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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Ratio in early Greek mathematics
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by D. H. Fowler PDF
Bull. Amer. Math. Soc. 1 (1979), 807-846
    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). 4. W. Burkert, Lore and science in ancient Pythagoreanism, translated by E. L. Minar, Jr., Harvard Univ. Press, Cambridge, Mass., 1972.
  • H. S. M. Coxeter, Geometry, Lectures on Modern Mathematics, Vol. III, Wiley, New York, 1965, pp. 58–94. MR 0178389
  • 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.
  • 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. MR 0075873
  • 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). 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. 12. C. F. Gauss, Werke, 12 vol., Olms, Hildesheim, reprint, 1973. 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.
  • Marshall Hall Jr., On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966–993. MR 22568, DOI 10.2307/1969389
  • 15. T. L. Heath, A history of Greek mathematics, 2 vol., Oxford Univ. Press, London, 1921. 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.
  • A. Ya. Khinchin, Continued fractions, University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
  • Felix Klein, Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis, Dover Publications, New York, N.Y., 1945. MR 0015349
  • 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. MR 0472317
  • Wilbur Richard Knorr, The evolution of the Euclidean elements, Synthese Historical Library, Vol. 15, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1975. A study of the theory of incommensurable magnitudes and its significance for early Greek geometry. MR 0472300, DOI 10.1007/978-94-010-1754-1
  • Wilbur R. Knorr, Archimedes and the measurement of the circle: a new interpretation, Arch. History Exact Sci. 15 (1975/76), no. 2, 115–140. MR 0497462, DOI 10.1007/BF00348496
  • Wilbur R. Knorr, Archimedes and the pre-Euclidean proportion theory, Arch. Internat. Hist. Sci. 28 (1978), no. 103, 183–244 (1979). MR 553444
  • Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456
  • 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. 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. 28. G. N. Raney, On continued fractions and finite automata, Math. Ann. 206 (1973), 265—283. 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.
  • Harold M. Stark, An introduction to number theory, MIT Press, Cambridge, Mass.-London, 1978. MR 514402
  • 31. A. E. Taylor, Forms and numbers: A study in Platonic metaphysics, Mind 35 (1926), 419—440; ibid., 36 (1927), 12-33. 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.
  • B. L. van der Waerden, Science awakening, P. Noordhoff Ltd., Groningen, 1954. English translation by Arnold Dresden. MR 0062670
  • 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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Additional Information
  • Journal: Bull. Amer. Math. Soc. 1 (1979), 807-846
  • MSC (1970): Primary 01A20; Secondary 10A32, 10F20
  • DOI:
  • MathSciNet review: 546311