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

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, mathematical structure, structure, space, Mathematica ...

. His most important contributions were within

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

and he advised the young, aspiring Norwegian 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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

Braunschweig Braunschweig () or Brunswick ( ; from Low German , local dialect: ) is a List of cities and towns in Germany, city in Lower Saxony, Germany, north of the Harz Mountains at the farthest navigable point of the river Oker, which connects it to the ...

Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...

, but the family moved to

Copenhagen Copenhagen ( ) is the capital and most populous city of Denmark, with a population of 1.4 million in the Urban area of Copenhagen, urban area. The city is situated on the islands of Zealand and Amager, separated from Malmö, Sweden, by the ...

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

Royal Danish Orchestra The Royal Danish Orchestra (Danish: ''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 Dani ...

. 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 ( , ; ), classically known in English as Elsinore ( ), is a coastal city in northeastern Denmark. Helsingør Municipality had a population of 63,953 on 1 January 2025, making it the 23rd most populated municipality in Denmark. Helsin ...

. He graduated from there in 1783 and continued at the

University of Copenhagen The University of Copenhagen (, KU) is a public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in Scandinavia, after Uppsala University. ...

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

classical language According to the definition by George L. Hart, a classical language is any language with an independent literary tradition and a large body of ancient written literature. Classical languages are usually extinct languages. Those that are still ...

philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

natural science Natural science or empirical 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 ...

s and in particular

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

Projekt Runeberg Project 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 by Lars Aronsson and ...

, 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 Thale ...

in each, Degen won the prize both in

theology Theology is the study of religious belief from a Religion, religious perspective, with a focus on the nature of divinity. It is taught as an Discipline (academia), academic discipline, typically in universities and seminaries. It occupies itse ...

and in mathematics. He was fluent in

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

Greek Greek may refer to: 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 of all kno ...

and

Hebrew Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...

, was well-acquainted with Romance and

Germanic languages The Germanic languages are a branch of the Indo-European languages, 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 spoke ...

and could read

Russian Russian(s) may refer to: *Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *A citizen of Russia *Russian language, the most widely spoken of the Slavic languages *''The Russians'', a b ...

and Polish. 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 Frederick, Hereditary Prince of Denmark, Hereditary Prin ...

. In 1798 Degen was made a

Doctor of Philosophy A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of Postgraduate education, graduate study and original resear ...

based on a thesis on

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

'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 The Royal Danish Academy of Sciences and Letters ({{Langx, da, Det Kongelige Danske Videnskabernes Selskab or ''Videnskabernes Selskab'') is a Danish academy of science. The Royal Danish Academy was established on 13 November 1742, and was create ...

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

and

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

at the

Odense Odense ( , , ) is the third largest city in Denmark (after Copenhagen and Aarhus) and the largest city on the island of Funen. As of 1 January 2025, the city proper had a population of 185,480 while Odense Municipality had a population of 210, ...

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 ...

. After a few years there he was appointed rector 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 tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

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 is a subregion#Europe, subregion of northern Europe, with strong historical, cultural, and linguistic ties between its constituent peoples. ''Scandinavia'' most commonly refers to Denmark, Norway, and Sweden. It can sometimes also ...

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 special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

which Degen had encouraged. He is buried on the

Assistens Kirkegård An Assistens Cemetery () is a cemetery that functions as an expansion of another, older cemetery often in relation to a city church. Already by the end of the 17th century, Danish authorities deemed that the conditions for inner-city cemeteries we ...

Nørrebro Nørrebro (, ) is one of the 10 official districts of Copenhagen Municipality, 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 N ...

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 a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

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 Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...

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

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

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

continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

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 y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

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 () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...

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

norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...

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 for ...

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. The algebra of quater ...

s discovered by

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

. In 1818 Degen presented to the Academy of Sciences in

St. Petersburg Saint Petersburg, formerly known as Petrograd and later Leningrad, is the second-largest city in Russia after Moscow. It is situated on the River Neva, at the head of the Gulf of Finland on the Baltic Sea. The city had a population of 5,601, ...

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 and

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

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, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

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, mathematical analysis, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...

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 ; ) 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 1,064,235 in 2022 ...

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

quintic equation In mathematics, 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 word ...

. None of his teachers or professors at the

University of Oslo The University of Oslo (; ) is a public university, public research university located in Oslo, Norway. It is the List of oldest universities in continuous operation#Europe, oldest university in Norway. Originally named the Royal Frederick Univ ...

could find anything wrong with his work. The astronomy professor Christopher Hansteen recommended then that the paper ought to be published by the Science Academy 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 an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of t ...

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 i ...

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 in Oslo he wrote that he had constructed

elliptic function In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

s by inverting the corresponding

integrals In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...

. 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