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The Pell Equation in the Pythagoreans, Theaetetus, and Hindu Mathematics

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

The reinterpretation of Plato’s philosophy in terms of periodic anthyphairesis, in fact of palindromically periodic anthyphairesis in the Politicus, and the reading of Book X of Euclid’s Elements under the new light, reveal deep mathematical contributions by Theaetetus, including a proof of the general Pell equation. Fascinating similarities of Theaeteus’ reconstructed proofs with the Hindus’ solution of the problem of Pell are noted.

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Bibliography

  • Acerbi F (2000) Plato: Parmenides 149a7-c3. A proof by complete induction? Arch Hist Exact Sci 55:57–76

    Article  MathSciNet  Google Scholar 

  • AVGI (2000) A mathematician throws new light on Plato-professor Stelios Negrepontis reads Plato anew, 30 July, 6 and 13 August 2000

    Google Scholar 

  • Bachet de Méziriac CG (1612) Problèmes plaisants et délectables qui se font par les nombres, avec leur demonstration, 2nd ed (1624). Pierre Rigaud, Lyon

    Google Scholar 

  • Bassiakou A (2004) Plato’s statesman and the palindromic periodicity of the Anthyphairesis of quadratic irrationals. MSc Thesis, University of Athens [in Greek]

    Google Scholar 

  • Bhanu Murthy IS (1992) A modern introduction to ancient Indian mathematics. Wiley Eastern, New Delhi

    Google Scholar 

  • Bhattacharyya RK (2009) Brahmagupta: the ancient Indian mathematician. In: Yadav BS, Mohan M (eds) Ancient Indian leaps into mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4695-0_12

    Chapter  Google Scholar 

  • Bombelli (1572) L’Algebra, Opera di Rafael Bombelli da Bologna, divisa in tre libri. Giovanni Rossi, Bologna

    Google Scholar 

  • Brokou M (2014) Pell’s equation in Greek and Hindu mathematics. MSc Thesis, University of Athens [in Greek]

    Google Scholar 

  • Burnyeat MF (1978) The philosophical sense in Theaetetus’ mathematics. Isis 69:489–513

    Article  MathSciNet  Google Scholar 

  • Colebrooke HT translated (1817) Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhascara, John Murray, Albemarle Street, London

    Google Scholar 

  • Datta B, Singh AN (1938) History of Hindu mathematics, a source book, part II, Algebra. Motilal Banarsidass, Lahore

    Google Scholar 

  • de Fermat P (1679) Varia opera Mathematica D.Petri de Fermat Senatoris Tolosani. Apud Joannem Pech, juxta collegium PP. Societatis Jesu, Tolosae

    Google Scholar 

  • Dutta AK (2010) Kuṭṭaka, Bhāvanā and Cakravāla In: Seshadri CS (ed) Studies in the history of Indian mathematics. Culture and history of mathematics, vol 5. Hindustan book agency (India), New Delhi, pp 145–199

    Google Scholar 

  • Emch GG, Srinivas MD, Sridharan R (eds) (2005) Contributions to the history of Indian mathematics. Culture and history of mathematics, vol 3. Hindustan Book Agency (India), New Delhi

    Google Scholar 

  • Euler L (1767) De usu novi algorithmi in problemate Pelliano solvendo. Novi Commentarii academiae scientiarum Petropolitanæ 11:28–66

    Google Scholar 

  • Fowler D (1987) The mathematics of Plato’s academy, a new reconstruction, 2nd edn (1999). Clarendon Press, Oxford

    Google Scholar 

  • Fowler D (1994) Could the Greeks have used mathematical induction? Did they use it? Physis 31:253–265

    Google Scholar 

  • Freudenthal H (1953) Zur Geschichte der Vollstandingen Induction. Archives Internationals d’ Histoire des Sciences 6:17–37

    Google Scholar 

  • Hankel H (1874) Zur Geschichte der Mathematik in Alterthum und Mittelalter. Teubner, Leipzig

    Google Scholar 

  • Hardy GH, Wright EM (1938) Introduction to the theory of numbers. Clarendon Press, Oxford

    Google Scholar 

  • Heath TL (1908) The thirteen books of Euclid’s elements, three volumes, 2nd edn (1926). Cambridge University Press, Cambridge

    Google Scholar 

  • Heiberg IL, Stamatis ES (eds) (1977) Euclidis elementa Vol. V Pars 2: Scholia in libros VI-XIII cum appendicibus, 2nd edn. Vieweg+Teubner Verlag [in German]

    Google Scholar 

  • Iamblichus (1894) In Nichomachi arithmeticam introductionem liber. Pistelli H (ed). Teubner BG, Leipzig

    Google Scholar 

  • Kahane JP (1985) La Théorie de Théodore des corps quadratiques réels. L’Enseignement Mathématique 31:85–92

    MathSciNet  Google Scholar 

  • Knorr WR (1975) The evolution of Euclidean elements: a study of the theory of incommensurable magnitudes and its significance for early Greek geometry. Reidel, Dordrecht

    Book  Google Scholar 

  • Knorr WR (1983) “La Croix des Mathématiciens”: the Euclidean theory of irrational lines. Bull Am Math Soc 9:41–69

    Article  Google Scholar 

  • Krishnaswamy Ayyangar AA (1929–30), New light on Bhaskara’s Chakravala or cyclic method of solving indeterminate equations of the second degree in two variables. J Indian Math Soc18:225–248

    Google Scholar 

  • Lagrange JL (1769a) Solution d’un problème d’arithmétique, Miscellanea Taurinensia, vol 4. Reprinted in: Serret JA (ed) (1867) Oeuvres de Lagrange, vol 1. Gauthier-Villars, Paris, pp 671–731

    Google Scholar 

  • Lagrange JL (1769b) Sur la solution des problèmes indeterminés du second degré, Mem. Acad. Roy. Berlin 23. Reprinted in: Serret JA (ed) (1869) Gauthier-VillarsOeuvres de Lagrange, vol 2, Paris, pp 377–535

    Google Scholar 

  • Lagrange JL (1770) Additions au mémoire sur la résolution des équations numériques. Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin 14:581–652

    Google Scholar 

  • Morrow GR (1970) Proclus: a commentary on the first book of Euclid’s elements, translated, introduction, notes. Princeton University Press, Princeton

    Google Scholar 

  • Mueller I (1981) Philosophy of mathematics and deductive structure in Euclid’s “elements”. MIT Press, Cambridge, MA

    Google Scholar 

  • Negrepontis S (1999) Plato’s dialectic under the anthyphairetic scrutiny. In: Syros V, Kouris A, Kalokairinou E (eds) Lectures, academic year 1996–97, pp 15–58. Philosophy Group of the University of Cyprus, Nicosia

    Google Scholar 

  • Negrepontis S (Notes by Kleftaki V) (2003) A reconstruction of the proof of the palindromic periodicity of the Anthyphairesis of commensurable in power only lines, employing only tools from book X. of Euclid’s elements. Manuscript (22 pages)

    Google Scholar 

  • Negrepontis S (2005) The Anthyphairetic nature of the platonic principles of infinite and finite. In: Proceedings of the 4th Mediterranean Conference on Mathematics Education, 28–30 January 2005, Palermo, pp 3–27

    Google Scholar 

  • Negrepontis S (2012) Plato’s theory of knowledge of forms by division and collection in the Sophistes is a philosophic analogue of periodic anthyphairesis (and modern continued fractions), arXiv:1207.2950, 12 Jul 2012

  • Negrepontis S (2018) The Anthyphairetic revolutions of the platonic ideas. In: Sialaros M (ed) Revolutions and continuity in Greek mathematics. Science, technology, and medicine in ancient cultures, vol 8. Walter De Gruyter, Berlin/Boston, pp 335–381

    Google Scholar 

  • Negrepontis S (2019) Plato on geometry and the geometers. In: Dani SG, Papadopoulos A (eds) Geometry in history. Springer Nature, Cham, pp 1–88

    Google Scholar 

  • Negrepontis S (n.d.) From Musical Intervals to Hapseis and Logoi: the Periodic Anthyphairetic Nature of the One of the Second Hypothesis in Plato’s Parmenides and Consequences (Indivisible Line, TMA, Zeno Anthyphairetic). In: Hascher X, Papadopoulos A (eds) Proceedings of the Conference Mathématiques et musique: des Grecs à Euler, Strasbourg, 10–11 September 2015. Hermann, Paris, to appear

    Google Scholar 

  • Negrepontis S, Farmaki V (2019) The history of ancient Greek mathematics, vol 1. Ekkremes, Athens. [in Greek]

    Google Scholar 

  • Negrepontis S, Farmaki V (2021) The paradoxical nature of mathematics. In: Papadopoulos A (ed) Topology and geometry: a collection of papers dedicated to Vladimir G. Turaev. European Mathematical Society Publishing House, Berlin

    Google Scholar 

  • Negrepontis S, Farmaki V, Kalisperi D (2021). Pre-Eudoxian Geometric Algebra. Ganita Bharati: Bulletin of the Indian Society for the History of Mathematics, vol 43

    Google Scholar 

  • Plato (1921) Theaetetus, Sophistes, Statesman, In: Plato in Twelve Volumes, vol 12 (trans: Fowler HN). Cambridge, MA, Harvard University Press; London, William Heinemann Ltd.

    Google Scholar 

  • Plato (1925) Parmenides, Phaedrus, Philebus. In: Plato in Twelve Volumes, vol 9 (trans: Fowler HN). Cambridge, MA, Harvard University Press; London, William Heinemann Ltd.

    Google Scholar 

  • Plato (1967 & 1968) Nomoi. In: Plato in Twelve Volumes, vols 10, 11 (trans: Bury RG). Cambridge, MA, Harvard University Press; London, William Heinemann Ltd.

    Google Scholar 

  • Plato (1969) Politeia. In: Plato in Twelve Volumes, vols 5, 6 (trans: Shorey P). Cambridge, MA, Harvard University Press; London, William Heinemann Ltd.

    Google Scholar 

  • Procli Diadochi (1873) In Primum Euclidis Elementorum Librum Commentarii. Friedlein G (ed), Teubner BG, Leipzig

    Google Scholar 

  • Procli Diadochi (1899–1901) In Platonis rem publicam commentarii. In: Kroll W (ed) Bibliotheca scriptorum Graecorum et Romanorum Teubneriana, 2 vols. Teubner BG, Leipzig

    Google Scholar 

  • Raghavan S (1997) Cakravala method. In: Selin H (ed) Encyclopaedia of the history of science, technology, and medicine in non-Western cultures. Springer, Berlin-New York, pp 1007–1010

    Google Scholar 

  • Selenius CO (1975) Rationale of the Chakravāla process of Jayadeva and Bhāskara II. Hist Math 2(2):167–184

    Article  MathSciNet  Google Scholar 

  • Shukla KS (1954) Acarya Jayadeva, the mathematician. Ganita 5:1–20

    Google Scholar 

  • Singh P (1983) Varga-prakṛti – the cakravāla method of its solution and the regular continued fractions. Indian J Hist Sci 19(1):1–17

    Google Scholar 

  • Srinivasiengar CN (1988) The history of ancient Indian mathematics. The World Press Private Ltd., Calcutta

    Google Scholar 

  • Taisbak CM (1982) Coloured Quadrangles. A Guide to the Tenth Book of Euclid’s Elements. Museum Tusculanum Press, Copenhagen

    Google Scholar 

  • Tannery P (1882) Sur la mesure du cercle d’Archimède. Mém Soc Sci Phys Nat Bordeaux 4:313–337

    Google Scholar 

  • Tarrant H (1985) Scepticism or Platonism?: the philosophy of the fourth academy. Cambridge University Press, Cambridge

    Google Scholar 

  • Theon Smyrnaeus (1878) Expositio rerum ad legendum Platonem utilium. Hiller E (ed). Teubner BG, Leipzig

    Google Scholar 

  • Unguru S (1991) Greek mathematics and mathematical induction. Physis 28:273–289

    MathSciNet  Google Scholar 

  • van der Waerden BL (1950) Ontwakende Wetenschap. Noordhoff, Groningen. English edition: van der Waerden BL (1954) Science Awakening (trans: Dresden A). Noordhoff, Groningen

    Google Scholar 

  • van der Waerden BL (1976) Уравнение Пелля в математике греков и индийцев. Uspeki Math. Nauk 31(5):57–70. English edition: van der Waerden BL (1976) Pell’s Equation in Greek and Hindu Mathematics. Russ Math Surv 31:210–225

    Google Scholar 

  • Wedberg A (1955) Plato’s philosophy of mathematics. Almqvist & Wiksell, Stockholm

    Google Scholar 

  • Weil A (1984) Number theory, An approach through history from Hammurapi to Legendre. Birkhäuser, Boston

    Google Scholar 

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Acknowledgments

We express our warm thanks to Athanase Papadopoulos for encouragement and extensive and essential discussions that have led to a substantial improvement of our work.

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Authors and Affiliations

  1. Department of Mathematics, Athens University, Athens, Greece

    Stelios Negrepontis, Vassiliki Farmaki & Marina Brokou

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  1. Stelios Negrepontis
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  2. Vassiliki Farmaki
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Correspondence to Stelios Negrepontis .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. University of Strasbourg, Strasbourg, France

    Athanase Papadopoulos

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Negrepontis, S., Farmaki, V., Brokou, M. (2021). The Pell Equation in the Pythagoreans, Theaetetus, and Hindu Mathematics. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_72-1

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  • DOI: https://doi.org/10.1007/978-3-030-19071-2_72-1

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