Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a
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 ...
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 ...
who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned
Greek mathematician since
antiquity.
Origins
Constantin Carathéodory was born in 1873 in
Berlin
Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...
to
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 ...
parents and grew up in
Brussels
Brussels, officially the Brussels-Capital Region, (All text and all but one graphic show the English name as Brussels-Capital Region.) is a Communities, regions and language areas of Belgium#Regions, region of Belgium comprising #Municipalit ...
. His father , a lawyer, served as the
Ottoman ambassador to
Belgium
Belgium, officially the Kingdom of Belgium, is a country in Northwestern Europe. Situated in a coastal lowland region known as the Low Countries, it is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeas ...
,
St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of
Chios
Chios (; , traditionally known as Scio in English) is the fifth largest Greece, Greek list of islands of Greece, island, situated in the northern Aegean Sea, and the List of islands in the Mediterranean#By area, tenth largest island in the Medi ...
. The Carathéodory family, originally from
Bosna, was well established and respected in
Constantinople
Constantinople (#Names of Constantinople, see other names) was a historical city located on the Bosporus that served as the capital of the Roman Empire, Roman, Byzantine Empire, Byzantine, Latin Empire, Latin, and Ottoman Empire, Ottoman empire ...
, and its members held many important governmental positions. His grandfather, the
Ottoman Greek
Ottoman Greeks (; ) were ethnic Greeks who lived in the Ottoman Empire (1299–1922), much of which is in modern Turkey. Ottoman Greeks were Greek Orthodox Christians who belonged to the Rum Millet (''Millet-i Rum''). They were concentrated in ...
physician
Constantinos Caratheodory, was the personal doctor to Sultan
Abdülmecit I.
The Carathéodory family spent 1874–75 in Constantinople, where Constantin's paternal grandfather lived, while his father Stephanos was on leave. Then in 1875 they went to Brussels when Stephanos was appointed there as Ottoman Ambassador. In Brussels, Constantin's younger sister Julia was born. The year 1879 was a tragic one for the family since Constantin's paternal grandfather died in that year, but much more tragically, Constantin's mother Despina died of
pneumonia
Pneumonia is an Inflammation, inflammatory condition of the lung primarily affecting the small air sacs known as Pulmonary alveolus, alveoli. Symptoms typically include some combination of Cough#Classification, productive or dry cough, ches ...
in
Cannes
Cannes (, ; , ; ) is a city located on the French Riviera. It is a communes of France, commune located in the Alpes-Maritimes departments of France, department, and host city of the annual Cannes Film Festival, Midem, and Cannes Lions Internatio ...
. Constantin's maternal grandmother took on the task of bringing up Constantin and Julia in his father's home in Belgium. They employed a German maid who taught the children to speak German. Constantin was already bilingual in French and Greek by this time.
Constantin began his formal schooling at a private school in Vanderstock in 1881. He left after two years and then spent time with his father on a visit to Berlin, and also spent the winters of 1883–84 and 1884–85 on the
Italian Riviera
The Italian Riviera or Ligurian Riviera ( ; ) is the narrow coastal strip in Italy which lies between the Ligurian Sea and the mountain chain formed by the Maritime Alps and the Apennines. Longitudinally it extends from the border with F ...
. Back in Brussels in 1885 he attended a grammar school for a year where he first began to become interested in mathematics. In 1886, he entered the high school Athénée Royal d'Ixelles and studied there until his graduation in 1891. Twice during his time at this school Constantin won a prize as the best mathematics student in Belgium.
At this stage Carathéodory began training as a military engineer. He attended the École Militaire de Belgique from October 1891 to May 1895 and he also studied at the École d'Application from 1893 to 1896. In 1897
a war broke out between the Ottoman Empire and Greece. This put Carathéodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire. Since he was a trained engineer he was offered a job in the British colonial service. This job took him to Egypt where he worked on the construction of the
Assiut
AsyutAlso spelled ''Assiout'' or ''Assiut''. ( ' ) is the capital of the modern Asyut Governorate in Egypt. It was built close to the ancient city of the same name, which is situated nearby. The modern city is located at , while the ancient city i ...
dam until April 1900. During periods when construction work had to stop due to floods, he studied mathematics from some textbooks he had with him, such as
Jordan's ''Cours d'Analyse'' and
Salmon's text on the analytic geometry of
conic sections
A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is ...
. He also visited the
Cheops pyramid and made measurements which he wrote up and published in 1901. He also published a book on Egypt in the same year which contained a wealth of information on the history and geography of the country.
Studies and university career

Carathéodory studied engineering in
Belgium
Belgium, officially the Kingdom of Belgium, is a country in Northwestern Europe. Situated in a coastal lowland region known as the Low Countries, it is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeas ...
at the
Royal Military Academy, where he was considered a charismatic and brilliant student.
University career
* 1900 Studies at
University of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
.
* 1902 Completed graduation at
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
(1904 Ph.D., 1905 Habilitation)
* 1908 Dozent at
Bonn
Bonn () is a federal city in the German state of North Rhine-Westphalia, located on the banks of the Rhine. With a population exceeding 300,000, it lies about south-southeast of Cologne, in the southernmost part of the Rhine-Ruhr region. This ...
* 1909 Ordinary Professor at
Hannover Technical High School.
* 1910 Ordinary Professor at
Breslau Technical High School.
* 1913 Professor following Klein at
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
.
* 1919 Professor at
University of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
* 1919 Elected to
Prussian Academy of Science.
* 1920 University Dean at
Ionian University of Smyrna (later,
University of the Aegean
The University of the Aegean (UA; ) is a public, multi-campus university located in Lesvos, Chios, Samos, Rhodes, Syros and Lemnos, Greece. It was founded on March 20, 1984, by the Presidential Act 83/1984 and its administrative headquarters ar ...
).
* 1922 Professor at
University of Athens
The National and Kapodistrian University of Athens (NKUA; , ''Ethnikó kai Kapodistriakó Panepistímio Athinón''), usually referred to simply as the University of Athens (UoA), is a public university in Athens, Greece, with various campuses alo ...
.
* 1922 Professor at
Athens Polytechnic.
* 1924 Professor following Lindemann at
University of Munich
The Ludwig Maximilian University of Munich (simply University of Munich, LMU or LMU Munich; ) is a public university, public research university in Munich, Bavaria, Germany. Originally established as the University of Ingolstadt in 1472 by Duke ...
.
* 1938 Retirement from professorship. Continued working at Bavarian Academy of Science
Doctoral students
Carathéodory had about 20 doctoral students among these being
Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892 – 7 February 1969) was a German-born American mathematician, known for work in mathematical analysis and number theory.
Biography
Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göt ...
, known for his work on analysis and number theory, and
Paul Finsler known for his creation of
Finsler space.
Academic contacts in Germany
Carathéodory had numerous contacts in Germany. They included such famous names as:
Hermann Minkowski
Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...
,
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
,
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
,
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
,
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
,
Hermann Amandus Schwarz, and
Lipót Fejér. During the difficult period of World War II, his close associates at the Bavarian Academy of Sciences were Perron and Tietze.
Einstein, then a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory for clarifications on the
Hamilton-Jacobi equation and
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as ''form invariance''. Although Hamilton's equations are preserved, it need not ...
s. He wanted to see a satisfactory derivation of the former and the origins of the latter. Einstein told Carathéodory his derivation was "beautiful" and recommended its publication in the ''Annalen der Physik.'' Einstein employed the former in a 1917 paper titled ''Zum Quantensatz von Sommerfeld und Epstein'' (On the Quantum Theorem of Sommerfeld and Epstein). Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to
E.T. Whittaker's ''
Analytical Dynamics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
''. Einstein was trying to solve the problem of "closed time-lines" or the geodesics corresponding to the closed trajectory of light and free particles in a static universe, which he introduced in 1917.
Landau and Schwarz stimulated his interest in the study of complex analysis.
Academic contacts in Greece
While in Germany, Carathéodory retained numerous links with the Greek academic world, details of which can be found in Georgiadou's book. He was directly involved with the reorganization of Greek universities. An especially close friend and colleague in Athens was Nicolaos Kritikos who had attended his lectures at Göttingen, later going with him to Smyrna, then becoming professor at Athens Polytechnic. Kritikos and Carathéodory helped the Greek topologist
Christos Papakyriakopoulos take a doctorate in topology at Athens University in 1943 under very difficult circumstances. While teaching at Athens University, Carathéodory had Evangelos Stamatis as an undergraduate student, who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.
Works
Calculus of variations
In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems.
Previously, between the mid-1700s to the mid-1800s,
Leonhard 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 ...
,
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...
, and
Carl Gustav Jacob Jacobi were able to establish necessary but insufficient conditions for the existence of a strong relative minimum. In 1879,
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
added a fourth that does indeed guarantee such a quantity exists.
Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as ''Carathéodory's method of equivalent variational problems'' or ''the royal road to the calculus of variations''.
A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations.
It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the
Euler-Lagrange equation and the Weierstrass condition. He published his ''Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung'' (Calculus of Variations and First-order Partial Differential Equations) in 1935.
More recently, Carathéodory's work on the calculus of variations and the Hamilton-Jacobi equation has been taken into the theory of
optimal control
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
and dynamic programming.
Convex geometry
Carathéodory's theorem in convex geometry states that if a point
of
lies in the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of a set
, then
can be written as the convex combination of at most
points in
. Namely, there is a subset
of
consisting of
or fewer points such that
lies in the convex hull of
. Equivalently,
lies in an
-
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
with vertices in
, where
. The smallest
that makes the last statement valid for each
in the convex hull of ''P'' is defined as the ''Carathéodory's number'' of
. Depending on the properties of
, upper bounds lower than the one provided by Carathéodory's theorem can be obtained.
He is credited with the authorship of the
Carathéodory conjecture claiming that a closed convex surface admits at least two
umbilic points. The conjecture was proven in 2024 by Brendan Guilfoyle and Wilhelm Klingenberg.
Real analysis
He proved an
existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
for the solution to ordinary differential equations under mild regularity conditions.
Another theorem of his on the derivative of a function at a point could be used to prove the
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
and the formula for the
derivative of inverse functions.
Complex analysis
He greatly extended the theory of
conformal transformation
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
proving his
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of
prime ends.
He exhibited an elementary proof of the
Schwarz lemma.
Carathéodory was also interested in the theory of functions of multiple complex variables. In his investigations on this subject he sought analogs of classical results from the single-variable case. He proved that a ball in
is not holomorphically equivalent to the bidisc.
Theory of measure
He is credited with the
Carathéodory extension theorem which is fundamental to modern measure theory. Later Carathéodory extended the theory from sets to
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
s.
Thermodynamics
Thermodynamics had been a subject dear to Carathéodory since his time in Belgium.
In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the second law of thermodynamics axiomatically, that is, without the use of Carnot engines and refrigerators and only by mathematical reasoning. This is yet another version of the second law, alongside the statements of
Clausius
Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Nicolas Léonard Sadi Ca ...
, and of
Kelvin and Planck. Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld.
According to Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic." Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.
However, Max Planck had some misgivings in that while he was impressed by Carathéodory's mathematical prowess, he did not accept that this was a fundamental formulation, given the statistical nature of the second law.
In his theory he simplified the basic concepts, for instance ''heat'' is not an essential concept but a derived one. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The
second law of thermodynamics
The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term
adiabatic accessibility.
Optics
Carathéodory's work in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
is closely related to his method in the calculus of variations. In 1926 he gave a strict and general proof that no system of lenses and mirrors can avoid
aberration, except for the trivial case of plane mirrors.
In his later work he gave the theory of the
Schmidt telescope. In his ''Geometrische Optik'' (1937), Carathéodory demonstrated the equivalence of Huygens' principle and Fermat's principle starting from the former using Cauchy's theory of characteristics. He argued that an important advantage of his approach was that it covers the integral invariants of
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
and
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...
and completes the
Malus law. He explained that in his investigations in optics,
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
conceived a minimum principle similar to that enunciated by
Hero of Alexandria
Hero of Alexandria (; , , also known as Heron of Alexandria ; probably 1st or 2nd century AD) was a Greek mathematician and engineer who was active in Alexandria in Egypt during the Roman era. He has been described as the greatest experimental ...
to study reflection.
Historical
During the Second World War Carathéodory edited two volumes of
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 ...
's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946.
The University of Smyrna

At the time, Athens was the only major educational centre in the wider area and had limited capacity to sufficiently satisfy the growing educational needs of the eastern part of the Aegean Sea and the
Balkans
The Balkans ( , ), corresponding partially with the Balkan Peninsula, is a geographical area in southeastern Europe with various geographical and historical definitions. The region takes its name from the Balkan Mountains that stretch throug ...
. Carathéodory, who was a professor at the
University of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
at the time, proposed the establishment of a new university - the difficulties regarding the establishment of a Greek university in
Constantinople
Constantinople (#Names of Constantinople, see other names) was a historical city located on the Bosporus that served as the capital of the Roman Empire, Roman, Byzantine Empire, Byzantine, Latin Empire, Latin, and Ottoman Empire, Ottoman empire ...
led him to consider three other cities:
Thessaloniki
Thessaloniki (; ), also known as Thessalonica (), Saloniki, Salonika, or Salonica (), is the second-largest city in Greece (with slightly over one million inhabitants in its Thessaloniki metropolitan area, metropolitan area) and the capital cit ...
,
Chios
Chios (; , traditionally known as Scio in English) is the fifth largest Greece, Greek list of islands of Greece, island, situated in the northern Aegean Sea, and the List of islands in the Mediterranean#By area, tenth largest island in the Medi ...
and
Smyrna
Smyrna ( ; , or ) was an Ancient Greece, Ancient Greek city located at a strategic point on the Aegean Sea, Aegean coast of Anatolia, Turkey. Due to its advantageous port conditions, its ease of defence, and its good inland connections, Smyrna ...
.
At the invitation of the Greek Prime Minister
Eleftherios Venizelos
Eleftherios Kyriakou Venizelos (, ; – 18 March 1936) was a Cretan State, Cretan Greeks, Greek statesman and prominent leader of the Greek national liberation movement. As the leader of the Liberal Party (Greece), Liberal Party, Venizelos ser ...
, he submitted a plan on 20 October 1919 for the creation of a new university at
Smyrna
Smyrna ( ; , or ) was an Ancient Greece, Ancient Greek city located at a strategic point on the Aegean Sea, Aegean coast of Anatolia, Turkey. Due to its advantageous port conditions, its ease of defence, and its good inland connections, Smyrna ...
in Asia Minor, to be named
Ionian University of Smyrna. In 1920 Carathéodory was appointed dean of the university and took a major part in establishing the institution, touring Europe to buy books and equipment. The university, however, never actually admitted students, due to the
War in Asia Minor which ended in the
Great Fire of Smyrna. Carathéodory managed to save books from the library and was only rescued at the last moment by a journalist who took him by rowboat to the battleship Naxos which was standing by.
["''His daughter Mrs Despina Rodopoulou – Carathéodory referred to this period: "He stayed to save anything he could: library, machines etc which were shipped in different ships hoping that one day they will arrive in Athens. My father stayed until the last moment. George Horton, consul of U.S.A. in Smyrni wrote a book... which was translated in Greek. In this book Horton notes: "One of the last Greek I saw on the streets of Smyrna before the entry of the Turks was Professor Carathéodory, president of the doomed University. With him departed the incarnation of Greek of culture and civilization on Orient." ''"] Carathéodory brought to Athens some of the university library and stayed there, teaching at the university and technical school until 1924.
In 1924 Carathéodory was appointed professor of mathematics at the University of Munich, and held this position until retirement in 1938. He later worked at the Bavarian Academy of Sciences until his death in 1950.
The new Greek university in the broader area of the Southeast Mediterranean region, as originally envisioned by Carathéodory, finally materialised with the establishment of the
Aristotle University of Thessaloniki
The Aristotle University of Thessaloniki ( AUTh; ), often called the University of Thessaloniki, is the second oldest tertiary education institution in Greece. Named after the philosopher Aristotle, who was born in Stageira, about east of Thessa ...
in 1925.
Linguistic and oratorical talents
Carathéodory excelled at languages, much like many members of his family.
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
French were his first languages, and he mastered
German with such perfection, that his writings composed in the German language are stylistic masterworks. Carathéodory also spoke and wrote
English,
Italian
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, a Romance ethnic group related to or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance languag ...
,
Turkish, and the
ancient languages without any effort. Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extended his fields of knowledge.
Much more than that, Carathéodory was a treasured conversation partner for his fellow professors in the Munich Department of Philosophy. The well-respected German
philologist
Philology () is the study of language in oral and written historical sources. It is the intersection of textual criticism, literary criticism, history, and linguistics with strong ties to etymology. Philology is also defined as the study of ...
and professor of ancient languages,
Kurt von Fritz
Karl Albert Kurt von Fritz (25 August 1900 in Metz – 16 July 1985 in Feldafing) was a German classical philologist.
Appointed to an extraordinary professorship for Greek at the University of Rostock in 1933, he was one of only two German prof ...
, praised Carathéodory on the grounds that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Von Fritz conducted numerous philosophical discussions with Carathéodory.
The mathematician sent his son Stephanos and daughter Despina to a German high school, but they also obtained daily additional instruction in Greek language and culture from a Greek priest, and at home he allowed them to speak Greek only.
Carathéodory was a talented public speaker, and was often invited to give speeches. In 1936, it was he who handed out the first ever
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
s at the meeting of the International Congress of Mathematicians in Oslo, Norway.
Legacy

In 2002, in recognition of his achievements, the University of Munich named one of the largest lecture rooms in the mathematical institute the Constantin-Carathéodory Lecture Hall.
In the town of Nea Vyssa, Caratheodory's ancestral home, a unique family museum is to be found. The museum is located in the central square of the town near to its church, and includes a number of Karatheodory's personal items, as well as letters he exchanged with Albert Einstein. More information is provided at the original website of the club, http://www.s-karatheodoris.gr.
At the same time, Greek authorities had long since intended to create a museum honoring Karatheodoris in
Komotini, a major town of the northeastern Greek region, more than 200 km away from his home town above. On 21 March 2009, the "Karatheodoris" Museum (Καραθεοδωρής) opened its gates to the public in Komotini.
The coordinator of the museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), has noted that the museum provides a home for original manuscripts of the mathematician running to about 10,000 pages, including correspondence with the German mathematician
Arthur Rosenthal for the algebraization of measure. At the showcase, visitors are also able to view the books ''" Gesammelte mathematische Schriften Band 1,2,3,4 ", "Mass und ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen "'', and a number of others. Handwritten letters by Carathéodory to
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
and
Hellmuth Kneser, as well as photographs of the Carathéodory family, are on display.
Efforts to furnish the museum with more exhibits are ongoing.
Publications
Journal articles
A complete list of Carathéodory's journal article publications can be found in his ''Collected Works''(''Ges. Math. Schr.''). Notable publications are:
* ''Über die kanonischen Veränderlichen in der Variationsrechnung der mehrfachen Integrale''
* ''Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen''
* ''Über die diskontinuierlichen Lösungen in der Variationsrechnung.'' Diss. Göttingen Univ. 1904; Ges. Math. Schr. I 3–79.
* ''Über die starken Maxima und Minima bei einfachen Integralen.'' Habilitationsschrift Göttingen 1905; Math. Annalen 62 1906 449–503; Ges. Math. Schr. I 80–142.
* ''Untersuchungen über die Grundlagen der Thermodynamik'', Math. Ann. 67 (1909) pp. 355–386; Ges. Math. Schr. II 131–166.
* ''Über das lineare Mass von Punktmengen – eine Verallgemeinerung des Längenbegriffs.'', Gött. Nachr. (1914) 404–406; Ges. Math. Schr. IV 249–275.
* ''Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen''. Schwarzsche Festschrift, Berlin 1914; Ges. Math. Schr.IV 249–275.
* ''Zur Axiomatic der speziellen Relativitätstheorie''. Sitzb. Preuss. Akad. Wiss. (1924) 12–27; Ges. Math. Schr. II 353–373.
* ''Variationsrechnung'' in Frank P. & von Mises (eds): ''Die Differential= und Integralgleichungen der Mechanik und Physik'', Braunschweig 1930 (Vieweg); New York 1961 (Dover) 227–279; Ges. Math. Schr. I 312–370.
* ''Entwurf für eine Algebraisierung des Integralbegriffs'', Sitzber. Bayer. Akad. Wiss. (1938) 27–69; Ges. Math. Schr. IV 302–342.
Books
* Reprinted 1968 (Chelsea)
* ''Conformal Representation'', Cambridge 1932 (Cambridge Tracts in Mathematics and Physics)
* ''Geometrische Optik'', Berlin, 1937
* ''Elementare Theorie des Spiegelteleskops von B. Schmidt'' (Elementary Theory of B. Schmidt's Reflecting Telescope), Leipzig Teubner, 1940 36 pp.; Ges. math. Schr. II 234–279
* ''Funktionentheorie I, II'', Basel 1950,
1961 (Birkhäuser). English translation: ''Theory of Functions of a Complex Variable'', 2 vols, New York, Chelsea Publishing Company, 3rd ed 1958
* ''Mass und Integral und ihre Algebraisierung'', Basel 1956. English translation, ''Measure and Integral and Their Algebraisation'', New York, Chelsea Publishing Company, 1963
* ''Variationsrechnung und partielle Differentialgleichungen erster Ordnung'', Leipzig, 1935. English translation next reference
* ''Calculus of Variations and Partial Differential Equations of the First Order'', 2 vols. vol. I 1965, vol. II 1967 Holden-Day.
* ''Gesammelte mathematische Schriften'' München 1954–7 (Beck) I–V.
See also
*
Domain (mathematical analysis)
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space or the complex coordinate space . A connected op ...
*
Nemytskii operator
*
Herbert Callen
Herbert Bernard Callen (July 1, 1919 – May 22, 1993) was an American physicist specializing in thermodynamics and statistical mechanics. He is considered one of the founders of the modern theory of irreversible thermodynamics, and is the author ...
, who also sought an axiomatic formulation of thermodynamics
Notes
References
Books
* Maria Georgiadou,
Constantin Carathéodory: Mathematics and Politics in Turbulent Times'' Berlin-Heidelberg: Springer Verlag, 2004. .
*
Themistocles M. Rassias (editor) (1991) ''Constantin Caratheodory: An International Tribute'', Teaneck, NJ: World Scientific Publishing Co., .
* Nicolaos K. Artemiadis; translated by Nikolaos E. Sofronidis
000
Triple zero, Zero Zero Zero, 0-0-0 or variants may refer to:
* 000 (emergency telephone number), the Australian emergency telephone number
* 000, the size of several small List of screw drives, screw drives
* 0-0-0, a Droid (Star Wars)#0-0-0, dro ...
2004), ''History of Mathematics: From a Mathematician's Vantage Point'', Rhode Island, USA: American Mathematical Society, pp. 270–4, 281, .
* ''Constantin Carathéodory in his...origins''. International Congress at Vissa-Orestiada, Greece, 1–4 September 2000. Proceedings: T Vougiouklis (ed.), Hadronic Press, Palm Harbor FL 2001.
Biographical articles
* C. Carathéodory, ''Autobiographische Notizen'', (In German) Wiener Akad. Wiss. 1954–57, vol.V, pp. 389–408. Reprinted in Carathéodory's Collected Writings vol.V. English translation in A. Shields, ''Carathéodory and conformal mapping'', The Mathematical Intelligencer 10 (1) (1988), 18–22.
* Oskar Perron, O. Perron, ''Obituary: Constantin Carathéodory'', Jahresberichte der Deutschen Mathematiker Vereinigung 55 (1952), 39–51.
* N. Sakellariou, ''Obituary: Constantin Carathéodory'' (Greek), Bull. Soc. Math. Grèce 26 (1952), 1–13.
* Heinrich Tietze, H Tietze, ''Obituary: Constantin Carathéodory'', Arch. Math. 2 (1950), 241–245.
* H. Behnke, ''Carathéodorys Leben und Wirken'', in A. Panayotopolos (ed.), Proceedings of C .Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 17–33.
* Bulirsch R., Hardt M., (2000): ''Constantin Carathéodory: Life and Work'', International Congress: "Constantin Carathéodory", 1–4 September 2000, Vissa, Orestiada, Greece
Encyclopaedias and reference works
* Chambers Biographical Dictionary (1997), ''Constantine Carathéodory'', 6th ed., Edinburgh: Chambers Harrap Publishers Ltd, pp 270–1, (also available
online.
* ''The New Encyclopædia Britannica'' (1992), ''Constantine Carathéodory'', 15th ed., vol. 2, USA: The University of Chicago, Encyclopædia Britannica, Inc., pp 842,
New edition Online entry* H. Boerner, Biography of ''Carathéodory'' in Dictionary of Scientific Biography (New York 1970–1990).
Conferences
* ''C. Carathéodory International Symposium'', Athens, Greece September 1973. Proceedings edited by A. Panayiotopoulos (Greek Mathematical Society) 1975
Online* Conference on ''Advances in Convex Analysis and Global Optimization (Honoring the memory of C. Carathéodory)'' June 5–9, 2000, Pythagorion, Samos, Greece
Online
* International Congress: ''Carathéodory in his ... origins'', September 1–4, 2000, Vissa Orestiada, Greece. Proceedings edited by Thomas Vougiouklis (Democritus University of Thrace), Hadronic Press FL USA, 2001. .
External links
*
*
*
Web site dedicated to Carathéodory*
club www.s-karatheodoris.gr*
{{DEFAULTSORT:Caratheodory, Constantin
1873 births
1950 deaths
20th-century German mathematicians
Greeks from the Ottoman Empire
German people of Greek descent
19th-century Greek mathematicians
Eastern Orthodox Christians from Germany
Complex analysts
Mathematical analysts
Members of the Prussian Academy of Sciences
Thermodynamicists
Mathematicians from Berlin
Scientists from Brussels
Occupation of Smyrna
University of Göttingen alumni
Members of the Academy of Athens (modern)
Variational analysts
19th-century Greek scientists
Measure theorists
Emigrants from the Ottoman Empire to Germany
People of the Burning of Smyrna