Anders Johan Lexell: Difference between revisions

| name = Anders Lexell

| image = Lexell.png

| caption = Silhouette by [[Johann Friedrich Anthing|F. Anting]] (1784)

| birth_date = {{Birth date|df=yes|1740|12|24}}

| birth_place = [[Turku|Åbo]], [[Sweden]] (now [[Finland]])

| death_date = {{Death date and age|1784|12|11|1740|12|24|df=y}}<br /><small><nowiki>[</nowiki>[[Adoption of the Gregorian calendar#Adoption in Eastern Europe|OS]]: 30 November 1784<nowiki>]</nowiki></small>

| death_place = [[St. Petersburg]], [[Russian Empire]]

| alma_mater = [[The Royal Academy of Turku]]

| doctoral_advisor = [[Jakob Gadolin]]

| academic_advisors = M. J. Wallenius

| doctoral_students = <!-- Martin Platzmann -->

| notable_students =

| known_for = ComputedCalculated the orbit of [[Lexell's Comet]]<br />

ComputedCalculated the orbit of [[Uranus]]

'''Anders Johan Lexell''' (24 December 1740 &ndash; {{OldStyleDate|11 December|1784|30 November}}) was a [[Swedish-speaking Finnspopulation of Finland|Finnish-Swedish]] [[astronomer]], [[mathematician]], and [[physicist]] who spent most of his life in [[Russian Empire|Imperial Russia]], where he was known as '''Andrei Ivanovich Leksel''' (Андрей Иванович Лексель).

Anders Johan Lexell was born in [[Turku]] to Johan Lexell, a goldsmith and local administrative officer, and Madeleine-Catherine née Björkegren. At the age of fourteen he enrolled at the [[The Royal Academy of Turku|UniversityAcademy of Åbo]] and in 1760 received his [[Doctor of Philosophy]] degree with a dissertation "''Aphorismi mathematico-physici"'' (academic advisor [[Jakob Gadolin]]). In 1763 Lexell moved to [[Uppsala]] and worked at [[Uppsala University]] as a mathematics lecturer. From 1766 he was a professor of mathematics at the Uppsala Nautical School.

In 1762, [[Catherine II of Russia|Catherine the Great]] ascended to the Russian throne and started the politics of [[enlightened absolutism]]. She was aware of the importance of science and ordered to offer [[Leonhard Euler]] to "state his conditions, as soon as he moves to St. Petersburg without delay".<ref name="euler">{{cite book|title=Leonhard Euler|author=A. Ya. Yakovlev|date=1983|publisher=Prosvesheniye|location=Moscow}}</ref> Soon after his return to Russia, Euler suggested that the director of the [[Russian Academy of Science]] should invite mathematics professor Anders Johan Lexell to study mathematics and its application to astronomy, especially [[spherical geometry]]. The invitation by Euler and the preparations that were made at that time to observe the [[Transit of Venus#1761 and 1769|1769 transit of Venus]] from eight locations in the vast [[Russian Empire]] made Lexell seek the opportunity to become a member of the [[St. Petersburg]] scientific community.

Lexell made a large contribution to [[Lunar theory]] and especially to determining the [[parallax]] of the [[Sun]] from the results of observations of the [[transit of Venus]]. He earned universal recognition and, in 1771, when the [[Russian Academy of Sciences]] affiliated new members, Lexell was admitted as an Astronomy [[academician]]., Hehe wasalso admittedbecame toa membershipmember inof the [[Royal Swedish Academy of Sciences|Academy of Stockholm]] and [[Royal Society of Sciences in Uppsala|Academy of Uppsala]] in 1773 and 1774, and became a [[corresponding member]] of the [[French Academy of Sciences|Paris Royal Academy of Sciences]].

In 1775, the [[Swedish King]] appointed Lexell to a [[Chair (academic)|chair]] of the mathematics department at the [[The Royal Academy of Turku|University of Åbo]] with permission to stay at [[St. Petersburg]] for another three years to finish his work there; this permission was later prolonged for two more years. Hence, in 1780, Lexell was supposed to leave St. Petersburg and return to Sweden, which would have been a great loss for the [[Russian Academy of Sciences]]. Therefore, Director [[:ru:Домашнев, Сергей Герасимович|Domashnev]] proposed that Lexell travel to [[Germany]], [[England]], and [[France]] and then to return to St. Petersburg via Sweden. Lexell made the trip and, to the [[Russian Academy of Sciences|Academy's]] pleasure, got a discharge from the [[Swedish King]] and returned to St. Petersburg in 1781, after more than a year of absence, very satisfied with his trip.

Lexell departed St. Petersburg in late July 1780 on a [[sailing ship]] and via [[Swinemünde]] arrived in [[Berlin]], where he stayed for a month and travelled to [[Potsdam]], seeking in vain for an [[Audience (meeting)|audience]] with King [[Frederick II of Prussia|Frederick II]]. In September he left for [[Bavaria]], visiting [[Leipzig]], [[Göttingen]], and [[Mannheim]]. In October he traveled to [[Strasbourg|Straßbourg]] and then to [[Paris]], where he spent the winter. In March 1781 he moved to [[London]]. In August he left London for [[Belgium]], where he visited [[Flanders]] and [[Duchy of Brabant|Brabant]], then moved to the [[Netherlands]], visited [[The Hague]], [[Amsterdam]], and [[Saardam]], and then returned to [[Germany]] in September. He visited [[Hamburg]] and then boarded a ship in [[Kiel]] to sail to Sweden; he spent three days in [[Kopenhagen]] on the way. In Sweden he spent time in his native city [[Åbo]], and also visited [[Stockholm]], [[Uppsala]], and the [[Åland Islands]]. In early December 1781 Lexell returned to St. Petersburg, after having travelled for almost a year and a half.

There are 28 letters in the archive of the Academyacademy that Lexell wrote during the trip to [[Johann Euler]], while the official reports that Euler wrote to the Director of the Academyacademy, [[:ru:Домашнев, Сергей Герасимович|Domashnev]], were lost. However, unofficial letters to Johann Euler often contain detailed descriptions of places and people whom Lexell had met, and his impressions.<ref name="lyub">{{cite journal|author=Lubimenko, Inna|date=1936|title=The foreign trip of Academician A. J. Lexell in 1780-1781|journal=Archiv Istorii Nauki i Techniki|volume=8|pages=327–358|author-link=Inna Lubimenko}}</ref>

After Euler's passing, Academy Director, [[Princess]] [[Yekaterina Romanovna Vorontsova-Dashkova|Dashkova]], appointed Lexell in 1783 toEuler's replace himsuccessor. Lexell became a corresponding member of the Turin Royal Academy, and the London [[Board of Longitude]] put him on the list of scientists receiving its proceedings.

When applying for a position at the Russian Academy of Sciences, Lexell submitted a paper called "Method of analysing some differential equations, illustrated with examples",<ref>{{cite journal|date=1769|author=A. J. Lexell|title=Methodus integrandi nonnulis aequationum differentialum exemplis illustrata|journal=Novi Commentarii Academia Scientarum Imperialis Petropolitanae|volume=14|issue=1|pages=238–248}}</ref> which was highly praised by Leonhard Euler in 1768. Lexell's method is as follows: for a given nonlinear [[differential equation]] (e.g. second order) we pick an intermediate integral—a first-order [[differential equation]] with undefined coefficients and exponents. After differentiating this intermediate integral we compare it with the original equation and get the equations for the coefficients and exponents of the intermediate integral. After we express the undetermined coefficients via the known coefficients we substitute them in the intermediate integral and get two particular solutions of the original equation. Subtracting one particular solution from another we get rid of the differentials and get a general solution, which we analyse at various values of constants. The method of [[Ordinary differential equation#Reduction to a first -order system|reducing the order of the differential equation]] was known at that time, but in another form. Lexell's method was significant because it was applicable to a broad range of linear differential equations with constant coefficients that were important for physics applications. In the same year, Lexell published another article "On integrating the differential equation ''a''^''n''''d''^''n''''y'' + ''ba''^''n-1''''d''^''m-1''''ydx'' + ''ca''^''n-2''''d''^''m-2''''ydx''^''2'' + ... + ''rydx''^''n'' = ''Xdx''^''n''"<ref>{{cite journal| date = 1769| author = A. J. Lexell |title = De integratione aequationis differentialis ''a''^''n''''d''^''n''''y'' + ''ba''^''n-1''''d''^''m-1''''ydx'' + ''ca''^''n-2''''d''^''m-2''''ydx''^''2'' + ... + ''rydx''^''n'' = ''Xdx''^''n''|journal=Novi Commentarii Academia Scientarum Imperialis Petropolitanae|volume=14|issue=1|pages=215–237}}</ref> presenting a general highly algorithmic method of solving higher order linear differential equations with constant coefficients.

Lexell also looked for criteria of integrability of differential equations. He tried to find criteria for the whole differential equations and also for separate differentials. In 1770 he derived a criterion for integrating differential function, proved it for any number of items, and found the integrability criteria for <math display=inline>\scriptstyle dx\int{Vdx}</math>, <math display=inline>\scriptstyle dx\int{dx\int{Vdx}}</math>, <math>\scriptstyle display=inline >dx\int{dx\int{dx\int{Vdx}}}</math>. His results agreed with those of Leonhard Euler but were more general and were derived without the means of [[calculus of variations]]. At Euler's request, in 1772 Lexell communicated these results to [[Lagrange]]<ref>{{cite book|date=1862|author=Lagrange J. L.|title=Oeuvres|volume=3|location=Paris}}</ref> and [[Johann Heinrich Lambert|Lambert]].<ref>{{cite journal|date=1924|author=Bopp K.|title=Leonhard Eulers und Johann Heinrich Lamberts Briefwechsel|journal=Abh. Preuss. Akad. Wiss|volume=2|pages=38–40}}</ref>

There is an unpublished Lexell paper in the archive of the Russian Academy of Sciences with the title "Methods of integration of some differential equations", in which a complete solution of the equation <math>x=y\phi(x')+\psi(x')</math>, now known as the [[:ru:Уравнение Д’Аламбера{{ill|Lagrange-dLagrange–d'Alembert equation]]|ru|Уравнение Д’Аламбера}}, is presented.<ref>{{cite journal|date=1990|author=V. I. Lysenko|title=Differential equations in the works of A. I. Leksel|journal=Istoriko-Matematicheskie Issledovaniya|issue=32–33|location=Moscow|publisher=Nauka}}</ref>

[[Polygonometry]] was a significant part of Lexell's work. He used the [[trigonometry|trigonometric]] approach using the advance in trigonometry made mainly by [[Leonhard Euler|Euler]] and presented a general method of solving [[simple polygon|simple]] [[polygon]]s in two articles "On solving rectilinear polygons".<ref>{{cite journal|date=1774|author=A. J. Lexell|title=De resolutione polygonorum rectilineorum. Dissertiatio prima|journal=Novi Commentarii Academia Scientarum Imperialis Petropolitanae|volume=19|pages=184–236}}{{cite journal|date=1775|author=A. J. Lexell|title=De resolutione polygonorum rectilineorum. Dissertiatio secunda|journal=Novi Commentarii Academia Scientarum Imperialis Petropolitanae|volume=20|pages=80–122}}</ref> Lexell discussed two separate groups of problems: the first had the polygon defined by its [[Edge (geometry)|sides]] and [[Polygon#Angles|angles]], the second with its [[diagonal]]s and angles between [[diagonals]] and [[Edge (geometry)|sides]]. For the problems of the first group Lexell derived two general formulas giving <math>n</math> equations allowing to solve a polygon with <math>n</math> sides. Using these theorems he derived explicit formulas for [[triangle]]s and [[tetragon]]s and also gave formulas for [[pentagon]]s, [[hexagon]]s, and [[heptagon]]s. He also presented a classification of problems for [[tetragon]]stetragons, [[pentagon]]spentagons, and [[hexagon]]shexagons. For the second group of problems, Lexell showed that their solutions can be reduced to a few general rules and presented a classification of these problems, solving the corresponding [[combinatorics|combinatorial]] problems. In the second article he applied his general method for specific [[tetragon]]stetragons and showed how to apply his method to a [[polygon]] with any number of sides, taking a [[pentagon]] as an example.

The successor of Lexell's trigonometric approach (as opposed to a [[Analytic geometry|coordinate]] approach]]) was [[Swiss people|Swiss]] mathematician [[Simon Antoine Jean L'Huilier|L'Huilier]]. Both L'Huilier and Lexell emphasized the importance of [[polygonometry]] for theoretical and practical applications.

After that, Lexell spent most of his effort on [[comet]] [[astronomy]] (though his first paper on computingcalculating the [[orbit]] of a comet is dated 1770). In the next ten years he computedcalculated the orbits of all the newly discovered comets, among them the comet which [[Charles Messier]] discovered in 1770. Lexell computedcalculated its orbit, showed that the comet had had a much larger [[perihelion]] before the encounter with [[Jupiter]] in 1767 and predicted that after encountering [[Jupiter]] again in 1779 it would be altogether expelled from the [[Solar system#Inner Solar System|inner Solar System]]. This comet was later named [[Lexell's Comet]].

Lexell also was the first to computecalculate the orbit of [[Uranus]] and to actually prove that it was a [[planet]] rather than a [[comet]].<ref>{{cite journal|date=1783|author=A. J. Lexell|title=Recherches sur la nouvelle planete, decouverte par M. Herschel & nominee Georgium Sidus|journal=Acta Academiae Scientiarum Imperialis Petropolitanae|issue=1|pages=303–329}}</ref> He made preliminary calculations while travelling in [[Europe]] in 1781 based on [[William Herschel|Hershel's]] and [[Nevil Maskelyne|Maskelyne's]] observations. Having returned to [[Russia]], he computedestimated the orbit more precisely based on new observations, but due to the long [[orbital period]] it was still not enough data to prove that the [[orbit]] was not [[Parabolic trajectory|parabolic]]. Lexell then found the record of a star observed in 1759 by [[Christian Mayer (astronomer)|Christian Mayer]] in [[Pisces (constellation)|Pisces]] that was neither in the [[Flamsteed]] catalogues nor in the sky by the time [[Johann Elert Bode|Bode]] sought it. Lexell presumed that it was an earlier sighting of the same [[astronomical object]] and using this data he calculated the exact orbit, which proved to be elliptical, and proved that the new object was actually a [[planet]]. In addition to calculating the parameters of the orbit Lexell also estimated the planet's size more precisely than his contemporaries using [[Mars]] that was in the vicinity of the new planet at that time. Lexell also noticed that the orbit of [[Uranus]] was being [[Perturbation (astronomy)|perturbed]]. He then stated that, based on his data on various [[comet]]s, the size of the [[Solar systemSystem]] can be 100 [[Astronomical unit|AU]] or even more, and that it could be other [[planet]]s there that [[Perturbation (astronomy)|perturb]] the [[orbit]] of [[Uranus]] (although the position of the eventual [[Neptune]] was not calculated until much later by [[Urbain Le Verrier]]).

{{Reflist|30em}}

Johan* Stén, Johan C.-E. (2015): ''A Comet of the Enlightenment.: Anders Johan Lexell's Life and Discoveries.'' Basel: Birkhäuser. {{ISBN|978-3-319-00617-8}}

{{DEFAULTSORT:Lexell, Anders Johan}}

[[Category:18th-century mathematicians from the Russian mathematiciansEmpire]]

[[Category:ImperialAstronomers from the Russian astronomersEmpire]]

[[Category:18th-century mathematiciansSwedish astronomers]]