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Ratio in early Greek mathematics. (English) Zbl 0425.01002


MSC:

01A20 History of Greek and Roman mathematics
11A55 Continued fractions
11-04 Software, source code, etc. for problems pertaining to number theory

Keywords:

theory of ratio
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Full Text: DOI

References:

[1] Alexander of Aphrodisias, In topica, edited by M. Wallies, Berlin, 1891.
[2] Aristotle, Topics, edited and translated by E. S. Forster, Loeb Classical Library Series, Heinemann, London and Harvard Univ. Press, Cambridge, Mass., 1960.
[3] O. Becker, Eudoxos-Studien. I-IV, and other articles, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2 (1933); 3 (1936). · JFM 62.0011.03
[4] W. Burkert, Lore and science in ancient Pythagoreanism, translated by E. L. Minar, Jr., Harvard Univ. Press, Cambridge, Mass., 1972.
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[6] H. Davenport, The higher arithmetic, Hutchinson, London, 1968.
[7] H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed., Weidmannsche, Berlin, 1951-1952.
[8] Euclid, The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III – IX. Vol. III: Books X – XIII and Appendix, Dover Publications, Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath; 2nd ed. · Zbl 0071.24203
[9] T. Fletcher, Approximating by vectors (1), (2), Math. Teaching 63 (1973), 4-9; ibid., 64 (1973), 42-44.
[10] D. H. Fowler, Book II of Euclid’s Elements and a pre-Eudoxan theory of ratio (to appear). · Zbl 0502.01004
[11] K. von Fritz, The discovery of incommensurability by Hippasus of Metapontum, Ann. of Math. (2) 46 (1945), 242-264; reprinted in Studies in pre-Socratic philosophy, vol. 1, edited by D. J. Furley and R. E. Allen, Routledge and Kegan Paul, London, 1970, pp.382-412.
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[13] R. W. Gosper, Continued fraction arithmetic, Item 101B in HAKMEM, Artificial Intelligence Memo No. 239, A. I. Laboratory, M. I. T., Cambridge, Mass., February 29, 1972.
[14] Marshall Hall Jr., On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966 – 993. · Zbl 0030.02201 · doi:10.2307/1969389
[15] T. L. Heath, A history of Greek mathematics, 2 vol., Oxford Univ. Press, London, 1921. · JFM 48.0046.01
[16] A. Hurwitz, Über die Kettenbruch-Entwicklung der Zahl e, Phys.-okon. Ges. Konigsberg, 1891; reprint, Mathematische Werke, Bd. 2, Birkhäuser, Basel, 1933, pp. 129-133.
[17] A. Ya. Khinchin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. · Zbl 0117.28601
[18] Felix Klein, Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis, Dover Publications, New York, N. Y., 1945.
[19] Jacob Klein, Greek mathematical thought and the origin of algebra, The M.I.T. Press, Cambridge, Mass.-London, 1968. Translated from the German by Eva Brann; With an appendix containing Vieta’s Introduction to the analytical art, translated from the Latin by J. Winfree Smith. · Zbl 0159.00302
[20] Wilbur Richard Knorr, The evolution of the Euclidean elements, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1975. A study of the theory of incommensurable magnitudes and its significance for early Greek geometry; Synthese Historical Library, Vol. 15. · Zbl 0362.01001
[21] Wilbur R. Knorr, Archimedes and the measurement of the circle: a new interpretation, Arch. History Exact Sci. 15 (1975/76), no. 2, 115 – 140. · Zbl 0334.01007 · doi:10.1007/BF00348496
[22] Wilbur R. Knorr, Archimedes and the pre-Euclidean proportion theory, Arch. Internat. Hist. Sci. 28 (1978), no. 103, 183 – 244 (1979). · Zbl 0403.01003
[23] Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0357.68001
[24] Pappus, The commentary of Pappus on Book X of Euclid’s Elements, edited and translated by G. Junge and W. Thompson, Harvard Univ. Press, Cambridge, Mass., 1930; reprint, Johnson Reprint Corporation, New York and London, 1968.
[25] Proclus, A commentary on the first book of Euclid’s Elements, translated by G. N. Morrow, Princeton Univ. Press, Princeton, N. J., 1970. · Zbl 0229.01002
[26] Plato, Loeb Classical Library Series, Charmides (W. R. M. Lamb, 1927); Gorgias (W. R. M. Lamb, 1925); Laws (R. G. Bury, 2 vol., 1926); Lesser Hippias (H. N. Fowler, 1926); Philebus (H. N. Fowler, 1925); Protagoras (W. R. M. Lamb, 1924); Republic (P. Shorey, 2 vol., 1930-1935); Theaetetus (H. N. Fowler, 1921), Heinemann, London and Harvard Univ. Press., Cambridge, Mass.
[27] E. B. Plooij, Euclid’s conception of ratio and his definition of proportional magnitudes as criticized by Arabian commentators, Rotterdam, 1950. · Zbl 0045.14601
[28] G. N. Raney, On continued fractions and finite automata, Math. Ann. 206 (1973), 265-283. · Zbl 0251.10024
[29] H. J. S. Smith, Notes on continued fractions, Messenger of Mathematics, Ser. II, vol. 6, 1876, 1-14; reprint, Collected Works, vol. II, Chelsea, New York, 1965, pp. 135-147.
[30] Harold M. Stark, An introduction to number theory, MIT Press, Cambridge, Mass.-London, 1978. · Zbl 0442.10001
[31] A. E. Taylor, Forms and numbers: A study in Platonic metaphysics, Mind 35 (1926), 419-440; ibid., 36 (1927), 12-33. · JFM 52.0007.01
[32] I. Thomas [= I. Bulmer-Thomas], editor and translator, Selections illustrating the history of Greek mathematics, 2 vol., Loeb Classical Library Series, Heinemann, London and Harvard Univ. Press, Cambridge, Mass., 1939-1941.
[33] D’A. W. Thompson, Excess and defect: Or the little more and the little less, Mind 38 (1929), 43-55.
[34] B. L. van der Waerden, Science awakening, P. Noordhoff Ltd., Groningen, 1954. English translation by Arnold Dresden. · Zbl 0056.24203
[35] E. C. Zeeman, Research ancient and modern, Bull. Inst. Math. Appl. 10 (1974), 272-281; reprinted in Catastrophe Theory-Selected Papers 1972-1977, Addison-Wesley, Reading, Mass., 1977.
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