cambridge University Press |page=[https://archive.org/details/isaacnewtonadven0000hall/page/232 232] |url=https://archive.org/details/isaacnewtonadven0000hall/page/232 |url-access=registration |author-link=Alfred Rupert Hall |isbn=9780631179023}}</ref>

Huygens returned to mathematical topics in his last years and observed the acoustical phenomenon now known as [[flanging]] in 1693.<ref>{{cite book|author=Curtis ROADS|title=The computer music tutorial|url=https://books.google.com/books?id=nZ-TetwzVcIC&pg=PA437|access-date=11 May 2013|date=1996|publisher=MIT Press|isbn=978-0-262-68082-0|page=437|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113110/https://books.google.com/books?id=nZ-TetwzVcIC&pg=PA437|url-status=live}}</ref> Two years later, on 8 July 1695, Huygens died in The Hague and was buried, like his father before him, in an unmarked grave at the [[Grote Kerk, The Hague|Grote Kerk]].<ref>{{cite web |url=http://www.grotekerkdenhaag.nl/ |title=GroteKerkDenHaag.nl |language=nl |publisher=GroteKerkDenHaag.nl |access-date=13 June 2010 |archive-url=https://web.archive.org/web/20170720102056/http://www.grotekerkdenhaag.nl/ |archive-date=20 July 2017 |url-status=dead }}</ref>

Huygens never married.<ref>{{Cite web|title=Christiaan Huygens|url=https://biography.yourdictionary.com/christiaan-huygens|access-date=2022-02-16|website=biography.yourdictionary.com|language=en}}</ref>

==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.<ref name=":15">Bos, H. J. M. (2004). Huygens and mathematics. ''Titan: From discovery to encounter'', pp. 67–80.[http://adsabs.harvard.edu/pdf/2004ESASP1278...67B] {{Webarchive|url=https://web.archive.org/web/20210831042659/http://adsabs.harvard.edu/pdf/2004ESASP1278...67B|date=31 August 2021}}</ref> Huygens's preferred method in his published works was that of Archimedes, though he made use of Descartes's analytic geometry and Fermat's [[Adequality|infinitesimal techniques]] more extensively in his private notebooks.<ref name="completedictionary" /><ref name=":32" />

=== Published works ===
==== ''Theoremata de Quadratura'' ====
[[File:Fig. 2. The second stage of the research on the topic of circle quadrature.tif|thumb|upright=.9|Huygens's first publication was in the field of [[Quadrature (geometry)|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 [[House of Elzevir|Elzeviers]] in [[Leiden]] in 1651.<ref name=":12">{{Cite thesis|last=Howard|first=N. C.|title=Christiaan Huygens: The construction of texts and audiences.|date=2003|degree=Master's|publisher=Indiana University|url=https://www.proquest.com/openview/73ddff1a6bcedaa2f3a8f9432dcae2e7/1?pq-origsite=gscholar&cbl=18750&diss=y}}</ref> The first part of the work contained theorems for computing the areas of hyperbolas, ellipses, and circles that paralleled Archimedes's work on conic sections, particularly his ''[[Quadrature of the Parabola]]''.<ref name=":10" /> The second part included a refutation to Grégoire de Saint-Vincent's claims on circle quadrature, which he had discussed with Mersenne earlier.

Huygens demonstrated that the centre of gravity of a segment of any [[hyperbola]], [[ellipse]], or [[circle]] was directly related to the area of that segment. He was then able to show the relationships between triangles inscribed in conic sections and the centre of gravity for those sections. By generalizing these theorems to cover all conic sections, Huygens extended classical methods to generate new results.<ref name="completedictionary" />

Quadrature and rectification were live issues in the 1650s and, through Mylon, Huygens participated in the controversy surrounding [[Thomas Hobbes]]. Persisting in highlighting his mathematical contributions, he made an international reputation.<ref>{{cite book|author=Schoneveld, Cornelis W|url=https://books.google.com/books?id=1s4UAAAAIAAJ&pg=PA41|title=Intertraffic of the Mind: Studies in Seventeenth-century Anglo-Dutch Translation with a Checklist of Books Translated from English Into Dutch, 1600–1700|date=1983|publisher=Brill Archive|isbn=978-90-04-06942-8|page=41|access-date=22 April 2013|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616220726/https://books.google.com/books?id=1s4UAAAAIAAJ&pg=PA41|url-status=live}}</ref>

==== ''De Circuli Magnitudine Inventa'' ====
Huygens's next publication was ''De Circuli Magnitudine Inventa'' (''New findings on the magnitude 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 a Circle|''Measurement of the Circle'']], showing that the ratio of the circumference to its diameter or [[Pi|pi ({{pi}})]] must lie in the first third of that interval.<ref name=":12" />

Using a technique equivalent to [[Richardson extrapolation]],<ref>{{Citation|last=Brezinski|first=C.|title=Some pioneers of extrapolation methods|date=2009|url=https://www.worldscientific.com/doi/10.1142/9789812836267_0001|work=The Birth of Numerical Analysis|pages=1–22|publisher=World Scientific|doi=10.1142/9789812836267_0001|isbn=978-981-283-625-0}}</ref> Huygens was able to shorten the inequalities used in Archimedes's method; in this case, by using the centre of the gravity of a segment of a parabola, he was able to approximate the centre of gravity of a segment of a circle, resulting in a faster and accurate approximation of the circle quadrature.<ref>{{Cite journal|last=Hardingham|first=C. H.|date=1932|title=On the Area of a Circle|url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-the-area-of-a-circle/FC23B51A22878BB3DD9EF86EEB0C8D1E|journal=The Mathematical Gazette|language=en|volume=16|issue=221|pages=316–319|doi=10.2307/3605535|jstor=3605535|s2cid=134068167 |access-date=27 August 2021|archive-date=27 August 2021|archive-url=https://web.archive.org/web/20210827044550/https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-the-area-of-a-circle/FC23B51A22878BB3DD9EF86EEB0C8D1E|url-status=live}}</ref> From these theorems, Huygens obtained two set of values for {{pi}}: the first between 3.1415926 and 3.1415927, and the second between 3.1415926533 and 3.1415926538.<ref>{{Cite journal|last=Milne|first=R. M.|date=1903|title=Extension of Huygens' Approximation to a Circular Arc|url=https://www.jstor.org/stable/3605128|journal=The Mathematical Gazette|volume=2|issue=40|pages=309–311|doi=10.2307/3605128|jstor=3605128|s2cid=125405606 |issn=0025-5572}}</ref>

Huygens also showed that, in the case of the [[hyperbola]], the same approximation with parabolic segments produces a quick and simple method to calculate [[logarithm]]s.<ref>{{Cite web|title=Christiaan Huygens {{!}} Encyclopedia.com|url=https://www.encyclopedia.com/people/science-and-technology/physics-biographies/christiaan-huygens|access-date=2021-03-13|website=www.encyclopedia.com|archive-date=26 August 2016|archive-url=https://web.archive.org/web/20160826174330/http://www.encyclopedia.com/topic/Christiaan_Huygens.aspx|url-status=live}}</ref> 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'').<ref name=":12" />

==== ''De Ratiociniis in Ludo Aleae'' ====
Huygens became interested in [[Game of chance|games of chance]] after he visited Paris in 1655 and encountered the work of Fermat, [[Blaise Pascal]] and [[Girard Desargues]] years earlier.<ref>{{cite book|author=Malcolm|first=N.|url=https://books.google.com/books?id=GYF_mBtgIVwC&pg=PA841|title=The Correspondence of Thomas Hobbes: 1660–1679|publisher=Oxford University Press|year=1997|isbn=978-0-19-823748-8|page=841|archive-url=https://web.archive.org/web/20160616195730/https://books.google.com/books?id=GYF_mBtgIVwC&pg=PA841|archive-date=16 June 2016|url-status=live}}</ref> 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'').<ref name=":4">{{Citation|last=Schneider|first=Ivo|title=Christiaan Huygens|date=2001|url=https://doi.org/10.1007/978-1-4613-0179-0_5|work=Statisticians of the Centuries|pages=23–28|editor-last=Heyde|editor-first=C. C.|archive-url=https://web.archive.org/web/20210902210759/https://link.springer.com/chapter/10.1007%2F978-1-4613-0179-0_5|place=New York, NY|publisher=Springer|language=en|doi=10.1007/978-1-4613-0179-0_5|isbn=978-1-4613-0179-0|access-date=2021-04-15|archive-date=2 September 2021|editor2-last=Seneta|editor2-first=E.|editor3-last=Crépel|editor3-first=P.|editor4-last=Fienberg|editor4-first=S. E.|url-status=live}}</ref><ref>p963-965, [[Jan Gullberg]], Mathematics from the birth of numbers, W. W. Norton & Company; {{ISBN|978-0-393-04002-9}}</ref> Frans van Schooten translated the original Dutch manuscript into Latin and published it in his ''Exercitationum Mathematicarum'' (1657).<ref>{{Cite journal|last=Whiteside|first=D. T.|date=March 1964|title=Book Review: Frans van Schooten der Jüngere (Boethius. Texte und Abhandlungen zur Geschichte der exakten Wissenschaften. Band II)|url=http://journals.sagepub.com/doi/10.1177/007327536400300116|journal=History of Science|language=en|volume=3|issue=1|pages=146–148|doi=10.1177/007327536400300116|s2cid=163762875|issn=0073-2753}}</ref><ref name=":25">{{Cite journal|last=Stigler|first=S. M.|date=2007|title=Chance Is 350 Years Old|url=http://www.tandfonline.com/doi/full/10.1080/09332480.2007.10722870|journal=Chance|language=en|volume=20|issue=4|pages=26–30|doi=10.1080/09332480.2007.10722870|s2cid=63701819|issn=0933-2480}}</ref>

The work contains early [[Game theory|game-theoretic]] ideas and deals in particular with the [[problem of points]].<ref name=":5" /><ref name=":25" /> 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.<ref>{{Citation|last1=Vovk|first1=V.|title=Game-theoretic probability|date=2014|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118763117.ch6|work=Introduction to Imprecise Probabilities|pages=114–134|publisher=John Wiley & Sons, Ltd|language=en|doi=10.1002/9781118763117.ch6|isbn=978-1-118-76311-7|last2=Shafer|first2=G.}}</ref> His success in applying algebra to the realm of chance, which hitherto seemed inaccessible to mathematicians, demonstrated the power of combining Euclidean synthetic proofs with the symbolic reasoning found in the works of Viète and Descartes.<ref>{{Cite journal |last=Schneider |first=I. |date=1996 |title=Christiaan Huygens' non-probabilistic approach to a calculus of games of chance |url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0017.php |journal=De Zeventiende Eeuw. Jaargang |volume=12 |pages=171–183}}</ref>

Huygens included five challenging problems at the end of the book that became the standard test for anyone wishing to display their mathematical skill in games of chance for the next sixty years.<ref>Hacking, I. (2006). ''The emergence of probability'' (p. 119). Cambridge University Press.</ref> People who worked on these problems included [[Abraham de Moivre]], Jacob Bernoulli, [[Johannes Hudde]], [[Baruch Spinoza]], and Leibniz.

=== Unpublished work ===
[[File:Huygens floating bodies.jpg|thumb|upright=1.1|Huygens's results for the stability of a floating rectangular [[parallelepiped]]]]
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 liquids''). 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 Huygens chose not to publish it, and at one point suggested it be burned.<ref name=":10" /><ref name=":11" /> Some of the results found here were not rediscovered until the eighteenth and nineteenth centuries.<ref name=":28" />

Huygens first re-derives Archimedes's solutions for the stability of the sphere and the paraboloid by a clever application of Torricelli's principle (i.e., that bodies in a system move only if their centre of gravity descends).<ref>{{Cite book|last=Gaukroger|first=S.|url=https://books.google.com/books?id=tCeQDwAAQBAJ&dq=Alan+Gabbey+%E2%80%98Huygens+and+Mechanics%E2%80%99&pg=PA425|title=The Emergence of a Scientific Culture: Science and the Shaping of Modernity 1210-1685|publisher=Clarendon Press|year=2008|isbn=978-0-19-156391-1|pages=424–425|language=en}}</ref> He then proves the general theorem that, for a floating body in equilibrium, the distance between its centre of gravity and its submerged portion is at a minimum.<ref name=":28" /> Huygens uses this theorem to arrive at original solutions for the stability of floating [[cone]]s, [[parallelepiped]]s, and [[cylinder]]s, in some cases through a full cycle of rotation.<ref>{{Citation|last1=Nowacki|first1=H.|title=Historical Roots of the Theory of Hydrostatic Stability of Ships|date=2011|url=https://doi.org/10.1007/978-94-007-1482-3_8|work=Contemporary Ideas on Ship Stability and Capsizing in Waves|pages=141–180|editor-last=Almeida Santos Neves|editor-first=M.|series=Fluid Mechanics and Its Applications|place=Dordrecht|publisher=Springer Netherlands|language=en|doi=10.1007/978-94-007-1482-3_8|isbn=978-94-007-1482-3|access-date=2021-10-26|last2=Ferreiro|first2=L. D.|editor2-last=Belenky|editor2-first=V. L.|editor3-last=de Kat|editor3-first=J. O.|editor4-last=Spyrou|editor4-first=K.}}</ref> His approach was thus equivalent to the principle of [[virtual work]]. Huygens was also the first to recognize that, for these homogeneous solids, their specific weight and their [[aspect ratio]] are the essentials parameters of [[Hydrostatics|hydrostatic stability]].<ref>{{Cite book |last=Capecchi |first=D. |url=https://books.google.com/books?id=8qUXV6lo8-wC&pg=PA2 |title=History of Virtual Work Laws: A History of Mechanics Prospective |publisher=Springer Science & Business Media |year=2012 |isbn=978-88-470-2056-6 |pages=187–188 |language=en}}</ref><ref>{{Cite book|last=Nowacki|first=H.|chapter=The Heritage of Archimedes in Ship Hydrostatics: 2000 Years from Theories to Applications |date=2010|title=The Genius of Archimedes -- 23 Centuries of Influence on Mathematics, Science and Engineering|series=History of Mechanism and Machine Science|volume=11|pages=227–249|doi=10.1007/978-90-481-9091-1_16|isbn=978-90-481-9090-4|s2cid=107630338}}</ref>

==Natural philosophy==
Huygens was the leading European natural philosopher between Descartes and Newton.<ref name="completedictionary" /><ref>{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C&pg=PA123|access-date=11 May 2013|date=25 February 2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=123|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617101132/https://books.google.com/books?id=pOQy6-qnVx8C&pg=PA123|url-status=live}}</ref> 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 [[Cartesianism|Cartesian philosophy]] of his time).<ref name=":6" /><ref name=":10">{{Cite web|last=Yoder|first=J.|date=1996|title='Following in the footsteps of geometry': The mathematical world of Christiaan Huygens|url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php|url-status=live|access-date=2021-05-12|website=DBNL|archive-date=12 May 2021|archive-url=https://web.archive.org/web/20210512223641/https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php}}</ref> 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]].<ref name="completedictionary" /><ref>{{cite book|author=William L. Harper|title=Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology|url=https://books.google.com/books?id=oKGlHjDzGjQC&pg=PA206|access-date=23 April 2013|date=8 December 2011|publisher=Oxford University Press|isbn=978-0-19-957040-9|pages=206–7|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113100/https://books.google.com/books?id=oKGlHjDzGjQC&pg=PA206|url-status=live}}</ref>

In common with [[Robert Boyle]] and [[Jacques Rohault]], Huygens advocated an experimentally oriented, mechanical natural philosophy during his Paris years.<ref>{{cite book|author1=R. C. Olby|author2=G. N. Cantor|author3=J. R. R. Christie|author4=M. J. S. Hodge|title=Companion to the History of Modern Science|url=https://books.google.com/books?id=NpGE5MDElvwC&pg=PA238|access-date=12 May 2013|date=1 June 2002|publisher=Taylor & Francis|isbn=978-0-415-14578-7|pages=238–40|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617101448/https://books.google.com/books?id=NpGE5MDElvwC&pg=PA238|url-status=live}}</ref> Already in his first visit to England in 1661, Huygens had learnt about Boyle's [[air pump]] experiments during a meeting at [[Gresham College and the formation of the Royal Society|Gresham College]]. Shortly afterwards, he reevaluated Boyle's experimental design and developed a series of experiments meant to test a new hypothesis.<ref name=":33">{{Cite book |last1=Gross |first1=Alan G. |url=https://books.google.com/books?id=uXbI1HEjY90C |title=Communicating Science: The Scientific Article from the 17th Century to the Present |last2=Harmon |first2=Joseph E. |last3=Reidy |first3=Michael S. |date=2002-04-11 |publisher=Oxford University Press |isbn=978-0-19-535069-2 |language=en}}</ref> It proved a yearslong process that brought to the surface a number of experimental and theoretical issues, and which ended around the time he became a Fellow of the Royal Society.<ref>{{cite book |author=David B. Wilson |url=https://books.google.com/books?id=53w2gMknsMYC&pg=PA19 |title=Seeking nature's logic |date=1 January 2009 |publisher=Penn State Press |isbn=978-0-271-04616-7 |page=19 |access-date=12 May 2013 |archive-url=https://web.archive.org/web/20160616192313/https://books.google.com/books?id=53w2gMknsMYC&pg=PA19 |archive-date=16 June 2016 |url-status=live}}</ref> Despite the [[Reproducibility|replication of results]] of Boyle's experiments trailing off messily, Huygens came to accept Boyle's view of the void against the Cartesian denial of it.<ref>{{cite book|author1=Shapin|first=S.|title=Leviathan and the Air Pump|title-link=Leviathan and the Air Pump|author2=Simon Schaffer|publisher=Princeton University Press|year=1989|isbn=0-691-02432-4|pages=235–256|author1-link=Stephen Shapin|author2-link=Simon Schaffer}}</ref>

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.<ref name="Redman1997">{{cite book|author=Deborah Redman|title=The Rise of Political Economy As a Science: Methodology and the Classical Economists|url=https://books.google.com/books?id=1faeMedY8k8C&pg=PA62|access-date=12 May 2013|date=1997|publisher=MIT Press|isbn=978-0-262-26425-9|page=62|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113758/https://books.google.com/books?id=1faeMedY8k8C&pg=PA62|url-status=live}}</ref>

===Laws of motion, impact, and gravitation===
==== Elastic collisions ====
[[File:Collision huygens.gif|thumb|A boating metaphor as a way to think about [[relative motion]], simplifying the theory of colliding bodies, from Huygens's ''Oeuvres Complètes'']]
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 limitations,<ref>{{cite book|author=Cao|first=T. Y.|url=https://books.google.com/books?id=l4PtgYXpb_oC&pg=PA25|title=Conceptual Developments of 20th Century Field Theories|publisher=Cambridge University Press|year=1998|isbn=978-0-521-63420-5|pages=24–26|archive-url=https://web.archive.org/web/20160616181021/https://books.google.com/books?id=l4PtgYXpb_oC&pg=PA25|archive-date=16 June 2016|url-status=live}}</ref> while Leibniz, his student in Paris, later abandoned it.<ref>{{Cite book|last1=Garber|first1=D.|url=https://books.google.com/books?id=BPlkkgIhUXIC|title=The Cambridge History of Seventeenth-century Philosophy|last2=Ayers|first2=M.|publisher=Cambridge University Press|year=1998|isbn=978-0-521-53720-9|pages=595–596|language=en}}</ref> Understanding the universe this way made the theory of collisions central to physics, as only explanations that involved matter in motion could be truly intelligible. While Huygens was influenced by the Cartesian approach, he was less doctrinaire.<ref>{{cite book|author=Peter Dear|title=The Intelligibility of Nature: How Science Makes Sense of the World|url=https://books.google.com/books?id=W1GAG3vJHpgC&pg=PA25|access-date=23 April 2013|date=15 September 2008|publisher=University of Chicago Press|isbn=978-0-226-13950-0|page=25|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616200622/https://books.google.com/books?id=W1GAG3vJHpgC&pg=PA25|url-status=live}}</ref> He studied [[elastic collision]]s in the 1650s but delayed publication for over a decade.<ref name="Dict470">Bunge et al. (2003), ''Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers,'' p. 470.</ref>

Huygens concluded quite early that [[Cartesian laws of motion|Descartes's laws]] for elastic collisions were largely wrong, and he formulated the correct laws, including the conservation of the product of mass times the square of the speed for hard bodies, and the conservation of quantity of motion in one direction for all bodies.<ref>''The Beginnings of Modern Science'', edited by Rene Taton, Basic Books, 1958, 1964.</ref> An important step was his recognition of the [[Galilean invariance]] of the problems.<ref>{{Cite book|last1=Garber|first1=D.|url=https://books.google.com/books?id=BPlkkgIhUXIC|title=The Cambridge History of Seventeenth-century Philosophy|last2=Ayers|first2=M.|publisher=Cambridge University Press|year=1998|isbn=978-0-521-53720-9|pages=666–667|language=en}}</ref> Huygens had worked out the laws of collision from 1652 to 1656 in a manuscript entitled ''De Motu Corporum ex Percussione'', though his results took many years to be circulated. In 1661, he passed them on in person to [[William Brouncker, 2nd Viscount Brouncker|William Brouncker]] and [[Christopher Wren]] in London.<ref>{{Cite book|last1=Garber|first1=D.|url=https://books.google.com/books?id=BPlkkgIhUXIC|title=The Cambridge History of Seventeenth-century Philosophy|last2=Ayers|first2=M.|publisher=Cambridge University Press|year=1998|isbn=978-0-521-53720-9|pages=689|language=en}}</ref> What Spinoza wrote to [[Henry Oldenburg]] about them in 1666, during the [[Second Anglo-Dutch War]], was guarded.<ref name="Israel2001">{{cite book|author=Jonathan I. Israel|title=Radical Enlightenment:Philosophy and the Making of Modernity 1650–1750|url=https://books.google.com/books?id=vMvlEweVPTsC&pg=RA3-PR62|access-date=11 May 2013|date=8 February 2001|publisher=Oxford University Press|isbn=978-0-19-162287-8|pages=lxii–lxiii|author-link=Jonathan I. Israel|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616195147/https://books.google.com/books?id=vMvlEweVPTsC&pg=RA3-PR62|url-status=live}}</ref> The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He later published them in the ''Journal des Sçavans'' in 1669.<ref name="Dict470"/>

==== Centrifugal force ====
In 1659 Huygens found the constant of [[gravitational acceleration]] and stated what is now known as the second of [[Newton's laws of motion]] in quadratic form.<ref>{{Cite book |last=Mach |first=E. |url=http://archive.org/details/scienceofmechani005860mbp |title=The Science Of Mechanics |publisher=The Open Court Publishing Co. |others=Universal Digital Library |year=1919 |pages=143–187}}</ref> He derived geometrically the now standard formula for the [[centrifugal force]], exerted on an object when viewed in a rotating [[frame of reference]], for instance when driving around a curve. In modern notation:

:<math>F_{c}={m\ \omega^2}{r}</math>

with ''m'' the [[mass (physics)|mass]] of the object, ''ω'' the [[angular velocity]], and ''r'' the [[radius]].<ref name=":28" /> Huygens collected his results in a treatise under the title ''De vi Centrifuga'', unpublished until 1703, where the kinematics of free fall were used to produce the first generalized conception of [[force]] prior to Newton.<ref>{{Cite book |last=Westfall |first=R. S. |url=https://books.google.com/books?id=nqYNAQAAIAAJ |title=Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century |publisher=Macdonald & Co. |year=1971 |isbn=978-0-356-02261-1 |pages=146–193 |language=en |chapter=Christiaan Huygens' Kinematics}}</ref> 

==== Gravitation ====
The general idea for the centrifugal force, however, was published in 1673 and 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.<ref>{{cite book|author=J. B. Barbour|title=Absolute Or Relative Motion?: The discovery of dynamics|url=https://books.google.com/books?id=ekA9AAAAIAAJ&pg=PA542|access-date=23 April 2013|date=1989|publisher=CUP Archive|isbn=978-0-521-32467-0|page=542|archive-date=5 July 2014|archive-url=https://web.archive.org/web/20140705153744/http://books.google.com/books?id=ekA9AAAAIAAJ&pg=PA542|url-status=live}}</ref> Yet, the interpretation of Newton's work on gravitation by Huygens differed 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 between particles.<ref>{{cite book|author=A.I. Sabra|title=Theories of light: from Descartes to Newton|url=https://books.google.com/books?id=nB84AAAAIAAJ&pg=PA166|access-date=23 April 2013|date=1981|publisher=CUP Archive|isbn=978-0-521-28436-3|pages=166–9|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617101008/https://books.google.com/books?id=nB84AAAAIAAJ&pg=PA166|url-status=live}}</ref>

The approach used by Huygens also missed some central notions of mathematical physics, which were not lost on others. In his work on pendulums Huygens came very close to the theory of [[simple harmonic motion]]; the topic, however, was covered fully for the first time by Newton in Book II of the ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' (1687).<ref name="Allen1999">{{cite book|author=Richard Allen|title=David Hartley on human nature|url=https://books.google.com/books?id=NCu6HhGlAB8C&pg=PA98|access-date=12 May 2013|date=1999|publisher=SUNY Press|isbn=978-0-7914-9451-6|page=98|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617113017/https://books.google.com/books?id=NCu6HhGlAB8C&pg=PA98|url-status=live}}</ref> In 1678 Leibniz picked out of Huygens's work on collisions the idea of [[Conservation law (physics)|conservation law]] that Huygens had left implicit.<ref>{{cite book|author=Nicholas Jolley|title=The Cambridge Companion to Leibniz|url=https://books.google.com/books?id=SnRis5Gdi8gC&pg=PA279|access-date=12 May 2013|date=1995|publisher=Cambridge University Press|isbn=978-0-521-36769-1|page=279|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617035641/https://books.google.com/books?id=SnRis5Gdi8gC&pg=PA279|url-status=live}}</ref>

=== Horology ===
==== Pendulum clock ====
[[File:Christiaan Huygens Clock and Horologii Oscillatorii.jpg|thumb|310px|Spring-driven pendulum clock, designed by Huygens and built by [[Salomon Coster]] (1657),<ref>{{cite web |url=http://www.museumboerhaave.nl/AAcollection/english/M03V20_V09853.html |title=Boerhaave Museum Top Collection: Hague clock (Pendulum clock) (Room 3/Showcase V20) |publisher=Museumboerhaave.nl |access-date=13 June 2010 |url-status=dead |archive-url=https://web.archive.org/web/20110219235601/http://www.museumboerhaave.nl/AAcollection/english/M03V20_V09853.html |archive-date=19 February 2011  }}</ref> with a copy of the ''Horologium Oscillatorium'' (1673),<ref>{{cite web |url=http://www.museumboerhaave.nl/AAcollection/english/M03V20_g13604.html |title=Boerhaave Museum Top Collection: Horologium oscillatorium, siue, de motu pendulorum ad horologia aptato demonstrationes geometricae (Room 3/Showcase V20) |publisher=Museumboerhaave.nl |access-date=13 June 2010 |url-status=dead |archive-url=https://web.archive.org/web/20110220041948/http://www.museumboerhaave.nl/AAcollection/english/M03V20_g13604.html |archive-date=20 February 2011  }}</ref> at [[Museum Boerhaave]], Leiden]]

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 almost 300 years until the 1930s.<ref>{{cite journal
 |last        = Marrison
 |first       = Warren
 |title       = The Evolution of the Quartz Crystal Clock
 |journal     = Bell System Technical Journal
 |year        = 1948
 |volume      = 27
 |issue = 3
 |pages       = 510–588
 |url         = http://www.ieee-uffc.org/freqcontrol/marrison/Marrison.html
 |doi         = 10.1002/j.1538-7305.1948.tb01343.x
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20070513175811/http://www.ieee-uffc.org/freqcontrol/marrison/Marrison.html
 |archive-date = 13 May 2007
}}</ref> The pendulum clock was much more accurate than the existing [[verge and foliot]] clocks and was immediately popular, quickly spreading over Europe. Clocks prior to this would lose about 15 minutes per day, whereas Huygens's clock would lose about 15 seconds per day.<ref>{{Cite web |title=Huygens Invents the Pendulum Clock, Increasing Accuracy Sixty Fold : History of Information |url=https://www.historyofinformation.com/detail.php?entryid=3506 |access-date=2023-11-15 |website=www.historyofinformation.com}}</ref> Although Huygens patented and contracted the construction of his clock designs to [[Salomon Coster]] in The Hague,<ref>{{Cite web |title=Salomon Coster the clockmaker of Christiaan Huygens. Clock. |url=http://www.antique-horology.org/invention/Coster-the-clockmaker-of-Huygens.HTM |access-date=2023-11-15 |website=www.antique-horology.org}}</ref> he 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.<ref>{{cite book|author=Epstein/Prak|title=Guilds, Innovation and the European Economy, 1400–1800|url=https://books.google.com/books?id=fXlALljcyMkC&pg=PA269|access-date=10 May 2013|publisher=Cambridge University Press|isbn=978-1-139-47107-7|pages=269–70|date=2010|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616190600/https://books.google.com/books?id=fXlALljcyMkC&pg=PA269|url-status=live}}</ref> The oldest known Huygens-style pendulum clock is dated 1657 and can be seen at the [[Museum Boerhaave]] in [[Leiden]].<ref name=":17">van den Ende, H., Hordijk, B., Kersing, V., & Memel, R. (2018). [http://www.antique-horology.org/Invention/ ''The invention of the pendulum clock: A collaboration on the real story.'']</ref><ref>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.</ref><ref>Hooijmaijers, Hans; Telling time – Devices for time measurement in museum Boerhaave – A Descriptive Catalogue; Leiden: Museum Boerhaave, 2005</ref><ref>No Author given; Chistiaan Huygens 1629–1695, Chapter 1: Slingeruurwerken; Leiden: Museum Boerhaave, 1988</ref>

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, 2nd Earl of Kincardine|Alexander Bruce]] entered the field in 1662, and Huygens called in Sir Robert Moray and the Royal Society to mediate and preserve some of his rights.<ref>{{Cite journal|last=Howard|first=N.|date=2008|title=Marketing Longitude: Clocks, Kings, Courtiers, and Christiaan Huygens|url=https://www.jstor.org/stable/30227413|journal=Book History|volume=11|pages=59–88|jstor=30227413|issn=1098-7371}}</ref><ref name=":17" /> Trials continued into the 1660s, the best news coming from a Royal Navy captain [[Robert Holmes (Royal Navy officer)|Robert Holmes]] operating against the Dutch possessions in 1664.<ref>{{cite book|author=Michael R. Matthews|title=Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy|url=https://books.google.com/books?id=vCtYnEuW7TIC&pg=PA137|access-date=12 May 2013|date=2000|publisher=Springer|isbn=978-0-306-45880-4|pages=137–8|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616212836/https://books.google.com/books?id=vCtYnEuW7TIC&pg=PA137|url-status=live}}</ref> [[Lisa Jardine]] doubts that Holmes reported the results of the trial accurately, as [[Samuel Pepys]] expressed his doubts at the time.<ref>{{cite book |author=Jardine |first=L. |url=https://archive.org/details/goingdutchhoweng0000jard |title=Going Dutch: How the English Plundered Holland's Glory |publisher=HarperPress |year=2008 |isbn=978-0007197323 |pages=263–290 |language=en |chapter= |url-access=registration}}</ref>

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.<ref>Bunge et al. (2003), ''Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers,'' p. 471.</ref>

==== ''Horologium Oscillatorium'' ====
[[Image:Evolute-parab-1-e.svg|thumb|Diagram showing the evolute of a curve]]
Sixteen years after the invention of the pendulum clock, in 1673, Huygens published his major work on horology entitled ''[[Horologium Oscillatorium|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 on mechanics where a physical problem is idealized by a set of parameters then analysed mathematically.<ref name=":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.<ref name=":29">{{cite conference |last=Mahoney |first=M. S. |date=1980 |title=Christian Huygens: The Measurement of Time and of Longitude at Sea |url=http://www.princeton.edu/~mike/articles/huygens/timelong/timelong.html#_N_13 |conference= |publisher=Swets |pages=234–270 |archive-url=https://web.archive.org/web/20071204152637/http://www.princeton.edu/~mike/articles/huygens/timelong/timelong.html#_N_13 |archive-date=4 December 2007 |access-date=7 October 2010 |book-title=Studies on Christiaan Huygens}}</ref> 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 curve|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''.<ref name=":7" /><ref name=":21">{{Cite book|last1=Huygens|first1=Christiaan|url=https://catalog.hathitrust.org/Record/000875808|title=Christiaan Huygens' the pendulum clock, or, Geometrical demonstrations concerning the motion of pendula as applied to clocks|last2=Blackwell|first2=R. J.|date=1986|publisher=Iowa State University Press|isbn=978-0-8138-0933-5|location=Ames}}</ref>

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 [[center of oscillation|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.<ref name=":7" /><ref>{{Cite journal|last1=Slisko|first1=J.|last2=Cruz|first2=A. C.|date=2019|title=String Tension in Pendulum and Circular Motions: Forgotten Contributions of Huygens in Today Teaching and Learning|url=https://eric.ed.gov/?id=EJ1299946|journal=European Journal of Physics Education|language=en|volume=10|issue=4|pages=55–68|issn=1309-7202}}</ref>

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

:<math>T = 2 \pi \sqrt{\frac{l}{g}}</math>

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]].<ref name="mach">[[Ernst Mach]], ''The Science of Mechanics'' (1919), e.g. pp. 143, 172, 187 https://archive.org/details/scienceofmechani005860mbp</ref>

Huygens also observed [[coupled oscillation]]s: 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 "[[odd sympathy|an odd kind of sympathy]]" in the Society's minutes.<ref>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).</ref> This concept is now known as [[Entrainment (physics)|entrainment]].<ref>{{Cite journal|last1=Spoor|first1=P. S.|last2=Swift|first2=G. W.|date=2000-07-27|title=The Huygens entrainment phenomenon and thermoacoustic engines|journal=The Journal of the Acoustical Society of America|language=en|volume=108|issue=2|pages=588–599|doi=10.1121/1.429590|pmid=10955624|bibcode=2000ASAJ..108..588S|issn=0001-4966|doi-access=free}}</ref>

==== Balance spring watch ====
[[File:Drawing of one of Huygens first balance springs, attached to a balance wheel.jpg|thumb|Drawing of a balance spring invented by Huygens]]
In 1675, while investigating the oscillating properties of the cycloid, Huygens was able to transform a cycloidal pendulum into a vibrating spring through a combination of geometry and higher mathematics.<ref>{{Cite journal |last=Meli |first=Domenico Bertoloni |date=2010-10-01 |title=Patterns of Transformation in Seventeenth-Century Mechanics1 |url=https://doi.org/10.5840/monist201093433 |journal=The Monist |volume=93 |issue=4 |pages=580–597 |doi=10.5840/monist201093433 |issn=0026-9662}}</ref> In the same year, Huygens designed a spiral [[balance spring]] and patented a [[pocket watch]]. These watches are notable for lacking a [[fusee (horology)|fusee]] for equalizing the mainspring torque. The implication is that Huygens thought his spiral spring would isochronize the balance in the same way that cycloid-shaped suspension curbs on his clocks would isochronize the pendulum.<ref name=":20" />

He later used spiral springs in more conventional watches, made for him by [[Thuret family|Thuret]] in Paris. 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]], which interfered with the isochronal properties of any form of balance spring, spiral or otherwise.<ref>{{Cite journal|last=Whitestone|first=S.|date=2012|title=Christian Huygens' Lost and Forgotten Pamphlet of his Pendulum Invention|url=https://doi.org/10.1080/00033790.2011.637470|journal=Annals of Science|volume=69|issue=1|pages=91–104|doi=10.1080/00033790.2011.637470|s2cid=143438492|issn=0003-3790}}</ref>

Huygens's design came around the same time as, though independently of, Robert Hooke's. Controversy over the priority of the balance spring 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.<ref>Nature – International Weekly Journal of Science, number 439, pages 638–639, 9 February 2006</ref><ref>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</ref>

=== Optics ===
==== Dioptrics ====
[[File:Huygens Aerial telescope, 1684.jpg|thumb|190px|Huygens's [[aerial telescope]] from ''Astroscopia Compendiaria'' (1684)]]
Huygens had a long-term interest in the study of [[Refraction|light refraction]] and lenses or [[dioptrics]].<ref name=":27">Bunge et al. (2003), ''Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers,'' p. 472.</ref> From 1652 date the first drafts of a Latin treatise on the theory of dioptrics, known as the ''Tractatus'', which contained a comprehensive and rigorous theory of the telescope. Huygens was one of the few to raise theoretical questions regarding the properties and working of the telescope, and almost the only one to direct his mathematical proficiency towards the actual instruments used in astronomy.<ref>Dijksterhuis, F. J. (2004). Huygens and optics. In ''Titan-From Discovery to Encounter'' (Vol. 1278, pp. 81-89).</ref>

Huygens repeatedly announced its publication to his colleagues but ultimately postponed it in favor of a much more comprehensive treatment, now under the name of the ''Dioptrica''.<ref name=":0" /> It consisted of three parts. The first part focused on the general principles of refraction, the second dealt with [[Spherical aberration|spherical]] and [[chromatic aberration]], while the third covered all aspects of the construction of telescopes and microscopes. In contrast to Descartes' dioptrics which treated only ideal (elliptical and hyperbolical) lenses, Huygens dealt exclusively with spherical lenses, which were the only kind that could really be made and incorporated in devices such as microscopes and telescopes.<ref name=":26">{{Cite journal |last=Kubbinga |first=H. |date=1995 |title=Christiaan Huygens and the foundations of optics |url=https://iopscience.iop.org/article/10.1088/0963-9659/4/6/004/meta |journal=Pure and Applied Optics: Journal of the European Optical Society Part A |volume=4 |issue=6 |pages=723–739|doi=10.1088/0963-9659/4/6/004 |bibcode=1995PApOp...4..723K }}</ref>

Huygens also worked out practical ways to minimize the effects of spherical and chromatic aberration, such as long focal distances for the objective of a telescope, internal stops to reduce the aperture, and a new kind of ocular known as the [[Huygenian eyepiece]].<ref name=":26" /> The ''Dioptrica'' was never published in Huygens’s lifetime and only appeared in press in 1703, when most of its contents were already familiar to the scientific world.

==== Lenses ====
Together with his brother Constantijn, Huygens began grinding his own lenses in 1655 in an effort to improve telescopes.<ref>{{cite book|author=Robert D. Huerta|title=Vermeer And Plato: Painting The Ideal|url=https://books.google.com/books?id=JGPSyTAFVtsC&pg=PA101|access-date=24 April 2013|date=2005|publisher=Bucknell University Press|isbn=978-0-8387-5606-5|page=101|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617112645/https://books.google.com/books?id=JGPSyTAFVtsC&pg=PA101|url-status=live}}</ref> He designed in 1662 what is now called the Huygenian eyepiece, a set of two planoconvex lenses used as a telescope ocular.<ref>{{cite book|author=Randy O. Wayne|title=Light and Video Microscopy|url=https://books.google.com/books?id=14_4OxSxlpYC&pg=PA72|access-date=24 April 2013|date=28 July 2010|publisher=Academic Press|isbn=978-0-08-092128-0|page=72|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616221447/https://books.google.com/books?id=14_4OxSxlpYC&pg=PA72|url-status=live}}</ref><ref name="Dictionary">Bunge et al. (2003), ''Dictionary of Seventeenth and Eighteenth-Century Dutch Philosophers,'' p. 473.</ref> Huygens's lenses were known to be of superb quality and polished consistently according to his specifications; however, his telescopes did not produce very sharp images, leading some to speculate that he might have suffered from [[near-sightedness]].<ref name=":31">{{Cite journal |last=Pietrow |first=A. G. M. |date=2023 |title=Did Christiaan Huygens need glasses? A study of Huygens' telescope equations and tables |url=https://royalsocietypublishing.org/doi/10.1098/rsnr.2022.0054 |journal=Notes and Records: The Royal Society Journal of the History of Science |volume=78 |issue=3 |language=en |pages=355–366 |doi=10.1098/rsnr.2022.0054 |s2cid=257233533 |issn=0035-9149|arxiv=2303.05170 }}</ref>

Lenses were also a common interest through which Huygens could meet socially in the 1660s with [[Baruch Spinoza|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.<ref>{{cite book |author=Margaret Gullan-Whur |title=Within Reason: A Life of Spinoza |date=1998 |publisher=Jonathan Cape |isbn=0-224-05046-X |pages=170–1}}</ref> He encountered the work of [[Antoni van Leeuwenhoek]], another lens grinder, in the field of [[microscopy]] which interested his father.<ref name=":7" /> Huygens also investigated the use of lenses in projectors. He is credited as the inventor of the [[magic lantern]], described in correspondence of 1659.<ref>{{cite book|author=Jordan D. Marché|title=Theaters Of Time And Space: American Planetariums, 1930–1970|url=https://books.google.com/books?id=olT1ipj-EboC&pg=PA11|access-date=23 April 2013|date=2005|publisher=Rutgers University Press|isbn=978-0-8135-3576-0|page=11|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617060138/https://books.google.com/books?id=olT1ipj-EboC&pg=PA11|url-status=live}}</ref> 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).<ref>{{cite book|author=C. D. Andriesse|title=Huygens: The Man Behind the Principle|url=https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA128|access-date=23 April 2013|date=25 August 2005|publisher=Cambridge University Press|isbn=978-0-521-85090-2|page=128|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616221653/https://books.google.com/books?id=6FTqA9fwxFMC&pg=PA128|url-status=live}}</ref>

==== ''Traité de la Lumière'' ====
[[File:Huyghens - Traité de la lumière - Fig. 12-13.svg|thumb|Refraction of a plane wave, explained using Huygens's principle in ''[[Treatise on Light|Traité de la Lumière]]'' (1690)]]

Huygens is especially remembered in optics for his [[wave]] theory of light, which he first communicated in 1678 to the Académie des sciences in Paris. Originally a preliminary chapter of his ''Dioptrica'', Huygens's theory was published in 1690 under the title ''[[Treatise on Light|Traité de la Lumière]]''<ref>Christiaan Huygens, [https://archive.org/details/bub_gb_X9PKaZlChggC ''Traité de la lumiere''...] (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1.</ref> (''Treatise on light''), and contains the first fully mathematized, mechanistic explanation of an unobservable physical phenomenon (i.e., light propagation).<ref name=":6" /><ref name="HuyThomp1912">C. Huygens (1690), translated by Silvanus P. Thompson (1912), ''[[iarchive:treatiseonlight031310mbp|Treatise on Light]]'', London: Macmillan, 1912; [http://www.gutenberg.org/ebooks/14725 Project Gutenberg edition] {{Webarchive|url=https://web.archive.org/web/20200520130041/http://www.gutenberg.org/ebooks/14725|date=20 May 2020}}, 2005; [http://www.grputland.com/2016/06/errata-in-various-editions-of-huygens-treatise-on-light.html Errata] {{Webarchive|url=https://web.archive.org/web/20170610070735/https://archive.org/details/treatiseonlight031310mbp|date=10 June 2017}}, 2016.</ref> Huygens refers to [[Ignace-Gaston Pardies]], whose manuscript on optics helped him on his wave theory.<ref>[https://archive.org/details/bub_gb_X9PKaZlChggC ''Traité de la lumiere''...] (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From [https://archive.org/details/bub_gb_X9PKaZlChggC/page/n29 page 18]</ref>

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 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 but was later able to explain it using his wavefront theory and concept of evolutes.<ref name="HuyThomp1912" /> He also developed ideas on [[caustic (optics)|caustics]].<ref name=":7" /> Huygens assumes that the [[speed of light]] is finite, based on a report by [[Ole Rømer|Ole Christensen Rømer]] in 1677 but which Huygens is presumed to have already believed.<ref name="Smith1987">{{cite book|author=A. Mark Smith|title=Descartes's Theory of Light and Refraction: A Discourse on Method|url=https://books.google.com/books?id=Ei8LAAAAIAAJ&pg=PA70|access-date=11 May 2013|date=1987|publisher=American Philosophical Society|isbn=978-0-87169-773-8|page=70 with note 10|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616221725/https://books.google.com/books?id=Ei8LAAAAIAAJ&pg=PA70|url-status=live}}</ref> Huygens's theory posits light as radiating [[wavefront]]s, 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 wave]]s being emitted at every point along the wave front (known today as the [[Huygens–Fresnel principle]]).<ref>{{Cite journal|last=Shapiro|first=A. E.|date=1973|title=Kinematic Optics: A Study of the Wave Theory of Light in the Seventeenth Century|url=https://www.jstor.org/stable/41133375|journal=Archive for History of Exact Sciences|volume=11|issue=2/3|pages=134–266|doi=10.1007/BF00343533|jstor=41133375|s2cid=119992103|issn=0003-9519}}</ref> It assumed an omnipresent [[luminiferous aether|ether]], with transmission through perfectly elastic particles, a revision of the view of Descartes. The nature of light was therefore a [[longitudinal wave]].<ref name="Smith1987" />

His 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 (waves)|polarization]] which cannot explain the observed birefringence. However, [[Thomas Young (scientist)|Thomas Young]]'s [[Young's interference experiment|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. [[Fresnel]] became aware of Huygens's work and in 1821 was able to explain birefringence as a result of light being not a longitudinal (as had been assumed) but actually a [[transverse wave]].<ref>{{cite book|author1=Darryl J. Leiter|author2=Sharon Leiter|title=A to Z of Physicists|url=https://books.google.com/books?id=Yz1CFkrZ8QMC&pg=PA108|access-date=11 May 2013|date=1 January 2009|publisher=Infobase Publishing|isbn=978-1-4381-0922-0|page=108|archive-date=16 June 2016|archive-url=https://web.archive.org/web/20160616185115/https://books.google.com/books?id=Yz1CFkrZ8QMC&pg=PA108|url-status=live}}</ref> The thus-named Huygens–Fresnel principle was the basis for the advancement of physical optics, explaining all aspects of light propagation until [[James Clerk Maxwell|Maxwell's]] [[History of electromagnetic theory|electromagnetic theory]] culminated in the development of [[quantum mechanics]] and the discovery of the [[photon]].<ref name=":26" /><ref>{{Cite journal|last=Enders|first=P.|date=2009|title=Huygens' Principle as Universal Model of Propagation|url=https://dialnet.unirioja.es/servlet/articulo?codigo=3688899|journal=Latin-American Journal of Physics Education|volume=3|issue=1|pages=4|issn=1870-9095}}</ref>

=== Astronomy ===
==== ''Systema Saturnium'' ====
[[File:Huygens Systema Saturnium.jpg|thumb|upright=.9|Huygens's explanation for the aspects of Saturn, ''Systema Saturnium'' (1659)]]
In 1655, Huygens discovered the first of Saturn's moons, [[Titan (moon)|Titan]], and observed and sketched the [[Orion Nebula]] using a [[refracting telescope]] with a 43x magnification of his own design.<ref name=":2" /><ref name=":3" /> Huygens succeeded in subdividing the nebula into different stars (the brighter interior now bears the name of the ''Huygenian region'' in his honour), and discovered several [[nebula|interstellar nebulae]] and some [[double star]]s.<ref>{{cite book |author=Antony Cooke |url=https://books.google.com/books?id=T64M5qlZTxoC&pg=PA67 |title=Visual Astronomy Under Dark Skies: A New Approach to Observing Deep Space |date=1 January 2005 |publisher=Springer |isbn=978-1-84628-149-5 |page=67 |access-date=24 April 2013 |archive-url=https://web.archive.org/web/20160617062337/https://books.google.com/books?id=T64M5qlZTxoC&pg=PA67 |archive-date=17 June 2016 |url-status=live}}</ref> He was also the first to propose that the [[rings of Saturn|appearance of Saturn]], which had baffled astronomers, was due to "a thin, flat ring, nowhere touching, and inclined to the ecliptic”.<ref>{{Cite web|last=Baalke|first=R.|date=2011|title=Historical Background of Saturn's Rings|url=https://solarviews.com/eng/saturnbg.htm|url-status=live|access-date=|website=solarviews.com|publisher=Later, it was determined that Saturn's rings were not solid but made of several smaller bodies.|archive-date=11 July 2021|archive-url=https://web.archive.org/web/20210711063716/https://solarviews.com/eng/saturnbg.htm}}</ref>

More than three years later, in 1659, Huygens published his theory and findings in ''Systema Saturnium''. It is considered the most important work on telescopic astronomy since Galileo's ''[[Sidereus Nuncius]]'' fifty years earlier.<ref name=":19">{{Cite journal |last=Chapman |first=A. |date=1995 |title=Christiaan huygens (1629–1695): astronomer and mechanician |url=https://dx.doi.org/10.1016/0160-9327%2895%2990076-4 |journal=Endeavour |language=en |volume=19 |issue=4 |pages=140–145 |doi=10.1016/0160-9327(95)90076-4 |issn=0160-9327}}</ref> Much more than a report on Saturn, Huygens provided measurements for the relative distances of the planets from the Sun, introduced the concept of the [[Filar micrometer|micrometer]], and showed a method to measure angular diameters of planets, which finally allowed the telescope to be used as an instrument to measure (rather than just sighting) astronomical objects.<ref>Van Helden, A. (1980). Huygens and the astronomers. In H.J.M. Bos, M.J.S. Rudwick, H.A.M. Snelders, & R.P.W. Visser (Eds.), ''Studies on Christiaan Huygens'' (pp. 147-165). Swets & Zeitlinger B.V.</ref> He was also the first to question the authority of Galileo in telescopic matters, a sentiment that was to be common in the years following its publication.

In the same year, Huygens was able to observe [[Syrtis Major Planum|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.<ref>{{cite web |url=http://www.marsdaily.com/reports/A_dark_spot_on_Mars_Syrtis_Major_999.html |website=www.marsdaily.com |title=A dark spot on Mars – Syrtis Major |date=February 3, 2012 |access-date=17 May 2016 |archive-date=21 September 2015 |archive-url=https://web.archive.org/web/20150921162641/http://www.marsdaily.com/reports/A_dark_spot_on_Mars_Syrtis_Major_999.html |url-status=live }}</ref>

==== 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 died in the interim and Huygens never got to deliver his planetarium to the [[French Academy of Sciences]] as the new minister, [[François-Michel le Tellier, Marquis de Louvois|François-Michel le Tellier]], decided not to renew Huygens's contract.<ref name=":1">van den Bosch, D. (2018). The application of continued fractions in Christiaan Huygens planetarium.[https://repository.tudelft.nl/islandora/object/uuid:34d16f38-51f4-4f55-854a-5f317a5e79af/datastream/OBJ/download] {{Webarchive|url=https://web.archive.org/web/20210413204446/https://repository.tudelft.nl/islandora/object/uuid:34d16f38-51f4-4f55-854a-5f317a5e79af/datastream/OBJ/download|date=13 April 2021}}</ref><ref>{{Cite web |url=http://www.irem.univ-mrs.fr/IMG/pdf/huygens-delft.pdf |title=Amin, H. H. N. (2008). Christiaan Huygens' planetarium |access-date=13 April 2021 |archive-date=14 April 2021 |archive-url=https://web.archive.org/web/20210414041401/https://www.irem.univ-mrs.fr/IMG/pdf/huygens-delft.pdf |url-status=live }}</ref>

In his design, Huygens made an ingenious use of [[continued fraction]]s 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).<ref name=":1" />

==== ''Cosmotheoros'' ====
[[File:Christiaan Huygens Cosmotheoros - Relative sizes of sun and planets, 1698.jpg|thumb|200px|Relative sizes of the Sun and planets in ''Cosmotheoros'' (1698)]]
Shortly before his death in 1695, Huygens completed his most speculative work entitled ''Cosmotheoros''. At his direction, it was to be published only posthumously by his brother, which Constantijn Jr. did in 1698.<ref>Aldersey-Williams, Hugh, ''[https://publicdomainreview.org/essay/the-uncertain-heavens The Uncertain Heavens] {{Webarchive|url=https://web.archive.org/web/20201021223711/https://publicdomainreview.org/essay/the-uncertain-heavens/ |date=21 October 2020 }}'', [[Public Domain Review]], October 21, 2020</ref> In this work, Huygens speculated on the existence of [[extraterrestrial life]], which he imagined 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, though without acknowledging Newton's laws of gravitation or the fact that planetary atmospheres are composed of different gases.<ref>{{cite book|author=Philip C. Almond|title=Adam and Eve in Seventeenth-Century Thought|url=https://books.google.com/books?id=Rx1el1IsopEC&pg=PA61|access-date=24 April 2013|date=27 November 2008|publisher=Cambridge University Press|isbn=978-0-521-09084-1|pages=61–2|archive-date=1 January 2014|archive-url=https://web.archive.org/web/20140101060832/http://books.google.com/books?id=Rx1el1IsopEC&pg=PA61|url-status=live}}</ref><ref>{{cite web |url=https://www.houstonpublicmedia.org/articles/shows/engines-of-our-ingenuity/engines-podcast/2017/04/05/194011/engines-of-our-ingenuity-1329-life-in-outer-space-in-1698/ |title=Engines of Our Ingenuity 1329: Life In Outer Space – In 1698 |website=www.houstonpublicmedia.org |date=5 April 2017 |publisher=[[University of Houston]] |access-date=9 April 2017 |archive-date=10 April 2017 |archive-url=https://web.archive.org/web/20170410052250/https://www.houstonpublicmedia.org/articles/shows/engines-of-our-ingenuity/engines-podcast/2017/04/05/194011/engines-of-our-ingenuity-1329-life-in-outer-space-in-1698/ |url-status=live |last1=Lienhard |first1=John }}</ref> ''Cosmotheoros,'' translated into English as ''The celestial worlds discover’d'', is fundamentally a [[utopia]]n work that owes some inspiration to the work of [[Peter Heylin]], and it was likely seen by contemporary readers as a piece of fiction in the tradition of [[Francis Godwin]], [[John Wilkins]], and [[Cyrano de Bergerac]].<ref>{{cite book|last=Postmus|first=Bouwe|chapter=Plokhoy's ''A way pronouned'': Mennonite Utopia or Millennium?|editor=Dominic Baker-Smith|editor2=Cedric Charles Barfoot|title=Between dream and nature: essays on utopia and dystopia|chapter-url=https://books.google.com/books?id=-bTyK6Yri4UC&pg=PA86|access-date=24 April 2013|date=1987|location=Amsterdam|publisher=Rodopi|isbn=978-90-6203-959-3|pages=86–8|archive-date=1 January 2014|archive-url=https://web.archive.org/web/20140101055555/http://books.google.com/books?id=-bTyK6Yri4UC&pg=PA86|url-status=live}}</ref><ref>{{cite book |author1=Markley |first=Robert |author-link=Robert Markley |title=Science, Literature, and Rhetoric in Early Modern England |date=2007 |publisher=Ashgate Publishing, Ltd. |isbn=978-0-7546-5781-1 |editor-last=Cummins |editor-first=Juliet |pages=194–5 |chapter=Global Analogies: Cosmology, Geosymmetry, and Skepticism in Some Works of Aphra Behn |access-date=24 April 2013 |editor-last2=Burchell |editor-first2=David |editor-link2=David Burchell |chapter-url=https://books.google.com/books?id=LdL8iy4-OwQC&pg=PA194 |archive-url=https://web.archive.org/web/20140101050142/http://books.google.com/books?id=LdL8iy4-OwQC&pg=PA194 |archive-date=1 January 2014 |url-status=live}}</ref><ref>{{Cite book |last=Guthke |first=Karl Siegfried |title=The Last Frontier: Imagining Other Worlds, from the Copernican Revolution to Modern Science Fiction |date=1990 |publisher=Cornell University Press |isbn=978-0-8014-1680-4 |location=Ithaca, New York |pages=239 |chapter=Authorities in Conflict: Fontenelle and Huygens |chapter-url=https://archive.org/details/lastfrontierimag0000guth/page/238/mode/2up<!-- Also available at https://books.google.com/books?id=cO2YDwAAQBAJ&pg=PA239 -->}}</ref>

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.<ref>{{cite web |url=http://www.brighthub.com/science/space/articles/50441.aspx |title=Johar Huzefa (2009) Nothing But The Facts – Christiaan Huygens |publisher=Brighthub.com |date=28 September 2009 |access-date=13 June 2010 |archive-date=27 November 2020 |archive-url=https://web.archive.org/web/20201127020012/http://www.brighthub.com/science/space/articles/50441.aspx |url-status=live }}</ref> 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.<ref name=":22">{{cite book|last=Jacob|first=M.|title=The Scientific Revolution|date=2010|publisher=Bedford/St. Martin's|location=Boston|pages=29, 107–114}}</ref>

It was also in this book that Huygens published his estimates for the relative sizes of the [[Solar System]] and his method for calculating [[stellar distance]]s.<ref name=":14" /> 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 (astronomy)|photometry]] remained in its infancy until the time of [[Pierre Bouguer]] and [[Johann Heinrich Lambert]].<ref name="Mccormmach2012">{{cite book|author=Russell Mccormmach|title=Weighing the World: The Reverend John Michell of Thornhill|url=https://books.google.com/books?id=9eMkgfKIdXIC&pg=PA129|access-date=12 May 2013|date=2012|publisher=Springer|isbn=978-94-007-2022-0|pages=129–31|archive-date=17 June 2016|archive-url=https://web.archive.org/web/20160617035654/https://books.google.com/books?id=9eMkgfKIdXIC&pg=PA129|url-status=live}}</ref>

== Legacy ==
Huygens has been called the first [[Theoretical physics|theoretical physicist]] and a founder of modern [[mathematical physics]].<ref name=":30" /><ref>{{Cite journal |last=Dijksterhuis |first=F. J. |date=2004 |title=Once Snell Breaks Down: From Geometrical to Physical Optics in the Seventeenth Century |url=https://doi.org/10.1080/0003379021000041884 |journal=Annals of Science |volume=61 |issue=2 |pages=165–185 |doi=10.1080/0003379021000041884 |s2cid=123111713 |issn=0003-3790}}</ref> Although his influence was considerable during his lifetime, it began to fade shortly after his death. His skills as a geometer and mechanical ingenuity elicited the admiration of many of his contemporaries, including Newton, Leibniz, [[Guillaume de l'Hôpital|l'Hôpital]], and the [[Bernoulli family|Bernoullis]].<ref name=":12" /> For his work in physics, Huygens has been deemed one of the greatest scientists in the Scientific Revolution, rivaled only by Newton in both depth of insight and the number of results obtained.<ref name=":24">{{Cite book|last=Simonyi|first=K.|title=A Cultural History of Physics|publisher=CRC Press|year=2012|isbn=978-1568813295|pages=240–255}}</ref><ref>{{Cite journal|last=Stan|first=Marius|date=2016|title=Huygens on Inertial Structure and Relativity|url=https://www.journals.uchicago.edu/doi/full/10.1086/684912|journal=Philosophy of Science|volume=83|issue=2|pages=277–298|doi=10.1086/684912|s2cid=96483477|issn=0031-8248}}</ref> Huygens also helped develop the institutional frameworks for scientific research on the [[Continental Europe|European continent]], making him a leading actor in the establishment of modern science.<ref>{{Cite book|last=Aldersey-Williams|first=H.|url=https://books.google.com/books?id=7n7VDwAAQBAJ&q=In+the+case+of+two+bodies+which+meet%2C+the+quantity+obtained+by+taking+the+sum+of+their+masses+multiplied+by+the+squares+of+their+velocities+will+be+found+to+beequal+before+and+after+the+collision.%E2%80%99&pg=PP86|title=Dutch Light: Christiaan Huygens and the Making of Science in Europe|date=2020|publisher=Pan Macmillan|isbn=978-1-5098-9332-4|language=en|access-date=28 August 2021|page=24}}</ref>

=== Mathematics and physics ===
[[File:Christiaan-huygens4.jpg|thumb|Portrait of Christiaan Huygens by [[Bernard Vaillant]] (1686)]]
In mathematics, Huygens mastered the methods of ancient [[Greek mathematics|Greek geometry]], particularly the work of Archimedes, and was an adept user of the analytic geometry and infinitesimal techniques of Descartes and Fermat.<ref name=":11">{{Cite journal|last=Dijksterhuis|first=E. J.|date=1953|title=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|url=https://pubmed.ncbi.nlm.nih.gov/13082531/|journal=Centaurus; International Magazine of the History of Science and Medicine|volume=2|issue=4|pages=265–282|doi=10.1111/j.1600-0498.1953.tb00409.x|pmid=13082531|bibcode=1953Cent....2..265D|access-date=12 August 2021|archive-date=12 August 2021|archive-url=https://web.archive.org/web/20210812151101/https://pubmed.ncbi.nlm.nih.gov/13082531/|url-status=live}}</ref> His mathematical style can be best described as geometrical infinitesimal analysis of curves and of motion. Drawing inspiration and imagery from mechanics, it remained pure mathematics in form.<ref name=":15" /> Huygens brought this type of geometrical analysis to a close, as more mathematicians turned away from classical geometry to the [[calculus]] for handling infinitesimals, limit processes, and motion.<ref name=":9" />

Huygens was moreover able to fully employ mathematics to answer questions of physics. Often this entailed introducing a simple [[Mathematical model|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'':<ref name=":10" />

{{Blockquote|text=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 system|axiomatic]] presentations of his results, which require rigorous methods of geometric demonstration: although he allowed levels of uncertainty in the selection of primary axioms and hypotheses, the proofs of theorems derived from these could never be in doubt.<ref name=":10" /> Huygens's style of publication exerted an influence in Newton's presentation of his own [[Philosophiæ Naturalis Principia Mathematica|major]] [[Opticks|works]].<ref>Elzinga, A. (1972). ''On a research program in early modern physics''. Akademiförlaget.</ref><ref>{{Cite journal |last=Cohen |first=I. B. |date=2001 |title=The Case of the Missing Author |url=https://direct.mit.edu/books/book/2424/chapter/64171/The-Case-of-the-Missing-Author |journal=Isaac Newton's Natural Philosophy |language=en |pages=15–45 |doi=10.7551/mitpress/3979.003.0005|isbn=9780262269490 }}</ref>

Besides the application of mathematics to physics and physics to mathematics, Huygens relied on mathematics as methodology, specifically its ability to generate new knowledge about the world.<ref name=":18">{{Cite journal|last=Gijsbers|first=V.|date=2003|title=Christiaan Huygens and the scientific revolution|url=http://adsabs.harvard.edu/full/2004ESASP1278..171C|url-status=live|journal=Titan - from Discovery to Encounter|volume=1278|publisher=ESA Publications Division|pages=171–178|bibcode=2004ESASP1278..171C|access-date=12 August 2021|archive-date=12 August 2021|archive-url=https://web.archive.org/web/20210812151101/http://adsabs.harvard.edu/full/2004ESASP1278..171C}}</ref> Unlike Galileo, who used mathematics primarily as rhetoric or synthesis, Huygens consistently employed mathematics as a way to discover and develop theories covering various phenomena and insisted that the reduction of the physical to the geometrical satisfy exacting standards of fit between the real and the ideal.<ref name=":29" /> In demanding such mathematical tractability and precision, Huygens set an example for eighteenth-century scientists such as [[Johann Bernoulli]], [[Jean le Rond d'Alembert]], and [[Charles-Augustin de Coulomb]].<ref name=":10" /><ref name=":30">{{Cite web |last=Smith |first=G. E. |date=2014 |title=Science Before Newton's Principia. |url=https://dl.tufts.edu/downloads/pk02cn70n?filename=r781wt28b.pdf |access-date=2022-11-02 |website=dl.tufts.edu}}</ref>

Although never intended for publication, Huygens made use of algebraic expressions to represent physical entities in a handful of his manuscripts on collisions.<ref name=":13" /> This would make him one of the first to employ mathematical formulae to describe relationships in physics, as it is done today.<ref name=":14" /> Huygens also came close to the modern idea of [[Limit (mathematics)|limit]] while working on his ''Dioptrica,'' though he never used the notion outside geometrical optics.<ref>Malet, A. (1996). ''From indivisibles to infinitesimals'' (pp. 20-22). Universitat Autonoma de Barcelona.</ref>

==== Later influence ====
Huygens's standing as the greatest scientist in Europe was eclipsed by Newton's at the end of the seventeenth century, despite the fact that, as [[Hugh Aldersey-Williams]] notes, "Huygens's achievement exceeds that of Newton in some important respects".<ref>{{Cite book |last=Aldersey-Williams |first=H. |url=https://books.google.com/books?id=7n7VDwAAQBAJ&q=In+the+case+of+two+bodies+which+meet%2C+the+quantity+obtained+by+taking+the+sum+of+their+masses+multiplied+by+the+squares+of+their+velocities+will+be+found+to+beequal+before+and+after+the+collision.%E2%80%99&pg=PP86 |title=Dutch Light: Christiaan Huygens and the Making of Science in Europe |date=2020 |publisher=Pan Macmillan |isbn=978-1-5098-9332-4 |page=14 |language=en |access-date=28 August 2021}}</ref> Although his journal publications anticipated the form of the modern [[Scientific literature|scientific article]],<ref name=":33" /> his persistent classicism and reluctance to publish his work did much to diminish his influence in the aftermath of the Scientific Revolution, as adherents of Leibniz’ calculus and Newton's physics took centre stage.<ref name=":9" /><ref name=":11" />

Huygens's analyses 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 curve|brachistochrone]], the sail curve, and the catenary.<ref name=":8" /><ref name=":16" /> His application of mathematics to physics, such as in his studies of impact and birefringence, would inspire new developments in mathematical physics and [[Classical mechanics|rational mechanics]] in the following centuries (albeit in the new language of the calculus).<ref name=":6" /> Additionally, Huygens developed the oscillating timekeeping mechanisms, the pendulum and the balance spring, that have been used ever since in mechanical [[watch]]es and [[clock]]s. These were the first reliable timekeepers fit for [[Scientific instrument|scientific use]] (e.g., to make accurate measurements of the [[Synodic day|inequality of the solar day]], which was not possible before).<ref name=":7" /><ref name=":29" /> His work in this area foreshadowed the union of [[applied mathematics]] with [[mechanical engineering]] in the centuries that followed.<ref name=":20">{{Cite thesis|title=Christiaan Huygens: a foreign inventor in the Court of Louis XIV, his role as a forerunner of mechanical engineering|url=https://oro.open.ac.uk/57983/|publisher=The Open University|date=1996|degree=Phd|language=en|first=M. H.|last=Marconell|access-date=30 August 2021|archive-date=30 August 2021|archive-url=https://web.archive.org/web/20210830013817/https://oro.open.ac.uk/57983/|url-status=live}}</ref>

===Portraits===
During his lifetime, Huygens and his father had a number of portraits commissioned. These included:

* 1639 – Constantijn Huygens in the midst of his five children by [[Adriaen Hanneman]], painting with medallions, [[Mauritshuis]], The Hague<ref name="Portraits">{{Cite web |url=https://www.leidenuniv.nl/fsw/verduin/stathist/huygens/acad1666/huygpor/ |title=Portraits of Christiaan Huygens (1629–1695) |last=Verduin |first=C.J. Kees |date=31 March 2009 |publisher=University of Leiden |access-date=12 April 2018 |archive-date=26 August 2017 |archive-url=https://web.archive.org/web/20170826023741/http://www.leidenuniv.nl/fsw/verduin/stathist/huygens/acad1666/huygpor/ |url-status=live }}</ref>
* 1671 – Portrait by [[Caspar Netscher]], [[Museum Boerhaave]], Leiden, loan from Haags Historisch Museum<ref name="Portraits"/>
* c.1675 – Depiction of Huygens in ''Établissement de l'Académie des Sciences et fondation de l'observatoire, 1666'' by [[Henri Testelin]]. Colbert presents the members of the newly founded Académie des Sciences to king Louis XIV of France. [[Château de Versailles|Musée National du Château et des Trianons de Versailles]], [[Versailles (commune)|Versailles]]<ref>{{Cite book |title=Titan – from discovery to encounter |last=Verduin |first=C.J. |publisher=ESA Publications Division |year=2004 |isbn=92-9092-997-9 |editor-last=Karen |editor-first=Fletcher |location=Noordwijk, Netherlands |pages=157–170 |chapter=A portrait of Christiaan Huygens together with Giovanni Domenico Cassini|volume=1278 |bibcode = 2004ESASP1278..157V}}</ref>
* 1679 – [[Locket|Medaillon]] portrait in [[relief]] by the French sculptor [[Jean-Jacques Clérion]]<ref name="Portraits"/>
* 1686 – Portrait in pastel by [[Bernard Vaillant]], [[Hofwijck|Museum Hofwijck]], [[Voorburg]]<ref name="Portraits"/>
* 1684 to 1687 – Engravings by [[G. Edelinck]] after the painting by [[Caspar Netscher]]<ref name="Portraits"/>
* 1688 – Portrait by [[Pierre Bourguignon (painter)]], [[Royal Netherlands Academy of Arts and Sciences]], [[Amsterdam]]<ref name="Portraits"/>

===Commemorations===
The [[European Space Agency]]'s probe aboard the [[Cassini–Huygens|Cassini spacecraft]] that landed on [[Titan (moon)|Titan]], [[Saturn]]'s largest moon, in 2005 was [[Huygens (spacecraft)|named after him]].<ref>{{cite web |title=Cassini-Huygens |url=http://www.esa.int/Our_Activities/Space_Science/Cassini-Huygens/The_mission |access-date=April 13, 2022 |publisher=European Space Agency}}</ref>

A number of monuments to Christiaan Huygens can be found across important cities in the Netherlands, including [[Rotterdam]], [[Delft]], and [[Leiden]].

====Image gallery====
<gallery>
File:Christiaan Huygens Statue Rotterdam.jpg|Rotterdam
File:Christiaan Huygens Statue Delft 1.jpg|Delft
File:Christiaan Huygens by Frank Letterie.jpg|Leiden
File:Tuin, standbeeld van Christiaan Huygens - Haarlem - 20097899 - RCE.jpg|Haarlem
File:Voorburg monument huygensmonument.jpg|Voorburg
</gallery>

==Works==
[[File:Huygens Oeuvres complètes I.jpg|thumb|Title page of ''Oeuvres Complètes'' I|255x255px]]
Source(s):<ref name="completedictionary" />
* 1650 – ''De Iis Quae Liquido Supernatant'' (''About parts floating above liquids''), unpublished.<ref>{{cite journal|doi=10.1038/076381a0|title=Christiaan Huygens, Traité: De iis quae liquido supernatant|journal=Nature|volume=76|issue=1972|pages=381|year=1907|last1=L|first1=H|bibcode=1907Natur..76..381L|s2cid=4045325|url=https://zenodo.org/record/1628789|doi-access=free|access-date=12 September 2019|archive-date=28 July 2020|archive-url=https://web.archive.org/web/20200728142648/https://zenodo.org/record/1628789|url-status=live}}</ref>
* 1651 – ''Theoremata de Quadratura Hyperboles, Ellipsis et Circuli'', republished in ''Oeuvres Complètes'', Tome XI.<ref name=":12" />
* 1651 – ''Epistola, qua diluuntur ea quibus 'Εξέτασις [Exetasis] Cyclometriae Gregori à Sto. Vincentio impugnata fuit'', supplement.<ref name="Yoder2013">{{cite book|last=Yoder|first=Joella|url=https://books.google.com/books?id=XGZlIvCOtFsC|title=A Catalogue of the Manuscripts of Christiaan Huygens including a concordance with his Oeuvres Complètes|date=17 May 2013|publisher=BRILL|isbn=9789004235656|access-date=12 April 2018|archive-date=16 March 2020|archive-url=https://web.archive.org/web/20200316011539/https://books.google.com/books?id=XGZlIvCOtFsC|url-status=live}}</ref>
* 1654 – ''De Circuli Magnitudine Inventa.''<ref name=":10" />
* 1654 – ''Illustrium Quorundam Problematum Constructiones'', supplement.<ref name="Yoder2013"/>
* 1655 – ''Horologium'' (''The clock''), short pamphlet on the pendulum clock.<ref name=":7" />
* 1656 – ''De Saturni Luna Observatio Nova'' (''About the new observation of the [[Titan (moon)|moon]] of [[Saturn]]''), describes the discovery of [[Titan (moon)|Titan]].<ref>{{Cite book |title=Titan – from discovery to encounter |last=Audouin |first=Dollfus |publisher=ESA Publications Division |year=2004 |isbn=92-9092-997-9 |editor-last=Karen |editor-first=Fletcher |location=Noordwijk, Netherlands |pages=115–132 |chapter=Christiaan Huygens as telescope maker and planetary observer|volume=1278 |bibcode = 2004ESASP1278..115D}}</ref>
* 1656 – ''De Motu Corporum ex Percussione'', published posthumously in 1703.<ref>{{Cite journal |last=Huygens |first=Christiaan |date=1977 |title=Christiaan Huygens' The Motion of Colliding Bodies |journal=Isis |volume=68 |issue=4 |pages=574–597|translator-first=Richard J. |translator-last=Blackwell|jstor=230011 |doi=10.1086/351876 |s2cid=144406041 }}</ref>
* 1657 – ''De Ratiociniis in Ludo Aleae'' (''Van [[probability|reeckening]] in spelen van geluck''), translated into Latin by Frans van Schooten.<ref name=":25" />
* 1659 – ''Systema Saturnium'' (''System of Saturn'').<ref name="Yoder2013" />
* 1659 – ''De vi Centrifuga'' (''Concerning the [[centrifugal force]]''), published posthumously in 1703.<ref name=":23" />
* 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.<ref name=":21" />
* 1684 – ''Astroscopia Compendiaria Tubi Optici Molimine Liberata'' (''Compound telescopes without a tube'').<ref name=":12" />
* 1685 – ''Memoriën aengaende het slijpen van glasen tot verrekijckers'', dealing with the grinding of lenses.<ref name=":6" />
* 1686 – ''Kort onderwijs aengaende het gebruijck der horologiën tot het vinden der lenghten van Oost en West'' (in Old [[Dutch language|Dutch]]), instructions on how to use clocks to establish the [[longitude]] at sea.<ref>{{Cite web |url=http://dbnl.nl/tekst/huyg003oeuv22_01/huyg003oeuv22_01_0194.php?q=Kort%2520onderwijs%2520aengaende%2520het%2520gebruijck%2520der%2520horologi%25C3%25ABn%2520tot%2520het%2520vinden%2520der%2520lenghten%2520van%2520Oost%2520en%2520West#hl12 |title=Christiaan Huygens, Oeuvres complètes. Tome XXII. Supplément à la correspondance |publisher=Digitale Bibliotheek Voor de Nederlandse Lettern |language=nl |access-date=12 April 2018 |archive-date=13 April 2018 |archive-url=https://web.archive.org/web/20180413043407/http://dbnl.nl/tekst/huyg003oeuv22_01/huyg003oeuv22_01_0194.php?q=Kort%2520onderwijs%2520aengaende%2520het%2520gebruijck%2520der%2520horologi%25C3%25ABn%2520tot%2520het%2520vinden%2520der%2520lenghten%2520van%2520Oost%2520en%2520West#hl12 |url-status=live }}</ref>
* 1690 – ''Traité de la Lumière'', dealing with the nature of light propagation.<ref name=":0" />
* 1690 – ''Discours de la Cause de la Pesanteur'' (''Discourse about gravity''), supplement.<ref name=":12" />
* 1691 – ''Lettre Touchant le Cycle Harmonique'', short tract concerning the [[31 equal temperament|31-tone system]].<ref name="Cohen1984" />
* 1698 – ''Cosmotheoros'', deals with the solar system, cosmology, and extraterrestrial life.<ref name=":22" />
* 1703 – ''Opuscula Posthuma'' including:<ref name=":12" />
** ''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.<ref name="Cohen1984" />
* 1728 – ''Christiani Hugenii Zuilichemii, dum viveret Zelhemii Toparchae, Opuscula Posthuma'' (alternate title: ''Opera Reliqua''), includes works in optics and physics.<ref name=":23">{{cite journal | url=http://www.gewina.nl/journals/tractrix/yoder91.pdf | title=Christiaan Huygens' Great Treasure | first=Joella | last=Yoeder | journal=Tractrix | volume=3 | year=1991 | pages=1–13 | access-date=12 April 2018 | archive-date=13 April 2018 | archive-url=https://web.archive.org/web/20180413044740/http://www.gewina.nl/journals/tractrix/yoder91.pdf | url-status=live }}</ref>
* 1888–1950 – ''Huygens, Christiaan. Oeuvres complètes.'' Complete works, 22 volumes. Editors [[D. Bierens de Haan]] (1–5), J. Bosscha (6–10), [[Diederik Johannes Korteweg|D.J. Korteweg]] (11–15), [[Albertus Antonie Nijland|A.A. Nijland]] (15), J.A. Vollgraf (16–22). The Hague:<ref name="Yoder2013" />
** ''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).

==See also==

=== Concepts ===

* [[Parallel axis theorem|Huygens–Steiner theorem]]
* [[Huygens–Fresnel principle|Huygens's principle]]
* [[Lemniscate of Gerono|Huygens's lemniscate]]
* [[Evolute]]
* [[Center of percussion|Centre of oscillation]]

=== People ===

* [[René Descartes]]
* [[Galileo Galilei]]
* [[Isaac Newton]]
* [[Gottfried Wilhelm Leibniz]]

=== Technology ===
* [[History of the internal combustion engine]]
* [[List of largest optical telescopes historically]]
* [[Adriaan Fokker#Musical instruments|Fokker Organ]]
*[[Seconds pendulum]]

==References==
{{Reflist|30em}}

=== Further reading ===
* [[Cornelis Dirk Andriesse|Andriesse, C.D.]] (2005). ''Huygens: The Man Behind the Principle''. Foreword by Sally Miedema. [[Cambridge University Press]].
*Aldersey-Williams, Hugh. (2020). ''Dutch Light: Christiaan Huygens and the Making of Science in Europe''. London: Picador. 
* Bell, A. E. (1947). ''Christian Huygens and the Development of Science in the Seventeenth Century''
* [[Carl Benjamin Boyer|Boyer, C.B.]] (1968). ''A History of Mathematics'', New York.
* [[Eduard Jan Dijksterhuis|Dijksterhuis, E. J.]] (1961). ''The Mechanization of the World Picture: Pythagoras to Newton''
* Hooijmaijers, H. (2005). ''Telling time – Devices for time measurement in Museum Boerhaave – A Descriptive Catalogue'', Leiden, Museum Boerhaave.
* [[Dirk Jan Struik|Struik, D.J.]] (1948). ''A Concise History of Mathematics''
* Van den Ende, H. et al. (2004). ''Huygens's Legacy, The golden age of the pendulum clock'', Fromanteel Ltd, Castle Town, Isle of Man.
* [[Joella Yoder|Yoder, J. G.]] (2005). "Book on the pendulum clock" in [[Ivor Grattan-Guinness]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 33–45.

==External links==
{{Commons}}
{{Wikiquote}}

===Primary sources, translations===
* {{Gutenberg author |id=5648| name=Christiaan Huygens}}:
** C. Huygens (translated by Silvanus P. Thompson, 1912), ''[http://www.gutenberg.org/ebooks/14725 Treatise on Light]''; [http://www.grputland.com/2016/06/errata-in-various-editions-of-huygens-treatise-on-light.html Errata].
* {{Internet Archive author |sname=Christiaan Huygens}}
* {{Librivox author |id=6084}}
* {{Cite EB1911|wstitle= Huygens, Christiaan | volume= 14 |last1= Clerke |first1= Agnes Mary |author1-link= Agnes Mary Clerke | pages = 21&ndash;22 |short=1}}
* {{EMLO}}
* [http://math.dartmouth.edu/~doyle/docs/huygens/huygens.pdf De Ratiociniis in Ludo Aleae or The Value of all Chances in Games of Fortune, 1657] Christiaan Huygens's book on probability theory. An English translation published in 1714. Text pdf file.
* ''[http://digital.library.cornell.edu/cgi/t/text/text-idx?c=kmoddl;cc=kmoddl;view=toc;subview=short;idno=kmod053 Horologium oscillatorium]'' (German translation, pub. 1913) or ''[http://www.17centurymaths.com/contents/huygenscontents.html Horologium oscillatorium]'' (English translation by Ian Bruce) on the pendulum clock
* ''[http://www.staff.science.uu.nl/~gent0113/huygens/huygens_ct_en.htm ΚΟΣΜΟΘΕΩΡΟΣ]'' (''Cosmotheoros''). (English translation of Latin, pub. 1698; subtitled ''The celestial worlds discover'd: or, Conjectures concerning the inhabitants, plants and productions of the worlds in the planets.'')
* C. Huygens (translated by Silvanus P. Thompson), ''Traité de la lumière'' or ''Treatise on light'', London: Macmillan, 1912, [https://archive.org/details/treatiseonlight031310mbp archive.org/details/treatiseonlight031310mbp]; New York: Dover, 1962; Project Gutenberg, 2005, [http://www.gutenberg.org/ebooks/14725 gutenberg.org/ebooks/14725]; [http://www.grputland.com/2016/06/errata-in-various-editions-of-huygens-treatise-on-light.html Errata]
* [http://www.sil.si.edu/DigitalCollections/HST/Huygens/huygens.htm Systema Saturnium 1659 text] a digital edition of Smithsonian Libraries
* ''[http://www.princeton.edu/~hos/mike/texts/huygens/centriforce/huyforce.htm On Centrifugal Force]'' (1703)
* [http://worldcat.org/identities/find?fullName=christiaan+huygens Huygens's work at WorldCat] {{Webarchive|url=https://web.archive.org/web/20201023010303/http://worldcat.org/identities/find?fullName=christiaan+huygens |date=23 October 2020 }}
* [http://emlo.bodleian.ox.ac.uk/blog/?catalogue=christiaan-huygens The Correspondence of Christiaan Huygens] in [http://emlo.bodleian.ox.ac.uk/home EMLO]
* [http://www.brighthub.com/science/space/articles/50441.aspx Christiaan Huygens biography and achievements]
* [http://www.leidenuniv.nl/fsw/verduin/stathist/huygens/acad1666/huygpor/ Portraits of Christiaan Huygens]
* Huygens's books, in digital facsimile from the [[Linda Hall Library]]:
** (1659) [http://lhldigital.lindahall.org/cdm/ref/collection/astro_early/id/11210 ''Systema Saturnium''] (Latin)
** (1684) [http://lhldigital.lindahall.org/cdm/ref/collection/astro_early/id/9630 ''Astroscopia compendiaria''] (Latin)
** (1690) [http://lhldigital.lindahall.org/cdm/ref/collection/color/id/16013 ''Traité de la lumiére''] (French)
** (1698) [http://lhldigital.lindahall.org/cdm/ref/collection/astro_early/id/11319 ''ΚΟΣΜΟΘΕΩΡΟΣ, sive De terris cœlestibus''] {{Webarchive|url=https://web.archive.org/web/20201003035136/http://lhldigital.lindahall.org/cdm/ref/collection/astro_early/id/11319 |date=3 October 2020 }} (Latin)

===Museums===
* [https://web.archive.org/web/20070716001341/http://www.hofwijck.nl/hofwijck/en/ Huygensmuseum Hofwijck] in Voorburg, Netherlands, where Huygens lived and worked.
* [http://www.sciencemuseum.org.uk/onlinestuff/stories/huygens_clocks.aspx?keywords=huygens Huygens Clocks] {{Webarchive|url=https://web.archive.org/web/20070929083426/http://www.sciencemuseum.org.uk/onlinestuff/stories/huygens_clocks.aspx?keywords=huygens |date=29 September 2007 }} exhibition from the Science Museum, London
* [http://hdl.handle.net/1887.1/item:1843482 Online exhibition] on Huygens in [[Leiden University Library]] {{in lang|nl}}

===Other===
* {{MacTutor Biography|id=Huygens}}
* [https://web.archive.org/web/20010307171657/http://www.xs4all.nl/~huygensf/english/huygens.html Huygens and music theory] [[Huygens–Fokker Foundation]] —on Huygens's [[31 equal temperament]] and how it has been used
* [http://www-personal.umich.edu/~jbourj/money1.htm Christiaan Huygens on the 25 Dutch Guilder banknote of the 1950s.] {{Webarchive|url=https://web.archive.org/web/20111213071452/http://www-personal.umich.edu/~jbourj/money1.htm |date=13 December 2011 }}
* {{MathGenealogy |id=125561}}
* [https://web.archive.org/web/20030205221441/http://frank.harvard.edu/~paulh/misc/huygens.htm How to pronounce "Huygens"]

{{Christiaan Huygens}}
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