**Anders Johan Lexell** was a Swedish-born Russian

astronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

,

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, and

physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

who spent most of his life in

RussiaRussia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

where he is known as

**Andrei Ivanovich Leksel** (Андрей Иванович Лексель).

Lexell made important discoveries in polygonometry and

celestial mechanicsCelestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

; the latter led to a

cometD/1770 L1, popularly known as Lexell's Comet after its orbit computer Anders Johan Lexell, was a comet discovered by astronomer Charles Messier in June 1770. It is notable for having passed closer to the Earth than any other comet in recorded history, approaching to a distance of only...

named in his honour.

La Grande EncyclopédieLa Grande Encyclopédie, inventaire raisonné des sciences, des lettres, et des arts is a 31-volume encyclopedia published in France from 1886 to 1902 by H...

states that he was the prominent mathematician of his time who contributed to the

spherical trigonometrySpherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...

with new and interesting solutions, which he took as a basis for his research of

cometA comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

and planet motion. His name was given to one of the theorems about spherical triangles.

Lexell was one of the most prolific members of the

Russian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

at that time, having published 66 papers in 16 years of his work there. A statement attributed to

Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

expresses high approval of Lexell's works: "Besides Lexell, such a paper could only be written by D'Alambert or me".

Daniel BernoulliDaniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

also praised his work, writing in a letter to

Johann EulerJohann Albrecht Euler was a Swiss-Russian astronomer and mathematician. Also known as Johann Albert Euler or John-Albert Euler, he was the first child born to the great Swiss mathematician Leonhard Euler who had emigrated [for the first time] to Saint-Petersburg on 17 May 1727...

"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".

Lexell did not have a family and kept up a close friendship with Leonhard Euler and his family. He witnessed Euler's death at his house and succeeded him to the chair of the mathematics department at the Russian Academy of Sciences, but died the following year. The asteroid

2004 Lexell2004 Lexell is a main-belt asteroid discovered on September 22, 1973 by N. Chernykh at Nauchnyj.- External links :*...

is named in his honour, as is the lunar crater

LexellLexell is a lunar crater that lies across the southeastern rim of the huge walled plain Deslandres, in the southern part of the Moon. To the northeast is the walled plain Walther, and to the south is Orontius, another walled plain....

.

### Early years

Anders Johan Lexell was born in

ÅboTurku is a city situated on the southwest coast of Finland at the mouth of the Aura River. It is located in the region of Finland Proper. It is believed that Turku came into existence during the end of the 13th century which makes it the oldest city in Finland...

to Johan Lexell, a local administrative officer, and Madeleine-Catherine Björkegren. At the age of fourteen he enrolled at the

University of ÅboThe Royal Academy of Turku was the first university in Finland, and the only university in present-day Finland to be founded when it was still a part of Sweden. In 1809, after Finland became a Grand Duchy under the suzerainty of the Russian Tzar, it was renamed the Imperial Academy of Turku...

and in 1760 received his

Doctor of PhilosophyDoctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...

degree with a dissertation "Aphorismi mathematico-physici" (academic advisor

Jakob GadolinJakob Gadolin was a Finnish Lutheran bishop, professor of physics and theology, politician and statesman....

). In 1763 he moved to

Uppsala- Economy :Today Uppsala is well established in medical research and recognized for its leading position in biotechnology.*Abbott Medical Optics *GE Healthcare*Pfizer *Phadia, an offshoot of Pharmacia*Fresenius*Q-Med...

and worked at

Uppsala UniversityUppsala University is a research university in Uppsala, Sweden, and is the oldest university in Scandinavia, founded in 1477. It consistently ranks among the best universities in Northern Europe in international rankings and is generally considered one of the most prestigious institutions of...

as a mathematics lecturer, and from 1766 as a professor of mathematics at the Uppsala Nautical School.

### St. Petersburg

In 1762,

Catherine the GreatCatherine II, also known as Catherine the Great , Empress of Russia, was born in Stettin, Pomerania, Prussia on as Sophie Friederike Auguste von Anhalt-Zerbst-Dornburg...

came to the throne and started the politics of

enlightened absolutismEnlightened absolutism is a form of absolute monarchy or despotism in which rulers were influenced by the Enlightenment. Enlightened monarchs embraced the principles of the Enlightenment, especially its emphasis upon rationality, and applied them to their territories...

. She was aware of the importance of science and ordered to offer

Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

to "state his conditions, as soon as he moves to St. Petersburg without delay". Soon after his return to Russia, Euler proposed to the director of the Russian Academy of Science to invite the mathematics professor Anders Johan Lexell to study mathematics and its application to astronomy, especially the

spherical geometrySpherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....

. The invitation by Euler and the preparations that were made at that time to observe the 1769 transit of Venus from eight locations in the vast

Russian EmpireThe Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...

made Lexell seek the opportunity to become a member of the St. Petersburg scientific community.

To be admitted to the

Russian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

, Lexell in 1768 wrote a paper on integral calculus calles "Methodus integrandi nonnulis aequationum exemplis illustrata".

AcademicianThe title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...

Euler, who was appointed to evaluate the paper, highly praised it, and

CountA count or countess is an aristocratic nobleman in European countries. The word count came into English from the French comte, itself from Latin comes—in its accusative comitem—meaning "companion", and later "companion of the emperor, delegate of the emperor". The adjective form of the word is...

Vladimir Orlov, who was the director of the

Russian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

at that time, sent the invitation to Lexell to take the position of mathematics adjunct, which Lexell willingly accepted. In the same year he got the permission from the Swedish king and immediately left for St. Petersburg.

His first occupation was getting used to the astronomical instruments to take part in the observations of the

transit of VenusA transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, becoming visible against the solar disk. During a transit, Venus can be seen from Earth as a small black disk moving across the face of the Sun...

, which he did in 1769 in St. Petersburg together with Christian Mayer, who was hired by the

AcademyThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

to work at the

observatoryAn observatory is a location used for observing terrestrial or celestial events. Astronomy, climatology/meteorology, geology, oceanography and volcanology are examples of disciplines for which observatories have been constructed...

while the Russian astronomers went to other locations.

Lexell made a large contribution to the

Lunar theoryLunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

and especially to determining the

parallaxParallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...

of the

SunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

from the results of observations of the

transit of VenusA transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, becoming visible against the solar disk. During a transit, Venus can be seen from Earth as a small black disk moving across the face of the Sun...

. He earned universal recognition and, in 1771, when the

Russian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

affiliated new members, Lexell was admitted as an Astronomy

academicianThe title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...

. He was admitted to membership in the

Academy of StockholmThe Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

and

Academy of UppsalaThe Royal Society of Sciences in Uppsala , is the oldest of the royal academies in Sweden. The society has, by royal decree of 1906, 50 Swedish fellows and 100 foreign....

in 1773 and 1774, and became a corresponding member of the

Paris Royal Academy of SciencesThe French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

.

### Foreign trip

In 1775, the Swedish King appointed him to a chair of the mathematics department at the

University of ÅboThe Royal Academy of Turku was the first university in Finland, and the only university in present-day Finland to be founded when it was still a part of Sweden. In 1809, after Finland became a Grand Duchy under the suzerainty of the Russian Tzar, it was renamed the Imperial Academy of Turku...

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

SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....

, which would have been a great loss for the

AcademyThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

, if director Domashnev had not offered Lexell to go for a trip to

GermanyGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

,

EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

, and

FranceThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

and then to return to St. Petersburg via

SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....

. Lexell went on a trip and, to the

Academy'sThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

great 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.

Sending academicians abroad was quite rare at that time (as opposed to the early years of the

Russian Academy of SciencesThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

), so Lexell willingly agreed to go for a trip. He was offered to write an instruction for himself which then without changes was signed by Domashnev. The aims were as follows: since Lexell would visit major observatories on his way, he should learn how they are 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 about

cartographyCartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

and try to get new geographic,

hydrographicHydrography is the measurement of the depths, the tides and currents of a body of water and establishment of the sea, river or lake bed topography and morphology. Normally and historically for the purpose of charting a body of water for the safe navigation of shipping...

,

militaryA military is an organization authorized by its greater society to use lethal force, usually including use of weapons, in defending its country by combating actual or perceived threats. The military may have additional functions of use to its greater society, such as advancing a political agenda e.g...

, and

mineralogicMineralogy is the study of chemistry, crystal structure, and physical properties of minerals. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.-History:Early writing...

mapA map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

s. He should also write letters to the

AcademyThe Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

regularly to report interesting news on science, arts, and literature.

Lexell left St. Petersburg in late July 1780 on a

sailing shipThe term sailing ship is now used to refer to any large wind-powered vessel. In technical terms, a ship was a sailing vessel with a specific rig of at least three masts, square rigged on all of them, making the sailing adjective redundant. In popular usage "ship" became associated with all large...

and via Swinemünde arrived in

BerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, where he stayed for a month and travelled to

PotsdamPotsdam is the capital city of the German federal state of Brandenburg and part of the Berlin/Brandenburg Metropolitan Region. It is situated on the River Havel, southwest of Berlin city centre....

, seeking in vain for an audience with King

Frederick IIFrederick II was a King in Prussia and a King of Prussia from the Hohenzollern dynasty. In his role as a prince-elector of the Holy Roman Empire, he was also Elector of Brandenburg. He was in personal union the sovereign prince of the Principality of Neuchâtel...

. In September he left for

BavariaBavaria, formally the Free State of Bavaria is a state of Germany, located in the southeast of Germany. With an area of , it is the largest state by area, forming almost 20% of the total land area of Germany...

, visiting

LeipzigLeipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

,

GöttingenGöttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...

, and

MannheimMannheim is a city in southwestern Germany. With about 315,000 inhabitants, Mannheim is the second-largest city in the Bundesland of Baden-Württemberg, following the capital city of Stuttgart....

. In October, via Strassburg he arrived in

ParisParis is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

, where he spent the winter, and in March 1781 he moved to

LondonLondon is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...

. After spending three months in

EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

, in August he left for

BelgiumBelgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...

, where he visited

FlandersFlanders is the community of the Flemings but also one of the institutions in Belgium, and a geographical region located in parts of present-day Belgium, France and the Netherlands. "Flanders" can also refer to the northern part of Belgium that contains Brussels, Bruges, Ghent and Antwerp...

and

BrabantThe Duchy of Brabant was a historical region in the Low Countries. Its territory consisted essentially of the three modern-day Belgian provinces of Flemish Brabant, Walloon Brabant and Antwerp, the Brussels-Capital Region and most of the present-day Dutch province of North Brabant.The Flag of...

, then moved to the

NetherlandsThe Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

, visited

The HagueThe Hague is the capital city of the province of South Holland in the Netherlands. With a population of 500,000 inhabitants , it is the third largest city of the Netherlands, after Amsterdam and Rotterdam...

,

AmsterdamAmsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...

, and Saardam, and then returned to

GermanyGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

in September. He visited

Hamburg-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

and then got aboard a ship in

KielKiel is the capital and most populous city in the northern German state of Schleswig-Holstein, with a population of 238,049 .Kiel is approximately north of Hamburg. Due to its geographic location in the north of Germany, the southeast of the Jutland peninsula, and the southwestern shore of the...

to sail to

SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....

; he spent three days in Kopenhagen on the way. In

SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....

he spent some time in his native city

ÅboAbo may refer to:* ABO blood group system, a human blood type and blood group system** ABO , enzyme encoded by the ABO gene that determines the ABO blood group of an individual* Abo of Tiflis , an Arab East Orthodox Catholic saint...

, and also visited

StockholmStockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...

,

Uppsala- Economy :Today Uppsala is well established in medical research and recognized for its leading position in biotechnology.*Abbott Medical Optics *GE Healthcare*Pfizer *Phadia, an offshoot of Pharmacia*Fresenius*Q-Med...

, and the

Åland IslandsThe Åland Islands form an archipelago in the Baltic Sea. They are situated at the entrance to the Gulf of Bothnia and form an autonomous, demilitarised, monolingually Swedish-speaking region of Finland...

. 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 to

Johann EulerJohann Albrecht Euler was a Swiss-Russian astronomer and mathematician. Also known as Johann Albert Euler or John-Albert Euler, he was the first child born to the great Swiss mathematician Leonhard Euler who had emigrated [for the first time] to Saint-Petersburg on 17 May 1727...

, 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 Lexell met and his impressions.

### Last years

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. Lexell helped Leonhard Euler greatly, especially in applying

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

to

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and

astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. He helped Euler write down calculations and papers. On 18 September 1783, after a lunch with his family, during a conversation with Lexell about the newly discovered

UranusUranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

and its

orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

, Euler felt sick. He died a few hours later.

The director of the Academy,

PrincessPrincess is the feminine form of prince . Most often, the term has been used for the consort of a prince, or his daughters....

DashkovaPrincess Yekaterina Romanovna Vorontsova-Dashkova was the closest female friend of Empress Catherine the Great and a major figure of the Russian Enlightenment...

, appointed Lexell in 1783 to replace the late Leonhard Euler. Lexell became a corresponding member of the Turin Royal Academy, and the London

Board of LongitudeThe Board of Longitude was the popular name for the Commissioners for the Discovery of the Longitude at Sea. It was a British Government body formed in 1714 to administer a scheme of prizes intended to encourage innovators to solve the problem of finding longitude at sea.-Origins:Navigators and...

put him on the list of scientists receiving its proceedings.

Lexell did not enjoy his position for long: he died in the evening on 30 November 1784.

## Contribution to science

Lexell is mainly known for his works in

astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and

celestial mechanicsCelestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

, but he also worked in almost all areas of mathematics:

algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

,

differential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, integral calculus,

geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

,

analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

,

trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, as well as

continuum mechanicsContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

. Being a

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

and working on the main problems of

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, he never missed the opportunity to look into specific problems in

applied scienceApplied science is the application of scientific knowledge transferred into a physical environment. Examples include testing a theoretical model through the use of formal science or solving a practical problem through the use of natural 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 are

Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

,

Johann EulerJohann Albrecht Euler was a Swiss-Russian astronomer and mathematician. Also known as Johann Albert Euler or John-Albert Euler, he was the first child born to the great Swiss mathematician Leonhard Euler who had emigrated [for the first time] to Saint-Petersburg on 17 May 1727...

,

Wolfgang Ludwig KrafftWolfgang Ludwig Krafft was a German astronomer and physicist. He is the namesake of the lunar crater Krafft which has a diameter of 51 km.-External links:*...

, Stephan Rumovski, and Christian Mayer.

### Differential equations

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", which was highly praised by Leonhard Euler in 1768. Lexell's method is as follows: for a given nonlinear

differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

(e.g. second order) we pick an intermediate integral—a first-order

differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

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 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}" 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 of

calculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

. By special desire of Euler, in 1772 Lexell communicated these results to

LagrangeLa Grange literally means the barn in French. Lagrange may refer to:- People :* Charles Varlet de La Grange , French actor* Georges Lagrange , translator to and writer in Esperanto...

and

LambertJohann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

.

Concurrently with Euler, Lexell worked on expanding the

integrating factorIn mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...

method to higher order differential equations. He developed the method of integrating differential equations with two or three variables by means of the

integrating factorIn mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...

. 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".

Of interest is also the integration of differential equations done by Lexell in his paper "On reducing integral formulas to rectification of ellipses and hyperbolae", which discusses elliptic integrals and their classification, and in his paper "Integrating one differential formula with logarithms and circular functions", which was reprinted in the transactions of the

Swedish Academy of SciencesThe Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

. He also integrated a few complicated differential equations in his papers on

continuum mechanicsContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, including a four-order partial differential equation in a paper about coiling a flexible plate to a circular ring.

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 the Lagrange-d'Alembert equation, is presented.

### Polygonometry

Polygonometry was a significant part of Lexell's work. He used the

trigonometricalTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

approach using the advance in

trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

made mainly by

EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

and presented a general method of solving

simple polygonIn geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....

s in two articles "On solving rectilinear polygons". Lexell discussed two separate groups of problems: the first had the polygon defined by its sides and angles, the second with its

diagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...

s 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 for

triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s and tetragons and also gave formulas for

pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

s,

hexagons, and

heptagons. He also presented a classification of problems for tetragons,

pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

s, 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 corresponding

combinatorialCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

problems. In the second article he applies his general method for specific tetragons and shows how to apply his method to a

polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

with any number of sides, taking a

pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

as an example.

The successor of Lexell's

trigonometricTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

approach (as opposed to coordinate one) was a Swiss

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

named

L'HuilierSimon Antoine Jean L'Huilier was a Swiss mathematician of French Hugenot descent...

. Both L'Huilier and Lexell emphasized the importance of polygonometry for theoretical and practical applications.

### Celestial mechanics and astronomy

Lexell's first work at the Russian Academy of Sciences was to analyse the data collected from the observation of the 1769 transit of Venus. He published four papers in "Novi Commentarii Academia Petropolitanae" and ended his work with a monograph on determining the

parallaxParallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...

of the

SunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, published in 1772.

He also helped

EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

finish his

Lunar theoryLunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

and was credited as a coauthor in

Euler'sLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

1772 "Theoria motuum Lunae".

After that, Lexell switched to

cometA comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

(though his first paper on computing the

orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

of a comet is dated 1770). In the next ten years he computed the orbits of all the newly discovered comets, among them the comet discovered in 1770 by

Charles MessierCharles Messier was a French astronomer most notable for publishing an astronomical catalogue consisting of deep sky objects such as nebulae and star clusters that came to be known as the 110 "Messier objects"...

. Lexell computed its orbit, showed that the comet had had a much larger perihelion before the encounter with

JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

in 1767 and predicted that after encountering

JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

again in 1779 it would be altogether expelled from the inner Solar System. This comet was later named after him as

Lexell's CometD/1770 L1, popularly known as Lexell's Comet after its orbit computer Anders Johan Lexell, was a comet discovered by astronomer Charles Messier in June 1770. It is notable for having passed closer to the Earth than any other comet in recorded history, approaching to a distance of only...

.

Lexell also was the first to compute the orbit of

UranusUranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

and to actually prove that it was a

planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

rather than a

cometA comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

. He made first computations when he was travelling in

EuropeEurope is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...

in 1781 based on

Hershel'sSir Frederick William Herschel, KH, FRS, German: Friedrich Wilhelm Herschel was a German-born British astronomer, technical expert, and composer. Born in Hanover, Wilhelm first followed his father into the Military Band of Hanover, but emigrated to Britain at age 19...

and

Maskelyne'sThe Reverend Dr Nevil Maskelyne FRS was the fifth English Astronomer Royal. He held the office from 1765 to 1811.-Biography:...

observations. Having returned to

RussiaRussia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

, he computed the orbit more precisely based on new observations, but due to the long

orbital periodThe orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...

it was still not enough data to prove that the

orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

was not

parabolicIn astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

. Lexell then found the record of a star observed in 1759 by Christian Mayer in

PiscesPisces is a constellation of the zodiac. Its name is the Latin plural for fish, and its symbol is . It lies between Aquarius to the west and Aries to the east...

that was neither in the Flamsteed catalogues nor in the sky by the time

BodeJohann Elert Bode was a German astronomer known for his reformulation and popularization of the Titius-Bode law. Bode determined the orbit of Uranus and suggested the planet's name.-Biography:...

sought it. Lexell presumed that it was an earlier sighting of the same

astronomical objectAstronomical objects or celestial objects are naturally occurring physical entities, associations or structures that current science has demonstrated to exist in the observable universe. The term astronomical object is sometimes used interchangeably with astronomical body...

and using this data he calculated the exact orbit, which proved to be elliptical, and proved that the new object was actually a

planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

. In addition to calculating the parameters of the orbit Lexell also estimated the planet's size more precisely than his contemporaries using

MarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...

that was in the vicinity of the new planet at that time. Lexell also noticed that the orbit of

UranusUranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

was being

perturbedPerturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

. He then stated that, based on his data on various

cometA comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

s, the size of the

Solar systemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

can be 100

A.U.An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

or even more, and that it could be other

planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s there that

perturbPerturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

the

orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

of

UranusUranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

(although the position of the eventual

NeptuneNeptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

was not calculated until much later by

Urbain Le Verrier).