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Alexis Clairaut

Alexis Claude Clairaut (French pronunciation: ​[alɛksi klod klɛʁo]; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had outlined in the Principia of 1687. Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton's theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as "Clairaut's theorem". He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moon's orbit. In mathematics he is also credited with Clairaut's equation and Clairaut's relation.

Alexis Claude Clairaut
Alexis Claude Clairaut
Born(1713-05-13)13 May 1713[1]
Paris
Died17 May 1765(1765-05-17) (aged 52)
Paris
NationalityFrench
Known forClairaut's theorem
Clairaut's theorem on equality of mixed partials
Clairaut's equation
Clairaut's relation
Apsidal precession
Scientific career
FieldsMathematics

Biography

Childhood and early life

Clairaut was born in Paris, France, to Jean-Baptiste and Catherine Petit Clairaut. The couple had 20 children, however only a few of them survived childbirth.[2] His father taught mathematics. Alexis was a prodigy – at the age of ten he began studying calculus. At the age of twelve he wrote a memoir on four geometrical curves and under his father's tutelage he made such rapid progress in the subject that in his thirteenth year he read before the Académie française an account of the properties of four curves which he had discovered.[3] When only sixteen he finished a treatise on Tortuous Curves, Recherches sur les courbes a double courbure, which, on its publication in 1731, procured his admission into the Royal Academy of Sciences, although he was below the legal age as he was only eighteen.He gave a path breaking formulae called the distance formulae which helps to find out the distance between any 2 points on the cartesian or XY plane.

Personal life and death

Clairaut was unmarried, and known for leading an active social life.[2] His growing popularity in society hindered his scientific work: "He was focused," says Bossut, "with dining and with evenings, coupled with a lively taste for women, and seeking to make his pleasures into his day to day work, he lost rest, health, and finally life at the age of fifty-two." Though he led a fulfilling social life, he was very prominent in the advancement of learning in young mathematicians.

He was elected a Fellow of the Royal Society of London on 27 October 1737.[4]

Clairaut died in Paris in 1765.

Mathematical and scientific works

The shape of the Earth

In 1736, together with Pierre Louis Maupertuis, he took part in the expedition to Lapland, which was undertaken for the purpose of estimating a degree of the meridian arc.[5] The goal of the excursion was to geometrically calculate the shape of the Earth, which Sir Isaac Newton theorised in his book Principia was an ellipsoid shape. They sought to prove if Newton's theory and calculations were correct or not. Before the expedition team returned to Paris, Clairaut sent his calculations to the Royal Society of London. The writing was later published by the society in the 1736–37 volume of Philosophical Transactions.[6] Initially, Clairaut disagrees with Newton's theory on the shape of the Earth. In the article, he outlines several key problems that effectively disprove Newton's calculations, and provides some solutions to the complications. The issues addressed include calculating gravitational attraction, the rotation of an ellipsoid on its axis, and the difference in density of an ellipsoid on its axes.[6] At the end of his letter, Clairaut writes that:

"It appears even Sir Isaac Newton was of the opinion, that it was necessary the Earth should be more dense toward the center, in order to be so much the flatter at the poles: and that it followed from this greater flatness, that gravity increased so much the more from the equator towards the Pole."[6]

 
Théorie de la Figure de la Terre, tirée des Principes de l’Hydrostatique, 1743

This conclusion suggests not only that the Earth is of an oblate ellipsoid shape, but it is flattened more at the poles and is wider at the centre. His article in Philosophical Transactions created much controversy, as he addressed the problems of Newton's theory, but provided few solutions to how to fix the calculations. After his return, he published his treatise Théorie de la figure de la terre (1743). In this work he promulgated the theorem, known as Clairaut's theorem, which connects the gravity at points on the surface of a rotating ellipsoid with the compression and the centrifugal force at the equator. This hydrostatic model of the shape of the Earth was founded on a paper by Colin Maclaurin, which had shown that a mass of homogeneous fluid set in rotation about a line through its centre of mass would, under the mutual attraction of its particles, take the form of an ellipsoid. Under the assumption that the Earth was composed of concentric ellipsoidal shells of uniform density, Clairaut's theorem could be applied to it, and allowed the ellipticity of the Earth to be calculated from surface measurements of gravity. This proved Sir Isaac Newton's theory that the shape of the Earth was an oblate ellipsoid.[2] In 1849 Stokes showed that Clairaut's result was true whatever the interior constitution or density of the Earth, provided the surface was a spheroid of equilibrium of small ellipticity.

Geometry

In 1741, Clairaut wrote a book called Éléments de Géométrie. The book outlines the basic concepts of geometry. Geometry in the 1700s was complex to the average learner. It was considered to be a dry subject. Clairaut saw this trend, and wrote the book in an attempt to make the subject more interesting for the average learner. He believed that instead of having students repeatedly work problems that they did not fully understand, it was imperative for them to make discoveries themselves in a form of active, experiential learning.[7] He begins the book by comparing geometric shapes to measurements of land, as it was a subject that most anyone could relate to. He covers topics from lines, shapes, and even some three dimensional objects. Throughout the book, he continuously relates different concepts such as physics, astrology, and other branches of mathematics to geometry. Some of the theories and learning methods outlined in the book are still used by teachers today, in geometry and other topics.[8]

Focus on astronomical motion

One of the most controversial issues of the 18th century was the problem of three bodies, or how the Earth, Moon, and Sun are attracted to one another. With the use of the recently founded Leibnizian calculus, Clairaut was able to solve the problem using four differential equations.[9] He was also able to incorporate Newton's inverse-square law and law of attraction into his solution, with minor edits to it. However, these equations only offered approximate measurement, and no exact calculations. Another issue still remained with the three body problem; how the Moon rotates on its apsides. Even Newton could account for only half of the motion of the apsides.[9] This issue had puzzled astronomers. In fact, Clairaut had at first deemed the dilemma so inexplicable, that he was on the point of publishing a new hypothesis as to the law of attraction.

 
Théorie de la Lune & Tables de la Lune, 1765

The question of the apsides was a heated debate topic in Europe. Along with Clairaut, there were two other mathematicians who were racing to provide the first explanation for the three body problem; Leonhard Euler and Jean le Rond d'Alembert.[9] Euler and d'Alembert were arguing against the use of Newtonian laws to solve the three body problem. Euler in particular believed that the inverse square law needed revision to accurately calculate the apsides of the Moon.

Despite the hectic competition to come up with the correct solution, Clairaut obtained an ingenious approximate solution of the problem of the three bodies. In 1750 he gained the prize of the St Petersburg Academy for his essay Théorie de la lune; the team made up of Clairaut, Jérome Lalande and Nicole Reine Lepaute successfully computed the date of the 1759 return of Halley's comet.[10] The Théorie de la lune is strictly Newtonian in character. This contains the explanation of the motion of the apsis. It occurred to him to carry the approximation to the third order, and he thereupon found that the result was in accordance with the observations. This was followed in 1754 by some lunar tables, which he computed using a form of the discrete Fourier transform.[11]

The newfound solution to the problem of three bodies ended up meaning more than proving Newton's laws correct. The unravelling of the problem of three bodies also had practical importance. It allowed sailors to determine the longitudinal direction of their ships, which was crucial not only in sailing to a location, but finding their way home as well.[9] This held economic implications as well, because sailors were able to more easily find destinations of trade based on the longitudinal measures.

Clairaut subsequently wrote various papers on the orbit of the Moon, and on the motion of comets as affected by the perturbation of the planets, particularly on the path of Halley's comet. He also used applied mathematics to study Venus, taking accurate measurements of the planet's size and distance from the Earth. This was the first precise reckoning of the planet's size.

Publications

  • Theorie de la figure de la terre, tirée des principes de l'hydrostatique (in French). Paris: Laurent Durand. 1743.
  • Théorie de la figure de la terre, tirée des principes de l'hydrostatique (in French). Paris: Louis Courcier. 1808.

See also

Notes

  1. ^ Other dates have been proposed, such as 7 May, which Judson Knight and the Royal Society report. Here is a discussion and argument for 13 May. Courcelle, Olivier (17 March 2007). "13 mai 1713(1): Naissance de Clairaut". Chronologie de la vie de Clairaut (1713-1765) (in French). Retrieved 26 April 2018.
  2. ^ a b c Knight, Judson (2000). "Alexis Claude Clairaut". In Schlager, Neil; Lauer, Josh (eds.). Science and Its Times. Vol. 4 1700-1799. pp. 247–248. Retrieved 26 April 2018.
  3. ^ Taner Kiral, Jonathan Murdock and Colin B. P. McKinney. "The Four Curves of Alexis Clairaut". MAA publications.
  4. ^ "Fellow Details: Clairaut; Alexis Claude (1713 - 1765)". Royal Society. Retrieved 26 April 2018.
  5. ^ O'Connor and, J. J.; E. F. Robertson (October 1998). "Alexis Clairaut". MacTutor History of Mathematics Archive. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 12 March 2009.
  6. ^ a b c Claude, Alexis; Colson, John (1737). "An Inquiry concerning the Figure of Such Planets as Revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface". Philosophical Transactions. 40: 277–306. doi:10.1098/rstl.1737.0045. JSTOR 103921.
  7. ^ Clairaut, Alexis Claude (1 January 1881). Elements of geometry, tr. by J. Kaines.
  8. ^ Smith, David (1921). "Review of Èléments de Géométrie. 2 vols". The Mathematics Teacher.
  9. ^ a b c d Bodenmann, Siegfried (January 2010). "The 18th century battle over lunar motion". Physics Today. 63 (1): 27–32. Bibcode:2010PhT....63a..27B. doi:10.1063/1.3293410.
  10. ^ Grier, David Alan (2005). "The First Anticipated Return: Halley's Comet 1758". When Computers Were Human. Princeton: Princeton University Press. pp. 11–25. ISBN 0-691-09157-9.
  11. ^ Terras, Audrey (1999). Fourier analysis on finite groups and applications. Cambridge University Press. ISBN 978-0-521-45718-7., p. 30

References

External links

alexis, clairaut, alexis, claude, clairaut, french, pronunciation, alɛksi, klod, klɛʁo, 1713, 1765, french, mathematician, astronomer, geophysicist, prominent, newtonian, whose, work, helped, establish, validity, principles, results, that, isaac, newton, outli. Alexis Claude Clairaut French pronunciation alɛksi klod klɛʁo 13 May 1713 17 May 1765 was a French mathematician astronomer and geophysicist He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had outlined in the Principia of 1687 Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton s theory for the figure of the Earth In that context Clairaut worked out a mathematical result now known as Clairaut s theorem He also tackled the gravitational three body problem being the first to obtain a satisfactory result for the apsidal precession of the Moon s orbit In mathematics he is also credited with Clairaut s equation and Clairaut s relation Alexis Claude ClairautAlexis Claude ClairautBorn 1713 05 13 13 May 1713 1 ParisDied17 May 1765 1765 05 17 aged 52 ParisNationalityFrenchKnown forClairaut s theoremClairaut s theorem on equality of mixed partialsClairaut s equation Clairaut s relationApsidal precessionScientific careerFieldsMathematics Contents 1 Biography 1 1 Childhood and early life 1 2 Personal life and death 2 Mathematical and scientific works 2 1 The shape of the Earth 2 2 Geometry 2 3 Focus on astronomical motion 3 Publications 4 See also 5 Notes 6 References 7 External linksBiography EditChildhood and early life Edit Clairaut was born in Paris France to Jean Baptiste and Catherine Petit Clairaut The couple had 20 children however only a few of them survived childbirth 2 His father taught mathematics Alexis was a prodigy at the age of ten he began studying calculus At the age of twelve he wrote a memoir on four geometrical curves and under his father s tutelage he made such rapid progress in the subject that in his thirteenth year he read before the Academie francaise an account of the properties of four curves which he had discovered 3 When only sixteen he finished a treatise on Tortuous Curves Recherches sur les courbes a double courbure which on its publication in 1731 procured his admission into the Royal Academy of Sciences although he was below the legal age as he was only eighteen He gave a path breaking formulae called the distance formulae which helps to find out the distance between any 2 points on the cartesian or XY plane Personal life and death Edit Clairaut was unmarried and known for leading an active social life 2 His growing popularity in society hindered his scientific work He was focused says Bossut with dining and with evenings coupled with a lively taste for women and seeking to make his pleasures into his day to day work he lost rest health and finally life at the age of fifty two Though he led a fulfilling social life he was very prominent in the advancement of learning in young mathematicians He was elected a Fellow of the Royal Society of London on 27 October 1737 4 Clairaut died in Paris in 1765 Mathematical and scientific works EditThe shape of the Earth EditIn 1736 together with Pierre Louis Maupertuis he took part in the expedition to Lapland which was undertaken for the purpose of estimating a degree of the meridian arc 5 The goal of the excursion was to geometrically calculate the shape of the Earth which Sir Isaac Newton theorised in his book Principia was an ellipsoid shape They sought to prove if Newton s theory and calculations were correct or not Before the expedition team returned to Paris Clairaut sent his calculations to the Royal Society of London The writing was later published by the society in the 1736 37 volume of Philosophical Transactions 6 Initially Clairaut disagrees with Newton s theory on the shape of the Earth In the article he outlines several key problems that effectively disprove Newton s calculations and provides some solutions to the complications The issues addressed include calculating gravitational attraction the rotation of an ellipsoid on its axis and the difference in density of an ellipsoid on its axes 6 At the end of his letter Clairaut writes that It appears even Sir Isaac Newton was of the opinion that it was necessary the Earth should be more dense toward the center in order to be so much the flatter at the poles and that it followed from this greater flatness that gravity increased so much the more from the equator towards the Pole 6 Theorie de la Figure de la Terre tiree des Principes de l Hydrostatique 1743 This conclusion suggests not only that the Earth is of an oblate ellipsoid shape but it is flattened more at the poles and is wider at the centre His article in Philosophical Transactions created much controversy as he addressed the problems of Newton s theory but provided few solutions to how to fix the calculations After his return he published his treatise Theorie de la figure de la terre 1743 In this work he promulgated the theorem known as Clairaut s theorem which connects the gravity at points on the surface of a rotating ellipsoid with the compression and the centrifugal force at the equator This hydrostatic model of the shape of the Earth was founded on a paper by Colin Maclaurin which had shown that a mass of homogeneous fluid set in rotation about a line through its centre of mass would under the mutual attraction of its particles take the form of an ellipsoid Under the assumption that the Earth was composed of concentric ellipsoidal shells of uniform density Clairaut s theorem could be applied to it and allowed the ellipticity of the Earth to be calculated from surface measurements of gravity This proved Sir Isaac Newton s theory that the shape of the Earth was an oblate ellipsoid 2 In 1849 Stokes showed that Clairaut s result was true whatever the interior constitution or density of the Earth provided the surface was a spheroid of equilibrium of small ellipticity Geometry Edit In 1741 Clairaut wrote a book called Elements de Geometrie The book outlines the basic concepts of geometry Geometry in the 1700s was complex to the average learner It was considered to be a dry subject Clairaut saw this trend and wrote the book in an attempt to make the subject more interesting for the average learner He believed that instead of having students repeatedly work problems that they did not fully understand it was imperative for them to make discoveries themselves in a form of active experiential learning 7 He begins the book by comparing geometric shapes to measurements of land as it was a subject that most anyone could relate to He covers topics from lines shapes and even some three dimensional objects Throughout the book he continuously relates different concepts such as physics astrology and other branches of mathematics to geometry Some of the theories and learning methods outlined in the book are still used by teachers today in geometry and other topics 8 Focus on astronomical motion Edit One of the most controversial issues of the 18th century was the problem of three bodies or how the Earth Moon and Sun are attracted to one another With the use of the recently founded Leibnizian calculus Clairaut was able to solve the problem using four differential equations 9 He was also able to incorporate Newton s inverse square law and law of attraction into his solution with minor edits to it However these equations only offered approximate measurement and no exact calculations Another issue still remained with the three body problem how the Moon rotates on its apsides Even Newton could account for only half of the motion of the apsides 9 This issue had puzzled astronomers In fact Clairaut had at first deemed the dilemma so inexplicable that he was on the point of publishing a new hypothesis as to the law of attraction Theorie de la Lune amp Tables de la Lune 1765 The question of the apsides was a heated debate topic in Europe Along with Clairaut there were two other mathematicians who were racing to provide the first explanation for the three body problem Leonhard Euler and Jean le Rond d Alembert 9 Euler and d Alembert were arguing against the use of Newtonian laws to solve the three body problem Euler in particular believed that the inverse square law needed revision to accurately calculate the apsides of the Moon Despite the hectic competition to come up with the correct solution Clairaut obtained an ingenious approximate solution of the problem of the three bodies In 1750 he gained the prize of the St Petersburg Academy for his essay Theorie de la lune the team made up of Clairaut Jerome Lalande and Nicole Reine Lepaute successfully computed the date of the 1759 return of Halley s comet 10 The Theorie de la lune is strictly Newtonian in character This contains the explanation of the motion of the apsis It occurred to him to carry the approximation to the third order and he thereupon found that the result was in accordance with the observations This was followed in 1754 by some lunar tables which he computed using a form of the discrete Fourier transform 11 The newfound solution to the problem of three bodies ended up meaning more than proving Newton s laws correct The unravelling of the problem of three bodies also had practical importance It allowed sailors to determine the longitudinal direction of their ships which was crucial not only in sailing to a location but finding their way home as well 9 This held economic implications as well because sailors were able to more easily find destinations of trade based on the longitudinal measures Clairaut subsequently wrote various papers on the orbit of the Moon and on the motion of comets as affected by the perturbation of the planets particularly on the path of Halley s comet He also used applied mathematics to study Venus taking accurate measurements of the planet s size and distance from the Earth This was the first precise reckoning of the planet s size Publications EditTheorie de la figure de la terre tiree des principes de l hydrostatique in French Paris Laurent Durand 1743 Theorie de la figure de la terre tiree des principes de l hydrostatique in French Paris Louis Courcier 1808 1743 copy of Theorie de la Figure de la Terre tiree des Principes de l Hydrostatique Introduction to Theorie de la Figure de la Terre tiree des Principes de l Hydrostatique 1765 copy of Theorie de la Lune amp Tables de la Lune Dedication to Theorie de la Lune amp Tables de la Lune Dedication to Theorie de la Lune amp Tables de la Lune First page of Theorie de la Lune amp Tables de la Lune See also EditClairaut s equation Clairaut s relation Clairaut s theorem Differential geometry Human computer Intermolecular force Symmetry of second derivativesNotes Edit Other dates have been proposed such as 7 May which Judson Knight and the Royal Society report Here is a discussion and argument for 13 May Courcelle Olivier 17 March 2007 13 mai 1713 1 Naissance de Clairaut Chronologie de la vie de Clairaut 1713 1765 in French Retrieved 26 April 2018 a b c Knight Judson 2000 Alexis Claude Clairaut In Schlager Neil Lauer Josh eds Science and Its Times Vol 4 1700 1799 pp 247 248 Retrieved 26 April 2018 Taner Kiral Jonathan Murdock and Colin B P McKinney The Four Curves of Alexis Clairaut MAA publications Fellow Details Clairaut Alexis Claude 1713 1765 Royal Society Retrieved 26 April 2018 O Connor and J J E F Robertson October 1998 Alexis Clairaut MacTutor History of Mathematics Archive School of Mathematics and Statistics University of St Andrews Scotland Retrieved 12 March 2009 a b c Claude Alexis Colson John 1737 An Inquiry concerning the Figure of Such Planets as Revolve about an Axis Supposing the Density Continually to Vary from the Centre towards the Surface Philosophical Transactions 40 277 306 doi 10 1098 rstl 1737 0045 JSTOR 103921 Clairaut Alexis Claude 1 January 1881 Elements of geometry tr by J Kaines Smith David 1921 Review of Elements de Geometrie 2 vols The Mathematics Teacher a b c d Bodenmann Siegfried January 2010 The 18th century battle over lunar motion Physics Today 63 1 27 32 Bibcode 2010PhT 63a 27B doi 10 1063 1 3293410 Grier David Alan 2005 The First Anticipated Return Halley s Comet 1758 When Computers Were Human Princeton Princeton University Press pp 11 25 ISBN 0 691 09157 9 Terras Audrey 1999 Fourier analysis on finite groups and applications Cambridge University Press ISBN 978 0 521 45718 7 p 30References EditGrier David Alan When Computers Were Human Princeton University Press 2005 ISBN 0 691 09157 9 Casey J Clairaut s Hydrostatics A Study in Contrast American Journal of Physics Vol 60 1992 pp 549 554 External links Edit Wikisource has the text of the 1911 Encyclopaedia Britannica article Clairault Alexis Claude Chronologie de la vie de Clairaut 1713 1765 O Connor John J Robertson Edmund F Alexis Clairaut MacTutor History of Mathematics archive University of St Andrews W W Rouse Ball A Short Account of the History of Mathematics Retrieved from https en wikipedia org w index php title Alexis Clairaut amp oldid 1143518267, wikipedia, wiki, book, books, library,

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