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Constantin Carathéodory

Constantin Carathéodory

Constantin Carathéodory (Greek: Κωνσταντίνος Καραθεοδωρή Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory. His work also includes important results in conformal representations and in the theory of boundary correspondence. In 1909, Carathéodory pioneered the Axiomatic Formulation of Thermodynamics along a purely geometrical approach.

Constantin Carathéodory
Born(1873-09-13)13 September 1873
Died2 February 1950(1950-02-02)(aged 76)
Munich, West Germany
Alma materUniversity of Berlin
University of Göttingen
Known forCarathéodory's extension theorem
Carathéodory theorems
Carathéodory conjecture
General theory of outer measures
Axiomatic formulation of thermodynamics
Scientific career
FieldsMathematics — Theory of functions of a complex variable, Real analysis, Measure theory, Calculus of variations
  • University of Bonn
  • Hannover Technical High School
  • Breslau Technical High School
  • University of Göttingen
  • University of Berlin
  • University of Munich
  • National Technical University of Athens
  • Ionian University of Smyrna
Doctoral advisorHermann Minkowski[1]
Doctoral studentsPaul Finsler
Hans Rademacher
Georg Aumann
Hermann Boerner
Ernst Peschl
Hans Rügemer
Wladimir Seidel
Nazım Terzioğlu[2]


Constantin Caratheodory with his father.

Constantin Caratheodory with his father.

Constantin Caratheodory (left) pictured sitting with his father, brother in law and sister, Carlsbad 1898

Constantin Caratheodory (left) pictured sitting with his father, brother in law and sister, Carlsbad 1898

Constantin Carathéodory was born in Berlin to Greek parents and grew up in Brussels. His father Stephanos, a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios. The Carathéodory family, originally from Bosnochori or Vyssa, was well established and respected in Constantinople, and its members held many important governmental positions.

The Carathéodory family spent 1874–75 in Constantinople (now Istanbul), 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 in Cannes. 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. 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 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 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. He also visited the Cheops pyramid and made measurements which he wrote up and published in 1901[3]. 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. [4]

Studies and university career

Carathéodory studied engineering in Belgium at the Royal Military Academy, where he was considered a charismatic and brilliant student.

University career

  • 1900 Studies at University of Berlin.

  • 1902 Completed graduation at University of Göttingen (1904 Ph.D, 1905 Habilitation)

  • 1908 Dozent at Bonn

  • 1909 Ordinary Professor at Hannover Technical High School.

  • 1910 Ordinary Professor at Breslau Technical High School.

  • 1913 Professor following Klein at University of Göttingen.

  • 1919 Professor at University of Berlin

  • 1919 Elected to Prussian Academy of Science.

  • 1920 University Dean at Ionian University of Smyrna (later, University of the Aegean).

  • 1922 Professor at University of Athens.

  • 1922 Professor at Athens Polytechnic.

  • 1924 Professor following Lindemann at University of Munich.

  • 1938 Retirement from Professorship. Continued working from Bavarian Academy of Science

Doctoral students

Carathéodory had about 20 doctoral students among these being Hans Rademacher, known for his work on analysis and number theory, and Paul Finsler known for his creation of Finsler space.

Academic contacts in Germany

Carathéodory's contacts in Germany were many and included such famous names as: Minkowski, Hilbert, Klein, Einstein, Schwarz, Fejér. During the difficult period of World War II his close associates at the Bavarian Academy of Sciences were Perron and Tietze.

Academic contacts in Greece

While in Germany Carathéodory retained numerous links with the Greek academic world about which detailed information may 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 Gŏttingen, 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 in Athens University Carathéodory had as undergraduate student Evangelos Stamatis who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.[5]


Calculus of variations

In his doctoral dissertation Carathéodory originated his method 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 the royal road to the calculus of variations.[6] More recently the same idea has been taken into the theory of optimal control.[7] The method can also be extended to multiple integrals.

Real analysis

He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions.

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 algebras.

Theory of functions of a complex variable

He greatly extended the theory of conformal transformation[8] proving his theorem about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of prime ends.


In 1909, Carathéodory published a pioneering work "Investigations on the Foundations of Thermodynamics"[9] in which he formulated the Laws of Thermodynamics axiomatically. It has been said that he was using only mechanical concepts and the theory of Pfaff's differential forms. But in reality he also relied heavily on the concept of an adiabatic process. The physical meaning of the term adiabatic rests on the concepts of heat and temperature. Thus, in Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical", as distinct from "thermodynamic".[10] Carathéodory's "first axiomatically rigid foundation of thermodynamics" was acclaimed by Max Born[11] but criticized by Max Planck.[12]

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 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.[13]


Carathéodory's work in optics is closely related to his method in the calculus of variations. In 1926[14] 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.


During the 2nd World War Carathéodory edited two volumes of Euler's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946.[15]

A conjecture

He is credited with the authorship of the Carathéodory conjecture claiming that a closed convex surface admits at least two umbilic points. As of 2007, this conjecture remained unproven despite having attracted a large amount of research.

The University of Smyrna

Photo of the Ionian University of Smyrna.

Photo of the Ionian 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 need of the eastern part of the Aegean Sea and the Balkans. Constantin Carathéodory, who was a Professor at the University of Berlin at the time, proposed the establishment of a new University [16] - the difficulties regarding the establishment of a Greek university in Constantinople led him to consider three other cities: Thessaloniki, Chios and Smyrna.[17]

At the invitation of the Greek Prime Minister Eleftherios Venizelos he submitted a plan on 20 October 1919 for the creation of a new University at 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 rowing boat to the battleship Naxos which was standing by.[18] Carathéodory brought to Athens some of the university library and stayed in Athens, 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 from 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 in 1925.[19]

Linguistic talent

Carathéodory excelled at languages, much like many members of his family did. Greek and French were his first languages, and he mastered German with such perfection, that his writings composed in the German language are stylistic masterworks.[20] Carathéodory also spoke and wrote English, Italian, 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 extend 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, professor of ancient languages Kurt von Fritz praised Carathéodory, saying that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Fritz had numerous philosophical discussions with Carathéodory.

The Greek language was spoken exclusively in Carathéodory's house – his son Stephanos and daughter Despina went to a German high school, but they obtained daily additional instruction in Greek language and culture from a Greek priest. At home, they were not allowed to speak any other language.


Constantin Caratheodory at a mature age.

Constantin Caratheodory at a mature age.

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. [21]

Known correspondence Carathéodory–Einstein can be seen as facsimile in Einstein Archives Online (11 items). Three letters concern mathematics and these are printed in vol.8 of Einstein's Collected Works (Princeton Univ. Press 1987) now freely available online.[22][23]

In the town of Nea Vyssa, where Caratheodory's family came from, there is the unique Caratheodory's family museum. The museum is located in the central square of the town nearby the church and there are many personal items of Constantin as well as letters that he had exchanged with A. Einstein, for more information visit the original website of the club http://www.s-karatheodoris.gr [46] . On the other hand, the Greek authorities intended for a long time to create a museum honoring Karatheodoris in Komotini, a major town of the northeastern Greek region, which is more than 200 km far away for the town of Nea Vyssa where his family came from. On 21 March 2009, the museum "Karatheodoris" (Καραθεοδωρής) opened its gates to the public, in Komotini.[24][25][26]

The coordinator of the Museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), noted that the museum gave home to original manuscripts of the mathematician of about 10,000 pages including correspondence of Carathéodory with the German mathematician Arthur Rosenthal for the algebraization of measure. Also visitors can view at the showcases the books " Gesammelte Mathematische Schriften Band 1,2,3,4 ", "Mass und Ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen " and many more. Handwritten letters of C.Carathéodory to Albert Einstein, Hellmuth Kneser and photographs of the Carathéodory family are on display.

The effort to furnish the museum with more exhibits is continuous.[27][28][29]


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[30]

  • Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen[31]

  • Über die diskontinuierlichen Lösungen in der Variationsrechung. 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.[32]

  • Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909) pp. 355–386; Ges. Math. Schr. II 131–166.[33]

  • 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.[34]

  • 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.


  • Carathéodory, Constantin (1918), Vorlesungen über reelle Funktionen [47] (3rd ed.), Leipzig: Teubner, ISBN 978-0-8284-0038-1, MR 0225940 [48] 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

  • Functionentheorie I, II, Basel 1950,[35] 1961 (Birkhäuser). English translation: Theory of Functions of a Complex Variable, 2 vols, New York, Chelsea Publishing Company, 1954

  • 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

  • Borel–Carathéodory theorem

  • Carathéodory–Jacobi–Lie theorem

  • Carathéodory metric

  • Carnot–Carathéodory metric

  • Carathéodory's theorem (convex hull)

  • Carathéodory's lemma

  • Carathéodory kernel theorem


Citation Linkgenealogy.math.ndsu.nodak.edu"The Mathematics Genealogy Project - Constantin Carathéodory". Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Retrieved 27 August 2017.
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Citation Linkgenealogy.math.ndsu.nodak.edu"The Mathematics Genealogy Project - Nazım Terzioğlu". Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Retrieved 27 August 2017.
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Citation Linkopenlibrary.orgBrussells 1901 (Hayez);Ges. math. Schr. V. 273-281
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Citation Linkopenlibrary.orgH. Boerner, Carathéodory und die Variationsrechnung, in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80–90.
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Citation Linkopenlibrary.orgBellman for his Dynamic programming in its continuous-time form used Carathéodory's work in the form of the Hamilton–Jacobi–Bellman equation. Kálmán also explicitly used Carathéodory's formulation in his initial papers on optimal control. See e.g. R. E. Kalman: Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana 1960
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Citation Linkopenlibrary.orgA. Shields: Carathéodory and Conformal Mapping Math. Intelligencer vol.10(1), 1988
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Citation Link//doi.org/10.1007%2Fbf01450409Carathéodory, Constantin (1909). Translated by Delphinich, D. H. "Untersuchungen ueber die Grundlagen der Thermodynamik" [Examination of the foundations of Thermodynamics] (PDF). Mathematische Annalen. 67 (3): 355–386. doi:10.1007/bf01450409.
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Citation Linkopenlibrary.orgBailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, Woodbury NY, ISBN 0-88318-797-3.
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Citation Linkopenlibrary.orgMax Born: The Born–Einstein Letters, MacMillan 1971
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Citation Linkweb.ist.utl.ptConstantin Carathéodory and the axiomatic thermodynamicsby Lionello Pogliani and Mario N. Berberan-Santos
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Citation Linkde.arxiv.orgadiabatic accessibility = adiabatische Erreichbarkeit; see also Elliott H. Lieb, Jakob Yngvason: The Physics and Mathematics of the Second Law of Thermodynamics, Phys. Rep. 310, 1–96 (1999) and Elliott H. Lieb, (editors: B. Nachtergaele, J.P. Solovej, J. Yngvason): Statistical Mechanics: Selecta of Elliott H. Lieb, 2005, ISBN 978-3-540-22297-2
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Citation Linkopenlibrary.orgÜber den Zusammenhang der Theorie der absoluten optischen Instrumente mit einem Satz der Variationsrechnung, Münchener Sitzb. Math. -naturw Abteilung 1926 1–18; Ges. Math. Schr. II 181–197.
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Citation Linkopenlibrary.orgEuler Opera Omnia, Series 1 (a) vol.24: Methodus inveniendi lineas curvas maximi minimive gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti. Lausanne & Geneva 1744 (M. Bousquet) ed. C. Carathéodory Zürich 1952 (Fuesli). (b) vol.25 Commentationes analyticae ad calculum variationum pertinentes. ed C. Carathéodory Zürich 1952 (Fuesli).
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Citation Linkwww.tovima.grConstantin Carathéodory: A Biography, newspaper article, 2000 "(...) Είχε γνωρίσει τον Ελευθέριο Βενιζέλο από το 1895, στην Κρήτη, και από το 1913 είχε προτείνει τη δημιουργία δεύτερου ελληνικού πανεπιστημίου στη Θεσσαλονίκη. Ο πόλεμος που ξεσπάει μεταθέτει τις αποφάσεις. Στην Ελλάδα θα επανέλθει το 1930-32, όταν θα αποδεχθεί τη θέση του κυβερνητικού επιτρόπου και θα οργανώσει τα πανεπιστήμια Αθήνας και Θεσσαλονίκης με τον νόμο 5343/32, ο οποίος ίσχυε μέχρι προσφάτως. Από τη θέση αυτή θα τον απολύσει η κυβέρνηση Παπαναστασίου που διαδέχεται τον Βενιζέλο το 1932 και εκεί θα σταματήσει η ενεργός ανάμειξή του στα κοινά της Ελλάδας." (greek)
Sep 26, 2019, 7:34 PM
Citation Linkwww.elemedu.upatras.gr"The importance of the foundation of the University of Smyrna(Essay)". Department of Primary Education, University of Patras. Archived from the original on 14 June 2012.
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Citation Linkwww.24grammata.com"Constantin Carathéodory: His life and work(Essay)" (PDF). National Technical University of Athens. Archived from the original (PDF) on 2017-12-22."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 genious of culture and civilization on Orient.”"
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Citation Linkwww.auth.gr"Brief History". Aristotle University of Thessaloniki. Retrieved 2012-12-02.
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Citation Linkopenlibrary.orgDenker, Forscher und Entdecker: eine Geschichte der Bayerischen Akademie By Dietmar Willoweit p.263
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