The story of two Greek mathematicians of “modern times” Maurolico & Carathéodory

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  • The story of two Greek mathematicians of “modern times”

  • Maurolico & Carathéodory

Greece through the ages

  • 3000 to 1400BC Minoan Crete

  • 1600 to 1100BC Mycenean Greeks; Bronze Age

  • 1100 to 800BC Pre-classic period; Iron Age

  • 800 to 500BC Classical period

  • 1100BC to 700AD Hellenic Civilization

  • 284AD to 1453AD Byzantine Civilization

  • 1453 to 1821 Ottoman Rule

  • 1821 to 1945 Building of Greek nation

  • 1920 to 1922 Greek-Turkish War

  • 1922 to 1945 Absorption of Asia Minor Refugees

  • Depression & the German occupation

  • 1945 to 1950 Greek Civil War

  • 1967 to 1974 Coup of Colonels; Military Junta

  • 1974 to present Republic of Greece

Francesko Maurolico (1494-1575) Φραγκίσκος Μαυρόλυκος Clarissimum Siciliae lumen

  • Born in Messina, Sicily.

  • Father: Antonios Maroulis - Greek physician who fled Constantinople; affluent, aristocrat.

  • Learned Greek, Math & Astronomy from his father and from

  • Constantinos Laskaris.

  • Means of support: personal, church, academia, government.

  • Scientific interests: Math, astronomy, optics.

Maurolico’s scientific work

  • Public lectures at the Univ. of Messina (mainly Elements of Euclid).

  • Appointed professor in 1569.

  • Published: Cosmographia, Aristotle’s Mechanical Problems,

  • Classical Greek Geometry.

  • Published works on music, the islands of the world,

  • discovered a star in 1572, involved in military engineering.

First complete inductive proof credited to Maurolico

  • Supported by writings of Pascal (letter to Carcavi):

  • Çela est aise par Maurolic

  • Also claimed in Polya’s Mathematical discovery and in Bourbaki’s Set Theory.

  • Arithmeticorum Libri Duo (1575):

  • The sum of the first n odd integers equals

  • the square of n

Constantin Carathéodory (1873-1950) Κωνσταντίνος Καραθεοδωρής

Constantin Carathéodory - Chronology

  • Born in Berlin (to Greek parents: his father was a Turkish diplomat at the time Greeks could attain high office).

  • Raised by his Grandmother in Brussels.

  • Educated in Brussels (civil engineer-Belgian officer).

  • Worked in a British dam project in Egypt, road planning in Greece.

  • 1900: Enters Univ. of Berlin to study mathematics.

  • 1902: Starts Ph.D. at Univ. of Göttingen

  • (under Hermann Minkowski). Receives degree in 1904.

  • 1904-1909: Univ. Of Hanover (Full Professor).

  • 1910-1913: Univ. of Breslau.

  • 1913-1918: Univ. of Göttingen.

  • 1918-1920: Univ. of Berlin.

Chronology continued…

  • 1919: Admitted to Prussian Academy of Sciences

  • (dedication by Max Plank).

  • 1920: Accepts post at the Univ. of Smyrna which the Greeks

  • under Eleftherios Venizelos were setting up in Anatolia

  • (now Izmir in Turkey).

  • When the Turks razed Smyrna in 1922, Carathéodory saved

  • the university library and moved it to Athens.

  • 1922-1924: Taught at the National Technical Univ. of Athens.

  • 1924-1950: Invited and returned to Germany: Univ. of Munich.

Mathematical achievements

  • Calculus of variations/theory of discontinuous solutions of ode’s.

  • Point set measure theory & probability theory.

  • Function theory: conformal representation of simply connected

  • regions on the unit circle; theory of boundary correspondence.

  • Thermodynamics.

  • Geometrical optics.

  • Helped develop Einstein’s theory of special relativity.

Correspondence with Einstein

  • September 1916

  • "Would you think a little bit about the problem of closed

  • time trajectories? Here lies the essence of this still unsolved part

  • of the space-time problem.

  • I wish you all the best from yours truly, A. Einstein.“

  • December 1916

  • "Dear colleague, the main points in the theory of

  • canonical substitutions can be most easily derived in my opinion

  • in the following way."

  • Mathematical expressions from Hamilton-Jacobi Theory follow.

Einstein’s letter (on display in Einstein’s museum in Jerusalem)

  • Dear colleague!

  • I find your derivation wonderful, now I understand everything. At first, the small writing mistakes

  • on the second page had caused me some difficulties. Now, however, I understand everything.

  • You should publish the theory in this new form in the Annals of Physics since the physicists do

  • not normally know anything about this subject as was also the case with me. With my letter I

  • must have come across to you like a Berliner who had just discovered Grunewald and wondered

  • whether people were already living there.

  • If you wouldn't mind also making the effort to present to me the canonical transformations, you'll

  • find in me a grateful and attentive audience. If you, however, answer the question about the

  • closed time trajectories, I will appear before you with my hands folded. The underlying truth,

  • though, is well worth some perspiration.

  • Best regards,

  • yours Albert Einstein.

Carathéodory’s legacy

  • Carathéodory-Finsler manifold

  • Carnot-Carathéodory metric/problem

  • Carathéodory-Fejer method

  • Carathéodory-Toeplitz theorem/method

  • Carathéodory criterion

  • Integer Carathéodory property

  • Carathéodory-Pesin structure

  • Carathéodory-von Neumann algebraic probability

  • Carathéodory topology

  • Carathéodory superposition of multivalued maps

  • Carathéodory matrix coefficient problem

  • Carathéodory-Schur interpolation problem

  • Osgood-Taylor-Carathéodory theorem

  • Carathéodory extension theorem

  • Julia-Carathéodory theorem

  • Carathéodory-Rieffen distance

  • Borel-Carathéodory inequality

  • 700 items in Math Reviews with Carathéodory in title!

  • 4090 items with Carathéodory


  • Let S be any set of points and directions in R^n, and

  • let C=conv S. Then x belongs to C if and only if x can be

  • expressed as a convex combination of n+1 of the

  • points and directions in S (not necessarily distinct).

Facts and anecdotes

  • The “birth, rise, development & fortunes of the theory & axiomatization of thermodynamics” is generally attributed to him.

  • Command of French, Greek,

  • German, English, Turkish, Italian.

  • Math Genealogy Project:

  • 6 students/286 descendants.

  • Retired from Chair of the dept

  • in Munich (1938). Long quarrel

  • arose as to who would replace him. He proposed Herglotz,

  • Van der Waerden or Siegel

  • (opposing certain Nazi sympathizers).

Some more facts…

  • Married with two children (Despina and Stephanos) .

  • Influenced the “Harvard school” (Birkhoffs, Marshal Stone, Ahlfors).

  • Was on the Fields committee that awarded a medal to Garrett Birkhoff.

  • “Carathéodory was completely free of the widespread faults of

  • vanity and jealousy found frequently in the academic world.

  • He felt pure joy for others who made great accomplishments.“

  • (Erhard Schmidt).

  • He was able to give several of his "non-Arian" colleagues a chance

  • for a future by arranging for them an opportunity to emigrate.

…February 2, 1950

  • Nobody could have said it as well as another famous member of the Bavarian Academy of Sciences , the Geheimrat Oskar Perron:

  • Carathéodory, one of the most magnificent mathematicians, substantially enriched and vitally influenced the sciences ... a man of unusually extensive education. As a member of the Greek nation, with his soaring spirit and restless

  • pursuit, he continued the recognition of the tradition and legacy of classical Greek culture.

References & sources

  • Greek Scientists 1453-1821 (in Greek), Spandagos and Travlou.

  • Convex Analysis, Rockafellar.

  • McTutor history site (


  • Galileo project (

  • The Mathematics Genealogy Project.

  • Mathematical Reviews (several articles w/ Carathéodory in title).

  • Google and other search engines.

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