Carl Ferdinand Degen (1 November 1766 – 8 April 1825) was a

Danish Danish may refer to: * Something of, from, or related to the country of Denmark People * A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark * Culture of Denmark * Danish people or Danes, people with a Danish a ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. His most important contributions were within

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and he advised the young, aspiring

Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe *Norwegians, both a nation and an ethnic group native to Norway *Demographics of Norway *The Norwegian language, including the ...

mathematician

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

in a decisive way. Degen has received much of the credit for the introduction of more modern and advanced

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

in the Danish-Norwegian school system. He was born in

Braunschweig Braunschweig () or Brunswick ( , from Low German ''Brunswiek'' , Braunschweig dialect: ''Bronswiek'') is a city in Lower Saxony, Germany, north of the Harz Mountains at the farthest navigable point of the river Oker, which connects it to the Nor ...

Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...

, but the family moved to

Copenhagen Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan ar ...

in 1771 when his father Johan Philip Degen got a position in the

Royal Danish Orchestra The Royal Danish Orchestra (''Det Kongelige Kapel'') is a Danish orchestra based in Copenhagen. The Danish name for the orchestra indicates its original function as an ensemble geared to supplying the music for court events. The Royal Danish Orch ...

. As a musician he had a low salary, but his son Carl Ferdinand received a fellowship so that he could go to school in

Helsingør Helsingør ( , ; sv, Helsingör), classically known in English as Elsinore ( ), is a city in eastern Denmark. Helsingør Municipality had a population of 62,686 on 1 January 2018. Helsingør and Helsingborg in Sweden together form the northern ...

. He graduated from there in 1783 and continued at the

University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in ...

. Instead of following the normal path of studies, the young Degen followed his own interests and read

classical language A classical language is any language with an independent literary tradition and a large and ancient body of written literature. Classical languages are typically dead languages, or show a high degree of diglossia, as the spoken varieties of the ...

philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

natural science Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...

s and in particular

.Salmonsens Konservationsleksikon, ''Carl Ferdinand Degen'',

Projekt Runeberg Project Runeberg ( sv, Projekt Runeberg) is a digital cultural archive initiative that publishes free electronic versions of books significant to the culture and history of the Nordic countries. Patterned after Project Gutenberg, it was founded ...

, digitalized 2. edition (1916). When the university in 1792 for the first time announced a prize essay contest in several different fields with an award of 40

riksdaler The svenska riksdaler () was the name of a Swedish coin first minted in 1604. Between 1777 and 1873, it was the currency of Sweden. The daler, like the dollar,''National Geographic''. June 2002. p. 1. ''Ask Us''. was named after the German Thaler. ...

in each, Degen won the prize both in

theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...

and in mathematics. He was fluent in

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

and

Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

, was well-acquainted with

Romance Romance (from Vulgar Latin , "in the Roman language", i.e., "Latin") may refer to: Common meanings * Romance (love), emotional attraction towards another person and the courtship behaviors undertaken to express the feelings * Romance languages, ...

and

Germanic languages The Germanic languages are a branch of the Indo-European language family spoken natively by a population of about 515 million people mainly in Europe, North America, Oceania and Southern Africa. The most widely spoken Germanic language, Engli ...

and could read

Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

and

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken *Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screenwr ...

. In this period he was tutor in mathematics for the young prince who later became king

Christian VIII of Denmark Christian VIII (18 September 1786 – 20 January 1848) was King of Denmark from 1839 to 1848 and, as Christian Frederick, King of Norway in 1814. Christian Frederick was the eldest son of Hereditary Prince Frederick, a younger son of King Frederic ...

. In 1798 Degen was made a

Doctor of Philosophy A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common Academic degree, degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields ...

based on a thesis on

Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German Philosophy, philosopher and one of the central Age of Enlightenment, Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemolo ...

's philosophyA. Stubhaug, ''Niels Henrik Abel and his Times'', Springer-Verlag, Berlin (2000). . and was elected to the

Royal Danish Academy of Sciences and Letters {{Infobox organization , name = The Royal Danish Academy of Sciences and Letters , full_name = , native_name = Det Kongelige Danske Videnskabernes Selskab , native_name_lang = , logo = Royal ...

in 1800. In 1802 Degen got his first academic position as head teacher in

and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

at the

Odense Odense ( , , ) is the third largest city in Denmark (behind Copenhagen and Aarhus) and the largest city on the island of Funen. As of 1 January 2022, the city proper had a population of 180,863 while Odense Municipality had a population of 20 ...

cathedral school Cathedral schools began in the Early Middle Ages as centers of advanced education, some of them ultimately evolving into medieval universities. Throughout the Middle Ages and beyond, they were complemented by the monastic schools. Some of these e ...

. After a few years there he was appointed

rector Rector (Latin for the member of a vessel's crew who steers) may refer to: Style or title *Rector (ecclesiastical), a cleric who functions as an administrative leader in some Christian denominations *Rector (academia), a senior official in an edu ...

at the corresponding school in Viborg. There he remained until 1814 when he became

professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...

in mathematics at the University of Copenhagen. Although his lectures were not so well organized, he was loved by his students and he infused the courses with new and more modern mathematics. At the same time he pursued his own research and published results in many different directions. All this made him the most esteemed mathematician in

Scandinavia Scandinavia; Sámi languages: /. ( ) is a subregion#Europe, subregion in Northern Europe, with strong historical, cultural, and linguistic ties between its constituent peoples. In English usage, ''Scandinavia'' most commonly refers to Denmark, ...

at that time. When

as a student visited Degen in Copenhagen, he described him as very kind, but a little strange, with a large, private library. Degen remained there until his death in 1825. For that reason he did not live to see the great fame the young Abel shortly afterwards obtained from his discovery of

elliptic functions In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...

which Degen had encouraged. He is buried on the Assistens Kirkegård at

Nørrebro Nørrebro (, ) is one of the 10 official districts of Copenhagen, Denmark. It is northwest of the city centre, beyond the location of the old Northern Gate (''Nørreport''), which, until dismantled in 1856, was near the current Nørreport station ...

in Copenhagen.

Mathematical contributions

Degen worked in many branches of what was then modern mathematics. Most of his contributions had to do with problems within

, but he also wrote papers on

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

The Pell equation

In 1817 Degen got printed his large work on the fundamental solutions (''x'', ''y'') of

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...

''x''² – ''ny''² = 1 where ''n'' is a positive integer.

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

had earlier shown that these could be systematically calculated with the use of

continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

s. Degen used this method and presented integer solutions for all The same calculations also gave approximate, but very accurate rational results for the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

of ''n''. In addition, he also found solutions of the adjoint equation with −1 on the right hand side for the ''n''-values when they existed. These tables of numerical results became in the following years a standard reference for the Pell equation.

The eight-square identity

While his work on the Pell equation can be considered a continuation of previous contributions made by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre to this problem, Degen's discovery of the eight-square identity was his most important and original discovery. Most probably it resulted from his attempts to generalize the Pell equation. The two-square identity :

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2

had been known from the times of

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

. At the end of the 17th century it explained why the

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

of the product of two

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s equals the product of their

s. Around the same time Euler showed that there is also a similar four-square identity. Later it turned out to be related to the norm of

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s discovered by

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

. In 1818 Degen presented to the

Academy of Sciences An academy of sciences is a type of learned society or academy (as special scientific institution) dedicated to sciences that may or may not be state funded. Some state funded academies are tuned into national or royal (in case of the Unite ...

St. Petersburg Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...

where Euler had worked, his eight-square identity of exactly the same structure as the two previous identities. The following year he was elected as a «corresponding member» to the same academic society. His work on the eight-square identity was first published in 1822.C.F. Degen, ''Adumbratio Demonstrationis Theorematis Arithmetici Maxime Universalis'', Mémoires de l’Académie Impériale des Sciences de St. Pétersbourg, pour les années 1817 et 1818, 8, 207–219 (1822). Almost thirty years later his identity was rediscovered by

John T. Graves John Thomas Graves (4 December 1806 – 29 March 1870) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions in October 1843 and then discover ...

and

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

as obeyed by the norm of

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...

s. These were an extension of Hamilton's quaternions. In 1898

Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...

proved that such identities involving 2^''k'' squares can exist only for .

The encounter with Abel

In 1821

was a very gifted student in his last year at the

Oslo Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...

. He was convinced that he had found a way to solve the

quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...

. None of his teachers or professors at the

University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...

could find anything wrong with his work. The astronomy professor

Christopher Hansteen Christopher Hansteen (26 September 1784 – 11 April 1873) was a Norwegian geophysicist, astronomer and physicist, best known for his mapping of Earth's magnetic field. Early life and career Hansteen was born in Christiania as the son of J ...

recommended then that the paper ought to be published by the

Science Academy An academy of sciences is a type of learned society or academy (as special scientific institution) dedicated to sciences that may or may not be state funded. Some state funded academies are tuned into national or royal (in case of the Unite ...

in Copenhagen. It thus came in the hands of Degen to be evaluated. He again could not pinpoint any mistakes, but asked that this new method should first be tried out on a practical example. In a letter to Hansteen he proposed the equation ''x''⁵ − 2''x''⁴ + 3''x''² − 4''x'' + 5 = 0. He ended the letter with the wish that This would soon turn out to be a very prophetic piece of advice. Abel himself soon discovered a mistake in his investigations of the quintic equation, but continued to work on the existence of solutions. Two years later he could prove that they in general have no

algebraic solution A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divisio ...

s. Degen's recommendation to concentrate instead on the

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...

had most probably made some impression on the young student. In the summer of 1823 Abel was on a short visit to Copenhagen where he met Degen. In a letter to his friend and former teacher

Bernt Michael Holmboe Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a ...

in Oslo he wrote that he had constructed

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

s by inverting the corresponding

integrals In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...

. The following year in a letter to Degen he could report that these new functions had two periods.O. Ore, ''Niels Henrik Abel – Mathematician Extraordinary'', AMS Chelsea Publishing, Providence, RI (2008). . Although this discovery marks the beginning of a new and very important branch of modern mathematics, Abel waited with the publication of his results. That happened first in 1827. Degen had in the meantime died and was therefore unaware of the beautiful discoveries Abel had made and which he had prophesied.

References

{{DEFAULTSORT:Degen, Carl Ferdinand 1766 births 1825 deaths Scientists from Braunschweig University of Copenhagen alumni Academic staff of the University of Copenhagen 19th-century Danish mathematicians 18th-century Danish mathematicians Emigrants from the Holy Roman Empire to Denmark–Norway