The Mathematics of Harmony. Proclus’ Hypothesis and New View on Euclid’s Elements and History of Mathematics Starting since Euclid

Author(s)
Alexey Stakhov

ABSTRACT

We are discussing one of the most unlikely hypotheses in the history of mathematics—*Proclus’ hypothesis*, which overturns a traditional view on Euclid’s *Elements* and the history of mathematics, starting since Euclid. According to Proclus, the main goal of Euclid, when writing the *Elements*, was to build a complete geometric theory of *Platonic solids* (Book XIII), associated in the ancient philosophy (Pythagoras, Plato) with the Universe harmony. To construct this theory, Euclid introduced in Book II the *problem of dividing a segment into extreme and mean ratio* (*the “golden section”*). It follows from Proclus’ hypothesis that Euclid’s Elements are the first attempt to create the “*Mathematical Theory of the Universe Harmony*”, based on Platonic solids and the “golden section”.

We are discussing one of the most unlikely hypotheses in the history of mathematics—

KEYWORDS

Proclus’ Hypothesis, Mathematics of Harmony, Euclid’s Elements, Pythagoras, Plato, Golden Section, Platonic Solids

Proclus’ Hypothesis, Mathematics of Harmony, Euclid’s Elements, Pythagoras, Plato, Golden Section, Platonic Solids

Cite this paper

Stakhov, A. (2014) The Mathematics of Harmony. Proclus’ Hypothesis and New View on Euclid’s Elements and History of Mathematics Starting since Euclid.*Applied Mathematics*, **5**, 3335-3352. doi: 10.4236/am.2014.521311.

Stakhov, A. (2014) The Mathematics of Harmony. Proclus’ Hypothesis and New View on Euclid’s Elements and History of Mathematics Starting since Euclid.

References

[1] Shestakov, V.P. (1973) Harmony as an Aesthetic Category. Science, Moscow (Russian).

[2] Soroko, E.M. (1984) Structural Harmony of Systems. Science and Technology, Minsk (Russian).

[3] Stakhov, A.P. (2009) The Mathematics of Harmony. From Euclid to Contemporary Mathemartics and Computer Science. World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 748 p.

[4] Harmony of Spheres. The Oxford Dictionary of Philosophy, Oxford University Press, 1994, 1996, 2005.

[5] Dimitrov, V. (2005) A New Kind of Social Science. Study of Self-Organization of Human Dynamics. Morrisville Lulu Press, Richmond, Australia.

[6] Stakhov, A.P. (1998) The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers, Kluwer Academic Publishers, 7, 393-399.

http://dx.doi.org/10.1007/978-94-011-5020-0_43

[7] Euclid’s Elements. Books I-VI. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1948 (Russian).

[8] Euclid’s Elements. Books VII-X. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1949 (Russian).

[9] Euclid’s Elements. Books XI-XV. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1950 (Russian).

[10] Bunin, V.A. (2009) The Code of Bio-Similarity. Ternary Code of Meta-Harmony as as Bio-Similarity of Technological Systems by the Criterion of the Objective Function. “Academy of Trinitarism”, Moscow, № 77-6567, publ.15669, 24.11.2009 (Russian).

[11] Herz-Fischler, R. (1998) A Mathematical History of the Golden Number. Dover Publications, Inc., New York.

[12] Kline, M. (1984) Mathematics: The Loss of Certainty. (transl. from Engl.). Mir, Мoscow. (Russian)

[13] Kolmogorov, A.N. (1991) Mathematics in Its Historical Development. Science, Moscow. (Russian)

[14] Academician Mitropolsky’s Commentary on the Scientific Research of the Ukrainian Scientist Doctor of Engineering Sciences Professor Alexey Stakhov. Preface to the book Alexey Stakhov “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science”, World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2009, 748 p.

[15] Kahn, C.H. (2001) Pythagoras and Pythagoreans: A Brief History. Hackett Publishing Co, Inc., Indianapolis.

[16] Zhmud, L. (2006) The Origin of the History of Science in Classical Antiquity. Walter de Gruyter, Berlin.

[17] Smorinsky, C. (2008) History of Mathematics: A Supplement. Springer, Berlin.

[18] Klein, F. (1956) Lectures on the Icosahedron. Courier Dover Publications, Mineola.

[19] Stakhov, A.P. (2011) Mathematization of Harmony and Harmonization of Mathematics. Academy of Trinitarism, Moscow. NO.77-6567, publ.16897, 16.10.2011 (Russian).

[1] Shestakov, V.P. (1973) Harmony as an Aesthetic Category. Science, Moscow (Russian).

[2] Soroko, E.M. (1984) Structural Harmony of Systems. Science and Technology, Minsk (Russian).

[3] Stakhov, A.P. (2009) The Mathematics of Harmony. From Euclid to Contemporary Mathemartics and Computer Science. World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 748 p.

[4] Harmony of Spheres. The Oxford Dictionary of Philosophy, Oxford University Press, 1994, 1996, 2005.

[5] Dimitrov, V. (2005) A New Kind of Social Science. Study of Self-Organization of Human Dynamics. Morrisville Lulu Press, Richmond, Australia.

[6] Stakhov, A.P. (1998) The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers, Kluwer Academic Publishers, 7, 393-399.

http://dx.doi.org/10.1007/978-94-011-5020-0_43

[7] Euclid’s Elements. Books I-VI. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1948 (Russian).

[8] Euclid’s Elements. Books VII-X. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1949 (Russian).

[9] Euclid’s Elements. Books XI-XV. Translated from the Greek and Comments D.D. Mordukhai-Boltovsky. Moscow-Leningrad: GITTL 1950 (Russian).

[10] Bunin, V.A. (2009) The Code of Bio-Similarity. Ternary Code of Meta-Harmony as as Bio-Similarity of Technological Systems by the Criterion of the Objective Function. “Academy of Trinitarism”, Moscow, № 77-6567, publ.15669, 24.11.2009 (Russian).

[11] Herz-Fischler, R. (1998) A Mathematical History of the Golden Number. Dover Publications, Inc., New York.

[12] Kline, M. (1984) Mathematics: The Loss of Certainty. (transl. from Engl.). Mir, Мoscow. (Russian)

[13] Kolmogorov, A.N. (1991) Mathematics in Its Historical Development. Science, Moscow. (Russian)

[14] Academician Mitropolsky’s Commentary on the Scientific Research of the Ukrainian Scientist Doctor of Engineering Sciences Professor Alexey Stakhov. Preface to the book Alexey Stakhov “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science”, World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2009, 748 p.

[15] Kahn, C.H. (2001) Pythagoras and Pythagoreans: A Brief History. Hackett Publishing Co, Inc., Indianapolis.

[16] Zhmud, L. (2006) The Origin of the History of Science in Classical Antiquity. Walter de Gruyter, Berlin.

[17] Smorinsky, C. (2008) History of Mathematics: A Supplement. Springer, Berlin.

[18] Klein, F. (1956) Lectures on the Icosahedron. Courier Dover Publications, Mineola.

[19] Stakhov, A.P. (2011) Mathematization of Harmony and Harmonization of Mathematics. Academy of Trinitarism, Moscow. NO.77-6567, publ.16897, 16.10.2011 (Russian).