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希腊数学的新看法:原始资料、问题及出版物

#### Wilbur R. Knorr,张雁

**摘要(Abstract)： **

尽管残存的古希腊数学文献是有限的,它的起源和后来的发展问题却得到了人们的热切关注。的确,确凿证据的缺乏促进了再现失传成就的新方法,激发了把数学与早期希腊思想的其它方面相联系的独创性。通过对晚期希腊文注释本和中世纪的阿拉伯文及拉丁文译本的研究,再现某些从原希腊语作品中失传的重要著作已成为可能。这些进展提供了有关希腊数学所有较早(但绝非陈旧)的权威性典据。

**关键词(KeyWords)：**

**基金项目(Foundation):**

**作者(Author):**
Wilbur R. Knorr,张雁

#### 参考文献(References)：

- 1． On Byzantine letters, see N. G. Wilson, Scholars of Byzantium, Baltimore, 1983; esp. chaps. 4-6． For a more detailed account see P. Lemerle, Le premier humanisme byzantin, Paris, 1971, (English translation, Canberra, 1986) .
- 2． I omit here consideration of the evidence from the ancient papyri. These come mostly from Hellenistic Egypt, from the 3rd century B.C. to the 7th century A.D., and represent
- predominantly the everyday culture, as distinct from the domain of letters. In mathematics, only two or three papyrus fragments hold excerpts from Euclid, and there is nothing more advanced than that. Otherwise, the Greek mathematical papyri contain only basic arithmetic and practical geometry, as taught in the lower schools. On the papyri in general, see E. G. Turner, Greek Papyri: An Introduction, Oxford, 1968; for an inventory and description of the mathematical papyri, see D. H. Fowler, Mathematics of Plato's Academy: A New Reconstruction, Oxford, 1987．
- 3． For accounts of the work of these commentators, with specimens of texts, see W. R. Knorr, Textual Studies in Ancient and Medieval Geometry, Boston/Basel/Berlin, 1989．
- 4． Heiberg produced the critical editions of Archimedes, Euclid (the Elements and optical writings), Apollonius, Ptolemy (the Alamagest and minor astronomical works), geometrical works of Hero of Alexandria (first century A.D.) and minor mathematical writers, as well as medical and philosophical works of others; see Knorr (ref. 3) , 2． Extensive accounts of his life and work are given by C. Hoeg, Jahresbericht uber die Fortschritte der klassichen Altertumswissenshaft, 1931, 57 (233) , "Nekroioge," 38-77, and by J. Bidez (with portrait) in Paul Tannery: Mémoires scientiques, vol. 9, Paris, 1929, ⅸ-ⅹⅹⅷ. Among other editors of classical mathematics, one should note F. Hultsch, editor of Pappus' Collection (1876-78) ; H. Menge, Heiberg's collaborator in the editing of Euclid, vols. 6 and 8 (1896, 1916) ; P. Tannery, editor of Diophantus' Arithmetica (1893-95) ; and H. Schone, editor of Hero's Metrica (1903) .
- 5． For a useful review of this recent editing, see G. J. Toomer, "Lost Greek Mathematical Works in Arabic Translation," Mathematical Intelligencer, 6 (1984) , 32-38．
- 6． Diocles: On Burning Mirrors, Berlin/Heidelberg/New York, 1976． A new edition of Diocles and other Arabic works on geometric optics is planned by R. Rashed for the series "Les belles lettres," Paris.
- 7． Apollonius: Conics Books Ⅴ to Ⅶ, Berlin/Heidelberg/New York, 1990． Forasummaryofthe Arabic sources for Apolionius, see R. Sezgin, Geschichte des arabischen Schrifttums, vol. 5, Leiden, 1974, 136-143． A minor work of Apollonius, On the Cutting Offofa Ratio, survives only in Arabic translation; it was edited in Latin translation by Halley (1706) , and has recently been translated into English by E. M. Macierowski (Fairfield, Conn., 1987) .
- 8． Sesiano, Books Ⅳ to Ⅶ of Diophantus' Arithmetica in the Arabic Translation, New York/Heidelberg/Berlin, 1982; Rashed, Diophante: Les arithmétiques, vols. 3-4 (of the projected complete edition of Diophantus in four volumes, being prepared in collaboration with A. Allard), Paris, 1984．
- 9． "On Euclid's Lost Porisms and Its Arabic Traces," Bollettino di Storia delle Scienze Matematiche, 7 (1987) , 93-115; "Greek and Arabic Constructions of the Regular Heptagon,"Archive for History of Exact Sciences, 30 (1984) , 197-330; "Arabic Traces of Lost Works of Apollonius," same journal, 35 (1986) , 187-253; related materials appear in his edition of ibn al-Haytham's Completion of the Conics, New York/Berlin/Heidelberg/Tokyo, 1985．
- 10 For a survey of the technical methods of Greek geometry with accounts of the more advanced efforts, see Knorr, The Ancient Tradition of Geometric Problems, Boston/Basel/Stuttgart, 1986．
- 11． In the method of"analysis" one seeks the construction ofa geometric figure by first assuming the figure to have been produced and then deriving from this properties that are already known to be produceable; the formal construction (called the "synthesis") inverts the order of the analysis. On the method and its applications, see Knorr (ref. 10) , esp. chap. 8． Book Ⅶ of Pappus has been newly edited, with extensive commentary, by A. Jones, Pappus of Alexandria: Book 7 of the Collection, New York/Berlin/Heidelberg/Tokyo, 1986．
- 12． The principal manuscript, prototype of the remaining copies, is the early 10th cent. Vatican ms. gr. 218 (this date is the estimate by Jones, altering the assignment to the 12th century commonly made prior authorities; cf ref. 11, 30) ; it now lacks the first book and the beginning of the second, as well as the end of the eighth. For an account of the manuscript and its history, see Jones, (ref. 11) , 30-62．
- 13． The most detailed survey of Archimedes' work, with complete synopses of all his propositions, is by A. J. Dijksterhuis, Archimedes, Copenhagen, 1956． The reprint edition (Princeton, 1987) included a bibliographical supplement by Knorr, "Archimedes after Dijksterhuis: A Guide to Recent Studies."
- 14． This three-proposition tract includes the proofofa rule for the area of the circle (prop. 1) , and a computation leading to rigorous bounds on the value of the ratio of the circumference to the diameter of the circle (the constant now denoted as π), namely 3(1/7) as an upper bound, and 3(10/71) as a lower bound. This is the earliest known use of 3(1/7) for circle measurement.
- 15． For a listing of studies, see the bibliography by Knorr, (ref. 13) , 431-440．
- 16． Heiberg published a provisional text of the Method in 1906． On the history ofthe Archimedes manuscript, see Dijksterhuis, (ref. 13) , chapter 2． The Istanbul manuscript was removed from Turkey during the Revolution in the 1920s, and at present, being held in a private collection, is not available for scholarly study.
- 17． See Knorr, "Archimedes' Lost Treatise on the Centers of Gravity of Solids," Mathematical lntelligencer, (1973) , 102-109． An important survey of references to Archimedes' lost mechanical writings is A. G. Drachmann, "Fragments from Archimedes in Heron's Mechanics." Centaurus, 8 (1963) , 91-146 For other studies, see the Archimedes bibliography by Knorr, (ref. 13) , 437-438．
- 18． For the argument and texts, see Knorr, Ancient Sources of the Medieval Tradition of Mechanics. Greek, Arabic and Latin Studies of the Balance, Florence (Annali dell'Istituto e Museo di Storia della Scienza, monografia 6) , 1982．
- 19． A short Archimedean tract on "Mutually Tangent Circles" has been edited from the Arabic (in a facsimile of the Arabic manuscript, with German translation) by Y. Dold-Samplonius, H. Hermelink, and M. Schramm in Archimedis Opera Omnia, vol. 4, Stuttgart, 1975, Dold-Samplonius has also edited the Arabic tract, Book of Assumptions by "Aqatun" (the corresponding Greek name has not been identified), which appears to include Archimedean materials (Ph.D. dissertation, Amsterdam, 1976) . For other efforts, attributed to Archimedes by Arabic authorities, see Sezgin, (ref. 7) , 121-136．
- 20． On the Arabic recension of Dimerision of the Circle, see Knorr, (ref. 3) , part 3, chapters 3-4; this includes remarks on Sphere and Cylinder, for which a more extensive survey has been compiled by R. Lorch, "The Arabic Transmission of Archimedes' Sphere and Cylinder and Eutocius' Commentary," Zeitschriftfur Geschichte der arabische-islamischen Wissenschaften, 5 (1989) , 94-114．
- 21． See Knorr, "Archimedes and the pre-Euclidean Proportion Theory," Archives internat-ionales d'histoire des sciences, 28 (1978) , 183-244．
- 22． For an account of Hippocrates' constructions, see Knorr, (ref. 10) , chapter 2． (Note that this geometer is not to be confused with his contemporary, the famous physician, Hippocrates of Cos.) Problems of interpretation are examined in depth by G. E. R. Lloyd, "The Alleged Fallacy of Hippocrates of Chios." Apeiron, 20 (1987) , 103-128．
- 23． For a discussion of this fragment, see Knorr, The Evolution of the Euclidean Elements, Dordrecht, 1975, chapter 7．
- 24． Eutocius preserves two accounts of the construction of the cube duplication, one by Archytas, the other by Menaechmus (ca. 350 B.C.), that may derive, albeit in a much edited form, from Eudemus' history; see Knorr, (ref. 3) , part 1, chapter 5; and (ref. 10) , chapter 3．
- 25． If this sounds overly critical, one should note that the present writer has contributed his fair share to this spculative effort, as particularly in (ref. 23) .
- 26． An early edition of the Rhind Papyrus was issued by A. Eisenlohr in 1877, and there have been several since; see the detailed account by R. G. Gillings, Mathematics in the Time of the Pharaohs, Cambridge, Mass./London, 1972． The papyrus, now in the British Museum, has been published in a photofacsimile edition by G. Robins and C. Shute, The Rhind Mathematical Papyrus: An Ancient Egyptian Text, London, 1987． For a survey of ancient Egyptian mathematics, see O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed., Providence, 1957, chapter 4; and B. L. van der Waerden, Science Awakening, (2nd English edition), Leiden/New York, 1961, chapter 1． For a summary of recent studies, see P. Campbell,"Egyptian Mathematics," in J. W. Dauben (ed.), The History of Mathematics from Antiquity to the Present: A Selective Bibliography, New York/London, 1985, 29-37．
- 27． There are good summary accounts by Neugebauer, (ref. 26) , chapter 2, and van der Waerden, (ref. 26) , chapter 3． For recent bibliography, see J. Friberg in Dauben (ed.), ref. 26, 37-51．
- 28． See Neugebauer, (ref. 26) , chapter 6, and van der Waerden, (ref. 26) , 118-126．
- 29． For a summary of his own position, in response to critics, see Unguru, "History of Ancient Mathematics: Some Reflections on the State of the Art," Isis, 70 (1979) , 555-565．
- 30． His most extensive account is in Geometry and Algebra in Ancient Civilizations, Berlin/Heidelberg/New York/Tokyo, 1983; this amplifies the preliminary proposals in his two articles, "On Pre-Babylonian Mathematics," in Archive for History of Exact Sciences, 23 (1980) , 1-26, 27-46．
- 31． Seidenberg has developed his position in a set of papers appearing in the Archive for Exact Sciences, beginning with "The Ritual Orion of Geometry," 1 (1962) , 488-527． The most recent, published posthumously, is "On the Volume of a Sphere," 39 (1988) , 97-119, which refers back to some of his earlier treatments. See also van der Waerden, Geometry and Algebra, (ref. 30) , 10-13, 24-25, 39, 172．
- 32． For a critique along these lines, see my review of van der Waerden's book (ref. 30) , in "The Geometer and the Archaeoastronomers: On the Prehistoric Origins of Mathematics," British Journal for the History of Science, 18 (1985) , 197-212．
- 33． Szabo's most extensive account is in Anfange der griechischen Mathematik, Munchen and Wien, 1969 (French trans., Paris, 1977; English trans., Dordrecht, 1978) , which brings together several of his papers from the preceding decade. He has pursued these studies in a sequel volume on aspects of early geometric science, with co-author E. Maula, Enklima: Untersuchuugen zur Fruhgeschichte der oriechischen Astronomie, Geographie und der Sehnentafeln, Athens, 1982 (French trans., Paris, 1986) .
- 34． A detailed critique is given by Knorr, "On the Early History of Axiomatics," in J. Hintikka, D. Gruender, E. Agazzi (eds.), Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Dordrecht (Synthese Library, vol. 145) , 1, 145-186; see also the response in Szabr's behalf by F. Franciosi, ibid., 187-191． For a more sympathetic review of Szabo's speculations, set in the framework of ideas of Kant and Popper, see S. Marcucci, "Epistemologia, filogia e storia della scienze in Arpad Szabo," Physis, 25 (1983) , 127-165．
- 35． A summary of the Pythagorean testimonia, with a cautious effort at reconstruction, is given by L. Zhmud, "Pythagoras as a Mathematician," Historia Mathematica, 16 (1989) , 249-268． The most ambitious efforts at Pythagorean reconstruction are by van der Waerden, as in his Die Pythagoreer: Religiose Bruderschaft und Schule der Wissenschafi, Zurich and Munchen, 1979． The classic conservative study i~ W. Burkert, Lore and Science in Ancient Pythagoreanism, Cambridge, Mass., 1972 (translated from the German edition of 1962) .
- 36． See Knorr, "Infinity and Continuity: The Interaction of Mathematics and Philosophy in Antiquity," in N. Kretzmann (ed.), infinity and Continuity in Ancient and Medieval Thought, Ithaca, N. Y. and London, 1982, 112-145． This position is presupposed in the accounts of early geometry in Knott, (ref. 10) , chapters 2-3．
- 37． Aristotle and the Mathematicians: Some Crosscurrents in the 4th Century, ph.D. dissertation, Stanford, 1986 (Dissertation Abstracts, 47 [1986] , 550-A). A revised version of this study is being prepared for publication.
- 38． Philosophy of Mathematics and Deductive Structure in Euclid's.Elements, Cambridge, Mass. and London, 1981．
- 39． See Knorr, (ref. 23) . The proposals, as they bear on the 4th century efforts, have been expanded in "La croix des mathrmaticiens: The Euclidean Theory of Irrational Lines," Bulletin of the American Mathematical Society, 9 (1983) , 41--69; of. also "Euclid's Tenth Book: An Analytic Survey," Historia Scientiarum (Tokyo), 29 (1985) , 17． -35．
- 40． Coloured Quadranoles: A Guide to the Tenth Book of Euclid's Elements, Copenhagen, 1982．
- 41． Knorr,(ref. 23) ,chapter 7． Note that this is not the same formofproportion theory as the one reconstructed in (ref. 21) .
- 42． Fowler, (ref. 2) , offers the most extensive account, consolidating over a dozen of his articles and preprints, See the detailed review by B. Artmann, 'Die Mathematik in der Akademie Platons', Mathematische Semesterberichte, 35(1988) , 162-182．
- 43． A Mathematical History of Division in Extreme and Mean Ratio, Waterloo (Ont.), Canada, 1987． Despite his thoroughgoing negative critique, however, Herz-Fischler's attempts at positive reconstruction are not, at least in my view, notably convincing.
- 44． "A Tentative Reconstruction of the Formation Process of Book XIII of Euclid's Elements," Commentarii Mathematici Universitatis Sancti Pauli (Tokyo), 38(1989) , 101-127．
- 45． See, in particular, three of his papers in the Archive for History of Exact Sciences: "Die ersten vier Bucher der Elemente Euklids," 9 (1973) , 325-380; "Beitrage zur Fruhgeschichte der griechischen Geometrie," 11(1973) , 127-133; and "Die stereometrischen Bucher der Elemente Euklids," 14 (1974) , 91-125．
- 46． Among the most recent of them, "Ueber voreuklisdische 'Elemente der Raumgeometrie' aus der Schule des Eudoxos," Archive for History of Exact Sciences, 39 (1988) , 121-135． and "Die Mathematik in der Akademie Platons," Mathematische Semesterberichte, 35(1988) , 162-182．
- 47． Toth's fullest exposition is in"Das Parallelenproblem im Corpus Aristotelicum," Archive for History of Exact Sciences, 3(1967) , 249-422; Toth has elaborated his view in a variety of contexts, particularly philosophical, in several later studies, e. g., "Geometria more ethico-Die alternative: euklidische oder niehteuklidische Geometric bei Aristoteles und die axiomatische Grundlegung der euklidischen Geometric," in Y. Maeyama and W. G. Saltzer (eds.), Prismata... Festschrift fur Willy Hartner, Wiesbaden, 1977, 395-415．
- 48． For a recent discussion of the view of early intimations of non-euclidean geometry see H. Freudenthal, "Nicheuklidsehe Geometric im Altertum?" in H. G. Steiner and H. Winter (eds.), Mathematikdidaktik, Bildunosoeschichte, Wissenschaftsgeschichte, Ⅱ, Koln (Untersuchungen zum Mathematikunterricht, 14) , 1988． For a detailed survey of the history of noneuclidean geometry, with substantial accounts of ancient efforts, see B. Rozenfeld, A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, New York/Berlin/Heidelberg/London/Paris/Tokyo, 1988．
- 49． Euklid (in Biographien hervorragender Naturwissensehaftler, Techniker und Mediziner, vol. 87) , Leipzig, 1987．
- 50． Archimedes: lnoenieur, Naturwissenschaftler und Mathematiker, Darstadt, 1979．
- 51． Die Lehre yon den Keoelschnitten im Altertum, Copenhagen, 1886; repr. Hildesheim, 1966．
- 52． "Apollonius of Perga," in Dictionary of Scientific Biooraphy, 1 (1970) , 179-193． See also Toomer's new edition of Conics, Books V-VII (ref. 7) .
- 53． For an introduction to the commentators, as well as a specimen of the first fruits of this collaborative effort, one should consult the volume of 20 essays, Aristotle Transformed: The Ancient Commentators and Their Influence, ed. Sorabji, Ithaca, N. Y., 1990．
- 54． Relative to Theon's commentaries on Ptolemy's "Handy Tables," see Le 'Grand Commentaire'... Livre I, ed. Mogenet and Tihon, (Studie Testi, 315) , Vatican City, 1985; and Le 'Petit Cormmentaire', ed. Tihon, (Studi e Testi, 282) , 1978．
- 55． For an introductory guide, one should consult the bibliographical essay on Islamic mathematics by R. Lorch et al., and the essay on Hebrew mathematics by B. R. Goldstein, in Dauben (ed.), ref. 26．
- 56． Clagett presents texts and commentaries in Archimedes in the Middle Ages, 5 vols., Madison, Wise. and Philadelphia, Penn., 1964-1984; for discussion, see Knorr, (ref. 3) , part 3． The Euclid versions by Gerard of Cremona, Adelard of Bath (the so-called "version I"), Hermann of Carinthia, and the anonymous Graeco-Latin translator have been edited by Busard, and the edition of the Adelardian "version Ⅱ" is due shortly from Busard and Folkerts. For a summary of the medieval Euclid, see Folkerts, Euclid in Medieval Europe, Winnipeg, (Man.), Can. (in the series of the Benjamin Catalogue for History of Science, ed. W. M. Stevens), 1989; see also the bibliographical essay by Busard, Kolkerts and E. Sylla in Dauben (ed.), ref. 26,100-125．
- 57． See S. Brentjes, "Die Entwicklung der antiken griechischen Mathematik im Lichte einiger Tendenzen in der gegenwartigen Forschung," in G. Wendel (ed.), Wissenschaft in der Antike, Berlin, 1986; J. L. Berggren, "History of Greek Mathematics: A Survey of Recent Research," Historia Mathematica, 11 (1984) , 394-410; and Berggren, "Greek Mathematics," in Dauben (ed.), ref. 26, 51-64．