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Home , Angle trisection, Anthemius of Tralles, Arithmetica, Catoptrics, Dinostratus, Diocles (mathematician)

Bibliography

The following list of cited works is divided into ancient, medieval and modern writings, where, for convenience, I take the 7th century to separate the ancient and medieval periods. Among the ancient and medieval titles I cite manuscripts when a published version is not available. When multiple entries occur for a given modern author, I have listed the titles in the order of publication.

Ancient Sources

Anonymus. Prolegomena tes tou Ptolemaiou megales syntaxeos: Vat. ms. gr. 1594 (Vati• can, 10th cent.), f. lr-9r; Marc. ms. gr. 313 (Venice, 10th cent.), f. lr-28r; B.N. ms. gr. 2390 (Paris, 13th cent.), f. lr-13v -. Introduction to the Syntaxis = Prolegomena ... Syntaxeos -. [Introduction, excerpt on isoperimetric figures]: see F. Hultsch, Pappi Collectio, III, pp. 1138-65 -. see also J. Mogenet (1956) . Peri paradoxon mechanematon (On Paradoxical Mechanisms, a fragment), ed. J.L. Heiberg, Mathematici Graeci Minores, Copenhagen: A.F. H~st & Son, 1927, pp. 78-87 -. see also G.L. Huxley; and R. Rashed Apollonius. Quae graece exstant, ed. J.L. Heiberg, 2 vol., Leipzig: Teubner, 1891-93 (repr. Stuttgart, 1974) -. Conicorum libri octo, ed. E. Halley, Oxford, 1710 -. see also H. Balsam; T.L. Heath (1896); P. Ver Eecke (1924) . Opera Omnia, cum Commentariis Eutocii, ed. lL. Heiberg, 2nd ed., 3 vol., Leipzig: Teubner, 1910-15 (repr. Stuttgart, 1972) -. see also A. Czwalina; T.L. Heath (1897); EJ. Dijksterhuis; C. Mugler; P. Ver Eecke (1921) . Opera (I-II, Aristoteles Graece, ed. I. Bekker, 1831; III, Aristoteles Latine, interpretibus variis, 1831; IV, Scholia in Aristotelem, ed. C.A. Brandis, 1836; V, Index 818 Textual Studies in Ancient and Medieval Geometry

Aristotelicus, ed. H. Bonitz, 1870), Berlin (Academia litterarum regia borussica): G. Reimer -. Aristoteles Latinus, G. Verbeke (series ed.), Brussels: Desclee de Brouwer and Leiden: Brill . De institutione libri ii, De institutione musica libri v, ed. G. Fried• lein, Leipzig: Teubner, 1867 -, trans. De sophisticis elenchis, ed. B.G. Dod, in Aristoteles Latinus, VI: 1-3, 1975 CAG = Commentaria in Aristotelem Graeca (Academia litterarum regia borussica), Berlin: G. Reimer Damianus. Kephalaia ton optikon hypotheseon (Chapters of the optical hypotheses), ed. R. Schone, Damianos: Schrift aber Optik, Berlin: Reichsdruckerei, 1897 . Vitae Isidori Reliquiae, ed. C. Zintzen, Hildesheim: G. Olms, 1967 . On Burning , ed. GJ. Toomer, Berlin/Heidelberg/New York: 1976 -. see also R. Rashed of . Opera Omnia, ed. P. Tannery, 2 vol., Leipzig: Teubner, 1893-95 -. see also R. Rashed (1984); J. Sesiano Domninus of Larissa. Encheiridion arithmetikes eisagoges (Manual of the Arithmetic Introduction): in J.F. Boissonade, Anecdota Graeca, Paris, 1832 (repr. Hildesheim: G. Olms, 1962), IV, pp. 413-429 -. see also P. Tannery (1906) -. Pos esti logon ek logou aphelein (fragment, "How to subtract a from a ratid'): see C.E. Ruelle . Elementa, ed. J.L. Heiberg, 4 vol., 1883-85, Leipzig: Teubner (2nd ed., ed. E.S. Stamatis, 4 vol., 1969-73) -. see also T.L. Heath (1926) -. Elementa XlV-XV, Scholia in Elementa, and "Prolegomena critica," ed. J.L. Heiberg (Euclidis Opera, vol. 5), Leipzig: Teubner, 1888 (2nd ed., in two parts, ed. E.S. Stama• tis, 1977) -. (Opera Omnia, vol. 6), ed. H. Menge, Leipzig: Teubner, 1896 -. see also S. Ito -. Optica, Opticorum recensio Theonis, Catoptrica (Opera Omnia, vol. 7), ed. J.L. Heiberg, Leipzig: Teubner, 1895 -. Phaenomena, Scripta musica (Opera Omnia, vol. 8), ed. H. Menge, Leipzig: Teubner, 1916 . Commentaria in Conica: in Apollonii quae graece exstant, ed. J.L. Heiberg, vol. II, Leipzig: Teubner, 1893 (repr. Stuttgart, 1974) -. Commentarii in Libros de Sphaera et Cylindro, Dimensionem Circuli, de Planorum Aequilibriis: in Archimedis Opera Omnia, ed. J.L. Heiberg, vol. III, Leipzig: Teubner, 1915 (repr. Stuttgart, 1972) . Opera, 5 vol. (I, Pneumatica, ed. W. Schmidt, 1899; II, Mechanica, ed. L. Nix and [Ptolomeil De speculis, ed. W. Schmidt, 1900; III, Mechanica and Diop• tra, ed. H. Schone, 1903; IV, Definitiones and Geometrica, ed. J.L. Heiberg, 1912; V, Stereometrica, ed. J.L. Heiberg, 1914), Leipzig: Teubner. -. Metrica: see also E.M. Bruins -. Belopoeica: see E.W. Marsden . Book XIV of the Elements: see Euclidis Opera Omnia, ed. J.L. Heiberg, vol. 5 of Chalcis. In Nicomachi Introductionem Arithmeticam Liber, ed. H. Pistelli, Leipzig: Teubner, 1894 Bibliography 819

-. In Platonis Dialogos Commentariorum Fragmenta, ed. 1.M. Dillon, Leiden: Brill, 1973 Joannes Philoponus. In Aristotelis Analytica Posteriora Commentaria, ed. M. Wallies (CAG, vol. 13, pt. 2), Berlin: Reimer, 1909 . Vita Procli, ed. 1.F. Boissonade (1814), repro with the English trans• lation by K.S. Guthrie (1925), in Marinos of Neapolis. The Extant Ubrks, ed. A.N. Oikonomides, Chicago: Ares, 1977 . Sphaerica, ed. E. Halley, Oxford, 1758 of Gerasa. Harmonikon Encheiridion, ed. K. von Jan, Musici Scriptores Graeci, no. 5, Leipzig: Teubner, 1895 (repr. Hildesheim: G. Oims, 1962) -. Introductionis arithmeticae libri ii, ed. R. Hoche, Leipzig: Teubner, 1866 . Collectionis quae supersunt, ed. F. Hultsch, 3 vol., Berlin: Weid- mann, 1876-78 (repr. Amsterdam: A.M. Hakkert, 1965) -. see also A. Jones; P. Ver Eecke (1933) -. Commentaries on : see A. (1931) -. Commentary on Euclid's Book X: see W Thomson and G. Junge of Byzantium. Belopoeica: see E.w. Marsden Philoponus: see Joannes Philoponus . Opera, ed. 1. Burnet, (Oxford Classical Texts), 5 vol., Oxford: Clarendon Press, 1900-07 (repr. 1977) of Chaeronea. Scripta Moralia, ed. WR. Paton, I. Wegehaupt, et aI., 7 vol., Leipzig: Teubner, 1925-67 . In Platonis Timaeum Commentariorum Fragmenta, ed. A.R. Sodano, : [no publ.], 1964 Diadochus. In Primum Euclidis Elementorum Librum Commentarii, ed. G. Fried• lein, Leipzig: Teubner, 1873 (repr. 1967) -. see also G.R. Morrow -. In Platonis Timaeum Commentaria, ed. E. Diehl, 3 vol., Leipzig: Teubner, 1903-06 (repr. Amsterdam: A.M. Hakkert, 1965) Ptolemy of Alexandria. Opera quae exstant omnia, ed. 1.L. Heiberg, 2 vol. (I, 2 pts., Syn• taxis mathematica, 1898); II, Opera astronomica minora: Phaseis, Hypotheseis, Inscrip• tio Canobi, Procheiron kanonon diataxis, Analemma, , etc., 1907), Leipzig: Teubner -. = Syntaxis mathematica -. see also K. Manitius; GJ. Toomer (1984) Quintilian. Institutionis oratoriae libri xii, ed. L. Radermacher, corr. V. Buchheit, Leip• zig: Teubner, 1959 Serenus of Antinoeia. Opuscula (De sectione cylindri, De sectione coni), ed. J.L. Heiberg, Leipzig: Teubner, 1896 Simplicius. In Aristotelis de Caelo Commentaria, ed. J.L. Heiberg (CAG 7), Berlin: Reimer, 1894 -. In Aristotelis Physica Commentaria, ed. H. Diels, 2 vol. (CAG 9-10), Berlin: Reimer, 1882-95 of Cyrene. Opuscula, ed. N. Terzaghi, Rome: Regia officina polygraphica, 1944 -. Hymnes, ed. C. Lacombrade, Paris: Les Belles Lettres, 1978 -. see also A. Fitzgerald . In Metaphysica Commentaria, ed. W Kroll (CAG 6, pt. 1), Berlin: G. Reimer, 1902 Themistius. In Aristotelis Physica Paraphrasis, ed. H. Schenkl (CAG 5, pt. 2), Berlin: G. Reimer, 1900 820 Textual Studies in Ancient and Medieval Geometry

-. In Libros Aristotelis De Caelo Paraphrasis (Hebrew and Latin), ed. S. Landauer (CAG 5, pt. 4), Berlin: G. Reimer, 1902 . In Claudii Ptolemaei Magnam Constructionem Commentariorum Lib. XI, ed. 1. Camerarius, Basel: 1. Walder, 1538 -. see also A. Rome (1936; 1943) -. Commentaries on Ptolemy's Handy Tables: see 1. Mogenet (1985); A. Tihon (1978) . Expositio rerum mathematicarum ad legendum Platonem utilium, ed. E. Hiller, Leipzig: Teubner, 1878 (repr. New York and London: Garland, 1987) Vitruvius. De Architectura, ed. V. Rose, 2nd ed., Leipzig: Teubner, 1899 -. see also M.H. Morgan

Medieval Sources

Abu Bakr al-Harawl. [On the trisection of an ], B.N. ms. arab. 2457, no. 45alt (Paris), f. 194r-195r: see Part II, text A Abu Jacfar al-Khazin. [On the trisection of an angle], Bod!. ms. Hunt. 237 (Oxford), f. 103v-104v: see Part II, text A3 -. Ff stikhraj kha~~ain bain kha~~ain (On the construction of two lines between two lines), B.N. ms. arab. 2457, no. 47 (Paris), f. 198v-199r: see Part II, text F -. see also R. Lorch (1986) Abu 'I-Rashid cAbd al-Hadi. Qaul mansub ila Arshimfdis fl misa~at al-da'ira (Writing attributed to Archimedes on the measurement ofthe ) [Recension of Dimension 0/ the Circle, with preface, " applicable to the book of Archimedes"]' Columbia University ms. Smith 45, no. 4 (New York): see Part III, chap. 7, Appendix I Adelard of Bath. [Latin translation of Euclid's Elements]: see H.L.L. Busard (1983) Al).mad ibn Musa. Qaul fl tathlfth al-zawiya (Writing on the trisection ofthe angle), Bod!. ms. Thurston 3 (Oxford), f. 131 v-132r; Bod!. ms. Marsh 720, f. 260v: see Part II, texts B,B* -. see also 1. P. Hogendijk (1981) Anonymus. [Arabic translation of Archimedes' Dimension o/the Circle], ms. Fatih 3414, no. 1 (), f. 2v-6v (cf. Sezgin, GAS, V, p. 131): see Part III, Appendices to chaps. 3 and 4 -. [Arabic translation of Archimedes' Sphere and Cylinder], ms. Fatih 3414, no. 2 (Istan• bul), f. 7r-8r, 9r-59r (cf. Sezgin, GAS, V, p. 129) Anonymus. [Hebrew translation of Archimedes' Dimension o/the Circle], Vat. ms. Hebr. 384, f. 412r-412v: see Part III, Appendices to chaps. 3 and 4 Banu Musa, Kitab macrifa misa~a al-ashkiil al-basf~a wa- 'I-kurfya (Book of the Know• ledge of Measurement of Plane and Spherical Figures), in the recension of al-Tusi: see al-Tusi, Rasa'il, II, no. 1 -. Verbafiliorum: in M. Clagett, Archimedes in the , I, chap. 4 -. see also H. Suter (1902); and Al).mad ibn Musa al-Bironi, Abu Rail).an Mul). b. Al).mad. Rasa'il. Hyderabad: Osmania University, 1948 -. Istikhraj al-awtar fi 'l-da'ira (The Construction ofthe Chords in the Circle), in Rasa'il, no. 1; see also Suter (1910-11) Gerard of Cremona. Liber Arsamithis de mensura circuli, or De quadratura circuli trac• tatus [Latin translation of Archimedes' Dimension o/the Circle]: see Part III, Appen• dices to chaps. 3 and 4; and M. Clagett, Archimedes in the Middle Ages, I, chap. 2, sect. 2 Bibliography 821

-. [Latin translation of Euclid's Elements]: see H.L.L. Busard (1984) Grosseteste, R.: see R.C. Dales ibn al-Haytham: see 1. al-Dabbagh; and 1.P. Hogendijk (1985) Johannes de Tinemue (John of Tynemouth). Liber de curvis superficiebus: see M. Clagett, Archimedes in the Middle Ages, I, chap. 6, sect. 2 Jordanus de Nemore: see B.B. Hughes al-Jurjani, Abu Sacid al-parir. Masa'il handasfya (Geometric Questions), Dar al-Kutub ms. riya<;1. 41m (Cairo), f. 69r-71 v (cf. Sezgin, GAS, V, 263) al-Kashi: see P. Luckey Pachymeres, Georgios. Quadrivium (sc. Syntagma ton tessaron mathematon), ed. P. Tan• nery, (Studi e Testi 94), Vatican City: Biblioteca Apostolica Vaticana, 1940 Plato of Tivoli. In quadratum circuli [Latin translation of Archimedes' Dimension of the Circle]: see Part III, Appendices to chaps. 3 and 4; and M. Clagett, Archimedes in the Middle Ages, I, chap. 2, sect. 1 Qalonymus b. Qalonymus. [Hebrew translation of Archimedes' Sphere and Cylinder], Bodl. ms. Laud or. 93 (Oxford), f. Ir-28v; Bodl. ms. hebr. d. 4, f. 108r-141v al-Quhi, Abu Sahl. Maqala fi stikhraj al-kha~~ain bain al-kha~~ain . .. wa-qismat al• zawiya bi-thalathat aqsam (Writing on the construction of the two lines between the two lines ... and the division of the angle in three parts), Dar al-Kutub ms. riya9. 40m (Cairo), f. 226v-227r: see Part II, text E -. see also A. Sayili al-Sijz!, cAbd ai-Jam. [On the trisection of the angle], B.N. ms. arab. 2457, no. 33 (Paris), f. 16Or: see Part II, text D -. Risala fi qismat al-zawiya al-mustaqimat al-kha{{ain bi-thalathat aqsam mutasawiya: trans. as "Traite de la trisection de l'angle rectiligne;' by F. Woepcke, in lltlgebre d'Omar Alkhayyami, pp. 117-127 Suidae Lexicon. [10th century Greek historical and literary dictionary], ed. A. Adler, 5 vol., Leipzig: Teubner, 1928-38 Thabit ibn Qurra. Qismat al-zawiya ... bi-thalathat aqsam (Division of the angle ... into three parts), B.N. ms. arab. 2457, no. 45 (Paris), f. 192v-194r: see Part II, text C -. Kitab fi 'l-Qaras{un: see K. Jaouiche; and W.R. Knorr (1982) al-Tus!, Na~!r aI-Din. Rasa'il, 2 vol., Hyderabad: Osmania University, 1939-40 -. [Recension of the Arabic translation of Archimedes' Sphere and Cylinder], in Rasa'i!, II, no. 5 -. [Recension of the Arabic translation of Archimedes' Dimension of the Circle], in Rasa 'ii, II, no. 5, pp. 127-133: see Part III, chap. 7, Appendix II -. [Scholium to Conics V 52, on the construction of two mean proportional lines], Bodl. ms. Marsh 667 (Oxford), f. 106r: see Part II, text M cUmar al-Khayyam. Maqalafi 'l-jabr wa-1-muqabala: see F. Woepcke (1851) . [Latin translations of Archimedes from the Greek]: see M. Clagett, Archimedes in the Middle Ages, II

Modern Studies

AMA = Archimedes in the Middle Ages: see M. Clagett Arendt, F. "Eine Interpolation des Eutokios in unserem Apolloniostext," Bibliotheca Mathematica, 1913-14, 14 (3 ser.): 97-98 -. "Zu Archimedes;' Bibliotheca Mathematica, 1913-14, 14 (3rd ser.): 289-311 822 Textual Studies in Ancient and Medieval Geometry

Asmus, R. " in Tradition und Dichtung;' Studien zur vergleichenden literatur• geschichte, 1907, 7:11-44 Aujac, G. "Le rapport di isou (Euclide V, definition 17): Definition, utilisation, transmis• sion;' Historia Mathematica, 1986, 13:370-386 Balsam, H. Des Apollonios von Perga sieben Bucher aber Kegelschnitte, Berlin: G. Reimer, 1861 Becker, 0. "Eine voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid" (Eudoxos-Studien I), Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, 1933,2 (Abt. B): 311-333 -. "Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik" (Eudoxos• Studien IV), Quellen und Studien, 1936,3 (Abt. B): 370-388 Berggren, 1.L. "Spurious Theorems in Archimedes' Equilibria ofPlanes, Book I;' Archive for History of Exact , 1976-77, 16: 87-103 -. ')\ Lacuna in Book I of Archimedes' Sphere and Cylinder;' Historia Mathematica, 1977,4: 1-5 -. ')\ Coincidence of Pappos' Book VIII with al-Binini's Taltdid;' Journal for the History of Arabic , 1978,2: 137-142 -. "The Correspondence of Abu Sahl al-Kuhi and Abu ISQaq al-~abi: A Translation with Commentaries;' Journalfor the History of Arabic Science, 1983, 7: 39-124 -. Episodes in the ofMedieval Islam, New YorklBerlin/Heidelberg/London/ Paris/Tokyo: Springer, 1986 Boll, H. "Heliodorus": see PW Brandis, C.: see Aristotle Bregman, 1. Synesius of Cyrene, -Bishop, Berkeley/Los Angeles/London: University of California Press, 1982 Bruins, E.M. Codex Constantinopolitanus (palatii veteris no. 1), I: Reproduction of the Manuscript, Leiden: Brill, 1964 (includes Greek text of Herds Metrica, Geomet• rica, etc.) Bulmer-Thomas, I. "Eutocius of Ascalon;' DSB, 4 (1971): 488-491 -. "Isidorus of ;' DSB, 7 (1973): 28-30 Burkert, W. Lore and Science in Ancient , Cambridge: Harvard University Press, 1972 (trans. from the German, ~isheit und Wissenschaft, Nuremberg: H. Carl, 1962) Busard, H.L.L. "Ein mitte1alterlicher Euklid-Kommentar, der zugeschrie• ben werden kann;' Archives internationales d'histoire des sciences, 1974, 24: 199-218 -. The Translation of the Elements of Euclid from the Arabic into Latin by Hermann of Carinthia (?), Books I-VI, Leiden: Brill, 1968; Books VII-XlI, Amsterdam: Mathematisch Centrum, 1977 -. "Der Traktat De isoperimetris, der unmittelbar aus dem Griechischen ins Lateinische iibersetzt worden ist;' Mediaeval Studies, 1980,42: 61-88 _. The First Latin Translation ofEuclid's Elements Commonly Ascribed to Adelard ofBath, Toronto: Pontifical Institute of Mediaeval Studies (Studies and Texts, 64), 1983 -. The Latin Translation ofthe Arabic Version ofEuclid's Elements Commonly Ascribed to Gerard of Cremona, Leiden: Brill, 1984 -. The Mediaeval Latin Translation of Euclid's Elements, Made Directly from the Greek, (Boethius, vol. 15), Stuttgart: F. Steiner-Verlag-Wiesbaden, 1987 Cantor, M. [Review of Nokk, Zenodorus' Abhandlung aber die isoperimetrischen Figu• ren], Zeitschriftfiir Mathematik und Physik (Literaturzeitung), 1861,6: 1-3 -. [Review of Hultsch, Pappi Collectio, II] , Zeitschrift for Mathematik und Physik (hist.• lit. Abth.), 1877,22: 173-179 Bibliography 823

Carruccio, E. Mathematics and Logic in History and in Contemporary Thought, London: Faber and Faber, 1964 Clagett, M. "The Medieval Latin Translations from the Arabic of the Elements of Euclid;' Isis, 1953, 44: 16-42 -. "King Alfred and the Elements of Euclid;' Isis, 1954,45: 269-277 -. "The De curvis superjiciebus Archimenidis," Osiris, 1954, 11: 295-358 -. The Science ofMechanics in the Middle Ages, Madison: University of Wisconsin Press, 1959 -. Archimedes in the Middle Ages: I, The Arabo-Latin Tradition (Madison: University of Wisconsin Press, 1964); II, The Translations from the Greek by William of Moerbeke (: American Philosophical Society, 1976); III, The Fate of the Medieval Archimedes 1300-1565 (Philadelphia, 1978); IV, A Supplement on the Medieval Latin Traditions of Conic Sections (Philadelphia, 1980); V, Quasi-Archimedean Geometry in the Thirteenth Century (Philadelphia, 1984) -. see also E.A. Moody and M. Clagett Coolidge, 1.L.L. A History ofGeometrical Methods, Oxford: Clarendon Press, 1940 (repr. New York: Dover, 1963) Courant, R. and H. Robbins. What is Mathematics? An Elementary Approach to Ideas and Methods, 4th ed. (repr.), Oxford/New York, etc.: Oxford University Press, 1978 Czwalina, A. Archimedes ~rke, 5 vol., Leipzig: Teubner, 1922-25 al-Dabbagh,1. ["Ibn al-Haytham. Tract on isoperimetric figures" (Russian translation and commentary)], Istoriko-matematicheskie Issledovaniya, 1966, 17: 399-448 Dales, R.C. RDberti Grosseteste ... Commentarius in VIII Libros Physicorum Aristotelis, Boulder: University of Colorado Press, 1963 Derenbourg, H. and H.PJ. Renaud, Les manuscrits arabes de l'Escurial, vol. 2, fasc. 3 ("Sciences exactes et sciences occultes"), Paris: Librairie Orientaliste P. Geuthner, 1941 Diels, H. and W. Kranz. Die Fragmente der Vorsokratiker, 6th ed., 3 vol., Dublin/Zurich: Weidmann, 1966 Dijksterhuis, EJ. Archimedes, Copenhagen: E. Munksgaard, 1956; repro with a bib• liographical supplement, '~rchimedes after Dijksterhuis;' by W.R. Knorr, Princeton: University Press, 1987 Dilgan, H. "Sur un theoreme isoperimetrique d'Ibn-i-Haitham;' Actes du IJC-! Congres International d'Histoire des Sciences (Barcelona and Madrid, 1-7 Sept. 1959), Barce• lona: Universidad de Barcelona and Paris: Hermann & Cie., I, pp. 453-460 Dold-Samplonius, Y. "al-Khazin, Abu Jacfar Mu~ammad ibn al-l:Iasan al-Khurasani;' DSB, 7 (1973): 334-335 Drachmann, A. "Philo of Byzantium;' DSB, 10 (1974): 586-589 -. " (Ktesibios);' DSB, 3 (1971): 491-492 DSB = Dictionary ofScientific Biography, ed. C.c. Gillispie, 16 vol., New York: C. Scrib• ner's Sons, 1970-80 Dumontier, 1.: see c.E. Ruelle Emden, A.B. A Biographical Register of the University of Oxford to A.D. 1500, 3 vol., Oxford: Clarendon Press, 1957-59 Etienne, E. and 1. Roels. "Deux aspects particuliers du probleme des moyennes dans Pappus d'Alexandrie," Revue des questions scientifiques, 1986, 157: 179-198 Fitzgerald, A. The Letters of Synesius of Cyrene (trans., with intro. and notes), Oxford: University Press and London: H. Milford, 1926 -. The Essays and Hymns of Synesius of Cyrene (trans., with intro. and notes), 2 vol., Oxford: University Press and London: H. Milford, 1930 824 Textual Studies in Ancient and Medieval Geometry

Foley, V.: see W. Soedel and V. Foley Folkerts, M. "Adelard's Versions of Euclid's Elements;' in C. Burnett, ed., Adelard ofBath: An English Scientist and Arabist ofthe Early Twelfth Century, London: Warburg Institute (Surveys and Texts 14), 1987, pp. 55-68 Fowler, D.H. "Ratio in Early ;' Bulletin ofthe American Mathematical Society, 1979, 1 (n.s.): 807-846 -. The Mathematics ofPlato's Academy: A New Reconstruction, Oxford: Clarendon Press, 1987 Gericke, H. "Zur Geschichte des isoperimetrischen Problems;' Mathematische Semester• berichte, 1982, 29: 160-187 Gibbs, S.L. Greek and Roman Sundials, London and New Haven: Yale University Press, 1976 Gow, A.S.F. and D.L. Page. The : Hellenistic Epigrams, Cambridge: University Press, 1965 Grant, E. A Source Book in Medieval Science, Cambridge: Harvard University Press, 1974 Gudemann. "Sporos": see PW Guthrie, W.K.C. A History ofGreek , 6 vol., Cambridge: Cambridge University Press, 1962-81 Hartner, W. "The Principle and Use of the ;' in A Survey ofPersian Art, ed. A.V. Pope, London and New York: Oxford University Press, 1939, pp. 2530-2554 (repr. in W. Hartner, Oriens-Occidens, Hildesheim: G. Oims, 1968, pp. 287-311) Haskins, C.H. Studies in the History of Mediaeval Science, Cambridge: Harvard Univer• sity Press, 1924; 2nd ed., 1927 (repr. New York: F. Ungar, 1960) -. The Renaissance of the Twelfth Century, Cambridge: Harvard University Press, 1927 (repr. Cleveland and New York: World Publishing Co., 1957) Heath, T.L. : on Conic Sections, Cambridge: Cambridge University Press, 1896 -. The Works of Archimedes, Cambridge: Cambridge University Press, 1897; Supple• ment: The Method of Archimedes, 1912 (repr. New York: Dover, n.d.) -. A History of Greek Mathematics, 2 vol., Oxford: Clarendon Press, 1921 (repr. New York: Dover, 1981) -. The Thirteen Books ofEuclid's Elements, 2nd ed., 3 vol., Cambridge: University Press, 1926 (repr. New York: Dover, 1956) -. Mathematics in Aristotle, ed. A.M. Heath, Oxford: Clarendon Press, 1949 Heiberg, 1.L. Quaestiones Archimedeae, Copenhagen: R. Klein, 1879 -. Philologische Studien zu griechischen Mathematikem, (from Supplementbiinde der lahrbiicher fUr classische Philologie): I, "Ueber Eutokios" (1880, 11: 357-384); II, "Ueber die Restitution der zwei Bucher des Archimedes Peri Sphairas kai Kylindrou" (1880, 11: 384-398); III, "Die Handschriften Georg Vallas von griechischen Mathematikern" (1881, 12: 377-402); IV, "Ueber den Dialekt des Archimedes" (1883, 13: 543-566); V, "Interpolationen in den Schriften des Archimedes" (1883, 13: 566-577) -. "Die arabische Tradition der Elemente Euklid's;' ZeitschriftfUr Mathematik und Physik (hist.-lit. Abth.), 1884,29: 1-22 -. "Eine mittelalterliche Ubersetzung der Syntaxis des Ptolemaios;' Hermes, 1910,45: 57-66 -. "Noch einmal die mittelalterliche Ptolemaios-Ubersetzung," Hermes, 1911, 46: 207-216 Hintikka,1. and V. Remes. The Method ofAnalysis: Its Geometrical Origin and Its General Significance, Dordrecht: Reidel, 1974 Bibliography 825

Hoeg, C. "Johann Ludvig Heiberg" (obituary), lahresbericht aber die Fortschritte der klassischenAltertumswissenschaJt, 1931,57 (233), pt. 4 (Nekrologe), pp. 38-77 Hogendijk, J.P. "On the trisection of an angle and the construction of a regular by means of conic sections in medieval Islamic geometry;' preprint nr. 113, University of Utrecht: Department of Mathematics, March 1979 -. "How Trisections of the Angle Were Transmitted from Greek to Islamic Geometry;' Historia Mathematica, 1981,8: 417-438 -. Ibn al-Haytham's Completion of the Conics, New YorklBerlin/Heidelberg/Tokyo: Springer, 1985 -. ':.\rabic Traces of Lost Works of Apollonius;' Archive for History of Exact Sciences, 1986, 35: 187-253 Hughes, B.B. The De Numeris Datis oflordanus de Nemore: A Critical Edition, Analysis, Evaluation and Translation, Ph.D. diss., Stanford University, 1970 Hultsch, F. "Domninos" and "Eutocius": see PW Huxley, G.L. Anthemius of Tralles: A Study in Later Greek Geometry (Greek, Roman and Byzantine Monographs, no. 1), Cambridge, Mass., 1959 Ito, S. The Medieval Latin Translation of the Data of Euclid, Tokyo: University of Tokyo Press and Boston/Basel/Stuttgart: Birkhauser, 1980 Jackson, D.E.P. "The Arabic Translation of a Greek Manual of Mechanics;' Islamic Quart• erly, 1972, 16: 96-lO3 -. "Towards a Resolution of the Problem of ta heni diastemati graphomena in Pappus' Col• lection Book VIII," Classical Quarterly, 1980, 30 (n.s.): 523-533 Jaouiche, K. Le livre du Qaraszun de Thabit ibn Qurra, Leiden: Brill, 1976 Jones, A. Pappus of Alexandria: Book 7 of the Collection, 2 vol., New York/Ber• lin/Heidelberg/Tokyo: Springer, 1986 King, D.A. "Notes on the Sources for the History of Early Islamic Mathematics" [essay review of F. Sezgin, Geschichte des arabischen Schrifttums, vol. 5], lournal of the American Oriental Society, 1979, 99: 450-459 -. A Catalogue of the Scientific Manuscripts in the Egyptian National Library, Cairo: General Egyptian Book Organization (in collaboration with the American Research Center in Egypt and the Smithsonian Institution), 1981 Kliem. "Sporos": see PW Knaack, G. "": see PW Knorr, W.R. The Evolution ofthe Euclidean Elements: A Study of the Theory ofIncommen• surable Magnitudes and Its Significance for Early Greek Geometry, Dordrecht: D. Reidel, 1975 -. ':.\rchimedes and the Measurement of the Circle: A New Interpretation;' Archive for History of Exact Sciences, 1976, 15: 115-140 -. ':.\rchimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus;' Archive for History of Exact Sciences, 1978, 19: 211-290 -. ':.\rchimedes and the pre-Euclidean Proportion Theory;' Archives internationales dhistoire des sciences, 1978,28: 183-244 -. Ancient Sources of the Medieval Tradition of Mechanics: Greek. Arabic and Latin Studies of the Balance, Florence: Istituto e Museo di Storia della Scienza (monograph no. 6 of the Annali dell7stituto), 1982 -. "The -Construction in the Conics, Book II: Ancient Variations on a The• orem of Apollonius," Centaurus, 1982, 25: 253-291 -. "Infinity and Continuity: The Interaction of Mathematics and Philosophy in Anti• quity;' in N. Kretzmann (ed.), Infinity and Continuity in Ancient and Medieval Thought, pp. 112-145, Ithaca and London: Cornell University Press, 1982 826 Textual Studies in Ancient and Medieval Geometry

-. "Techniques of Fractions in Ancient Egypt and ;' Historia Mathematica, 1982, 9: 133-171 -. ''The Geometry of Burning-Mirrors in Antiquity;' Isis, 1983,74: 53-73 -. "On the Transmission of Geometry from Greek into Arabic;' Historia Mathematica, 1983,10: 71-78 -. ')\ncient Versions of Two Trigonometric Lemmas;' Classical Quarterly, 1985, 35: 362-391 -. ')\rchimedes and the pseudo-Euclidean : Early Stages in the Ancient Geo• metric Theory of Mirrors;' Archives internationales d'histoire des sciences, 1985, 35: 28-105 -. "Euclid's Tenth Book: An Analytic Survey;' Historia Scientiarum (Tokyo), 1985,29: 17-35 -. [rev. of l Sesiano, Books Wto VII ofDiophantus' Arithmetica . .. ], American Mathe• matical Monthly, 1985, 92:150-154 -. The Ancient Tradition of Geometric Problems, Boston/Basel/Stuttgart: Birkhiiuser, 1986 -. ')\rchimedes' Dimension of the Circle: A View of the Genesis of the Extant Text;' Archive for History of Exact Sciences, 1986,35: 281-324 -. "The Medieval Tradition of a Greek Mathematical Lemma;' Zeitschrift fUr Geschichte der arabisch-islamischen Wissenschaften, 1986, 3: 230-261 -. "On Two Archimedean Rules for the Circle and the Sphere;' Bollettino di Storia delle Scienze Matematiche, 1986,6: 145-158 -. "The Medieval Tradition of Archimedes' Sphere and Cylinder;' in E. Grant and lE. Murdoch (eds.), Mathematics and its Applications to Science and Natural Philosophy in the Middle Ages: Essays in Honor of Marshall Clagett, Cambridge: University Press, 1987,pp.3-42 -. ')\rchimedes after Dijksterhuis: A Guide to Recent Studies": see EJ. Dijksterhuis (1987) -. "Paraphrase Editions of Latin Mathematical Texts: De Figuris Ysoperimetris;' Medi• aeval Studies (to appear 1990) -. ''John of Tynemouth alias John of London: Emerging Portrait of a Singular Medieval " (available in preprint, 1989) Kozhukhova, G.M. ["The Arabic Version of Archimedes' Dimension ofthe Circle' (in Rus• sian)], Istoriko-matematicheskie Issledovaniya, 1980,25: 315-316 Kramer, E. "Hypatia of Alexandria;' DSB, 6 (1972): 615-616 Krause, M. "Stambuler Handschriften islamischer Mathematiker;' Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, 1936,3 (Abt. B): 437-532 Kunitzsch, P. [Ibn al-$ala~]. Zur Kritik der Koordinatenilberlieferung im Sternkatalog des Almagest, Gottingen: Akademie der Wissenschaften (Abhandlungen, phil.-hist. Klasse, 3rd ser., 94), 1975 Lacombrade, C.: see Synesius Lang, M. "Herodotos and the Abacus," Hesperia, 1957,26: 271-287 Lemay, R. "Gerard ofCremona;' DSB, 15 (1978): 173-192 Lloyd, A.C. "The Later Neoplatonists;' Part IV of The Cambridge History ofLater Greek and Early Medieval Philosophy, ed. A.H. Armstrong, Cambridge: University Press, 1967 Lloyd, G.E.R. "The Alleged Fallacy of of Chios;' Apeiron, 1987, 20: 103-128. Lorch, R. "Some Remarks on the Arabic-Latin Euclid;' in C. Burnett, ed., Adelard of Bibliography 827

Bath: An English Scientist and Arabist of the Early Twelfth Century, London: Warburg Institute (Surveys and Texts 14), 1987, pp. 45-54 -. 'l\biJ Jacfar al-Khazin on Isoperimetry and the Archimedean Tradition," Zeitschriftfiir Geschichte der arabisch-islamischen Wissenschaften, 1986,3: 150-229 Lorent, H. "Sur les traces des pas d1\rchimede:' Mathesis, 64, suppl. I, 1955 LSJ = H.G. Liddell, R. Scott and H.S. Jones, A Greek-English Lexicon, Oxford: Claren• don Press, 1983 (repr. of 9th ed., 1940, with Supplement of 1968) Luckey, P. [al-Kash!]. Der Lehrbrief aber den Kreisumfang, Berlin: Akademie der Wissenschaften (Abhandlungen, Klasse f. Math. u. algem. Naturwissen., 1950, nr. 6), 1953 Manitus, K. Des Claudius Ptolemiius Handbuch der Astronomie, 2 vol., Leipzig: Teubner, 1912-13 Marsden, E.W. Greek and Roman Artillery: Technical , Oxford: Clarendon Press, 1971 (includes Greek text with translation of Hero's Belopoeica, Philo's Belopoeica, etc.) Mercier, R. "Studies in the medieval conception of precession:' Archives internationales dhistoire des sciences, part I, 1976,26: 197-220; part II. 1977,27: 33-71 Mogenet, 1. 17ntroduction a lltlmageste, Brussels: Academie Royale de Belgique (Memoires, 51, fasc. 2), 1956 -. 'THistoire des isoperimetres chez les Grecs," in Scrinium Lovaniense: Melanges histo• riques Etienne van Cauwenbergh, Universite de Louvain: Recueil de travaux dhistoire et de philologie, 4 ser., fase. 24, 1961. pp. 69-78 -. Le "Grand Commentaire" de Theon dltlexandrie aux Tables Faciles de Ptolemee, Livre I (critical edition, revised and completed with a commentary by A. Tihon), Studi e Testi 315, Vatican City: Biblioteca Apostolica Vatieana, 1985 Moody, E.A. and M. Clagett. The Medieval Science of Weights (Scientia de ponderibus) , Madison: University of Wisconsin Press, 1960 Morgan, M.H. Vitruvius: The Ten Books on Architecture, Cambridge: Harvard University Press, 1914 (repr. New York: Dover. 1960) Morrow, G.R. Proclus: A Commelltarv Oil the First Book of Euclid:5 Elements, Princeton: University Press, 1970. Mueller, I. Philosophy of Mathematics and Deductive Structure in Euclid:5 Elements, Cam• bridge: MIT Press, 1981 Muller, W. "Das isoperimetrische Problem im Altertum ," Sudh(iffs Archiv, 1953, 37: 39-71 Mugler, C. Les Oeuvres dltrchimMe, 4 vol., Paris: Les Belles Lettres, 1970-74 Murdoch, lE. "Euclides Graeco-Latinus. A Hitherto Unknown Medieval Latin Transla• tion of the Elements of Euclid Made Directly from the Greek:' Harvard Studies in Clas• sical Philology, 1966, 71: 249-302 -. "The Medieval Euclid: Salient Aspects of the Translations of the Elements by Adelard of Bath:' Revue de Synthese, 1968,89: 67-94 -. "Euclid: Transmission of the Elements," DSB, 4 (1971): 437-459 Neuenschwander, E. "Die ersten vier Bucher der Elemente Euklids." Archive for History of Exact Sciences, 1973,9: 325-380 -. "Zur Ueberlieferung der -Uisung des delischen Problems," Celltaurus, 1974, 18: 1-5 Neugebauer, 0. "The Early History of the Astrolabe," Isis, 1949,40: 240-256 (repr. in and History: Selected EssQ\'s. New York/Berlin/Heidelberg/Tokyo: Springer, 1983, pp. 278-294) 828 Textual Studies in Ancient and Medieval Geometry

-. A History of Ancient Mathematical Astronomy, 3 vol., Berlin/Heidelberg/New York: Springer, 1975 Nokk, [no init.]. Zenodorus'Abhandlung iiber die isoperimetrischen Figuren, nach den Ausziigen, welche uns die Alexandriner Theon und Pappus aus derselben iiberliefert haben, Beilag z. d. Freiburger Lyceums-Programme, 1860 Parker, R.A. Demotic Mathematical Papyri, Providence: Brown University Press, 1972 Pedersen, 0. A Survey of the Almagest, Odense University Press, 1974 Pendlebury, R. "On a Method of Finding Two Mean Proportionals," Messenger of Mathematics, 1873 (2, ser. 2): 166-169 Pfeiffer, R. History of Classical Scholarship: From the Beginnings to the End of the Hellenistic Age, Oxford: Clarendon Press, 1968 Powell, 1U. Collectanea Alexandrina, Oxford: Clarendon Press, 1925 (repr. 1970) Praechter, K. "Hypatia" and "Syrianos": see PW PW = Pauly-Wissowa: Paulys Real-Encyclopiidie der classischen Altertumswissenschaft, 2nd ed., ed. G. Wissowa et aI., 24 vol. in 43, Stuttgart: 1B. Metzler, 1894-1963; 2nd ser. 10 vol. in 19, 1914-72; for the following articles: "Domninos;' Hultsch, 5 (1905): 1521-25 "Eratosthenes von Kyrene;' Knaack, 6 (1909), 358-389 "Eutocius;' Hultsch, 6 (1909): 1518 "Heliodoros;' Boll, 8, pt. 1 (1912): 18-19 "Hypatia;' Praechter, 9, pt. 1 (1914): 242-249 "Parmenion;' H. Riemann, 18, pt. 4 (1949): 1567-69 "Sporos, griechischer Grammatiker;' Gudemann, 2nd ser., 3, pt. 2 (1929): 1879-81; "S. von Nikaia, Mathematiker;' Kiiem, ibid., 1881-83 "Syrianos;' K. Praechter, 2nd ser., 4, pt. 2 (1932): 1728-75 "Theon von Alexandria;' K. Ziegler, 2nd ser., 5, pt. 2 (1934): 2075-80 Rashed, R. Diophante. Les Arithmetiques, vol. 3-4 (of projected 4 voL), Paris: Les Belles Lettres, 1984 -. Diocles, Anthemius de Tralles, Didyme . .. : Sur les miroirs ardents (edition of the Arabic texts with French translation), Paris: Les Belles Lettres, in preparation. Rawlins, D. "Eratosthenes' Geodesy Unraveled: Was There a High-Accuracy Hellenistic Astronomy?" Isis, 1982, 73: 259-265 Remes, u.: see 1 Hintikka and U. Remes Reynolds, L.D. and N.G. Wilson, Scholars and Scribes, 2nd ed., Oxford: Clarendon Press, 1974 Riemann, H. "Parmenion": see PW Rist, 1M. "Hypatia;' Phoenix, 1965, 19: 214-225 Roels, 1: see E. Etienne and 1 Roels Rome, A. Commentaires de Pappus et de Theon d:4lexandrie sur 1:4lmageste: I, Pappus, Commentaire sur les livres 5 et 6, Rome: Biblioteca Apostolica Vaticana, 1931; II, Theon, Commentaire sur les livres 1 et 2, Vatican City, 1936; III, Theon, Commentaire sur les livres 3 et 4, Vatican City, 1943 Ruelle, C.E. "Texte inedit de Domninus de Larisse sur l'arithmetique;' (with explanatory note by 1 Dumontier), Revue de Philologie, 1883, 7 (n.s.): 82-94 Saito, K. "Compounded Ratio in Euclid and Apollonius;' Historia Scientiarum (Tokyo), 1986, 31: 25-59 Saliba, G. "The Meaning of al-jabr wa'l-muqabalah;' Centaurus, 1972, 17: 189-204 Sambursky, S. "," DSB, 7 (1973): l34-l39 Sams6, 1 "aI-Khazin, Abu DjaCfar ... al-Khurasani;' Encyclopedia of Islam, 2nd ed., Leiden: Brill, 1978, IV, pp. 1182-83 Bibliography 829

Sarfatti, G.B. Mathematical Terminology in Hebrew Scientific Literature of the Middle Ages (Hebrew with English summary), Jerusalem: Magnes Press (The Hebrew Univer• sity), 1968 Sato, T. ':.\rchimedes' On the Proposition 1: An Attempt at Reconstruction," Japanese Studies in the (= Historia Scientiarum, Tokyo), 1979, 18: 83-99 -. "Quadrature of the Surface of the Sphere in the -Johannes de Tinemue and Bam! Musa;' Historia Scientiarum, 1985,28: 61-90 Sayili, A. "The Trisection ofthe Angle by Abu Sahl Way jan ibn Rustam AI Kuhl" (Arabic text with Turkish and English translations), Belleten (Ankara: Societe d'histoire turque), 1962,26: 693-700 Schmidt, W. "Zur Geschichte der Isoperimetrie im AItertum;' Bibliotheca Mathematica, 1901,2 (3rd ser.): 5-8 Schneider, I. Archimedes: Ingenieur. Naturwissenschaftler und Mathematiker, Darmstadt: Wissenschaftliche Buchgesellschaft, 1979 Schoy, C. "Graeco-Arabische Studien," Isis, 1926,8: 21-40 Sesiano,1. Books IV to VII ofDiophantus' Arithmetica in the Arabic Translation Attributed to Qus~a ibn LCtqa, New York/Heidelberg/Berlin: Springer, 1982 Sezgin, F. Geschichte des arabischen Schrifttums, V (Mathematik bis ca. 430 H.), Leiden: Brill, 1974 de Slane, Ie Baron. Catalogue des manuscrits arabes (Bibliotheque Nationale, Departe• ment des Manuscrits), Paris: Imprimerie Nationale, 1883-95 Soedel, W. and V. Foley, "Ancient Catapults;' Scientific American, March 1979, 240: 150-160 Sorabji, R. (ed.). Philoponus and the Rejection ofAristotelian Science, London and Ithaca, N.Y.: Cornell University Press, 1987 Steiner,1. "Einfache Beweise der isoperimetrischen Hauptsatze," Crelle's Journal, 1836, 18: 281-296 (repr. in Gesammelte ~rke, ed. K. Weierstrass, Berlin: G. Reimer, 1882, II, pp. 75-91) Steinschneider, M. Die hebraeischen Obersetzungen des Mittelalters und die Juden als Dolmetscher, Berlin: Bibliographisches Bureau, 1893 (repr. Graz: Akademische Druck• und Verlagsanstalt, 1956) -. Die Mathematik bei den Juden, Hildesheim: G. Olms, 1964 (repr. of editions Ber• lin/Leipzig, 1893-99 and Frankfurt, 1901) Sturm, A. Das delische Problem [in three parts], XXIX-XXXI Program des k. k. Ober• Gymnasiums der Benedictiner zu Seitenstetten, Linz, 1895-97 Suter, H. Die Mathematiker und Astronomen der Araber und ihre ~rke, Leipzig: Teubner, 1900 -. "Uber die Geometrie der S6hne des Musa ben Schakir;' Bibliotheca Mathematica, 1902, 3 (ser. 3): 259-272 -. "Das Buch der Auffindung der Sehnen im Kreise von AbU'I-RaiJ.!an MuJ.!. el-Blruni," Bibliotheca Mathematica, 1910-11, II (3rd ser.): 11-78 Tannery, P. La Geometrie grecque. comment son histoire nous est parvenue et ce que nous en savons, Paris: Gauthier-Villars, 1887 -. Memoires scientifiques: "Sciences exactes dans l'antiquite;' 3 vol., ed. 1.L. Heiberg and H.G. Zeuthen, Toulouse: Privat and Paris: Gauthier-Villars, 1912-15, for the fol• lowing papers: "Larticle de Suidas sur Hypatia" (1880), I, pp. 74-79 "I.:arithmetique des Grecs dans Pappus" (1880), I, pp. 80-105 "I.:article de Suidas sur Ie philosophe Isidore" (1881), I, pp. 111-117 830 Textual Studies in Ancient and Medieval Geometry

"Sur les fragments d'Eudeme de relatifs a l'histoire des mathematiques" (1882), I,pp.168-177 "Sur Sporos de Nicee" (1882), I, pp. 178-184 "Sur la mesure du cercie df\rchimede" (1882), I, pp. 226-253 "Serenus df\ntissa" (1883), I, pp. 290-299 "Domninos de Larissa" (1884), II, pp. 105-117 "Eutocius et ses contemporains" (1884), II, pp. 118-136 "Notes critiques sur Domninos" (1885), II, pp. 211-222 "Un fragment des Metriques de Heron" (1894), II, pp. 447-451 "Sur un fragment inedit des Metriques de Heron df\1exandrie" (1894), II, pp. 452-454 "Sur la religion des derniers mathematiciens de l'antiquite" (1896), II, pp. 527-539 "Le manuel d'introduction arithmetique du philosophe Domninos de Larissa" (1906), III, pp. 255-281 Taran, L. "Nicomachus of Gerasa;' DSB, 10 (1974): 111-114 Thaer, C. "Die Wiirfelverdoppelung des Apollonios;' Deutsche Mathematik, 1940, 5: 241-243 Theisen, W. "Liber de visu: The Greco-Latin Translation of Euciid's ;' Mediaeval Studies, 1979, 41: 44-105 Thomas, I. Greek Mathematical Works, 2 vol. (Loeb Classical Library), Cambridge, Mass.: Harvard University Press and London: W. Heinemann, 1939-41 (repr. 1957) -. see also I. Bulmer-Thomas Thomson, W. and G. Junge, The Commentary of Pappus on Book X of Euclid's Elements, ed. and trans. from the Arabic, Cambridge: Harvard University Press, 1930 Tihon, A. Le "Petit Commentaire" de Theon d:4.lexandrie aux Tables Faciles de Ptolemee, (Studi e Testi 282), Vatican City: Biblioteca Apostolica Vaticana, 1978 -. "B1oge: Joseph Mogenet;' Isis, 1981,72: 265-266 -. see also 1 Mogenet (1985) Toomer, G.l "The Mathematician Zenodorus;' Greek, Roman and Byzantine Studies, 1972,13: 177-192 -. "Theon of Alexandria," DSB, 13 (1976): 321-325 -. Ptolemy's Almagest, New YorkiBeriin/HeidelbergiTokyo: Springer, 1984 Unguru, S. "Pappus in the Thirteenth Century in the Latin West;' Archive for History of Exact Sciences, 1974, 13: 307-324 Uri, 1 Eibliothecae Bodleianae codicum manuscriptorum orientalium ... catalogus, Oxford: Clarendon Press, 1787 Ver Eecke, P. Les Oeuvres completes d:4.rchimede, Paris/Brussels: Desciee, 1921 (2nd ed. , 2 vol., Liege: Vaillant-Carmanne, 1960) -. Les Coniques d:4.pollonius de Perge, Bruges: Desciee, De Brouwer, 1924 -. Pappus d:4.lexandrie: La Collection mathematique, 2 vol., Paris/Bruges: Desciee, De Brouwer, 1933 Victor, S.K. Practical Geometry in the High Middle Ages: Artis cuiuslibet consummatio and the Pratike de geometrie, Philadelphia: American Philosophical Society (Memoirs 134), 1979 van der Waerden, B.L. Science Awakening, Groningen: P. Noordhoff, 1954 (repr. New York: 1 Wiley & Sons, 1963) Waithe, M.E. "Hypatia of Alexandria;' in A History of Women , I, ed. M.E. Waithe, pp. 169-195, Dordrecht: M. Nijhoff, 1987 Weaver, lH. "The Duplication Problem," American Mathematical Monthly, 1916, 23: 106-113 Bibliography 831

-. "Pappus. Introductory Paper;' Bulletin of the American Mathematical Society, 1916-17, 23: 127-135 Wehr, H. A Dictionary of Modern Written Arabic, 3rd ed., ed. J.M. Cowan, Ithaca, NY.: Spoken Language Services, Inc., 1976 (from the German edition, Wiesbaden: O. Har• rassowitz, 1971) von Wilamowitz-Moellendorff, U. "Ein Weihgeschenk des Eratosthenes," Nachrichten der k. Gesellschaft der Wissenschaften zu Gottingen, phil.-hist. Klasse, 1894, pp. 15-35 (repr. in Kleine Schriften, Berlin: Akademie Verlag and Amsterdam: A.M. Hakkert, 1971, II, pp. 48-70) Wilson, N .G. Scholars of Byzantium, London: Duckworth and Baltimore: Johns Hopkins University Press, 1983 -. see also L.D. Reynolds and N.G. Wilson Woepcke, F. l:4lgebre d'Omar Alkhayyami, Paris: B. Duprat, 1851 -. "Sur une me sure de la circonference du cercle, due aux astronomes arabes, et fondee sur un calcul d'Aboul Waffi.;' Journal Asiatique, 1860, 15: 281-320 Wright, J.K. "Notes on the knowledge oflatitudes and longitudes in the middle ages;' Isis, 1923,5: 75-98 Youschkevitch [Iushkevich], A.P. Les Mathematiques arabes (VIw-xve Siecles), [French translation of Part III of the Russian edition, with the author's new notes and bib• liographical citations], Paris: 1. Vrin, 1976 Ziegler, K. "Eutokios von Askalon;' in Der Kleine Pauly, Lexikon der Antike, Munich: A. Druckenmiiller, 2 (1975): 468 -. "Theon": see PW Index

References are to the pages where the specified items are discussed. Entries with suffix "n" or "nn" followed by one or more numerals refer to the note or notes, respectively, so numbered, appearing on the indicated pages. In cases where a whole chapter is devoted to the discussion of an item, the part and (or let• ter) of the chapter are given; for instance "III9" refers to Part III, chapter 9. The symbol "-+" is used for cross-referencing. Since for most subject headings only those discussions are cited in which the actual terms ofthe heading appear, it will in general be essential to consult the cross-referenced headings as well. Where a subheading also appears elsewhere as a heading, cross-referencing is often omit• ted; e.g., in the entry "Pappus, on cube duplication;' one should understand "-+ cube duplication" as an implicit cross-reference. The elements '~bu", "al-" and "ibn" are not counted in the alphabetization of Arabic names, nor is the word "De" in that of Latin titles. iifortiori (pol/oi) 428, 514-519, 619, Albert of Saxony 618, 649, 676, 632-633, 643; medieval terminology 678(nn22,24), 810; ---> Quadratura 445, 447-448 circuli, Questio of Albert of Saxony abacus 759, 796(nn26-27) Alexandria 5-6, 27n22, 41, 63, 143, 145, Abraham bar l:Iiyya 449 167, 176(nn81,84), 177n87, 227, Abraham ibn Ezra 449 229,754,770,813 Adelard of Bath 597-598 Ammonius 6, 27n22, 155, 167, 169n5, Adelardian versions of Elements 612n25; 177n87, 229-230, 243n39, 527, 754, Adelard I 798n66; Adelard II 641, 796n35 658, 678(nn22,24), 680n51, 681n67, 754 683n93; Adelard IlIA 597, 611nI9, analysis and synthesis 85-87, 114, 685n127, 687n144, 688n153; 213-217,220,226, 238, 239n2, 288, Adelard IlIB 688n154; ---> Euclid, 297, 303-304, 308-309, 751n91; Elements, medieval versions in Arabic 229, 242n25 A~mad ibn Musa 249,262; ---> angle of Diocles' solid problem 116 trisection in Diocles 151nll 834 Textual Studies in Ancient and Medieval Geometry analysis and synthesis (cont.) 83,114, 124n97, 176(nn85-86), in Pappus 91, 223n7, 744n40 177n87, 229-230, 318, 521, 799n78 in 94-99 Antiphon 114, 128n149 of 213, 272, 274 Apollonius 33-34, 46, 50, 75n40; of Hero-Apollonian cube duplication analytic works of (lost) 214-215, 36-38 240n12, 241n19, 242n26 of hyperbola construction 213, 232 in Arabic 229, 248-249 of regular solids construction 121n57 commentaries on 5-6, 235-236, analysis, = raising of order 244n74, 755, 765, 767, 770; of 760, 784-786, 792 Eutocius 226, 228-230, 245n82, Anaritius (al-Nayrizi) 670, 685n127 765-771; of Hypatia 406, 755, 765, angle trisection, 769-770; -+ Pappus, lemmas on the of A\:Imad ibn Musil lIB; 217-218, 228, Conics 283-289; text 267-270, 326, 328, on compound ratio 158 330, 358-360 Conics, proposition of of Archimedes 214,216,219,274, 125n102; suppressed first edition of 295-296 231, 244n53; in Arabic 218,222, in Arabic 219-220 259-260, 306; variants in mss. 286 of Banu Musil 80, 536 Conics II 4: 284; text 331,333, via conchoid 214,219, 223n11 367-368 [-+ Eutocius, hyperbola via hyperbola 214, 220-221; in Pappus construction; angle trisection, of 308-309; text 303-304, 340-341 Thiibit]; II 8: text 257, 326, 356; II via neusis, in Pappus lIB; 249, 283, 12: 279-280, 335, 369; IV 30 (35): 287-288; text 267-270, 327, 329, text 257, 327, 357 331; -+ of A\:Imad ibn Musil; of on constructions 55-56, 118 Archimedes; of Thiibit ibn Qurra cube duplication, in Philoponus (AJ) of Nicomedes 274; -+ via conchoid Il, (PK) 13; in Eutocius (AE) 81; via in Pappus (survey) 18; 233, 295 conics 54, 75n40, 77; -+ cube dupli• of al-Quhi lIE; 220-222, 238, cation, of Apollonius 294-300; text 301-303, 339-340 On Plane Loci 797n45 of al-Sijzi, lID; 220, 305; text sundial of 28n34, 53 293-294, 338, 370 approximation 75n36; via solid 214-215 of cubic roots 536 of Thilbit ibn Qurra lIe; 216, 218, 228, of 121n39 295-296; text 277-279, 330, of roots 156, 522-523, 332-334, 361-364 540-542, 546; Hero's rule for annulus (itys) 497-498, 507n22 493n15, 592n42; values for 492n9 anonymous, Peri isoperimetrbn (On [-+ Archimedes, on square roots] Isoperimetric Figures) (AI) IIIlO; of ratio of to diameter of contents 692; summary of compari• circle (1t) 230, 544; Archimedean sons 723-725; text 738-741 values of 127n133 [-+ Archimedes, mss. 741n2, 743n18 Dim. circ., prop. 3]; alternative prototype (AI*) 723, 725-726, 728, values 541,550, 591n29, 730-734, 753, 774-776, 778-780; 593(nn52-57), 816n2; of Ptolemy provenance 776, 779-780; relation to 397n22 CS*, DC* 733-734 93, 123(nn87,89) -+ Introduction to the Almagest arbelos 498 anonymous, Introduction to Ptolemy's Syn• 157,159,165-166,168, taxis [-+ Introduction to the 170n15, 178,230,417,689 Almagest]; Mogenet's thesis on Archimedes 58n24; authorship of 17 [-+ Eutocius; Moge• axiom on convergence (')\rchimedean• net] Euclidean axiom") 408, 418n26; Anthemius of Tralles 6, 26n21, 58n33, Latin variant 684n104 Index 835 axiom on including (convex) arcs 63, on approximations 492n9, 72n6, 386, 417n9, 603, 682n80 504 [-+ rectification] style 508n33, 509n38 axiom on least distance 382, 399n41, synthenti summation lemma for 406 744n33 on balances 496, 506n12 rule 51On40 centers of gravity 243n45, 594n58 Archimedes, Dimension of Circle, survey chronology of works 418n15 of textual tradition 11112; table 806 citations of Euclid 381-382, 399n37, as extant in Greek (DC) 495-497, 408-409, 418n33 503-505, 537-538, 541; date cited, in AI 159, 167, 186,700-702, 453n35; as known to Eutocius 1II6 717-719,726-728; in 52; [-+ Eutocius]; style 403, 527, in Proclus 729; in Simplicius 730, 771-774; defects 375,477-478,496, 734, 737; in Theon 742n14 514, 548 [-+ Hero; Pappus; Proclus] prototype (OC*) 1II6, 422-423, commentaries on 6; Eutocius 226, 495-496,503-505; date 608; 229-230 provenance 753-756, 774, 779-780; on compound ratio 158 relation to AI/AI* 689, 702, 733-734 conical solids ( surfaces) of 598-60 1, older prototype (A*) 495-496, 747n57,776 502-505,806-807; relation to Sphere on conics 223n16 and Cylinder 527, 808 corpus of writings, evolution of 531 n51 prop. I (circle ) as extant (OC) crown problem of 51On40 III2; text 411-414; 422-435, and cube duplication (Heronian) 53 536-538,547,700,702; prototype editing of corpus 409-410 (A*) 1II2; text 411-413; prototype problem 307 (OC*) 424-425, 431-435; versions in on infinite 737 Pappus (P) and Theon (T) 1111; cita• on inscribed 735-736, tions of circle theorem 376-380 749(nn82-83,86) prop. 2 (area of circle), as extant 1II4, and isoperimetrics 502-503,729-730, 397n23, 506n7; text 484; prototype 733-738 (OC*) 479-483; citations of 478 on measurement of sphere (Sphere and prop. 3 (approximation of ratio of cir• Cylinder) 377, 503, 538, 596, cumference and diameter) III4, 605-606; rule for volume of sphere 397n23, 497, 503-505, 536-541; text 493n16; on spherical sectors 485-489; for 511n48; 612n30 rounding off technique for 504, mechanical writings of 531n51 512n50; alternative values 397n23, on mechanisms 129n152 502; citations of 480-481 medieval text of Sphere and Cylinder medieval translations III3-4, 615n58, 528, 532n55; in Arabic 442-450; in 805; text of Arabic (in English trans• Hebrew 442-450; in Latin 598; adap• lation) 436-438, 484-489; Hebrew tations 610 [-+ De curvis superjicie• and Latin variants 422,438-441, bus; BanO MOsa] 489-491; facsimiles, of Arabic (AF) on parabolic segment, area of 377, 455-463, of Hebrew (H) 464-465, 498-501,508n33 of Latin (LP, LG) 466-475 on ratio of cylinder and cube 208n9, [-+ Gerard of Cremona; Plato of 605 [-+ indivisibles] Tivoli; Quadratura circuli]; tradition sector theorem 431, 435 [-+ Hero, on of AF 544-551; Arabic adaptations Archimedes]; medieval form 422 III7 solid problem of 85-87, 101, 116-117, Archytas 26n22, 113, 128(nnI51-152), 222, 223nI4,230-231, 722, 743n26, 142, 147, 150, 158; cube duplication 746n56 of 53, 79-80, 110; -+ cube duplica• 95 tion, of Archytas 836 Textual Studies in Ancient and Medieval Geometry area of circular sectors 505n3; -- Archi• Autolycus 592n43, 698 medes, sector theorem axis 105, 107 area of circular segments 507n24; -- Hero, on circular segments Arendt, F. 243n49, 493n14 Babylonian mathematics 507n24, 763 Aristaeus 72n7, 75n40, 77, 123n83, 215, Baghdad 6 223n15, 228, 237, 240n12, 241n21 balances 151n9, 171n19, 496, 506n12, Aristarchus 28n34, 530n38, 592n43, 698 816(nn1,3) Ariston 144 Balsam, H. 243n49 Aristotle, on applied science 74~n70; on Baltzer, R. 125n106 astronomy 128n150; cited 682n76; Bam} Musa 6, 79-80, 809; commentaries on 6; on isoperimet• on Archimedes' Dimension of Circle rics 730-731, 734, 737-738, 748n67, 535-543 811; locus problem 57nIl; on parts on Archimedes' measurement of sphere (mathematical) 57n12; on zoology (VF) 538, 596, 598-599 93, 123n89, 747n64, 751n9l and Archytas' cube duplication [Aristotle], Mechanics 496, 506n8 101-110, 11~ arithmetic, relative to geometry 157-159, convergence method of 598, 601-602, 171(nn19,21,24), 181, 21On48, 550, 613n38 593n47; verbal expression of and De curvis superficiebus 598-603, numerals 593n50 608 arithmetic operations 155, l72n34; and Eutocius 809 division by table 757-759,761-764; expression for ratio of circumference text from Theon 780-784,786-787; and diameter (1t) 531n43, 539, text from Introduction to Almagest 591n12,6OO 787-793 on geometry (tract) 119n16, 809 expression and manipulation of large in al-Khazin 750n86 numbers 539-540, 545-546, 591n18, lemmas on polygons 536-537 592n40; in Quadratura circuli (Ger• and Menelaus 53,219,289 ardian, Florence and Glasgow ver• on ratio of circumference and diameter sions) 665-667; reduction of ratios 497, 506n15, 602 667 style of 536,551 long division 757, 759-760; text from Bashmakova, I. 795n14 Theon 784-786, 802-804 Becker, 0. 418n21 long multiplication 522-523, 591n19 bees 93, 748n64, 751n9l sexagesimal methods of 550,757,759, Berggren, 1.L. 228, 264, 591n32, 781,783,786,788, 796n31 [-- divi• 592(nn34-35), 594n58, 816(nnl-2) sion by table; long division]; notation Berosus 28n34 764, 790-791 Bible (Vulgate) 673-674, 678n18, tabular methods of 763, 789-791, 687n141 796(nn22,26) al-Bhun) 127n135, 242n24, 296, 593n54 Artis cuiuslibet consummatio 672-676, 418n26, 631; -- convergence, 811 Euclidean; Euclid, Elements X 1 astrolabe 699, 795n11. 814, 816n12; Boethius 170n17, 673, 676, -- Theon of Alexandria, commen• 687(nnI41-142) tary, On Small Astrolabe Bradwardine, T. 644, 669, 671, 682n73 astronomical corpus, minor 698; -- inter• Bregman, 1. 755, 794(nn9-1O), 796n36, mediate corpus 816(nn6, 11) astronomical tables 670, 687n149 Bruins, E.M. 494n25 astronomy 592n43; -- Aristotle; Bulmer-Thomas, I. 240(nn4-5), 242n34; Eudoxus; Plato -- Thomas 5, 166-168, 176n81, l77n87, 526 Burkert, W. 171n23 August, E.F. 798n66 Burnett, C. 612n25 Aujac, G. 678n23 burning mirrors 318; -- Diocles; mirrors Index 837

Busard, H.L.L. 611n22, 612(nn24-25), -> Hypatia; Eutocius; Pappus; Theon 614(nn45-46), 615n57, 669-671, of Alexandria 680n46, 685(nnI24-125,127), commentum 677, 813 687n147, 689, 741nl 18-19, 117; for drawing para• Byzantine scholarship 531 n48 bolas 99, 115, 120n38 [-> ] conchoids 30n34, 56, 75n40, 78, III, Cameron, A. 794nlO, 795n18, 807-808 219,316-318; -> neusis, for angle Campanus 669-670, 682n76, trisection; Nicomedes 685(nnI27-128),688n155 cone, as surface of revolution 765, 767; Cantor, M. 742n14, 748n75 equated to sphere 717-719; isoperi• Carruccio, E. 742n13 metric 720-722, 745n53, 746n54 catapults 13,41,53, 75n40, 137, 141, conics, early theory of 120n38, 128n146, 143, 148 215, 223(nn14-16), 241n21, 308, center of gravity 506n5, 592n36, 594n58, 771; terminology of (Apollonian) [ --> 816nl; -> Archimedes terminology] chords (in ) 103, 107,619,635, 6, 526, 531 n48, 813 642, 645, 656, 664; construction, product of segments 34-36 [--> Euclid, via conics 213 Elements III 35-36] tracing apparatus 109-110. 117, construction of, in Archimedes and 119n12, Pappus 590n6 as existence proof 747n58 lemma on ratios of 682n80, 735, geometric vs. instrumental 262-264, 742n16; -> tangents 295-296, 316-318 use for computation 541-542,550, graphical 70, 75n37, 117 591n29 mechanical 117-118, 120n32, 213; of circle, as infinite 749n84 cube duplication 79-80, 132-133, circle quadrature 620, 644, 647, 135-136 [-> cube duplication, of 649-650, 654, 662, 677n3; early Eratosthenes; of Nicomedes; efforts on 93,108,114,501, 51On39; attributed to Plato] via 88, 223n10, 227 physical 316 [-> quadratrix] planar 117-118, 122n72, 241n19 "cissoid" 122n76, 129n154; -> Diocles, pointwise 83,94, 118, 121n39, 290 curve of solid 117 -118 Clagett, M. 2, 396n2, 421, 451n4, continually (aei) 448-449; medieval ter• 452n29, 595-598, 61O(nn1 ,4), minology for 434, 449 611(nn8, 10-11,13,19-20,23), continuity and divisibility 646, 653 612n25, 613(nn37,44), convergence, 614(nn51-52,55), 617-618, 625, Archimedean (Sph. Cyl. I 1-6) 385, 628,638,640-641, 644, 648-649, 407-408,497,602,612n36 655, 663-666, 669-672, Euclidean 383-385, 402-403, 406-407, 677(nn7, 15), 678n16, 501-502, 507nl9 [-> Euclid, Ele• 679(nn37,39,43-44), ments Xl; XII 2]; in Latin 683n93 680(nn45,51-52,56,59), Euclidean alternative form (Elements, 681(nn67,73),682n80, XII 16) 497, 506n16, 507n19, 683(nn85,90,95,97),684n107, 537-538, 601-602, 613n38-39; in 685n 121, 688nl55 Latin 598-600, 646 [--> Bam) Musil.; collation (paranagignoskein) 521, 756- De curvis superficiebusl 757,762, 795(nnI6,18), 807-808 Eudoxean 406-407 Commandinus, F. 243n49, 272, 742nl4 in Pappus (alternative) 744n38 commentaries, ancient format of 231. convergence enhancement 51 On42, 236,244n72 591n29 commentators (ancient), style of Coolidge, J.L.L. 241n17, 742n13, 225-226,551,809.812-816; 744(nn33-34) 838 Textual Studies in Ancient and Medieval Geometry

Ctesibius of Alexandria 41-42, 56n5, area expressions in 453n40 129n152, 145 arithmetic estimate of sphere volume cube duplication, 604-606 anonymous, in Pappus 14; 112, 228 convergence method in 538, 613n38 of Apollonius (in Eutocius, AE) II; 78, influence of 595-596 113; (in Philoponus, AJ) II; (in lemma on sphere volume 603-604 Philoponus, PK) 249, 259-261; text mss. 679(nn28-29) 251-253, 321, 323 on proportion of and of Archytas 100-110, 289 diameters 497, 506n16, 507n19, 538, of Diodes 39n8, 113-114; and Sporus 613n40 89-90 and De quadratura circuli 614n55, 617, of Eratosthenes 16; Ill, 115, 228 620-627, 635-637, 646, 672; of Eudoxus 108, 112 citations of De quad. circ. 606, of Abu Bakr al-Harawl lIA; 220, 222, 615n58 225,271,287,302,305-307; text on rectification of curves and surfaces 251-253, 320, 322, 364-366 613n44; -> rectification hypothesis of Hero II (in Belopoeica, HB) 41, 45, relation to Archimedes' Sph. Cyl. 596, 51, 119n8; (in Mechanica, HM) II, 598-603, 609-610 51; (in Eutocius, HE) II, 45, 51-52; relation to Baml Musa (VF) 596, (in Pappus, HP) II 598-603, 608 in Eutocius (survey) 15, 769-770 style 596-598,621-623, 678n18, of al-Khazin 254, 262-264; text 776-780 254-255, 322-324, 352-354 cylinders, isoperimetric 720-722, of Menaechmus 39n8, 94-100, 745n53, 746n56 114-115, 225 cylindric 104, 107 of Nicomedes 108, Ill, 115, 245n83, Cyril, 754 249 of Nicomedes (in Eutocius) 316-318; text 312-315, 342-345; (in al• al-Dabbagh, 1. 749n86 Khazin) IIF; text 311-315,341- Dales, R.C. 688n150 342 Damascius 794n4 of Pappus 87-93, 114, 227, 238; -> Damianus 731, 748n73 attributed to Plato Delian orade 17,45, 57n18, 140, 147 of Philo of Byzantium 113; in Demetrius of Alexandria 223n17 Belopoeica (PB) 13, 25nlO; in di'isou 605-606, 626, 678n23, 695-696; Eutocius (PE) 13, 78; in Philoponus and compound ratio 171n24 (PJ) 13; the same, attributed to Apol• diagrams, lonius (PK) 13, 36-38 for Archimedean cirde theorem, com• attributed to Plato 36-38, 78-80, 109, pared 432, 433, 529n22 112-113, 536; and Pappus 89-93 lettering of 15, 19,48,50,96,218, of al-Quhl lIE, 220; text 301-303, 244n63, 262, 271-272, 383, 432, 339-340 519,521, 529n19, 545, 548, 650, of Sporus 87-93, 114, 124n92; -> of 694-695,698,700,703, 705-706, Diodes 708-709,713-714; medieval of al-Tus! 264-265, 302; text 255-256, 451(nnI2,l5); lettering written out in 324-325, 355 Arabic 318-319 curves, as solid sections 104, 107-108, Diels, H. 53 128n150; on surfaces 128n150 Dijksterhuis, E.1. 396(nn2-3), 418n26, De curvis superficiebus (CS) Ill8; 492n3, 612n36, 744n37 adaptations of (medieval) 628-631, Dilgan, H. 749n86 672-673; -> De piramidibus Dillon, 1. 751 n87 ancient provenance (as prototype CS*) Dimension of the Circle; -> Archimedes; 497, 507n17, 538, 595-596, Quadratura circuli 608-610, 776-780, 809-810; relation 216, 223n18 to AI 608-610,689,718, 733-734 Diodes 26n21, 308; Index 839

on Archimedes' solid problem 85-87, Epiphanius 754, 793n2 96, 116,222, 223n14, 743n26 Eratosthenes 30, 33, 75n40; on burning mirrors 81, 94, 113, on cube duplication, account of 124n97 (authenticity) 16 [---> Wilamowitz]; on conics 221, 223nl6 early history of 128n151, 131, 142, cube duplications of, via special curve 147 81-85; via two 94, 100, cube duplication method of 16; 71, 115; ---> cube duplication 73n14; and anonymous (in Pappus) curve of 82-84, 89, 113-114; ---> "cis• 64, 67-68, 70 [---> Pappus]; in soid" Eutocius 78 [---> Eutocius] in Eutocius 151nll date 145, 153n38 problem of two means 32 epigram 99,131-134,144-145, 150, Diogenes Laertius 128nl52 151n15, 153n36 Dionysius 153n36 epitaph to 153n36 28n34, 222, 743n26 on loci 240nl2 Diophantus 5, 57n14, 158, 169n2, measure of earth 481; commentaries on 209n17; in Arabic 592n40; com• 481-482 mentaries on 406, 755, 765, 795nl4 mesolabe 111-112; ---> mesograph [---> Hypatia]; on ratios 173n36 Platonicus 45, 57n18, 79, 113 [Diophantus], on Archimedes' rule for value for obliquity of ecliptic 530n38 area of circle 492n7; on polygons Erycinus, paradoxes of 63 492n9 Euclid 592n43; diorism 87, 722 on application of 749n81 "divine" (theios) 174n46, on astronomy 698 176(nn79,81-82,85); "divine on balances 25n6 Eudoxus" 150; "divine men" (theioi cited, by Archimedes 381-382, andres) 167, 186 399n37, 408-409, 418n33; by division, of sexagesimals 176n78; Eutocius 174n42; by Latin geometers ---> arithmetic operations 678n26, 684nl07, 686nl39 Dold-Samplonius, Y. 254 commentaries on 5, 227 Domninus of Larissa 166-168, 176n77 , on conics 223n15; ---> conics, early 177n89 theory of Doric dialect 8,86,101, 122n62, on fourth proportional 764, 789 171n23, 410, 452n17, 528, 532n57, on loci 215, 240n12, 244n75 807-808, 816nl lost works 240n12 Dositheus 120n37, 398n31 Optics. Data, Catoptrics (Graeco-Latin doubling of cube (restricted form of translations of) 614n57 problem) 71, 137, 147; ---> Delian Phaenomena 417n 1 oracle style 35, 158-159, 407, 596, Dover, K.1. 796n37 80ln107 Drachmann, A. 26nl2 Euclid, Elements: Durnheimer, W. 680n59, 683n90 122 (on construction of ) 704, dynamis (power, mathematical) 43, 708 599-600, 612n33, 704, 725, 743n28; 147 (identity for right triangles) 661, (force, mechanical) 496, 506n9 708 II 8 (difference of identity) 507n22 Egyptian mathematics 507n24, 51On39 III 7-8 (on secants) 768-770, 85-86, 245n79; area of 507n20 798(nn60,66), 799n75 ellipsoid 83 III 15 (on chords) 798n55 Emden, A.B. 611(nnI3-14) III 35-36 (on segments of chords and encyclopedias 52; ---> Geminus; secants) 34-36, 39n25, 50, 55, 58n22, Menelaus 260 epicycle and eccentric 764 V def. 3 (definition of ratio) \59, 163, epigram 153n36; ---> Eratosthenes 185 840 Textual Studies in Ancient and Medieval Geometry

Euclid, Elements (cont.) commentaries 155-156; on Apollonius VI def. 5 (definition of compound 222, 224n39, 231, 237-238, 245n81, ratio) 157-63, 170n16, 171n25, 177- 446,765-771; on Archimedes 63, 178, 186, 202, 204, 21O(nn36,47) 410, 445-447, 482-483, 757; on VI 3 (on angle bisectors) 504, 561n15 Ptolemy, alleged 164-166, 169n5, X (on irrationals) 799n75 175n7l,230 X 1 (on convergence via bisection); on compound ratio 17 Latin citations 631,641,646, on cube duplications 15; 39n16, 217, 658-660, 681n67; subtraction proce• 222, 262-264, 295, 300, 798n68; of dure 653-654, 660; --> convergence, Apollonius (AE) II; of Philo (PE) 13, Euclidean 41; --> cube duplication XII 2 (on ratios of circles) 381-385, date 26n22, 229-230 425,429,496,501,527,646, on Diocles 308 796n30 editing of Archimedes commentaries XII 16 (on chords) 637; --> conver• 99, 807 [--> Isidore of Miletus] gence, Euclidean alternative form on Eratosthenes 16; 67-68, mss. 612n25, 678n26 73(nn9-1O), 74n23, 115-116 medieval versions: Graeco-Latin on hyperbola construction, 284, 614n57; Arabic and Arabo-Latin 743n26; text 280-281, 336-337 597, 612n25 [--> Adelard; Gerard of on loci 228, 236, 241n20 Cremona; Hermann]; citations in on Nicomedes 29-33 Latin 623,625, 627, 673 on rectification of circle 613n44 [Euclid], on polygons 492nn8-9 religion of 167, 176(nn85-86), 177n87 Euclidean convergence (subtraction, or sources of 18, 24, 26(nn21-22), 45, bisection) procedure: --> conver• 51-53,77-78,81, 100, 128n146, gence; Euclid, Elements X 1 133, 149, 244n69, 262, 531n51, 100-10 1, 106-109, 797n45,798n62 114-115, 123n85, 126n124, style 19; 18, 27(nn26-27), 45, 50, 128n146, 149, 289 57(nnI5-16), 74n21, 84-86, 88, 98, Eudoxus ofCnidus 26n22, 113, 128n151, 100-101,107,111-118, 418n21; cited by Eratosthenes 142, 119(nnlO-ll), 124nloo, 126n128, 147, 150; by Hero 51On40; in 127n144, 134-144, 160-161, Introduction to Almagest 159, 167, 164-165, 175n70, 765-771, 773, 186; astronomy of 126n124, 797n48, 798(nn68-70); modern 128n150; cube duplication of 78-89; assessments of 226, 240n4 sundial of 28n34 text of Archimedes III6; 410, 418n33, Eulalius 794n2 447, 494(nn32-33), 753, 807-808 Eutocius 6, 11, 254, 615; example (hypodeigma) 159, 162-163, alleged author of Introduction to 174n58, 180, 187; --> induction Almagest 17; 67-68, 788, 801n11O; exhaustion theory 418n21 [--> Euclid, --> Mogenet Elements X 1; XII 2]; variant form and Ammonius 167, 169n5 418n22 [--> Euclid, Elements XII in Arabic 127n132, 262, 447, 809; in 16] al-Tusi547, 549; possible use by Baml Musil 539-543 on Archimedes' circle theorem (Dim. Jalsigraphus 597, 6lln19; and sophista, Circ., prop. I) 424, 432-433, sophisticus 637, 687n141 452(nn21,32), 607 Fitzgerald, A. 796n37 on Archimedes' estimates of circle ratio focus-directrix construction, of conics (n)494n38, 504-505, 510n41 215, 223n15; of 94,100, on Archimedes' Sph. Cyl. 16: 418n24, 120n38, 124n97 445-447,516; on 110: 516-520; text Folkerts, M. 612n25 533 forgery 16; 131, 134, 139, 141, citations: of Conics 306; of Ptolemy 145-146; --> methodology of stylistic commentaries 417 analysis Index 841 formalism (Euclidean) 766-767, Gudemann, [n.i.] 123(nn87,88) 770-771,773,779-780 formulaic repetition: ---+ repetition al-I:Jajjaj 442, 612n25 fourth proportional 189, 209n24, 21On41, al-Harawi, Abu Bakr 6,41, 54-55, 77, 789 249; on conics 290-291; cube dupli• Fowler, D.H. 530n38 cation of IIA [---+ cube duplication] fractions: continued 512n49, 530n38; al-Harawi, Abu '1-I:Jasan 296 mixed 525, 530n43; notations for Hartner, W. 795n 11 522-523, 529n29, 530(nn32,34); Haskins, C.H. 611n12, 614n57, 687n147, operations with 530n31; reduction of 688n150, 741nl 524-525,546; truncation of 540; unit ibn al-Haytham (Alhazen) 8nll, 318, 748n77 736. 749n86 Heath, T.L. 39n17, 150n4. 229, 239n3. Galen 748n64 240n4. 241n17, 244n53, 245n83, Gellius, Aulus 123n87 396n2, 492n3, 508n32. Geminus of Rhodes 52, 58n24, 81,108, 742(nnI3-14),744(nn33,37), 120n24, 228, 262, 273, 797n45; in 745n52, 747n62, 796n28 Eutocius 230, 241n19 Heiberg, J.L. 2, 8(nn4-7), 121n56, generality 72, 76n48, 84,142, 157, 159, 122n68, 123n85, 131. 153n47, 231, 162,171025, 172n30, 179-181, 186, 238. 242(nn34-35), 243n42, 190, 214, 300; of proof 707, 244(nn53,73), 245n83, 375-376, 711-713, 721-722 396n2, 418(nn31-33), 424, 446-447, geometric 453n37 451n6, 452n17, 494n26, 508n33, geometry, analytic 158 522. 530n31, 532n56. 611n10, Gerard of Brussels 595, 61On7, 636-637, 612n25. 615n57. 741nl, 747n63, 672-676, 810-811 795n18. 798(nn62-63,66), 799n73, Gerard of Cremona 6, 119n 16, 396n2, 807-808, 816n4; on interpolations in 397n7,421; texts 8n7, 39n20, 170nI6, 208n4, style of 127n139 409-410, 418n24. 482, 523-524, trans. of Archimedes' Dim. eire. 528, 528nl, 532n56, 799n89 452n18, 595, 606-607, 671, 676; Heliodorus 155, 168, 169n5, 731, 748n73 text 467,469,471,473,475; vari• hemicylinder 140, 147; of Archytas 102, ants 422, 438-441, 489-491; 104, 108, 110 ---+ Quadratura eirculi Heraclides 58n24, 230, 483, 494n38 trans. of Euclid's Elements 597, Hermann of Carinthia 597-598, 612n25 798n66 Hero of Alexandria 5, 75n40, 793nl; trans. of Banu Musil (Verba filiorum) on analysis 242n27 119n16, 127n139, 536, 542, 549, on arches (Kamarika, lost) 9 598,601,611n23 and Archimedes, citations of 230, Gerbert 673 805-807; for approximation of circle Gericke, H. 742n13, 749n79 ratio (re) 397n23, 481. 483, 493n18, Gervase of Essex 595, 635-636 502; for more accurate value Gibbs, S. 28n34, 53 (attributed) 496-497; for rule for "givens": ---+ terminology area of circle (cf. Dim. eire., prop. 2) Gordanus 638, 676, 680n59, 683n90; dat- 397n23, 477-478, 493n14, 615n58; ing of 670; ---+ Quadratura circuli for circle theorem (cf. Dim. eire., Grabmann, M. 612n27 prop. I) 377, 379, 397n23, 405; for graphical construction: ---+ construction sector theorem 377, 379, 495, 503; Greaves, D. 190 for measurement of parabola 397n15, Gregory, D. 798n66 509n35 Gregory, 1. 591n29 on circular segments 377, 497-502 groove (of instrument) 30, 79, 132, 146, cube duplication of 71, 81; in Mechan• 150n2; ---+ construction, mechanical; ica (HM) and Belopoeica (HB) 11; in terminology Eutocius (HE); ---+ cube duplication Grosseteste: ---+ Robert Grosseteste on curvilinear areas 507n20 842 Textual Studies in Ancient and Medieval Geometry

Hero of Alexandria (cont.) 381-382, 710, 743n24, on fractions 530n43 747(nn61-62), 778, 800n95, and Philo 41-42; on catapults 143 80 1(nnl04-106) on polygons 492n9 Huygens, C. 591n29 practical geometry in mechanics Hypatia of Alexandria IIIll; 5, 177n87, 152n31 244n74, 406, 723, 808, 811-812, rule for triangle 536; -> Archimedes 814; life 754, 794(nn4-5); sources 51Onn40-41, 531 n51 philosophy 754-755, 763, on sphere, arithmetic estimate for 794(nn8-1O); science 795(nnll, 16); 493n16, 599, 604, 612n31, 614n50 mathematics 755-756,762-763; on square root approximation 156, 741n3 cited in Synesius 794n3; style style 499-501 756-764 Heron of Alexandria (teacher of Proclus) hyperbola 85; 754, 793nl for angle trisection 56, 297 Heronas 157, 170n17, 173n36, 178 for auxiliary problem of al-Sijzi 297 Heronian metrical writers, construction of 213-214, 217, on Archimedes' circle ratio estimate (7t) 223(nn5,14), 230-234, 236-237, 481, 483, 492n8 274, 287-288; in Apollonius (Conics on Archimedes' circle theorem (Dim. II 4) 231, 233, 281-282, 284; text eire., prop. 1) 378 331,333; in Eutocius 230-234,249; on Archimedes' rule for area of circle text 336-337; in al-Khiizin 263, (Dim. eire., prop. 2) 478, 492n8 289-291,317; in Pappus 230-234; in on area of circular segments, aUr. to Thabit 283-285; text 277-278, 330, Archimedes 509n34 332 excerpts of Proclus in 453 (nn34-35) for cube duplication 54-55 [-> cube on isoperimetrics 747n63 duplication, of al-Harawi] on length of circular arcs 507n25 in isoperimetric cylinders problem on sphere, arithmetic rule for 493n16 746n56 scholia 494 (nn25-26) as locus of constant product 125n103 Hierocles 754 with Menaechmus 94-100 Hiller, E. 131, 153n47 for solution of problems 213, 222 al-l:Iimsi 256, 277, 284 Hypsicles 73n7, 592n43, 745n43, 750n86 Hintikka, 1. and U. Remes 242n27 417nl 32, 56nll, 108, Iarnblichus 5,167, 176n79, 751n87, 114, 126n124, 128n149, 147, 796n36 400n56, 504, 678n40 Ibrahim ibn Sinan 229 hippopede 128n150 impossible (adynaton, atopon) 384, 403, Hjelmslev,1. 418n26 417n6, 427, 636; Latin terminology Hogendijk, 1.p. 224(nnI9,23,31-32,34), 685n127; -> proofs, indirect 229, 242n26, 267, 272, 275, incommensurables 158; -> Euclid, Ele- 294-295, 297, 300, 302, 307 ments X; irrationals hoi peri 25n3 lndi 686n138 167 indivisibles 209n9, 605, 614n49 homogeneity 159, 162, 164, 171n26, 186 induction 74n26, 157, 159, 164, 166, homonymy 173n36; -> paronymy 178, 201, 592n36; -> example hoposagfmon 383,389 instrumental construction: -> compass; Horace 596,674 construction: graphical, mechanical, Hughes, B.B. 686n130 physical; motion (geometric); neusis; Hultsch, F. 8n7, 153n40, 169n3, 272, ruler 417n7, 511n46, 614n45, 689-691, intermediate corpus (curriculum) 547, 723, 728, 732, 741(nnl,7), 551, 592n43; -> astronomical 742(nn13-14), 743n22, 745n52, corpus, minor 748n75; on interpolations in texts interpolation 128n151, 153(nn43-45), 122(nn67,72), 153n41, 242n33, 172n32, 209n15, 210(nn47-48), 218, Index 843

230, 245n76, 286-287, 381-382, in Arabic 749n86 384, 397nl0, 399n37, 402, 408-410, and bees (in Pappus) 93, 748n64; and 417n12, 418n27, 433, 445-447, swallows (in Themistius) 747n64 452(nn25,27), 453(nn34-35), 506n4, circle 811; relative to regular polygons 523-525, 530n37, 591n20, 614n50, 699-702 686n139, 742n15, 743n23, 778-779, circular segments 387, 400n56, 497, 800(nnl0l-l02), 808-809: 502-503, 716 -> Heiberg; Hultsch cones and cylinders 720-722 Introduction to Ptolem/I' Syntaxis in Introduction to Almagest (AI) lIllO, (Almagest), 175n68: -> Introduction to Almagest assigned to Eutocius (Mogenet thesis) in Pappus () III 10: 380, 386-387, 17; alternatively, assigned to Arcadius 398n30 166, 168,230 rectilinear figures: triangles 703-713; on Archimedes' circle theorem (Dim. polygons 713-716: regular polygons eire., prop. 1) 376, 379, 424, 432, 693-697 452n32, 513, 607 sphere relative to solids 387, 716-717, on Archimedes' estimate of the circle 811: relative to Archimedean conical ratio (n) 482, 494n29 solids 717; relative to regular solids date 397n13, 615n59 716,718-720 on isoperimetric figures (AI) III 10; spherical segments 503 595,603-604, 615n60, 811; on in Theon (TI) IIIIO: 379-381, 386-387 isoperimetric solids 51On44; Ito, S. 615n57 -> anonymous, On Isoperimetric Figures lemma on sphere volume 603-604 Jackson, D.E.P. 75, 112 mss. 791, 796n34; portions currently Jacob of Cremona 596 edited 169n4 John of Tynemouth 6, 595, 597, 611 n 13, on Pappus' measurement of the sphere 635, 637, 663, 672, 674-676, 614n50 687n147, 688n153, 776, 809-811, on Pappus' method of division 813: style 676-677; -> De cun'is 764-765; text 787-793 superficiebus style 160-161, 164-165, 168-169, Jones, A. 72n2, 75n49, 169n5, 229, 175n68, 505, 608-610, 689-690, 240(nn3,5,11-12),241(nnI6,22), 741n3, 763-764, 774-776. 779, 788, 242(nn28,30-33), 244n73, 245n76, 792-793 511n46 irrationality of circumference-diameter Jordanus de Nemore 670-671, 680n59, ratio (n) 542 682n80, 686n130 irrationals 158, 171 n26, 178: -> incom• 177n87, 755 mensurables al-Jurjani 228, 242n25 Is~aq ibn Hunain 441-442,448, 450nl; trans. of Archimedes 547 Isidore of Miletus 6, 58n33, 99, al-Kashi 593(nn52,57), 816n2 126n117, 229-230, 243n38, 521, al-Khazin, AbG JaCfar 6, 54, 249, 254; 526-528, 532n56, 753, 762, 795n18, and al-Harawl 259,262-264 [-> cube 798n70, 807-808, 811; parabolic duplication]; compiler on conics 222, compass of 99-100, 120n38, 243n36; 272, 286, 295; on cube duplication of (disciple of, as editor of Eutocius) Nicomedes IIF; on isoperimetrics 99-100, 114, 243n38, 521-528, 750n86 591n19,757 al-Kindi 318 Isidore, disciple of Proclus 794n4 King, D.A. 248 isoperimetric figures 169n3, 227, 399 Kliem, [n.i.] 123n88 (nn54-56); Knaack, G. 150n4, 152n34 in ancient cosmological commentaries koilogonion 711, 728, 747(nn60-62) 728-738; -> Proclus; Simplicius; Kozhukhova, G.M. 451, 592n45 Themistius Kramer, E. 794n4 844 Textual Studies in Ancient and Medieval Geometry

Krause, M. 451n7 Menelaus of Alexandria 5,52,79-81, Kunitzsch, P. 249 101, 108, 115, 127n135, 120n24, 225-226, 273, 318, 536, 592n43; compiler 217,219,262,288-289; Lacombrade, C 755, 794n8, 796n36 cited in Pappus 215, 223n17 Lang, M. 796n27 Mercier, R. 687n148 latitudo 646, 653, 682n82 mesograph 133, 140, 142, 150, 151n9; latus rectum: --+ terminology of conics, --+ mesolabe Apollonian mesolabe 75n40, Ill, 133; Lemay, R. 687n147 --+ Eratosthenes; mesograph lemma to cube duplication (Hero• methodology, Apollonian method) 12; 21, 22-24; of criticism 592n36 in Nicomedes 32-33 of source analysis 603-604, 654-655, lemmas, order of 492n5, 702, 743 699; hypothesis of single source 520, (nn26-27); --+ Archimedes, Dim. 529n23 Circ., defects of of stylistic analysis 134, 138-142, 145, , the philosopher 531(nn48,51) 156, 161, 164,623-624,628, 634, Leontius 123n87, 243n38 670-671, 756, 761-762, 779-780, Lesbian architecture 83 810-812 Lloyd, A.C 793nl, 794n6, 796n35 of text analysis 4, 410, 691, 693, Lloyd, G.E.R. 126n124 742n13,808 loci, classification of 24ln20; solid of text reconstruction 40 1-402 l23n83, 245n79 use of evidence from logistic 158, 179 410; from medieval sources 528, Lorch, R. 612n25, 750n86 532n55 Lorent, H. 745n43 Minos 147 Luckey, P. 593n57, 816n2 mirrors, burning 81,83,87, 114; --+ Dio• cles Moerbeke, W. 229, 396n2, 421, 451n6, 662,665 596, 598, 611n9, 612n27; trans. of Magnus 230, 512n53, 541, 591n24 Archimedes' Sph. Cyl. 625-626, Malalas 794n4 655-656 al-Ma'mfin 6, 216 Mogenet, 1 17; 242n34, 376, Marinus of Neapolis 168, 177n88, 397(nnI2-13), 615n59, 689, 741n2, 241n22, 793nl 801; on isoperimetrics 751n92 Marsden, E.W. 41, 153n39 Mo~he ibn Tibbon 6, 442, 450, Martin, CT. 611n14 451(nnI2,14) mathematics, Arabic 225 [--+ medieval motion, celestial 128n150, 737, 748n67; tradition]; 16th century 158; 17th in geometry 104-105, 107 [--+ neu• century 121n39 sis; ruler]; of mechanical devices Maurolico, F. 595 132, 135, 146, 149 means, arithmetic 493n17; harmonic 63; Mueller, W. 742n13 geometric, general construction Mugler, C 121n62, 169n5, 242n34, 71-72, 75n48 243(nn35,37) mechanical: --+ construction, mechanical murabbac (square) 25n6 medieval tradition (Arabic), style 551, Murdoch, lE. 612n25, 614n57 591n26, 617, 809; sources 599,610; myriads 176n77, 494n35, 512n53, (Latin), style III9, 809-810, 816n3 529n28, 546, 591n18, 737; --+ arith• medieval translations (Arabic, Hebrew, metic operations, ... of large num• Latin), style 422, 434-435, 445, bers 448-450, 451(nnI2-13), 452n27 Menaechmus 26n22, 113, 142, 147; and mechanical construction 128n151; 152n24, 166-168, --+ cube duplication 176(nn81 ,84), 177n87, 794n6; Index 845

Neoplatonists 169n6, 176(nn81,84), on Apollonius' cube duplication 54; on 177n87, 227, 729-730, 737, 751n87, his Neuses (lost) 288 754-755,815 in Arabic 217,228-229,273; -+ Jack• Neuenschwander, E. 126(nnI26-127) son Neugebauer,o. 169n5, 743(nnI9,2I), on Archimedes 805-807; approxima• 795nll, 816n12 tion to circle ratio (n) 378, 480, 483; neusis constructions 38, 117-118, circle theorem (cf. Dim. Circ., prop. 120n32; I) IIIl, 401-403, 405, 495, 511n46, alternative formulations of 219, 607; text P 387-392; sector theorem 285-288 377, 382, 384-387, 495-497; text S for angle trisection 213-219,287 394-396; measurement of sphere relative to circle and 228 716 via conchoid 30-31, 311 cited in Eutocius 178, 236, 765, via conics 34; via hyperbola 272 798n62; in Introduction to Almagest and cube duplication 55-56; of Hero 155,160,187,202,789,792-793 53; of Pappus 91, 114; of Sporus 89; classification of problems 71 effected via hyperbola 260-261 commentaries on Ptolemy 227, 691; on for spirals 95 Book I (lost) 380-381, 385-387; Nicomachus of Gerasa 157-158, 170n 17 , cited by Eutocius 157-159, 165 173n36, 178, 208n5, 21On42, 211n48 on compound ratio 160, 162 Nicomedes 33-34, 50, 52-53, 56, 75n4O, on construction by instrument 318 77-79,214-216,219,223nll on cube duplication 14; 71, 217, On Conchoids 111-112,115,228, 240(nn13-16); own method 71-72, 511n46; -+ conchoids 87-93 cube duplication of 29, 31-32, 71; on division 175n76, 763-764, -+ cube duplication 787-793 neusis of 285 on Eratosthenes 131-136 Nix, L. 448 and Geminus 52 Nokk, [n.i.] 742nl4 on hyperbola construction 230-234 on isoperimetric figures (PI) 11110, 497,502-503, 615n6O, 774-776, 811; contents 692; compared with Olympiodorus 176n84 other accounts 723-725; lost version Omar (CUmar) al-Khayyam 224n19, 300 (PI*) 691, 693, 697-698, 701-702, optative mood 27n26 704,710-711,715,725-728,730, Oresme, N. 682n82 734, 736; on isoperimetric solids Origenes 794n2 51On44 originality 238-239, 812; -+ Pappus lemma on circumference-diameter Ovid 596, 674 proportion 497, 511n46, 602, Oxford 663,671-672,674-675 613n42 lemma on volume of sphere 614n50 lemmas on Apollonius' Conics 228, Pachymeres, G. 173n38 234-238,410, 767-768, 797n45, Paeonius 795n11 798n58 Pallas 596-598 lemmas on Euclid's Optics 242n29 Pappus 5, 77; Mechanics (cf. Collection VIII) 75, alternative to Archimedes' Sph. Cyl. 87-88, 112, 228-229, 235, 273 (Collection V) 744n38 on neusis for Archimedes' Spirals 95 on analysis and synthesis 86, 237, on Nicomedes 29-33,50 751n91; -+ analysis and synthesis; originality 88, 92-93; modern assess- angle trisection ments of 226, 239n3 on angle trisection 18; lIB (text on rule for volume of sphere 493nl6 268-270, 327-331); lIe; lIE (text sources 92, 531n51, 716 303-304, 340-341); 56 on square roots 156 846 Textual Studies in Ancient and Medieval Geometry

Pappus (cont.) superficiebus) 637, 673, 676; -> Ger• style 19; IIIID; 63, 68-70, 111-112, ard of Brussels 118, 150n6, 215-216, 382-384, pivot 14-15, 26n14, 30, 79, 91-92; 511n46, 767, 774-776, -> construction, mechanical; termi• 798(nn59-60), 812-813 nology parabola, in cylinder maximization planar constructions 34, 38, 56; in rela• problem 746n56; in Menaechmus' tion to cube duplication 63, 69-70 cube duplication 94-100; -> conics Plato, on astronomy 126n124; on geome• parabolic segment, Archimedes' measure• try 79, 113, 128n151, 147, 749n81; ment of 498-501 [-> Archimedes; Timaeus 5,167,728-729,731,811; Hero]; relation to circular segment commentaries on 728-733,736-737, 500, 509n36 747n63; cube duplication paraboloid 83 (attributed): -> cube duplication paradoxes [-> Erycinus]; relative to Plato of Tivoli 6, 396n2, 421, 614n51; isoperimetrics 749n81 trans. of Dim. circ. (In quadratum Paris 674-675, 687n145 circuli), text 466, 468, 470, 472; var• Parker, R.A. 507n24 iants 422,438-441,489-491; rela• 167 tion to Arabic (AF) 482-483 Parmenion 16, 19,21,24, 28n34, 41, 46, 167, 176n84, 755 50-51,53-54, 58n30, 261 Plutarch 57,109, 128(nnI51-152), 140 paronymy (of ratios and numbers) 158, 72n7, 116, 119nI5, 814 160, 171n26, 173n36, 178, 208n5, polygons, area rule for 639-640, 642, 21On42; -> homonymy 644-645,650, 656-657, 664, part (mathematical term) 57n12, 654, 696 681n70,684n117 Pedersen, 0. 687n148 polyhedra: -> solids 128n150 Porphyry 176n84, 751n87, 755 Peyrard, F. 398n36 Powell, 1. V. 151 Philo of Byzantium 14, 41-42, 75n40; practical applications of geometry date 145; Belopoeica 143-145; cube 142-143, 147-148 duplication (PB) 41-45,51-52; in practical construction: -> construction Eutocius (PE) 44-45, 48-52 [-> cube practical geometry: -> Artis cuiuslibet duplication]; on mechanisms consummatio 129n152; style 43-44,143-145 Praechter, K. 167 Philo of Gadara 230,505,541, 591n24 precession of equinoxes 675, 687n148 Philo of Tyana 223n 17 Price, D. de S. 28n34 Philoponus, John (Joannes) 6, 11, 28n30, problems, 113; of angle division, general 214; -> angle and Ammonius 169n5 trisection on Apollonius' cube duplication, Hero• classification of71, 213 nian form (AJ) II; Philonian form of inscribing hexagons 124n92 (PK) 13, IIA; alternative (PJ) 13 locus of constant product 95, 125nID3; on astrolabe 699 -> hyperbola, construction of commentaries on Aristotle 21, 45, of means, general 142, 150; of two 796n35 means, in Diocles 84, 89-90; in date 26n22 Pappus 91; in Sporus 90; -> cube on instrumental construction 16, 318 duplication on Delian oracle 57n18, 140 solid 75(nn34,40), 112 on isoperimetrics 730-731, 735 Proclus Diadochus 5, 176n84, 242n22, sources 21, 26n22, 51 531,754, 793nl; style 46, 798n69 on Archimedes 419n33 philosopher 722-723, 756, 762, on Archimedes' circle theorem (Dim. 794(nn3-4), 796n36, 814 eire., prop. 1) 376, 397nlO, 417, 794n4 432, 452(nn33-34), 494n34, 525-526 De piramidibus (adaptation of De curvis on circle quadrature 433, 452(nn33-34) Index 847

Commentary on Euclid, mss. of 453n35 [--> Introduction to Almagest]; Latin and Geminus 52, 58n24 translation 614n57; citations in Latin on isoperimetrics 728-731, 733, 735, 645, 682n80, 686(nnI38-139) 748n71,749n81 approximation of circumference• on Menaechmus 126(nnI20-121) diameter ratio (rr) 550, 593n53; on on paradoxes 72n5, 749n81 Archimedean values 480, 483, 503, on Plato (Timaeus) 167, 176n79, 505n4 728-729,732,734, 737. 751n87, on Archimedean circle theorem (Dim. 811 eire., prop. 1) 378, 505n4 sources 100, 126nl24 on chords 645, 682n80; table of 75n33, and Syrianus 168 541-543,550, 591n29: --> chords on terminology 126n126 commentaries on: --> Arcadius; on Zenodorus 728. 747(nn60.62) Eutocius; Hypatia; Pappus; Theon product (geometric formulation) On Gravities 230 402-403,425,430. 495, 497. 527. on isoperimetrics 690, 736 536,547, 591n25, 599-602, on mean motion of planets 757-758. 605-608,612n33; 780, 786-788 for planes (area) 28n29, 106-107, on obliquity of ecliptic 530n38 505n4, 697. 700-702, 745n48 on planisphere 699 for solids (volume) 717. 720, 733. on proportionality of arcs and sectors 745(nn43,49-50), 800n97 505n4 terminology, in Greek (to h~po tr3n ... ) on spherics 175n61 106-107, 378-380; in Arabic 284, 176, 813; Pythagorean 425. 434, 449; in Hebrew 449; in theorem: --> Euclid, Elements I 47 Latin (ductus) 621,647,656-657. 677n15, 683(nnl01-I03) proofs, alternative 94, 96, 98-99. 114, Qalonymos b. Qalonymos 6. 421, 441. 125n114. 272-273. 300, 446, 497; 450, 451(nn3,14) general (contrasted to inductive) Q.E.D. (hoper edei deixai) 159, 189, [--> induction]; indirect 383. 399n45, 21On31, 452n27 538, 592n38. 597, 600-602, 611n19, quadratrix 58n26, 92-93, 114; for angle 636, 686n129, 713, 7641--> impossi• division 214; for circle quadrature: ble] --> Sporus proportion theory, early (Eudoxean) 158. De quadratura circuli, LG (translation of 399n53, 407, 418(nnI8.20-21). 496. Dim. eire. by Gerard of Cremona) 506n5. 613n41, 816nl 617,619,623-624,627,641. proportionality, of circular arcs and sec• 648-649. 663, 665. 667-668, tors 386, 398n36, 399n49, 407, 682n73. 687n147, 810; text 467, 495-497, 502, 505n4; of circumfer• 469.471,473,475; variants from ences and diameters 496-497, 502. Arabic 438-441,489-491; titles in 506(nn6,16), 507n19, 536. 538-539, mss.614n52 590n8, 602, 613n40; of areas and adaptations of III9; overview 617-618, perimeters of regular polygons 668-677, 810 684n 117; of cylinders and cubes 605 Abbreviated Version (AV) 648-649; proportions, terminological variants in --> Roger Bacon Greek 696; Arabic 285; Hebrew 449; Cambridge Version (C) 625-631; and Latin 622, 630-631 other versions: Naples 635; Corpus Ptolemy (king) Ill, 115, 131-132, 134. Christi 655-663; and De cun'is 144, 147,150; III (Euergetes) 144. superficiehus 628-631; and De 152n34; IV (Philopator) 144-145; V ysoperimetris 680n46 (Epiphanes) 145, 153n38 Corpus Christi Version (Cor) 655-663; Ptolemy (Klaudios Ptolemaios) 5, and Vatican Version 645 152n24,417nl; Florence Version (F) 614n54, 618-624, Almagest (Syntaxis), Book I 155 799n85; and other versions: Cam- 848 Textual Studies in Ancient and Medieval Geometry

De quadratura circuli (cant.) ratio, Florence Version (F) (cant.) compound 122n70, 208nl; in Domni• bridge 626-631; Naples 631-635; nus 201-207; in Eutocius 155-159, Corpus Christi 655-663; Glasgow 162-164, 177-185; in Introduction to 665-668; and De curvis superfieiebus Almagest 155-156, 159-164, 620-624; --+ John of Tynemouth 185-201; relation to di'isou 171n25 Glasgow Version (Gg) 663-668 duplicate 122n70 Gordanus Version (Go) 638-640; and epimorios 208n2 other versions: Munich 640-643, in relation to number 158, 171n27, 648; Vatican 644-648,650,654-655 174n53, 208n5, 531n44, 539, 600 Munich Version (M) 640-643 subtraction of 159, 161, 175(nn61,63), Naples Version (N) 631-637; and other 185, 187-190, 202-204, 21On35; versions: Gordanus 638-640; Munich rule for 209(nn2l ,27), 21On43 640-643; --+ Gerard of Brussels synthenti, form in Pappus and Archi- in Questio of Albert of Saxony (QA) medes 711 649-655 --+ terminology Vatican (ps.-Bradwardine) Version (V) Rawlins, D. 530n38 644-648; and Abbreviated Version rectification of curved lines and surfaces, 648-649; and Questio of Albert hypothesis of 541-542, 602-603, 649-655 613n44, 625-626, 629,645, 656, mss. of LG and adaptations 618, 664, 674, 679(nn32, 35-36), 682n81, 624-625, 628, 635-636, 638, 641, 683n97 648, 655, 663, 669, 677(nn5,7), religion, in late antiquity 152n24, 167, 681n73, 683(nn85,90), 684nl13, 176n86, 177n87, 754-755, 793nl, 685nI20,686n131 814 quadrivium 674 repetition (formulaic) 758-768, 779-780, al-Quhi 6, 54-55, 216, 249; 796n31; --+ Hypatia, style of angle trisection of lIE; --+ angle trisec• rhomboid 722 tion; al-Sijzi rhombus, solid (double pyramid) of on approximation of circumference• Archimedes 600, 777-778, 800nl00; diameter ratio (n) 544, 592n36 medieval terminology 448 on Archimedes' Dim. eire., doubts Richard of Fournival 61On7, 637, 675 543-544 Rist, 1. 754, 763, 794(nn4-6,9-1O) on Archimedes' heptagon problem 307 Robert Grosseteste 595, 61On7, 663, 672, on centers of gravity 592n36, 594n58 675,688n151 citation of Conics 222, 245n81, 306 Roger Bacon 663,669-671, 685nl27, cube duplication of lIE; --+ cube dupli- 688n152 cation Roger of Hereford 688n150 and al-Harawi 259 Rome, A. 240n7, 381, 396n7, Quintilian 732, 748n64, 749n81, 811 398(nn29,35), 399n40, 417nlO, Qust,li ibn Luqa 6, 11,447-450,809; 452n27, 511n46, 690, 709, 741n2, proposed as translator of Dim. eire. 742(nnlO-11,14),743n23, 535, 549, 551 747(nn60-61), 756-757, 780, 782-783, 795n16, 796n23, 797n40 rota Aristotelis 683n97 Rashed, R. 318,448, 592n40, 795n14 Rozenfe1d, B.A. 795n14 Abu 'I-Rashid, cAbd al-Hadi 6, 450n2, Ruelle, C.E. 202 451n7, 494n35, 617, 671, 799n85, ruler (kanan, kanonion): 809-810; question of identity 543; in accounts of cube duplications, of version of Dim. eire. (AR) 421, Apollonius 16-17; of Banu Musil 79; 543-546; text 552-576; possible use of Hero 12-14; of Pappus 91-92; of by al-Tusi 547,549-550; style Philo 42-44; of Plato 79, 92; of 544-546, 551 Sporus 89 Index 849

flexible 83 al-Sijzi 6, 54, 248-249, 263; on angle of Nicomedes' conchoid 30 trisection (survey tract) 216-217, rod or guideline (Eratosthenes) 135, 219,222,289,294-295; (own 140, 142, .146 method) lID; and al-Quhi lIE; on sliding or moving 12-18, 22, 42-44, Apollonius 229 263; ---> neusis Simplicius 6; and Ammonius 169n5 on Archimedes' approximation of circle al-~abi, Abu Is~aq 592(nn34,36) ratio (n) 482, 494n30 al-~aghani 296 and Iamblichus 751n87 Saito, K. 171n25 on isoperimetrics 730-731, 734, 737, Saliba, G. 242n27 748(nn64,67), 811 salinon 498 on Plato and Aristotle 176n81 al-Samau'al 242n27 and Proclus 748n69 Sams6, J. 254 sources 100, 114, 126n124, Samson and Delilah 673 128(nnI46,149) Sarfatti, G. 449, 451n3 size (pelikotes) of ratio 157, 160-164, Sato, T. 396n7, 423, 451n16, l70(nnI6-17,26), 171n24, 172n32, 452(nn17-18),613n37 l73n36, 178, 185, 21On33; (pelikon) Sayili, A. 301 187; (megethos) 189 schesis (relation), and ratio 208n3 Slavutin, E. 795n14 Schneider, I. 396n2 Snel, W. 591n29 scholastic 239, 617, 640-641, 648-649, (historian) 794n4 654-655, 663, 669, 681n63, 810 Sodano, A.R. 751n87 scholia (marginalia) 127n128, 433, 447, solid loci: ---> loci 482, 543, 742n17, 797n45, 798n63; solids, in De curvis superficiebus 61On7; to of revolution 104, 107, 599-601, 717, Euclid 171 n26; to Heronian corpus 744n38, 746n55; Archimedean coni• 492n7, 493n16, 494(nn25-26), cal 776-777 509n34; in Abu 'I-Rashid 561-562; polyhedral 719-720; regular 63, (as commentary) 155, 164-165, 121n57, 308, 510n44, 716, 718-719, 172n34, 178,231,235-238, 244n72, 736, 744n39, 745n43, 750n86; semi• 495,767-768 regular 692,716,736, 744n37, Schoy, C. 307-308 745(nn44-45); ---> isoperimetric secant product 34-36, 39n25; ---> Euclid, figures Elements III 36 sections of: ---> sections sections, of planes 105; of solids 104, 107 Sosigenes 126n124 sectors, exterior 417n7, 529n16, sphere, surface and volume 601-602 632-633, 680(nn52,54), 681n71; [ ..... Archimedes]; lemma on volume similar 429, 51On45 603-604, 614n50, 717-718; arith• segment of circle, area approximation metic estimate for 599, 604-606 498, 500-502; ---> Hero [---> Hero] Serapeion 28n34, 53 592n43, 697-698; Serenus 244n74, 245n81, 799n73 ---> Theodosius Sesiano, J. 122n63, 592n40, 795n14 spirals (helikes) 128n150; Archimedean Severus Sebokht 699 51On45; use for angle division 214; Sezgin, F. 2, 224, 254, 450nl, 591n30, use for circle quadrature 494n38 592(nn43,44), 751n86 Sporus of Nicaea 483, 541; Shemuel ibn Tibbon 449 cube duplication 87-93, 114, 124n92 Shewring, W.H. 796n37 date 88, 93, 123n87 Shiraz 248, 297 Keria Aristotelika 58n26, 93, 108, Sicilian translators 614n57, 675, 114-115, 123(nn85,87,88), 124n93, 687n147,741nl 126n124, 230, 504-505, 799n81; on 850 Textual Studies in Ancient and Medieval Geometry

Sporus of Nicaea (eont.) of conics, Apollonian 86,94-97,99, Keria Aristotelika (eont.) 116,214-215,220-221,233, Archimedes' estimate for circle ratio 241n19, 263, 297, 771; Arabic (n) 494n38 284-286, 307; early 125n108 on circle quadrature 88, 123(nn82,86), of "givens" 85, 629-630 124n93, 128(nn146,149), 215, mechanical 112, 132, 135-137, 140, 223n17; ...... quadratrix 143-144, 146, 149; ...... groove; pivot; Steiner, 1. 723, 742n13 ruler Stein schneider, M. 441, 451 (nn3, 14) of isoperimetrics 511n47 Sturm, A. 245n83 of logical sequence (gar, etc.) 711, 766, style (editorial aims and methods, charac• 769, 772, 774, 778-780, teristic terminology, level of compe• 797(nn53-54), 800n89, tence, originality, etc.): 801(nn103, 106, 110) ...... commentators, ancient; medieval for numbers (large): ...... myriads tradition and translations; scholastic; for numerals in mss. 489, 591n17 and individual authors of ratios 160, 166, 173n38, 180, 201, Suidas 743n21, 754-755, 765, 203, 208n5,209nI7, 480, 493n21, 794(nn4-5) 531n43,606 sundials 24, 53 used in translations: Arabic 107, Suter, H. 80, 450n2, 543 127n139, 429, 434, 445, 446-450, symmetry (of exposition) 35, 383, 430, 544, 547-549, 561-562, 583-584; 432,664, 715, 767-778, Hebrew 429,434,445,449-450; 799(nn75,87); ...... Hypatia, style Graeco-Latin 597-598, 608, Synesius of Cyrene 754-755, 763, 611(nn16,23); Arabo-Latin III9, 794(nn3-5,1O), 795nll, 796n37, l27n139, 429, 434, 597-598, 606, 811; on scholarship 813-815; on 611n23, 612n24 isoperimetrics 731, 735; style of Thiibit ibn Qurra 6, 101, 216-222, 249, 763 272; syntagma (collection, composition) 166, on balances 592n38 381,388, 400n56, 401, 417nl,483, editions: of Archimedes 218, 421, 495, 813 441-442,448-449, 450n1, 539, 547, Syrianus 5, 155, 166-168, 549, 551, 809; of Euclid 442; of 176(nn78,81-82,84), 177n88, 202, Eutocius 311; of Menelaus 127n135, 723 217, 262, 288-289 style 285-287 Thaer, C. 58 Theisen, W. 615n57 tangents, lemma on ratios of 697-699, Themistius 747n64, 748n66, 735, 742(nnI6-17), 749n86; 751(nn90-91),796n37 ...... chords Theodosius 28n34, 511n46, 592n43, Tannery, P. 26n22, 58n24, 108, 123n88, 698 124n93, 126n124, 128n146, 167, Theon of Alexandria 5, 169n2; 169n5, 176n86, 201-202, 21On48, on Archimedes' circle theorem (T; cf. 241n20, 242n34, 243n38, 245n83, Dim. eire., prop. I) IIII; 401-406, 492n3, 493n15, 755, 794n4, 423-435, 511n46, 513, 607,805; 795(nn13-14), 798n62, 799n73, text 387-391, 392-393 808 on Archimedes' estimate of circle Tee, GJ. 452n16 ratio (n) 479, 480-482, terminology, 493(nnI8,20) of Apollonius 127n128, 176n77; ...... of on area of circle, arithmetic estimate conics (cf. Dim. eire., prop. 2) 493n14, archaic (Greek) 43, 100-101, 115, 479-480, 483, 503, 614n48, 753 128n46 cited, by Eutocius 157-159, 165, 178; Index 851

in Introduction to Almagest 155, triads, Menaechmean 99-100, 115, 698-699, 723, 741n3 126n120, 142, 150 commentaries: On Small Astrolabe trigonometry 593n54 698-699, 743n21; On Small al-Tusi 6, 54, 119n16, 248-249, 254, Astronomer 698-699, 723; On Hero's 417n9, 617; Metrica 243n44, 479, 493n15, cube duplication: --> cube duplication 741n3; On Ptolemy's Almagest IIII, edition of Archimedes 421, 434-435, lIllO, 227, 754, 756; On Ptolemy's 441, 448, 450nl, 451n16, 452n28, Tables 175n63, 417nl, 793n2, 532(nn52,55), 543, 546-551, 809; 795n13 text 577-590; mss. 592n43 on compound ratio 170n15, 17ln20, edition of Banu Musa 79-80, 103-105, 174n60, 175n61, 21On48 119n16, 127n139, 536, 549-550, editions of Euclid 398n36, 452n 17, 601, 611n23 748n73, 816n4 edition of intermediate curriculum on Hypatia 756, 762, 808 546-547 on isoperimetric figures (TI) II11O; and al-Harawi 259, 262-265 169n3, 615n60, 774-776, 811; con• on rectification of circle, hypothesis of tents 692; summary of comparisons 542, 613n44 723-725; on solids 51On44; variants style 264-265, 547-551 in mss. 709, 747n61 lemma to Archimedes' circle theorem (T) 403-405, 427-428, 518-520, Unguru, S. 242n29 529n22, 544, 753; text 414-416, 534 unrolling (anaploun, exaploun) 379, on long division 757-764, 770-771; 398n25 texts 780-787, 802-804 on proportionality of arcs and sectors 506n4 relation to Pappus 227, 240n7, 398n36, Ver Eecke, P. 243n35, 492n3, 482, 796n38 742(nnI3-14),745n52 on square roots 156, 175(nn68,75) Victor, S.K. 686(nn135-140), style, IIIlO; 382-384, 406, 409, 423, 687(nnI43,145-146) 481, 758-764, 774-776, 779, 787, Vieta, F. 158 796n31, 812-813, 816n4 Vitruvius 24,42, 128n152, 51On40 on volume of sphere, arithmetic esti• van der Waerden, B.L. 240(nn3,4), mate 493n16, 530n43, 599, 604-606, 796n28 612n31,614n48 on Zenodorus l24n9l, 379, 381, 690-691, 741n7 Abu 'l-Wafii' 549,582, 593(nn52,57), 809 Theon of Smyrna 45, 57n18, 140; and Waithe, M.E. 794n4, 795n16 Theon of Alexandria 493n14; on Wallies, M. 48 ratios 173n36 Weaver, J.H. 122n74 Thomas, I. 169n5; --> Bulmer-Thomas von Wilamowitz-Moellendorff, U. 16; Tihon, A. 169n5, 793n2, 795n13 74n23 , III, 115, 245n83 Tinemue, = Tynemouth 595,597; vari- Wilson, N.G. 531n28 ants 61On5, 611(nnI4-15) Witelo 229 Tiphys 596-598 Woepcke, F. 224(nnI9,31), 294, 307, Toledo Tables 687n149 593n57 Toomer, GJ. 81, 120(nn25-26,29), Wright, J.K. 688nl49 121n41, 122n63, 240(nn4,7), 318, 398(nn31,36), 748n79, 797n38 torus 104, 108, 110 tradition 6-7,238-239,551,815-816 Ya'akov b. Makhir 6,442,450, 451nl4 Treweek, A.P. 240nll, 273 Youschkevitch, A.P. 750n86 852 Textual Studies in Ancient and Medieval Geometry

De ysoperimetris 595, 614(nn45,57), 510(nn43-44), 690-691, 710, 723, 680n46, 689, 741n1, 743n18; adapta• 728, 730, 734-737, 741n7, 811-812; tions 686n132, 687n147, date 398n31, 742n14, 748n79, 688(nn154,157), 741n1, 810, 816n3; 751n92; On /soperimetric Figures, and John of Tynemouth 676; citation lost tract (Z*) 697-698, 701-702, of Archimedes' circle theorem 607 704, 710, 715-716, 724-728, 732, 734; relation to extant accounts 724-728; on koilogonion 747(nn60,62) al-Zarqali 687n149 Zerachiah b. Yitshaq Ha-Sefardi 751n90 Zenodorus 124n91, 169n3, 227, Zeuthen, H.G. 238, 240nlO, 245n83, 379-381, 386-387, 398n29, 747n58 400(nn55-56), 402, 405, 502, Ziegler, K. 242n34, 243n37, 796n38