Anders Lexell | |
|---|---|
 Silhouette byF. Anting (1784) | |
| Born | (1740-12-24)24 December 1740 |
| Died | 11 December 1784(1784-12-11) (aged 43) [OS: 30 November 1784] |
| Alma mater | Royal Academy of Åbo |
| Known for | Calculated the orbit ofLexell's Comet Calculated the orbit ofUranus |
| Scientific career | |
| Fields | |
| Institutions | Uppsala Nautical School Imperial Russian Academy of Sciences |
| Doctoral advisor | Jakob Gadolin |
| Other academic advisors | M. J. Wallenius |
Anders Johan Lexell (24 December 1740 – 11 December [O.S. 30 November] 1784) was aFinnish-Swedishastronomer,mathematician, andphysicist who spent most of his life inImperial Russia, where he was known asAndrei Ivanovich Leksel (Андрей Иванович Лексель).
Lexell made important discoveries inpolygonometry andcelestial mechanics; the latter led to acomet named in his honour.La Grande Encyclopédie states that he was the prominent mathematician of his time who contributed tospherical trigonometry with new and interesting solutions, which he took as a basis for his research ofcomet andplanet motion. His name was given to a theorem ofspherical triangles.
Lexell was one of the most prolific members of theRussian Academy of Sciences at that time, having published 66 papers in 16 years of his work there. A statement attributed toLeonhard Euler expresses high approval of Lexell's works: "Besides Lexell, such a paper could only be written byD'Alambert or me".[1]Daniel Bernoulli also praised his work, writing in a letter toJohann Euler "I like Lexell's works, they are profound and interesting, and the value of them is increased even more because of his modesty, which adorns great men".[2]
Lexell was unmarried, and kept up a close friendship with Leonhard Euler and his family. He witnessed Euler's death at his house and succeeded Euler to thechair of the mathematics department at the Russian Academy of Sciences, but died the following year. The asteroid2004 Lexell is named in his honour, as is the lunar craterLexell.
Anders Johan Lexell was born inÅbo (Turku) to Johan Lexell, a goldsmith and local administrative officer, and Madeleine-Catherine née Björkegren. At the age of fourteen he enrolled at theRoyal Academy of Åbo and in 1760 received hisDoctor of Philosophy degree with a dissertationAphorismi mathematico-physici (academic advisorJakob Gadolin). In 1763 Lexell moved toUppsala and worked atUppsala University as a mathematics lecturer. From 1766 he was a professor of mathematics at the Uppsala Nautical School.
In 1762,Catherine the Great ascended to the Russian throne and started the politics ofenlightened absolutism. She was aware of the importance of science and ordered to offerLeonhard Euler to "state his conditions, as soon as he moves to St. Petersburg without delay".[3] Soon after his return to Russia, Euler suggested that the director of theRussian Academy of Science should invite Lexell to study mathematics and its application to astronomy, especiallyspherical geometry. The invitation by Euler and the preparations that were made at that time to observe the1769 transit of Venus from eight locations in the vastRussian Empire made Lexell seek the opportunity to become a member of theSt. Petersburg scientific community.
To be admitted to theRussian Academy of Sciences, Lexell in 1768 wrote a paper onintegral calculus called "Methodus integrandi nonnulis aequationum exemplis illustrata". Euler was appointed to evaluate the paper and highly praised it, andCountVladimir Orlov, director of theRussian Academy of Sciences, invited Lexell to the position of mathematics adjunct, which Lexell accepted. In the same year he received permission from theSwedish king to leave Sweden, and moved toSt. Petersburg.
His first task was to become familiar with theastronomical instruments that would be used in the observations of thetransit of Venus. He participated in observing the 1769 transit atSt. Petersburg together withChristian Mayer, who was hired by theAcademy to work at theobservatory while the Russian astronomers went to other locations.
Lexell made a large contribution toLunar theory and especially to determining theparallax of theSun from the results of observations of thetransit of Venus. He earned universal recognition and, in 1771, when theRussian Academy of Sciences affiliated new members, Lexell was admitted as an Astronomyacademician, he also became a member of theAcademy of Stockholm andAcademy of Uppsala in 1773 and 1774, and became acorresponding member of theParis Royal Academy of Sciences.
In 1775, theSwedish King appointed Lexell to achair of the mathematics department at theUniversity of Åbo with permission to stay atSt. 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 theRussian Academy of Sciences. Therefore, DirectorDomashnev proposed that Lexell travel to Germany, England, and France and then to return to St. Petersburg via Sweden. Lexell made the trip and, to theAcademy's pleasure, got a discharge from theSwedish King and returned to St. Petersburg in 1781, after more than a year of absence, very satisfied with his trip.
Sending academicians abroad was quite rare at that time (as opposed to the early years of theRussian Academy of Sciences), so Lexell willingly agreed to make the trip. He was instructed to write his itinerary, which without changes was signed byDomashnev. The aims were as follows: since Lexell would visit major observatories on his way, he should learn how they were built, note the number and types of scientific instruments used, and if he found something new and interesting he should buy the plans and design drawings. He should also learn everything aboutcartography and try to get newgeographic,hydrographic,military, andmineralogicmaps. He should also write letters to theAcademy regularly to report interesting news on science, arts, and literature.[4]
Lexell departed St. Petersburg in late July 1780 on asailing ship and viaSwinemünde arrived inBerlin, where he stayed for a month and travelled toPotsdam, seeking in vain for anaudience with KingFrederick II. In September he left forBavaria, visitingLeipzig,Göttingen, andMannheim. In October he traveled toStraßbourg and then toParis, where he spent the winter. In March 1781 he moved toLondon. In August he left London for Belgium, where he visitedFlanders andBrabant, then moved to the Netherlands, visitedThe Hague,Amsterdam, andSaardam, and then returned to Germany in September. He visitedHamburg and then boarded a ship inKiel to sail to Sweden; he spent three days inCopenhagen on the way. In Sweden he spent time in his native cityÅbo, and also visitedStockholm,Uppsala, andÅland. 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 academy that Lexell wrote during the trip toJohann Euler, while the official reports that Euler wrote to the Director of the academy,Domashnev, were lost. However, unofficial letters to Johann Euler often contain detailed descriptions of places and people whom Lexell had met, and his impressions.[5]
Lexell became very attached to Leonhard Euler, who lost his sight in his last years but continued working using his elder son Johann Euler to read for him. Lexell helped Leonhard Euler greatly, especially in applyingmathematics tophysics andastronomy. He helped Euler to write calculations and prepare papers. On 18 September 1783, after a lunch with his family, during a conversation with Lexell about the newly discoveredUranus and itsorbit, Euler felt sick. He died a few hours later.[3]
After Euler's passing, Academy Director,PrincessDashkova, appointed Lexell in 1783 Euler's successor. Lexell became a corresponding member of the Turin Royal Academy, and the LondonBoard of Longitude put him on the list of scientists receiving its proceedings.
Lexell did not enjoy his position for long: he died on 30 November 1784.
Lexell is mainly known for his works inastronomy andcelestial mechanics, but he also worked in almost all areas of mathematics:algebra,differential calculus,integral calculus,geometry,analytic geometry,trigonometry, andcontinuum mechanics. Being amathematician and working on the main problems ofmathematics, he never missed the opportunity to look into specific problems inapplied science, allowing for experimental proof of theory underlying the physical phenomenon. In 16 years of his work at the Russian Academy of Sciences, he published 62 works, and 4 more with coauthors, among whom areLeonhard Euler,Johann Euler,Wolfgang Ludwig Krafft,Stephan Rumovski, andChristian Mayer.[5]
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",[6] which was highly praised by Leonhard Euler in 1768. Lexell's method is as follows: for a given nonlineardifferential equation (e.g. second order) we pick an intermediate integral—a first-orderdifferential 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 ofreducing 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 equationandny +ban-1dm-1ydx +can-2dm-2ydx2 + ... +rydxn =Xdxn"[7] 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,,. His results agreed with those of Leonhard Euler but were more general and were derived without the means ofcalculus of variations. At Euler's request, in 1772 Lexell communicated these results toLagrange[8] andLambert.[9]
Concurrently with Euler, Lexell worked on expanding theintegrating factor method to higher order differential equations. He developed the method of integrating differential equations with two or three variables by means of theintegrating factor. He stated that his method could be expanded for the case of four variables: "The formulas will be more complicated, while the problems leading to such equations are rare in analysis".[10]
Also of interest is the integration of differential equations in Lexell's paper "On reducing integral formulas to rectification of ellipses and hyperbolae",[11] which discusseselliptic integrals and their classification, and in his paper "Integrating one differential formula with logarithms and circular functions",[12] which was reprinted in the transactions of theSwedish Academy of Sciences. He also integrated a few complicated differential equations in his papers oncontinuum mechanics, including a four-order partial differential equation in a paper about coiling a flexible plate to a circular ring.[13]
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, now known as theLagrange–d'Alembert equation [ru], is presented.[14]
Polygonometry was a significant part of Lexell's work. He used thetrigonometric approach using the advance in trigonometry made mainly byEuler and presented a general method of solvingsimplepolygons in two articles "On solving rectilinear polygons".[15] Lexell discussed two separate groups of problems: the first had the polygon defined by itssides andangles, the second with itsdiagonals and angles between diagonals and sides. For the problems of the first group Lexell derived two general formulas giving equations allowing to solve a polygon with sides. Using these theorems he derived explicit formulas fortriangles andtetragons and also gave formulas forpentagons,hexagons, andheptagons. He also presented a classification of problems for tetragons, pentagons, and hexagons. 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 correspondingcombinatorial problems. In the second article he applied his general method for specific tetragons 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 acoordinate approach) wasSwiss mathematicianL'Huilier. Both L'Huilier and Lexell emphasized the importance of polygonometry for theoretical and practical applications.
Lexell's first work at the Russian Academy of Sciences was to analyse data collected from the observation of the1769 transit of Venus. He published four papers in "Novi Commentarii Academia Petropolitanae" and ended his work with a monograph on determining theparallax of theSun, published in 1772.[16]
Lexell aided Euler in finishing hisLunar theory, and was credited as a co-author in Euler's 1772 "Theoria motuum Lunae".[17]
After that, Lexell spent most of his effort oncometastronomy (though his first paper on calculating theorbit of a comet is dated 1770). In the next ten years he calculated the orbits of all the newly discovered comets, among them the comet whichCharles Messier discovered in 1770. Lexell calculated its orbit, showed that the comet had had a much largerperihelion before the encounter withJupiter in 1767 and predicted that after encounteringJupiter again in 1779 it would be altogether expelled from theinner Solar System. This comet was later namedLexell's Comet.
Lexell also was the first to calculate the orbit ofUranus and to actually prove that it was aplanet rather than acomet.[18] He made preliminary calculations while travelling in Europe in 1781 based onHershel's andMaskelyne's observations. Having returned to Russia, he estimated the orbit more precisely based on new observations, but due to the longorbital period it was still not enough data to prove that theorbit was notparabolic. Lexell then found the record of a star observed in 1759 byChristian Mayer inPisces that was neither in theFlamsteed catalogues nor in the sky by the timeBode sought it. Lexell presumed that it was an earlier sighting of the sameastronomical object and using this data he calculated the exact orbit, which proved to be elliptical, and proved that the new object was actually aplanet. In addition to calculating the parameters of the orbit Lexell also estimated the planet's size more precisely than his contemporaries usingMars that was in the vicinity of the new planet at that time. Lexell also noticed that the orbit ofUranus was beingperturbed. He then stated that, based on his data on variouscomets, the size of theSolar System can be 100AU or even more, and that it could be otherplanets there thatperturb theorbit ofUranus (although the position of the eventualNeptune was not calculated until much later byUrbain Le Verrier).
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