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 (Vati­can, 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) Anthemius of Tralles. 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) Archimedes. 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) Aristotle. 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

Boethius. De institutione arithmetica libri ii, De institutione musica libri v, ed. G. Fried­lein, 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 Damascius. Vitae Isidori Reliquiae, ed. C. Zintzen, Hildesheim: G. Olms, 1967 Diocles. On Burning Mirrors, ed. GJ. Toomer, Berlin/Heidelberg/New York: 1976 -. see also R. Rashed Diophantus of Alexandria. 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 ratio from a ratid'): see

C.E. Ruelle Euclid of Alexandria. 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. Stama­tis, 1977)

-. Data (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 Eutocius of Ascalon. 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)

Hero of Alexandria. 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 Diop­tra, 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 Hypsicles. Book XIV of the Elements: see Euclidis Opera Omnia, ed. J.L. Heiberg,

vol. 5 Iamblichus 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 Marinus of Neapolis. 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

Menelaus of Alexandria. Sphaerica, ed. E. Halley, Oxford, 1758 Nicomachus 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 Pappus of Alexandria. 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 Ptolemy: see A. Rome (1931) -. Commentary on Euclid's Book X: see W Thomson and G. Junge Philo of Byzantium. Belopoeica: see E.w. Marsden Philoponus: see Joannes Philoponus Plato. Opera, ed. 1. Burnet, (Oxford Classical Texts), 5 vol., Oxford: Clarendon Press,

1900-07 (repr. 1977) Plutarch of Chaeronea. Scripta Moralia, ed. WR. Paton, I. Wegehaupt, et aI., 7 vol.,

Leipzig: Teubner, 1925-67 Porphyry. In Platonis Timaeum Commentariorum Fragmenta, ed. A.R. Sodano, Naples:

[no publ.], 1964 Proclus 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, Inscrip­tio Canobi, Procheiron kanonon diataxis, Analemma, Planisphaerium, etc., 1907), Leipzig: Teubner

-. Almagest = 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 Synesius of Cyrene. Opuscula, ed. N. Terzaghi, Rome: Regia officina polygraphica, 1944 -. Hymnes, ed. C. Lacombrade, Paris: Les Belles Lettres, 1978 -. see also A. Fitzgerald Syrianus. 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

Theon of Alexandria. 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) Theon of Smyrna. 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 angle], 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 circle) [Recension of Dimension 0/ the Circle, with preface, "Theorems 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 (Istanbul), 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 (Istan­bul), 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 Know­ledge 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 Middle Ages, 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, Appen­dices 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 al­zawiya 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) William of Moerbeke. [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. "Hypatia in Tradition und Dichtung;' Studien zur vergleichenden literatur­geschichte, 1907, 7:11-44

Aujac, G. "Le rapport di isou (Euclide V, definition 17): Definition, utilisation, transmis­sion;' 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" (Eudoxos­Studien IV), Quellen und Studien, 1936,3 (Abt. B): 370-388

Berggren, 1.L. "Spurious Theorems in Archimedes' Equilibria of Planes, Book I;' Archive for History of Exact Sciences, 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 Science, 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 Mathematics of Medieval Islam, New YorklBerlin/Heidelberg/London/ Paris/Tokyo: Springer, 1986

Boll, H. "Heliodorus": see PW Brandis, C.: see Aristotle Bregman, 1. Synesius of Cyrene, Philosopher-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, Geomet­rica, etc.)

Bulmer-Thomas, I. "Eutocius of Ascalon;' DSB, 4 (1971): 488-491 -. "Isidorus of Miletus;' DSB, 7 (1973): 28-30 Burkert, W. Lore and Science in Ancient Pythagoreanism, 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 Roger Bacon zugeschrie­ben 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 of Euclid's Elements Commonly Ascribed to Adelard of Bath, Toronto: Pontifical Institute of Mediaeval Studies (Studies and Texts, 64), 1983

-. The Latin Translation of the Arabic Version of Euclid'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 Figu­ren], 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 of Mechanics 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 (Philadelphia: 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 of Geometrical 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 bib­liographical 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), Barce­lona: 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 -. "Ctesibius (Ktesibios);' DSB, 3 (1971): 491-492 DSB = Dictionary of Scientific 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.,

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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 of Bath:

An English Scientist and Arabist of the Early Twelfth Century, London: Warburg Institute (Surveys and Texts 14), 1987, pp. 55-68

Fowler, D.H. "Ratio in Early Greek Mathematics;' Bulletin of the American Mathematical Society, 1979, 1 (n.s.): 807-846

-. The Mathematics of Plato's Academy: A New Reconstruction, Oxford: Clarendon Press, 1987

Gericke, H. "Zur Geschichte des isoperimetrischen Problems;' Mathematische Semester­berichte, 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 Greek Anthology: 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 of Greek Philosophy, 6 vol., Cambridge: Cambridge University

Press, 1962-81 Hartner, W. "The Principle and Use of the Astrolabe;' in A Survey of Persian 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 Univer­sity 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. Apollonius of Perga: Treatise on Conic Sections, Cambridge: Cambridge University Press, 1896

-. The Works of Archimedes, Cambridge: Cambridge University Press, 1897; Supple­ment: 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 of Euclid'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

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828 Textual Studies in Ancient and Medieval Geometry

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830 Textual Studies in Ancient and Medieval Geometry

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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 number (or let­ter) 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 omit­ted; 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, 632-633, 643; medieval terminology 445, 447-448

abacus 759, 796(nn26-27) Abraham bar l:Iiyya 449 Abraham ibn Ezra 449 Adelard of Bath 597-598 Adelardian versions of Elements 612n25;

Adelard I 798n66; Adelard II 641, 658, 678(nn22,24), 680n51, 681n67, 683n93; Adelard IlIA 597, 611nI9, 685n127, 687n144, 688n153; Adelard IlIB 688n154; ---> Euclid, Elements, medieval versions

A~mad ibn Musa 249,262; ---> angle trisection

Albert of Saxony 618, 649, 676, 678(nn22,24), 810; ---> Quadratura circuli, Questio of Albert of Saxony

Alexandria 5-6, 27n22, 41, 63, 143, 145, 167, 176(nn81,84), 177n87, 227, 229,754,770,813

Ammonius 6, 27n22, 155, 167, 169n5, 177n87, 229-230, 243n39, 527, 754, 796n35

Ammonius Saccas 754 analysis and synthesis 85-87, 114,

213-217,220,226, 238, 239n2, 288, 297, 303-304, 308-309, 751n91;

in Arabic 229, 242n25 of Diocles' solid problem 116 in Diocles 151nll

834 Textual Studies in Ancient and Medieval Geometry

analysis and synthesis (cont.) in Pappus 91, 223n7, 744n40 in Menaechmus 94-99 of angle trisection 213, 272, 274 of Hero-Apollonian cube duplication

36-38 of hyperbola construction 213, 232 of regular solids construction 121n57

analysis, = raising of sexagesimal order 760, 784-786, 792

Anaritius (al-Nayrizi) 670, 685n127 angle trisection,

of A\:Imad ibn Musil lIB; 217-218, 228, 283-289; text 267-270, 326, 328, 330, 358-360

of Archimedes 214,216,219,274, 295-296

in Arabic 219-220 of Banu Musil 80, 536 via conchoid 214,219, 223n11 via hyperbola 214, 220-221; in Pappus

308-309; text 303-304, 340-341 via neusis, in Pappus lIB; 249, 283,

287-288; text 267-270, 327, 329, 331; -+ of A\:Imad ibn Musil; of Archimedes; of Thiibit ibn Qurra

of Nicomedes 274; -+ via conchoid in Pappus (survey) 18; 233, 295 of al-Quhi lIE; 220-222, 238,

294-300; text 301-303, 339-340 of al-Sijzi, lID; 220, 305; text

293-294, 338, 370 via solid locus 214-215 of Thilbit ibn Qurra lIe; 216, 218, 228,

295-296; text 277-279, 330, 332-334, 361-364

annulus (itys) 497-498, 507n22 anonymous, Peri isoperimetrbn (On

Isoperimetric Figures) (AI) IIIlO; contents 692; summary of compari­sons 723-725; text 738-741

mss. 741n2, 743n18 prototype (AI*) 723, 725-726, 728,

730-734, 753, 774-776, 778-780; provenance 776, 779-780; relation to CS*, DC* 733-734

-+ Introduction to the Almagest anonymous, Introduction to Ptolemy's Syn­

taxis [-+ Introduction to the Almagest]; Mogenet's thesis on authorship of 17 [-+ Eutocius; Moge­net]

Anthemius of Tralles 6, 26n21, 58n33,

83,114, 124n97, 176(nn85-86), 177n87, 229-230, 318, 521, 799n78

Antiphon 114, 128n149 Apollonius 33-34, 46, 50, 75n40;

analytic works of (lost) 214-215, 240n12, 241n19, 242n26

in Arabic 229, 248-249 commentaries on 5-6, 235-236,

244n74, 755, 765, 767, 770; of Eutocius 226, 228-230, 245n82, 765-771; of Hypatia 406, 755, 765, 769-770; -+ Pappus, lemmas on the Conics

on compound ratio 158 Conics, proposition numbers of

125n102; suppressed first edition of 231, 244n53; in Arabic 218,222, 259-260, 306; variants in mss. 286

Conics II 4: 284; text 331,333, 367-368 [-+ Eutocius, hyperbola construction; angle trisection, of Thiibit]; II 8: text 257, 326, 356; II 12: 279-280, 335, 369; IV 30 (35): text 257, 327, 357

on constructions 55-56, 118 cube duplication, in Philoponus (AJ)

Il, (PK) 13; in Eutocius (AE) 81; via conics 54, 75n40, 77; -+ cube dupli­cation, of Apollonius

On Plane Loci 797n45 sundial of 28n34, 53

approximation 75n36; of cubic roots 536 of curves 121n39 of square roots 156, 522-523,

540-542, 546; Hero's rule for 493n15, 592n42; values for 492n9 [-+ Archimedes, on square roots]

of ratio of circumference to diameter of circle (1t) 230, 544; Archimedean values of 127n133 [-+ Archimedes, Dim. circ., prop. 3]; alternative values 541,550, 591n29, 593(nn52-57), 816n2; of Ptolemy 397n22

Aratus 93, 123(nn87,89) arbelos 498 Arcadius 157,159,165-166,168,

170n15, 178,230,417,689 Archimedes 58n24;

axiom on convergence (')\rchimedean­Euclidean axiom") 408, 418n26; Latin variant 684n104

axiom on including (convex) arcs 63, 72n6, 386, 417n9, 603, 682n80 [-+ rectification]

axiom on least distance 382, 399n41, 406

on balances 496, 506n12 centers of gravity 243n45, 594n58 chronology of works 418n15 citations of Euclid 381-382, 399n37,

408-409, 418n33 cited, in AI 159, 167, 186,700-702,

717-719,726-728; in Geminus 52; in Proclus 729; in Simplicius 730, 734, 737; in Theon 742n14 [-+ Hero; Pappus; Proclus]

commentaries on 6; Eutocius 226, 229-230

on compound ratio 158 conical solids ( surfaces) of 598-60 1 ,

747n57,776 on conics 223n16 corpus of writings, evolution of 531 n51 crown problem of 51On40 and cube duplication (Heronian) 53 editing of corpus 409-410 heptagon problem 307 on infinite 737 on inscribed polygons 735-736,

749(nn82-83,86) and isoperimetrics 502-503,729-730,

733-738 on measurement of sphere (Sphere and

Cylinder) 377, 503, 538, 596, 605-606; rule for volume of sphere 493n16; on spherical sectors 612n30

mechanical writings of 531n51 on mechanisms 129n152 medieval text of Sphere and Cylinder

528, 532n55; in Arabic 442-450; in Hebrew 442-450; in Latin 598; adap­tations 610 [-+ De curvis superjicie­bus; BanO MOsa]

on parabolic segment, area of 377, 498-501,508n33

on ratio of cylinder and cube 208n9, 605 [-+ indivisibles]

sector theorem 431, 435 [-+ Hero, on Archimedes]; medieval form 422

solid problem of 85-87, 101, 116-117, 222, 223nI4,230-231, 722, 743n26, 746n56

on spirals 95

Index 835

on square root approximations 492n9, 504

style 508n33, 509n38 synthenti summation lemma for ratios

744n33 triangle rule 51On40

Archimedes, Dimension of Circle, survey of textual tradition 11112; table 806

as extant in Greek (DC) 495-497, 503-505, 537-538, 541; date 453n35; as known to Eutocius 1II6 [-+ Eutocius]; style 403, 527, 771-774; defects 375,477-478,496, 514, 548

prototype (OC*) 1II6, 422-423, 495-496,503-505; date 608; provenance 753-756, 774, 779-780; relation to AI/AI* 689, 702, 733-734

older prototype (A*) 495-496, 502-505,806-807; relation to Sphere and Cylinder 527, 808

prop. I (circle theorem) as extant (OC) III2; text 411-414; 422-435, 536-538,547,700,702; prototype (A*) 1II2; text 411-413; prototype (OC*) 424-425, 431-435; versions in Pappus (P) and Theon (T) 1111; cita­tions of circle theorem 376-380

prop. 2 (area of circle), as extant 1II4, 397n23, 506n7; text 484; prototype (OC*) 479-483; citations of 478

prop. 3 (approximation of ratio of cir­cumference and diameter) III4, 397n23, 497, 503-505, 536-541; text 485-489; algorithm for 511n48; rounding off technique for 504, 512n50; alternative values 397n23, 502; citations of 480-481

medieval translations III3-4, 615n58, 805; text of Arabic (in English trans­lation) 436-438, 484-489; Hebrew and Latin variants 422,438-441, 489-491; facsimiles, of Arabic (AF) 455-463, of Hebrew (H) 464-465, of Latin (LP, LG) 466-475 [-+ Gerard of Cremona; Plato of Tivoli; Quadratura circuli]; tradition of AF 544-551; Arabic adaptations III7

Archytas 26n22, 113, 128(nnI51-152), 142, 147, 150, 158; cube duplication of 53, 79-80, 110; -+ cube duplica­tion, of Archytas

836 Textual Studies in Ancient and Medieval Geometry

area of circular sectors 505n3; -- Archi­medes, sector theorem

area of circular segments 507n24; -- Hero, on circular segments

Arendt, F. 243n49, 493n14 Aristaeus 72n7, 75n40, 77, 123n83, 215,

223n15, 228, 237, 240n12, 241n21 Aristarchus 28n34, 530n38, 592n43, 698 Ariston 144 Aristotle, on applied science 74~n70; on

astronomy 128n150; cited 682n76; commentaries on 6; on isoperimet­rics 730-731, 734, 737-738, 748n67, 811; locus problem 57nIl; on parts (mathematical) 57n12; on zoology 93, 123n89, 747n64, 751n9l

[Aristotle], Mechanics 496, 506n8 arithmetic, relative to geometry 157-159,

171(nn19,21,24), 181, 21On48, 550, 593n47; verbal expression of numerals 593n50

arithmetic operations 155, l72n34; division by table 757-759,761-764;

text from Theon 780-784,786-787; text from Introduction to Almagest 787-793

expression and manipulation of large numbers 539-540, 545-546, 591n18, 592n40; in Quadratura circuli (Ger­ardian, Florence and Glasgow ver­sions) 665-667; reduction of ratios 667

long division 757, 759-760; text from Theon 784-786, 802-804

long multiplication 522-523, 591n19 sexagesimal methods of 550,757,759,

781,783,786,788, 796n31 [-- divi­sion by table; long division]; notation 764, 790-791

tabular methods of 763, 789-791, 796(nn22,26)

Artis cuiuslibet consummatio 672-676, 811

astrolabe 699, 795n11. 814, 816n12; -- Theon of Alexandria, commen­tary, On Small Astrolabe

astronomical corpus, minor 698; -- inter­mediate corpus

astronomical tables 670, 687n149 astronomy 592n43; -- Aristotle;

Eudoxus; Plato Athens 5, 166-168, 176n81, l77n87, 526 August, E.F. 798n66 Aujac, G. 678n23

Autolycus 592n43, 698 axis 105, 107

Babylonian mathematics 507n24, 763 Baghdad 6 balances 151n9, 171n19, 496, 506n12,

816(nn1,3) Balsam, H. 243n49 Baltzer, R. 125n106 Bam} Musa 6, 79-80, 809;

on Archimedes' Dimension of Circle 535-543

on Archimedes' measurement of sphere (VF) 538, 596, 598-599

and Archytas' cube duplication 101-110, 11~

convergence method of 598, 601-602, 613n38

and De curvis superficiebus 598-603, 608

and Eutocius 809 expression for ratio of circumference

and diameter (1t) 531n43, 539, 591n12,6OO

on geometry (tract) 119n16, 809 in al-Khazin 750n86 lemmas on polygons 536-537 and Menelaus 53,219,289 on ratio of circumference and diameter

497, 506n15, 602 style of 536,551

Bashmakova, I. 795n14 Becker, 0. 418n21 bees 93, 748n64, 751n9l Berggren, 1.L. 228, 264, 591n32,

592(nn34-35), 594n58, 816(nnl-2) Berosus 28n34 Bible (Vulgate) 673-674, 678n18,

687n141 al-Bhun) 127n135, 242n24, 296, 593n54 bisection 418n26, 631; -- convergence,

Euclidean; Euclid, Elements X 1 Boethius 170n17, 673, 676,

687(nnI41-142) Bradwardine, T. 644, 669, 671, 682n73 Bregman, 1. 755, 794(nn9-1O), 796n36,

816(nn6, 11) Bruins, E.M. 494n25 Bulmer-Thomas, I. 240(nn4-5), 242n34;

-- Thomas Burkert, W. 171n23 Burnett, C. 612n25 burning mirrors 318; -- Diocles; mirrors

Busard, H.L.L. 611n22, 612(nn24-25), 614(nn45-46), 615n57, 669-671, 680n46, 685(nnI24-125,127), 687n147, 689, 741nl

Byzantine scholarship 531 n48

Cameron, A. 794nlO, 795n18, 807-808 Campanus 669-670, 682n76,

685(nnI27-128),688n155 Cantor, M. 742n14, 748n75 Carruccio, E. 742n13 catapults 13,41,53, 75n40, 137, 141,

143, 148 center of gravity 506n5, 592n36, 594n58,

816nl; -> Archimedes chords (in circles) 103, 107,619,635,

642, 645, 656, 664; product of segments 34-36 [--> Euclid,

Elements III 35-36] construction of, in Archimedes and

Pappus 590n6 lemma on ratios of 682n80, 735,

742n16; -> tangents use for computation 541-542,550,

591n29 circle, as infinite polygon 749n84 circle quadrature 620, 644, 647,

649-650, 654, 662, 677n3; early efforts on 93,108,114,501, 51On39; via quadratrix 88, 223n10, 227 [-> quadratrix]

"cissoid" 122n76, 129n154; -> Diocles, curve of

Clagett, M. 2, 396n2, 421, 451n4, 452n29, 595-598, 61O(nn1 ,4), 611(nn8, 10-11,13,19-20,23), 612n25, 613(nn37,44), 614(nn51-52,55), 617-618, 625, 628,638,640-641, 644, 648-649, 655, 663-666, 669-672, 677(nn7, 15), 678n16, 679(nn37,39,43-44), 680(nn45,51-52,56,59), 681(nn67,73),682n80, 683(nn85,90,95,97),684n107, 685n 121, 688nl55

collation (paranagignoskein) 521, 756-757,762, 795(nnI6,18), 807-808

Commandinus, F. 243n49, 272, 742nl4 commentaries, ancient format of 231.

236,244n72 commentators (ancient), style of

225-226,551,809.812-816;

Index 837

-> Hypatia; Eutocius; Pappus; Theon of Alexandria

commentum 677, 813 compass 18-19, 117; for drawing para­

bolas 99, 115, 120n38 [-> Isidore of Miletus]

conchoids 30n34, 56, 75n40, 78, III, 219,316-318; -> neusis, for angle trisection; Nicomedes

cone, as surface of revolution 765, 767; equated to sphere 717-719; isoperi­metric 720-722, 745n53, 746n54

conics, early theory of 120n38, 128n146, 215, 223(nn14-16), 241n21, 308, 771; terminology of (Apollonian) [ --> terminology]

Constantinople 6, 526, 531 n48, 813 construction,

via conics 213 curve tracing apparatus 109-110. 117,

119n12, as existence proof 747n58 geometric vs. instrumental 262-264,

295-296, 316-318 graphical 70, 75n37, 117 mechanical 117-118, 120n32, 213; of

cube duplication 79-80, 132-133, 135-136 [-> cube duplication, of Eratosthenes; of Nicomedes; attributed to Plato]

physical 316 planar 117-118, 122n72, 241n19 pointwise 83,94, 118, 121n39, 290 solid 117 -118

continually (aei) 448-449; medieval ter­minology for 434, 449

continuity and divisibility 646, 653 convergence,

Archimedean (Sph. Cyl. I 1-6) 385, 407-408,497,602,612n36

Euclidean 383-385, 402-403, 406-407, 501-502, 507nl9 [-> Euclid, Ele­ments Xl; XII 2]; in Latin 683n93

Euclidean alternative form (Elements, XII 16) 497, 506n16, 507n19, 537-538, 601-602, 613n38-39; in Latin 598-600, 646 [--> Bam) Musil.; De curvis superficiebusl

Eudoxean 406-407 in Pappus (alternative) 744n38

convergence enhancement 51 On42, 591n29

Coolidge, J.L.L. 241n17, 742n13, 744(nn33-34)

838 Textual Studies in Ancient and Medieval Geometry

Ctesibius of Alexandria 41-42, 56n5, 129n152, 145

cube duplication, anonymous, in Pappus 14; 112, 228 of Apollonius (in Eutocius, AE) II; 78,

113; (in Philoponus, AJ) II; (in Philoponus, PK) 249, 259-261; text 251-253, 321, 323

of Archytas 100-110, 289 of Diodes 39n8, 113-114; and Sporus

89-90 of Eratosthenes 16; Ill, 115, 228 of Eudoxus 108, 112 of Abu Bakr al-Harawl lIA; 220, 222,

225,271,287,302,305-307; text 251-253, 320, 322, 364-366

of Hero II (in Belopoeica, HB) 41, 45, 51, 119n8; (in Mechanica, HM) II, 51; (in Eutocius, HE) II, 45, 51-52; (in Pappus, HP) II

in Eutocius (survey) 15, 769-770 of al-Khazin 254, 262-264; text

254-255, 322-324, 352-354 of Menaechmus 39n8, 94-100,

114-115, 225 of Nicomedes 108, Ill, 115, 245n83,

249 of Nicomedes (in Eutocius) 316-318;

text 312-315, 342-345; (in al­Khazin) IIF; text 311-315,341-342

of Pappus 87-93, 114, 227, 238; ->

attributed to Plato of Philo of Byzantium 113; in

Belopoeica (PB) 13, 25nlO; in Eutocius (PE) 13, 78; in Philoponus (PJ) 13; the same, attributed to Apol­lonius (PK) 13, 36-38

attributed to Plato 36-38, 78-80, 109, 112-113, 536; and Pappus 89-93

of al-Quhl lIE, 220; text 301-303, 339-340

of Sporus 87-93, 114, 124n92; -> of Diodes

of al-Tus! 264-265, 302; text 255-256, 324-325, 355

curves, as solid sections 104, 107-108, 128n150; on surfaces 128n150

De curvis superficiebus (CS) Ill8; adaptations of (medieval) 628-631,

672-673; -> De piramidibus ancient provenance (as prototype CS*)

497, 507n17, 538, 595-596, 608-610, 776-780, 809-810; relation to AI 608-610,689,718, 733-734

area expressions in 453n40 arithmetic estimate of sphere volume

604-606 convergence method in 538, 613n38 influence of 595-596 lemma on sphere volume 603-604 mss. 679(nn28-29) on proportion of circumferences and

diameters 497, 506n16, 507n19, 538, 613n40

and De quadratura circuli 614n55, 617, 620-627, 635-637, 646, 672; citations of De quad. circ. 606, 615n58

on rectification of curves and surfaces 613n44; -> rectification hypothesis

relation to Archimedes' Sph. Cyl. 596, 598-603, 609-610

relation to Baml Musa (VF) 596, 598-603, 608

style 596-598,621-623, 678n18, 776-780

cylinders, isoperimetric 720-722, 745n53, 746n56

cylindric line 104, 107 Cyril, patriarch of Alexandria 754

al-Dabbagh, 1. 749n86 Dales, R.C. 688n150 Damascius 794n4 Damianus 731, 748n73 Delian orade 17,45, 57n18, 140, 147 Demetrius of Alexandria 223n17 di'isou 605-606, 626, 678n23, 695-696;

and compound ratio 171n24 diagrams,

for Archimedean cirde theorem, com­pared 432, 433, 529n22

lettering of 15, 19,48,50,96,218, 244n63, 262, 271-272, 383, 432, 519,521, 529n19, 545, 548, 650, 694-695,698,700,703, 705-706, 708-709,713-714; medieval 451(nnI2,l5); lettering written out in Arabic 318-319

Diels, H. 53 Dijksterhuis, E.1. 396(nn2-3), 418n26,

492n3, 612n36, 744n37 Dilgan, H. 749n86 Dillon, 1. 751 n87 Dimension of the Circle; -> Archimedes;

Quadratura circuli Dinostratus 216, 223n18 Diodes 26n21, 308;

on Archimedes' solid problem 85-87, 96, 116,222, 223n14, 743n26

on burning mirrors 81, 94, 113, 124n97

on conics 221, 223nl6 cube duplications of, via special curve

81-85; via two parabolas 94, 100, 115; ---> cube duplication

curve of 82-84, 89, 113-114; ---> "cis­soid"

in Eutocius 151nll problem of two means 32

Diogenes Laertius 128nl52 Dionysius Cyzicus 153n36 Dionysodorus 28n34, 222, 743n26 Diophantus 5, 57n14, 158, 169n2,

209n17; in Arabic 592n40; com­mentaries on 406, 755, 765, 795nl4 [---> Hypatia]; on ratios 173n36

[Diophantus], on Archimedes' rule for area of circle 492n7; on polygons 492n9

diorism 87, 722 "divine" (theios) 174n46,

176(nn79,81-82,85); "divine Eudoxus" 150; "divine men" (theioi andres) 167, 186

division, of sexagesimals 176n78; ---> arithmetic operations

Dold-Samplonius, Y. 254 Domninus of Larissa 166-168, 176n77 ,

177n89 Doric dialect 8,86,101, 122n62,

171n23, 410, 452n17, 528, 532n57, 807-808, 816nl

Dositheus 120n37, 398n31 doubling of cube (restricted form of

problem) 71, 137, 147; ---> Delian oracle

Dover, K.1. 796n37 Drachmann, A. 26nl2 Durnheimer, W. 680n59, 683n90 dynamis (power, mathematical) 43,

599-600, 612n33, 704, 725, 743n28; (force, mechanical) 496, 506n9

Egyptian mathematics 507n24, 51On39 ellipse 85-86, 245n79; area of 507n20 ellipsoid 83 Emden, A.B. 611(nnI3-14) encyclopedias 52; ---> Geminus;

Menelaus epicycle and eccentric 764 epigram 153n36; ---> Eratosthenes

Epiphanius 754, 793n2 Eratosthenes 30, 33, 75n40;

Index 839

on cube duplication, account of (authenticity) 16 [---> Wilamowitz]; early history of 128n151, 131, 142, 147

cube duplication method of 16; 71, 73n14; and anonymous (in Pappus) 64, 67-68, 70 [---> Pappus]; in Eutocius 78 [---> Eutocius]

date 145, 153n38 epigram 99,131-134,144-145, 150,

151n15, 153n36 epitaph to 153n36 on loci 240nl2 measure of earth 481; commentaries on

481-482 mesolabe 111-112; ---> mesograph Platonicus 45, 57n18, 79, 113 value for obliquity of ecliptic 530n38

Erycinus, paradoxes of 63 Euclid 592n43;

on application of areas 749n81 on astronomy 698 on balances 25n6 cited, by Archimedes 381-382,

399n37, 408-409, 418n33; by Eutocius 174n42; by Latin geometers 678n26, 684nl07, 686nl39

commentaries on 5, 227 on conics 223n15; ---> conics, early

theory of on fourth proportional 764, 789 on loci 215, 240n12, 244n75 lost works 240n12 Optics. Data, Catoptrics (Graeco-Latin

translations of) 614n57 Phaenomena 417n 1 style 35, 158-159, 407, 596,

80ln107 Euclid, Elements:

122 (on construction of triangles) 704, 708

147 (identity for right triangles) 661, 708

II 8 (difference of squares identity) 507n22

III 7-8 (on secants) 768-770, 798(nn60,66), 799n75

III 15 (on chords) 798n55 III 35-36 (on segments of chords and

secants) 34-36, 39n25, 50, 55, 58n22, 260

V def. 3 (definition of ratio) \59, 163, 185

840 Textual Studies in Ancient and Medieval Geometry

Euclid, Elements (cont.) VI def. 5 (definition of compound

ratio) 157-63, 170n16, 171n25, 177-178, 186, 202, 204, 21O(nn36,47)

VI 3 (on angle bisectors) 504, 561n15 X (on irrationals) 799n75 X 1 (on convergence via bisection);

Latin citations 631,641,646, 658-660, 681n67; subtraction proce­dure 653-654, 660; --> convergence, Euclidean

XII 2 (on ratios of circles) 381-385, 425,429,496,501,527,646, 796n30

XII 16 (on chords) 637; --> conver­gence, Euclidean alternative form

mss. 612n25, 678n26 medieval versions: Graeco-Latin

614n57; Arabic and Arabo-Latin 597, 612n25 [--> Adelard; Gerard of Cremona; Hermann]; citations in Latin 623,625, 627, 673

[Euclid], on polygons 492nn8-9 Euclidean convergence (subtraction, or

bisection) procedure: --> conver­gence; Euclid, Elements X 1

Eudemus of Rhodes 100-10 1, 106-109, 114-115, 123n85, 126n124, 128n146, 149, 289

Eudoxus ofCnidus 26n22, 113, 128n151, 418n21; cited by Eratosthenes 142, 147, 150; by Hero 51On40; in Introduction to Almagest 159, 167, 186; astronomy of 126n124, 128n150; cube duplication of 78-89; sundial of 28n34

Eulalius 794n2 Eutocius 6, 11, 254, 615;

alleged author of Introduction to Almagest 17; 67-68, 788, 801n11O; --> Mogenet

and Ammonius 167, 169n5 in Arabic 127n132, 262, 447, 809; in

al-Tusi547, 549; possible use by Baml Musil 539-543

on Archimedes' circle theorem (Dim. Circ., prop. I) 424, 432-433, 452(nn21,32), 607

on Archimedes' estimates of circle ratio (n)494n38, 504-505, 510n41

on Archimedes' Sph. Cyl. 16: 418n24, 445-447,516; on 110: 516-520; text 533

citations: of Conics 306; of Ptolemy commentaries 417

commentaries 155-156; on Apollonius 222, 224n39, 231, 237-238, 245n81, 446,765-771; on Archimedes 63, 410, 445-447, 482-483, 757; on Ptolemy, alleged 164-166, 169n5, 175n7l,230

on compound ratio 17 on cube duplications 15; 39n16, 217,

222, 262-264, 295, 300, 798n68; of Apollonius (AE) II; of Philo (PE) 13, 41; --> cube duplication

date 26n22, 229-230 on Diocles 308 editing of Archimedes commentaries

99, 807 [--> Isidore of Miletus] on Eratosthenes 16; 67-68,

73(nn9-1O), 74n23, 115-116 on hyperbola construction, 284,

743n26; text 280-281, 336-337 on loci 228, 236, 241n20 on Nicomedes 29-33 on rectification of circle 613n44 religion of 167, 176(nn85-86), 177n87 sources of 18, 24, 26(nn21-22), 45,

51-53,77-78,81, 100, 128n146, 133, 149, 244n69, 262, 531n51, 797n45,798n62

style 19; 18, 27(nn26-27), 45, 50, 57(nnI5-16), 74n21, 84-86, 88, 98, 100-101,107,111-118, 119(nnlO-ll), 124nloo, 126n128, 127n144, 134-144, 160-161, 164-165, 175n70, 765-771, 773, 797n48, 798(nn68-70); modern assessments of 226, 240n4

text of Archimedes III6; 410, 418n33, 447, 494(nn32-33), 753, 807-808

example (hypodeigma) 159, 162-163, 174n58, 180, 187; --> induction

exhaustion theory 418n21 [--> Euclid, Elements X 1; XII 2]; variant form 418n22 [--> Euclid, Elements XII 16]

Jalsigraphus 597, 6lln19; and sophista, sophisticus 637, 687n141

Fitzgerald, A. 796n37 focus-directrix construction, of conics

215, 223n15; of parabola 94,100, 120n38, 124n97

Folkerts, M. 612n25 forgery 16; 131, 134, 139, 141,

145-146; --> methodology of stylistic analysis

formalism (Euclidean) 766-767, 770-771,773,779-780

formulaic repetition: ---+ repetition fourth proportional 189, 209n24, 21On41,

789 Fowler, D.H. 530n38 fractions: continued 512n49, 530n38;

mixed 525, 530n43; notations for 522-523, 529n29, 530(nn32,34); operations with 530n31; reduction of 524-525,546; truncation of 540; unit 748n77

Galen 748n64 Gellius, Aulus 123n87 Geminus of Rhodes 52, 58n24, 81,108,

120n24, 228, 262, 273, 797n45; in Eutocius 230, 241n19

generality 72, 76n48, 84,142, 157, 159, 162,171025, 172n30, 179-181, 186, 190, 214, 300; of proof 707, 711-713, 721-722

geometric algebra 453n37 geometry, analytic 158 Gerard of Brussels 595, 61On7, 636-637,

672-676, 810-811 Gerard of Cremona 6, 119n 16, 396n2,

397n7,421; style of 127n139 trans. of Archimedes' Dim. eire.

452n18, 595, 606-607, 671, 676; text 467,469,471,473,475; vari­ants 422, 438-441, 489-491; ---+ Quadratura eirculi

trans. of Euclid's Elements 597, 798n66

trans. of Banu Musil (Verba filiorum) 119n16, 127n139, 536, 542, 549, 598,601,611n23

Gerbert 673 Gericke, H. 742n13, 749n79 Gervase of Essex 595, 635-636 Gibbs, S. 28n34, 53 "givens": ---+ terminology Gordanus 638, 676, 680n59, 683n90; dat-

ing of 670; ---+ Quadratura circuli Grabmann, M. 612n27 graphical construction: ---+ construction Greaves, D. 190 Gregory, D. 798n66 Gregory, 1. 591n29 groove (of instrument) 30, 79, 132, 146,

150n2; ---+ construction, mechanical; terminology

Grosseteste: ---+ Robert Grosseteste

Index 841

Gudemann, [n.i.] 123(nn87,88)

al-I:Jajjaj 442, 612n25 al-Harawi, Abu Bakr 6,41, 54-55, 77,

249; on conics 290-291; cube dupli­cation of IIA [---+ cube duplication]

al-Harawi, Abu '1-I:Jasan 296 Hartner, W. 795n 11 Haskins, C.H. 611n12, 614n57, 687n147,

688n150, 741nl ibn al-Haytham (Alhazen) 8nll, 318,

736. 749n86 Heath, T.L. 39n17, 150n4. 229, 239n3.

240n4. 241n17, 244n53, 245n83, 396n2, 492n3, 508n32. 742(nnI3-14),744(nn33,37), 745n52, 747n62, 796n28

Heiberg, J.L. 2, 8(nn4-7), 121n56, 122n68, 123n85, 131. 153n47, 231, 238. 242(nn34-35), 243n42, 244(nn53,73), 245n83, 375-376, 396n2, 418(nn31-33), 424, 446-447, 451n6, 452n17, 494n26, 508n33, 522. 530n31, 532n56. 611n10, 612n25. 615n57. 741nl, 747n63, 795n18. 798(nn62-63,66), 799n73, 807-808, 816n4; on interpolations in texts 8n7, 39n20, 170nI6, 208n4, 409-410, 418n24. 482, 523-524, 528, 528nl, 532n56, 799n89

Heliodorus 155, 168, 169n5, 731, 748n73 hemicylinder 140, 147; of Archytas 102,

104, 108, 110 Heraclides 58n24, 230, 483, 494n38 Hermann of Carinthia 597-598, 612n25 Hero of Alexandria 5, 75n40, 793nl;

on analysis 242n27 on arches (Kamarika, lost) 9 and Archimedes, citations of 230,

805-807; for approximation of circle ratio (re) 397n23, 481. 483, 493n18, 502; for more accurate value (attributed) 496-497; for rule for area of circle (cf. Dim. eire., prop. 2) 397n23, 477-478, 493n14, 615n58; for circle theorem (cf. Dim. eire., prop. I) 377, 379, 397n23, 405; for sector theorem 377, 379, 495, 503; for measurement of parabola 397n15, 509n35

on circular segments 377, 497-502 cube duplication of 71, 81; in Mechan­

ica (HM) and Belopoeica (HB) 11; in Eutocius (HE); ---+ cube duplication

on curvilinear areas 507n20

842 Textual Studies in Ancient and Medieval Geometry

Hero of Alexandria (cont.) on fractions 530n43 and Philo 41-42; on catapults 143 on polygons 492n9 practical geometry in mechanics

152n31 rule for triangle 536; -> Archimedes sources 51Onn40-41, 531 n51 on sphere, arithmetic estimate for

493n16, 599, 604, 612n31, 614n50 on square root approximation 156, 741n3 style 499-501

Heron of Alexandria (teacher of Proclus) 754, 793nl

Heronas 157, 170n17, 173n36, 178 Heronian metrical writers,

on Archimedes' circle ratio estimate (7t) 481, 483, 492n8

on Archimedes' circle theorem (Dim. eire., prop. 1) 378

on Archimedes' rule for area of circle (Dim. eire., prop. 2) 478, 492n8

on area of circular segments, aUr. to Archimedes 509n34

excerpts of Proclus in 453 (nn34-35) on isoperimetrics 747n63 on length of circular arcs 507n25 on sphere, arithmetic rule for 493n16 scholia 494 (nn25-26)

Hierocles 754 Hiller, E. 131, 153n47 al-l:Iimsi 256, 277, 284 Hintikka, 1. and U. Remes 242n27 Hipparchus 417nl Hippocrates of Chios 32, 56nll, 108,

114, 126n124, 128n149, 147, 400n56, 504, 678n40

hippopede 128n150 Hjelmslev,1. 418n26 Hogendijk, 1.p. 224(nnI9,23,31-32,34),

229, 242n26, 267, 272, 275, 294-295, 297, 300, 302, 307

hoi peri 25n3 Homer 167 homogeneity 159, 162, 164, 171n26, 186 homonymy 173n36; -> paronymy hoposagfmon 383,389 Horace 596,674 Hughes, B.B. 686n130 Hultsch, F. 8n7, 153n40, 169n3, 272,

417n7, 511n46, 614n45, 689-691, 723, 728, 732, 741(nnl,7), 742(nn13-14), 743n22, 745n52, 748n75; on interpolations in texts 122(nn67,72), 153n41, 242n33,

381-382, 710, 743n24, 747(nn61-62), 778, 800n95, 80 1 (nnl04-106)

Huygens, C. 591n29 Hypatia of Alexandria IIIll; 5, 177n87,

244n74, 406, 723, 808, 811-812, 814; life 754, 794(nn4-5); philosophy 754-755, 763, 794(nn8-1O); science 795(nnll, 16); mathematics 755-756,762-763; cited in Synesius 794n3; style 756-764

hyperbola 85; for angle trisection 56, 297 for auxiliary problem of al-Sijzi 297 construction of 213-214, 217,

223(nn5,14), 230-234, 236-237, 274, 287-288; in Apollonius (Conics II 4) 231, 233, 281-282, 284; text 331,333; in Eutocius 230-234,249; text 336-337; in al-Khiizin 263, 289-291,317; in Pappus 230-234; in Thabit 283-285; text 277-278, 330, 332

for cube duplication 54-55 [-> cube duplication, of al-Harawi]

in isoperimetric cylinders problem 746n56

as locus of constant product 125n103 with Menaechmus 94-100 for solution of problems 213, 222

Hypsicles 73n7, 592n43, 745n43, 750n86

Iarnblichus 5,167, 176n79, 751n87, 796n36

Ibrahim ibn Sinan 229 impossible (adynaton, atopon) 384, 403,

417n6, 427, 636; Latin terminology 685n127; -> proofs, indirect

incommensurables 158; -> Euclid, Ele-ments X; irrationals

lndi 686n138 indivisibles 209n9, 605, 614n49 induction 74n26, 157, 159, 164, 166,

178, 201, 592n36; -> example instrumental construction: -> compass;

construction: graphical, mechanical, physical; motion (geometric); neusis; ruler

intermediate corpus (curriculum) 547, 551, 592n43; -> astronomical corpus, minor

interpolation 128n151, 153(nn43-45), 172n32, 209n15, 210(nn47-48), 218,

230, 245n76, 286-287, 381-382, 384, 397nl0, 399n37, 402, 408-410, 417n12, 418n27, 433, 445-447, 452(nn25,27), 453(nn34-35), 506n4, 523-525, 530n37, 591n20, 614n50, 686n139, 742n15, 743n23, 778-779, 800(nnl0l-l02), 808-809: -> Heiberg; Hultsch

Introduction to Ptolem/I' Syntaxis (Almagest),

assigned to Eutocius (Mogenet thesis) 17; alternatively, assigned to Arcadius 166, 168,230

on Archimedes' circle theorem (Dim. eire., prop. 1) 376, 379, 424, 432, 452n32, 513, 607

on Archimedes' estimate of the circle ratio (n) 482, 494n29

date 397n13, 615n59 on isoperimetric figures (AI) III 10;

595,603-604, 615n60, 811; on isoperimetric solids 51On44; -> anonymous, On Isoperimetric Figures

lemma on sphere volume 603-604 mss. 791, 796n34; portions currently

edited 169n4 on Pappus' measurement of the sphere

614n50 on Pappus' method of division

764-765; text 787-793 style 160-161, 164-165, 168-169,

175n68, 505, 608-610, 689-690, 741n3, 763-764, 774-776. 779, 788, 792-793

irrationality of circumference-diameter ratio (n) 542

irrationals 158, 171 n26, 178: -> incom­mensurables

Is~aq ibn Hunain 441-442,448, 450nl; trans. of Archimedes 547

Isidore of Miletus 6, 58n33, 99, 126n117, 229-230, 243n38, 521, 526-528, 532n56, 753, 762, 795n18, 798n70, 807-808, 811; parabolic compass of 99-100, 120n38, 243n36; (disciple of, as editor of Eutocius) 99-100, 114, 243n38, 521-528, 591n19,757

Isidore, disciple of Proclus 794n4 isoperimetric figures 169n3, 227, 399

(nn54-56); in ancient cosmological commentaries

728-738; -> Proclus; Simplicius; Themistius

Index 843

in Arabic 749n86 and bees (in Pappus) 93, 748n64; and

swallows (in Themistius) 747n64 circle 811; relative to regular polygons

699-702 circular segments 387, 400n56, 497,

502-503, 716 cones and cylinders 720-722 in Introduction to Almagest (AI) lIllO,

175n68: -> Introduction to Almagest in Pappus (PI) III 10: 380, 386-387,

398n30 rectilinear figures: triangles 703-713;

polygons 713-716: regular polygons 693-697

sphere relative to solids 387, 716-717, 811: relative to Archimedean conical solids 717; relative to regular solids 716,718-720

spherical segments 503 in Theon (TI) IIIIO: 379-381, 386-387

Ito, S. 615n57

Jackson, D.E.P. 75, 112 Jacob of Cremona 596 John of Tynemouth 6, 595, 597, 611 n 13,

635, 637, 663, 672, 674-676, 687n147, 688n153, 776, 809-811, 813: style 676-677; -> De cun'is superficiebus

Jones, A. 72n2, 75n49, 169n5, 229, 240(nn3,5,11-12),241(nnI6,22), 242(nn28,30-33), 244n73, 245n76, 511n46

Jordanus de Nemore 670-671, 680n59, 682n80, 686n130

Julian 177n87, 755 al-Jurjani 228, 242n25

al-Kashi 593(nn52,57), 816n2 al-Khazin, AbG JaCfar 6, 54, 249, 254;

and al-Harawl 259,262-264 [-> cube duplication]; compiler on conics 222, 272, 286, 295; on cube duplication of Nicomedes IIF; on isoperimetrics 750n86

al-Kindi 318 King, D.A. 248 Kliem, [n.i.] 123n88 Knaack, G. 150n4, 152n34 koilogonion 711, 728, 747(nn60-62) Kozhukhova, G.M. 451, 592n45 Kramer, E. 794n4

844 Textual Studies in Ancient and Medieval Geometry

Krause, M. 451n7 Kunitzsch, P. 249

Lacombrade, C 755, 794n8, 796n36 Lang, M. 796n27 latitudo 646, 653, 682n82 latus rectum: --+ terminology of conics,

Apollonian Lemay, R. 687n147 lemma to cube duplication (Hero­

Apollonian method) 12; 21, 22-24; parallel in Nicomedes 32-33

lemmas, order of 492n5, 702, 743 (nn26-27); --+ Archimedes, Dim. Circ., defects of

Leon, the philosopher 531(nn48,51) Leontius 123n87, 243n38 Lesbian architecture 83 Lloyd, A.C 793nl, 794n6, 796n35 Lloyd, G.E.R. 126n124 loci, classification of 24ln20; solid

l23n83, 245n79 logistic 158, 179 Lorch, R. 612n25, 750n86 Lorent, H. 745n43 Luckey, P. 593n57, 816n2

Macrobius 662,665 Magnus 230, 512n53, 541, 591n24 Malalas 794n4 al-Ma'mfin 6, 216 Marinus of Neapolis 168, 177n88,

241n22, 793nl Marsden, E.W. 41, 153n39 Martin, CT. 611n14 mathematics, Arabic 225 [--+ medieval

tradition]; 16th century 158; 17th century 121n39

Maurolico, F. 595 means, arithmetic 493n17; harmonic 63;

geometric, general construction 71-72, 75n48

mechanical: --+ construction, mechanical medieval tradition (Arabic), style 551,

591n26, 617, 809; sources 599,610; (Latin), style III9, 809-810, 816n3

medieval translations (Arabic, Hebrew, Latin), style 422, 434-435, 445, 448-450, 451(nnI2-13), 452n27

Menaechmus 26n22, 113, 142, 147; and mechanical construction 128n151; --+ cube duplication

Menelaus of Alexandria 5,52,79-81, 101, 108, 115, 127n135, 120n24, 225-226, 273, 318, 536, 592n43; compiler 217,219,262,288-289; cited in Pappus 215, 223n17

Mercier, R. 687n148 mesograph 133, 140, 142, 150, 151n9;

--+ mesolabe mesolabe 75n40, Ill, 133;

--+ Eratosthenes; mesograph methodology,

of criticism 592n36 of source analysis 603-604, 654-655,

699; hypothesis of single source 520, 529n23

of stylistic analysis 134, 138-142, 145, 156, 161, 164,623-624,628, 634, 670-671, 756, 761-762, 779-780, 810-812

of text analysis 4, 410, 691, 693, 742n13,808

of text reconstruction 40 1-402 use of evidence from late antiquity

410; from medieval sources 528, 532n55

Minos 147 mirrors, burning 81,83,87, 114; --+ Dio­

cles Moerbeke, W. 229, 396n2, 421, 451n6,

596, 598, 611n9, 612n27; trans. of Archimedes' Sph. Cyl. 625-626, 655-656

Mogenet, 1 17; 242n34, 376, 397(nnI2-13), 615n59, 689, 741n2, 801; on isoperimetrics 751n92

Mo~he ibn Tibbon 6, 442, 450, 451(nnI2,14)

motion, celestial 128n150, 737, 748n67; in geometry 104-105, 107 [--+ neu­sis; ruler]; of mechanical devices 132, 135, 146, 149

Mueller, W. 742n13 Mugler, C 121n62, 169n5, 242n34,

243(nn35,37) murabbac (square) 25n6 Murdoch, lE. 612n25, 614n57 myriads 176n77, 494n35, 512n53,

529n28, 546, 591n18, 737; --+ arith­metic operations, ... of large num­bers

Neoplatonism 152n24, 166-168, 176(nn81 ,84), 177n87, 794n6;

Neoplatonists 169n6, 176(nn81,84), 177n87, 227, 729-730, 737, 751n87, 754-755,815

Neuenschwander, E. 126(nnI26-127) Neugebauer,o. 169n5, 743(nnI9,2I),

795nll, 816n12 neusis constructions 38, 117-118,

120n32; alternative formulations of 219,

285-288 for angle trisection 213-219,287 relative to circle and chord 228 via conchoid 30-31, 311 via conics 34; via hyperbola 272 and cube duplication 55-56; of Hero

53; of Pappus 91, 114; of Sporus 89; effected via hyperbola 260-261

for spirals 95 Nicomachus of Gerasa 157-158, 170n 17 ,

173n36, 178, 208n5, 21On42, 211n48 Nicomedes 33-34, 50, 52-53, 56, 75n4O,

77-79,214-216,219,223nll On Conchoids 111-112,115,228,

511n46; -+ conchoids cube duplication of 29, 31-32, 71;

-+ cube duplication neusis of 285

Nix, L. 448 Nokk, [n.i.] 742nl4

Olympiodorus 176n84 Omar (CUmar) al-Khayyam 224n19, 300 optative mood 27n26 Oresme, N. 682n82 Origenes 794n2 originality 238-239, 812; -+ Pappus Ovid 596, 674 Oxford 663,671-672,674-675

Pachymeres, G. 173n38 Paeonius 795n11 Pallas 596-598 Pappus 5, 77;

alternative to Archimedes' Sph. Cyl. (Collection V) 744n38

on analysis and synthesis 86, 237, 751n91; -+ analysis and synthesis; angle trisection

on angle trisection 18; lIB (text 268-270, 327-331); lIe; lIE (text 303-304, 340-341); 56

Index 845

on Apollonius' cube duplication 54; on his Neuses (lost) 288

in Arabic 217,228-229,273; -+ Jack­son

on Archimedes 805-807; approxima­tion to circle ratio (n) 378, 480, 483; circle theorem (cf. Dim. Circ., prop. I) IIIl, 401-403, 405, 495, 511n46, 607; text P 387-392; sector theorem 377, 382, 384-387, 495-497; text S 394-396; measurement of sphere 716

cited in Eutocius 178, 236, 765, 798n62; in Introduction to Almagest 155,160,187,202,789,792-793

classification of problems 71 commentaries on Ptolemy 227, 691; on

Book I (lost) 380-381, 385-387; cited by Eutocius 157-159, 165

on compound ratio 160, 162 on construction by instrument 318 on cube duplication 14; 71, 217,

240(nn13-16); own method 71-72, 87-93

on division 175n76, 763-764, 787-793

on Eratosthenes 131-136 and Geminus 52 on hyperbola construction 230-234 on isoperimetric figures (PI) 11110,

497,502-503, 615n6O, 774-776, 811; contents 692; compared with other accounts 723-725; lost version (PI*) 691, 693, 697-698, 701-702, 704,710-711,715,725-728,730, 734, 736; on isoperimetric solids 51On44

lemma on circumference-diameter proportion 497, 511n46, 602, 613n42

lemma on volume of sphere 614n50 lemmas on Apollonius' Conics 228,

234-238,410, 767-768, 797n45, 798n58

lemmas on Euclid's Optics 242n29 Mechanics (cf. Collection VIII) 75,

87-88, 112, 228-229, 235, 273 on neusis for Archimedes' Spirals 95 on Nicomedes 29-33,50 originality 88, 92-93; modern assess-

ments of 226, 239n3 on rule for volume of sphere 493nl6 sources 92, 531n51, 716 on square roots 156

846 Textual Studies in Ancient and Medieval Geometry

Pappus (cont.) style 19; IIIID; 63, 68-70, 111-112,

118, 150n6, 215-216, 382-384, 511n46, 767, 774-776, 798(nn59-60), 812-813

parabola, in cylinder maximization problem 746n56; in Menaechmus' cube duplication 94-100; -> conics

parabolic segment, Archimedes' measure­ment of 498-501 [-> Archimedes; Hero]; relation to circular segment 500, 509n36

paraboloid 83 paradoxes [-> Erycinus]; relative to

isoperimetrics 749n81 Paris 674-675, 687n145 Parker, R.A. 507n24 Parmenides 167 Parmenion 16, 19,21,24, 28n34, 41, 46,

50-51,53-54, 58n30, 261 paronymy (of ratios and numbers) 158,

160, 171n26, 173n36, 178, 208n5, 21On42; -> homonymy

part (mathematical term) 57n12, 654, 696 Pedersen, 0. 687n148 Perseus 128n150 Peyrard, F. 398n36 Philo of Byzantium 14, 41-42, 75n40;

date 145; Belopoeica 143-145; cube duplication (PB) 41-45,51-52; in Eutocius (PE) 44-45, 48-52 [-> cube duplication]; on mechanisms 129n152; style 43-44,143-145

Philo of Gadara 230,505,541, 591n24 Philo of Tyana 223n 17 Philoponus, John (Joannes) 6, 11, 28n30,

113; and Ammonius 169n5 on Apollonius' cube duplication, Hero­

nian form (AJ) II; Philonian form (PK) 13, IIA; alternative (PJ) 13

on astrolabe 699 commentaries on Aristotle 21, 45,

796n35 date 26n22 on instrumental construction 16, 318 on Delian oracle 57n18, 140 on isoperimetrics 730-731, 735 sources 21, 26n22, 51 style 46, 798n69

philosopher 722-723, 756, 762, 794(nn3-4), 796n36, 814

Philostorgius 794n4 De piramidibus (adaptation of De curvis

superficiebus) 637, 673, 676; -> Ger­ard of Brussels

pivot 14-15, 26n14, 30, 79, 91-92; -> construction, mechanical; termi­nology

planar constructions 34, 38, 56; in rela­tion to cube duplication 63, 69-70

Plato, on astronomy 126n124; on geome­try 79, 113, 128n151, 147, 749n81; Timaeus 5,167,728-729,731,811; commentaries on 728-733,736-737, 747n63; cube duplication (attributed): -> cube duplication

Plato of Tivoli 6, 396n2, 421, 614n51; trans. of Dim. circ. (In quadratum circuli), text 466, 468, 470, 472; var­iants 422,438-441,489-491; rela­tion to Arabic (AF) 482-483

Plotinus 167, 176n84, 755 Plutarch 57,109, 128(nnI51-152), 140 polemics 72n7, 116, 119nI5, 814 polygons, area rule for 639-640, 642,

644-645,650, 656-657, 664, 681n70,684n117

polyhedra: -> solids Porphyry 176n84, 751n87, 755 Powell, 1. V. 151 practical applications of geometry

142-143, 147-148 practical construction: -> construction practical geometry: -> Artis cuiuslibet

consummatio Praechter, K. 167 precession of equinoxes 675, 687n148 Price, D. de S. 28n34 problems,

of angle division, general 214; -> angle trisection

classification of71, 213 of inscribing hexagons 124n92 locus of constant product 95, 125nID3;

-> hyperbola, construction of of means, general 142, 150; of two

means, in Diocles 84, 89-90; in Pappus 91; in Sporus 90; -> cube duplication

solid 75(nn34,40), 112 Proclus Diadochus 5, 176n84, 242n22,

531,754, 793nl; on Archimedes 419n33 on Archimedes' circle theorem (Dim.

eire., prop. 1) 376, 397nlO, 417, 432, 452(nn33-34), 494n34, 525-526

on circle quadrature 433, 452(nn33-34)

Commentary on Euclid, mss. of 453n35 and Geminus 52, 58n24 on isoperimetrics 728-731, 733, 735,

748n71,749n81 on Menaechmus 126(nnI20-121) on paradoxes 72n5, 749n81 on Plato (Timaeus) 167, 176n79,

728-729,732,734, 737. 751n87, 811

sources 100, 126nl24 and Syrianus 168 on terminology 126n126 on Zenodorus 728. 747(nn60.62)

product (geometric formulation) 402-403,425,430. 495, 497. 527. 536,547, 591n25, 599-602, 605-608,612n33;

for planes (area) 28n29, 106-107, 505n4, 697. 700-702, 745n48

for solids (volume) 717. 720, 733. 745(nn43,49-50), 800n97

terminology, in Greek (to h~po tr3n ... ) 106-107, 378-380; in Arabic 284, 425. 434, 449; in Hebrew 449; in Latin (ductus) 621,647,656-657. 677n15, 683(nnl01-I03)

proofs, alternative 94, 96, 98-99. 114, 125n114. 272-273. 300, 446, 497; general (contrasted to inductive) [--> induction]; indirect 383. 399n45, 538, 592n38. 597, 600-602, 611n19, 636, 686n129, 713, 7641--> impossi­ble]

proportion theory, early (Eudoxean) 158. 399n53, 407, 418(nnI8.20-21). 496. 506n5. 613n41, 816nl

proportionality, of circular arcs and sec­tors 386, 398n36, 399n49, 407, 495-497, 502, 505n4; of circumfer­ences and diameters 496-497, 502. 506(nn6,16), 507n19, 536. 538-539, 590n8, 602, 613n40; of areas and perimeters of regular polygons 684n 117; of cylinders and cubes 605

proportions, terminological variants in Greek 696; Arabic 285; Hebrew 449; Latin 622, 630-631

Ptolemy (king) Ill, 115, 131-132, 134. 144, 147,150; III (Euergetes) 144. 152n34; IV (Philopator) 144-145; V (Epiphanes) 145, 153n38

Ptolemy (Klaudios Ptolemaios) 5, 152n24,417nl;

Almagest (Syntaxis), Book I 155

Index 847

[--> Introduction to Almagest]; Latin translation 614n57; citations in Latin 645, 682n80, 686(nnI38-139)

approximation of circumference­diameter ratio (rr) 550, 593n53; on Archimedean values 480, 483, 503, 505n4

on Archimedean circle theorem (Dim. eire., prop. 1) 378, 505n4

on chords 645, 682n80; table of 75n33, 541-543,550, 591n29: --> chords

commentaries on: --> Arcadius; Eutocius; Hypatia; Pappus; Theon

On Gravities 230 on isoperimetrics 690, 736 on mean motion of planets 757-758.

780, 786-788 on obliquity of ecliptic 530n38 on planisphere 699 on proportionality of arcs and sectors

505n4 on spherics 175n61

Pythagoras 176, 813; Pythagorean theorem: --> Euclid, Elements I 47

Qalonymos b. Qalonymos 6. 421, 441. 450, 451(nn3,14)

Q.E.D. (hoper edei deixai) 159, 189, 21On31, 452n27

quadratrix 58n26, 92-93, 114; for angle division 214; for circle quadrature: --> Sporus

De quadratura circuli, LG (translation of Dim. eire. by Gerard of Cremona) 617,619,623-624,627,641. 648-649. 663, 665. 667-668, 682n73. 687n147, 810; text 467, 469.471,473,475; variants from Arabic 438-441,489-491; titles in mss.614n52

adaptations of III9; overview 617-618, 668-677, 810

Abbreviated Version (AV) 648-649; --> Roger Bacon

Cambridge Version (C) 625-631; and other versions: Naples 635; Corpus Christi 655-663; and De cun'is superficiehus 628-631; and De ysoperimetris 680n46

Corpus Christi Version (Cor) 655-663; and Vatican Version 645

Florence Version (F) 614n54, 618-624, 799n85; and other versions: Cam-

848 Textual Studies in Ancient and Medieval Geometry

De quadratura circuli (cant.) Florence Version (F) (cant.)

bridge 626-631; Naples 631-635; Corpus Christi 655-663; Glasgow 665-668; and De curvis superfieiebus 620-624; --+ John of Tynemouth

Glasgow Version (Gg) 663-668 Gordanus Version (Go) 638-640; and

other versions: Munich 640-643, 648; Vatican 644-648,650,654-655

Munich Version (M) 640-643 Naples Version (N) 631-637; and other

versions: Gordanus 638-640; Munich 640-643; --+ Gerard of Brussels

in Questio of Albert of Saxony (QA) 649-655

Vatican (ps.-Bradwardine) Version (V) 644-648; and Abbreviated Version 648-649; and Questio of Albert 649-655

mss. of LG and adaptations 618, 624-625, 628, 635-636, 638, 641, 648, 655, 663, 669, 677(nn5,7), 681n73, 683(nn85,90), 684nl13, 685nI20,686n131

quadrivium 674 al-Quhi 6, 54-55, 216, 249;

angle trisection of lIE; --+ angle trisec­tion; al-Sijzi

on approximation of circumference­diameter ratio (n) 544, 592n36

on Archimedes' Dim. eire., doubts 543-544

on Archimedes' heptagon problem 307 on centers of gravity 592n36, 594n58 citation of Conics 222, 245n81, 306 cube duplication of lIE; --+ cube dupli-

cation and al-Harawi 259

Quintilian 732, 748n64, 749n81, 811 Qust,li ibn Luqa 6, 11,447-450,809;

proposed as translator of Dim. eire. 535, 549, 551

Rashed, R. 318,448, 592n40, 795n14 Abu 'I-Rashid, cAbd al-Hadi 6, 450n2,

451n7, 494n35, 617, 671, 799n85, 809-810; question of identity 543; version of Dim. eire. (AR) 421, 543-546; text 552-576; possible use by al-Tusi 547,549-550; style 544-546, 551

ratio, compound 122n70, 208nl; in Domni­

nus 201-207; in Eutocius 155-159, 162-164, 177-185; in Introduction to Almagest 155-156, 159-164, 185-201; relation to di'isou 171n25

duplicate 122n70 epimorios 208n2 in relation to number 158, 171n27,

174n53, 208n5, 531n44, 539, 600 subtraction of 159, 161, 175(nn61,63),

185, 187-190, 202-204, 21On35; rule for 209(nn2l ,27), 21On43

synthenti, form in Pappus and Archi-medes 711

--+ terminology Rawlins, D. 530n38 rectification of curved lines and surfaces,

hypothesis of 541-542, 602-603, 613n44, 625-626, 629,645, 656, 664, 674, 679(nn32, 35-36), 682n81, 683n97

religion, in late antiquity 152n24, 167, 176n86, 177n87, 754-755, 793nl, 814

repetition (formulaic) 758-768, 779-780, 796n31; --+ Hypatia, style of

rhomboid 722 rhombus, solid (double pyramid) of

Archimedes 600, 777-778, 800nl00; medieval terminology 448

Richard of Fournival 61On7, 637, 675 Rist, 1. 754, 763, 794(nn4-6,9-1O) Robert Grosseteste 595, 61On7, 663, 672,

675,688n151 Roger Bacon 663,669-671, 685nl27,

688n152 Roger of Hereford 688n150 Rome, A. 240n7, 381, 396n7,

398(nn29,35), 399n40, 417nlO, 452n27, 511n46, 690, 709, 741n2, 742(nnlO-11,14),743n23, 747(nn60-61), 756-757, 780, 782-783, 795n16, 796n23, 797n40

rota Aristotelis 683n97 Rozenfe1d, B.A. 795n14 Ruelle, C.E. 202 ruler (kanan, kanonion):

in accounts of cube duplications, of Apollonius 16-17; of Banu Musil 79; of Hero 12-14; of Pappus 91-92; of Philo 42-44; of Plato 79, 92; of Sporus 89

flexible 83 of Nicomedes' conchoid 30 rod or guideline (Eratosthenes) 135,

140, 142, .146 sliding or moving 12-18, 22, 42-44,

263; ---> neusis

al-~abi, Abu Is~aq 592(nn34,36) al-~aghani 296 Saito, K. 171n25 Saliba, G. 242n27 salinon 498 al-Samau'al 242n27 Sams6, J. 254 Samson and Delilah 673 Sarfatti, G. 449, 451n3 Sato, T. 396n7, 423, 451n16,

452(nn17-18),613n37 Sayili, A. 301 schesis (relation), and ratio 208n3 Schneider, I. 396n2 scholastic 239, 617, 640-641, 648-649,

654-655, 663, 669, 681n63, 810 scholia (marginalia) 127n128, 433, 447,

482, 543, 742n17, 797n45, 798n63; in De curvis superficiebus 61On7; to Euclid 171 n26; to Heronian corpus 492n7, 493n16, 494(nn25-26), 509n34; in Abu 'I-Rashid 561-562; (as commentary) 155, 164-165, 172n34, 178,231,235-238, 244n72, 495,767-768

Schoy, C. 307-308 secant product 34-36, 39n25; ---> Euclid,

Elements III 36 sections, of planes 105; of solids 104, 107 sectors, exterior 417n7, 529n16,

632-633, 680(nn52,54), 681n71; similar 429, 51On45

segment of circle, area approximation 498, 500-502; ---> Hero

Serapeion 28n34, 53 Serenus 244n74, 245n81, 799n73 Sesiano, J. 122n63, 592n40, 795n14 Severus Sebokht 699 Sezgin, F. 2, 224, 254, 450nl, 591n30,

592(nn43,44), 751n86 Shemuel ibn Tibbon 449 Shewring, W.H. 796n37 Shiraz 248, 297 Sicilian translators 614n57, 675,

687n147,741nl

Index 849

al-Sijzi 6, 54, 248-249, 263; on angle trisection (survey tract) 216-217, 219,222,289,294-295; (own method) lID; and al-Quhi lIE; on Apollonius 229

Simplicius 6; and Ammonius 169n5 on Archimedes' approximation of circle

ratio (n) 482, 494n30 and Iamblichus 751n87 on isoperimetrics 730-731, 734, 737,

748(nn64,67), 811 on Plato and Aristotle 176n81 and Proclus 748n69 sources 100, 114, 126n124,

128(nnI46,149) size (pelikotes) of ratio 157, 160-164,

l70(nnI6-17,26), 171n24, 172n32, l73n36, 178, 185, 21On33; (pelikon) 187; (megethos) 189

Slavutin, E. 795n14 Snel, W. 591n29 Socrates (historian) 794n4 Sodano, A.R. 751n87 solid loci: ---> loci solids,

of revolution 104, 107, 599-601, 717, 744n38, 746n55; Archimedean coni­cal 776-777

polyhedral 719-720; regular 63, 121n57, 308, 510n44, 716, 718-719, 736, 744n39, 745n43, 750n86; semi­regular 692,716,736, 744n37, 745(nn44-45); ---> isoperimetric figures

sections of: ---> sections Sosigenes 126n124 sphere, surface and volume 601-602

[ ..... Archimedes]; lemma on volume 603-604, 614n50, 717-718; arith­metic estimate for 599, 604-606 [---> Hero]

spherical geometry 592n43, 697-698; ---> Theodosius

spirals (helikes) 128n150; Archimedean 51On45; use for angle division 214; use for circle quadrature 494n38

Sporus of Nicaea 483, 541; cube duplication 87-93, 114, 124n92 date 88, 93, 123n87 Keria Aristotelika 58n26, 93, 108,

114-115, 123(nn85,87,88), 124n93, 126n124, 230, 504-505, 799n81; on

850 Textual Studies in Ancient and Medieval Geometry

Sporus of Nicaea (eont.) Keria Aristotelika (eont.)

Archimedes' estimate for circle ratio (n) 494n38

on circle quadrature 88, 123(nn82,86), 124n93, 128(nn146,149), 215, 223n17; ...... quadratrix

Steiner, 1. 723, 742n13 Stein schneider, M. 441, 451 (nn3, 14) Sturm, A. 245n83 style (editorial aims and methods, charac­

teristic terminology, level of compe­tence, originality, etc.): ...... commentators, ancient; medieval tradition and translations; scholastic; and individual authors

Suidas 743n21, 754-755, 765, 794(nn4-5)

sundials 24, 53 Suter, H. 80, 450n2, 543 symmetry (of exposition) 35, 383, 430,

432,664, 715, 767-778, 799(nn75,87); ...... Hypatia, style

Synesius of Cyrene 754-755, 763, 794(nn3-5,1O), 795nll, 796n37, 811; on scholarship 813-815; on isoperimetrics 731, 735; style of 763

syntagma (collection, composition) 166, 381,388, 400n56, 401, 417nl,483, 495, 813

Syrianus 5, 155, 166-168, 176(nn78,81-82,84), 177n88, 202, 723

tangents, lemma on ratios of 697-699, 735, 742(nnI6-17), 749n86; ...... chords

Tannery, P. 26n22, 58n24, 108, 123n88, 124n93, 126n124, 128n146, 167, 169n5, 176n86, 201-202, 21On48, 241n20, 242n34, 243n38, 245n83, 492n3, 493n15, 755, 794n4, 795(nn13-14), 798n62, 799n73, 808

Tee, GJ. 452n16 terminology,

of Apollonius 127n128, 176n77; ...... of conics

archaic (Greek) 43, 100-101, 115, 128n46

of conics, Apollonian 86,94-97,99, 116,214-215,220-221,233, 241n19, 263, 297, 771; Arabic 284-286, 307; early 125n108

of "givens" 85, 629-630 mechanical 112, 132, 135-137, 140,

143-144, 146, 149; ...... groove; pivot; ruler

of isoperimetrics 511n47 of logical sequence (gar, etc.) 711, 766,

769, 772, 774, 778-780, 797(nn53-54), 800n89, 801(nn103, 106, 110)

for numbers (large): ...... myriads for numerals in mss. 489, 591n17 of ratios 160, 166, 173n38, 180, 201,

203, 208n5,209nI7, 480, 493n21, 531n43,606

used in translations: Arabic 107, 127n139, 429, 434, 445, 446-450, 544, 547-549, 561-562, 583-584; Hebrew 429,434,445,449-450; Graeco-Latin 597-598, 608, 611(nn16,23); Arabo-Latin III9, l27n139, 429, 434, 597-598, 606, 611n23, 612n24

Thiibit ibn Qurra 6, 101, 216-222, 249, 272;

on balances 592n38 editions: of Archimedes 218, 421,

441-442,448-449, 450n1, 539, 547, 549, 551, 809; of Euclid 442; of Eutocius 311; of Menelaus 127n135, 217, 262, 288-289

style 285-287 Thaer, C. 58 Theisen, W. 615n57 Themistius 747n64, 748n66,

751(nn90-91),796n37 Theodosius 28n34, 511n46, 592n43,

698 Theon of Alexandria 5, 169n2;

on Archimedes' circle theorem (T; cf. Dim. eire., prop. I) IIII; 401-406, 423-435, 511n46, 513, 607,805; text 387-391, 392-393

on Archimedes' estimate of circle ratio (n) 479, 480-482, 493(nnI8,20)

on area of circle, arithmetic estimate (cf. Dim. eire., prop. 2) 493n14, 479-480, 483, 503, 614n48, 753

cited, by Eutocius 157-159, 165, 178;

in Introduction to Almagest 155, 698-699, 723, 741n3

commentaries: On Small Astrolabe 698-699, 743n21; On Small Astronomer 698-699, 723; On Hero's Metrica 243n44, 479, 493n15, 741n3; On Ptolemy's Almagest IIII, lIllO, 227, 754, 756; On Ptolemy's Tables 175n63, 417nl, 793n2, 795n13

on compound ratio 170n15, 17ln20, 174n60, 175n61, 21On48

editions of Euclid 398n36, 452n 17, 748n73, 816n4

on Hypatia 756, 762, 808 on isoperimetric figures (TI) II11O;

169n3, 615n60, 774-776, 811; con­tents 692; summary of comparisons 723-725; on solids 51On44; variants in mss. 709, 747n61

lemma to Archimedes' circle theorem (T) 403-405, 427-428, 518-520, 529n22, 544, 753; text 414-416, 534

on long division 757-764, 770-771; texts 780-787, 802-804

on proportionality of arcs and sectors 506n4

relation to Pappus 227, 240n7, 398n36, 482, 796n38

on square roots 156, 175(nn68,75) style, IIIlO; 382-384, 406, 409, 423,

481, 758-764, 774-776, 779, 787, 796n31, 812-813, 816n4

on volume of sphere, arithmetic esti­mate 493n16, 530n43, 599, 604-606, 612n31,614n48

on Zenodorus l24n9l, 379, 381, 690-691, 741n7

Theon of Smyrna 45, 57n18, 140; and Theon of Alexandria 493n14; on ratios 173n36

Thomas, I. 169n5; --> Bulmer-Thomas Tihon, A. 169n5, 793n2, 795n13 Tinemue, = Tynemouth 595,597; vari-

ants 61On5, 611(nnI4-15) Tiphys 596-598 Toledo Tables 687n149 Toomer, GJ. 81, 120(nn25-26,29),

121n41, 122n63, 240(nn4,7), 318, 398(nn31,36), 748n79, 797n38

torus 104, 108, 110 tradition 6-7,238-239,551,815-816 Treweek, A.P. 240nll, 273

Index 851

triads, Menaechmean 99-100, 115, 126n120, 142, 150

trigonometry 593n54 al-Tusi 6, 54, 119n16, 248-249, 254,

417n9, 617; cube duplication: --> cube duplication edition of Archimedes 421, 434-435,

441, 448, 450nl, 451n16, 452n28, 532(nn52,55), 543, 546-551, 809; text 577-590; mss. 592n43

edition of Banu Musa 79-80, 103-105, 119n16, 127n139, 536, 549-550, 601, 611n23

edition of intermediate curriculum 546-547

and al-Harawi 259, 262-265 on rectification of circle, hypothesis of

542, 613n44 style 264-265, 547-551

Unguru, S. 242n29 unrolling (anaploun, exaploun) 379,

398n25

Ver Eecke, P. 243n35, 492n3, 742(nnI3-14),745n52

Victor, S.K. 686(nn135-140), 687(nnI43,145-146)

Vieta, F. 158 Vitruvius 24,42, 128n152, 51On40 van der Waerden, B.L. 240(nn3,4),

796n28

Abu 'l-Wafii' 549,582, 593(nn52,57), 809 Waithe, M.E. 794n4, 795n16 Wallies, M. 48 Weaver, J.H. 122n74 von Wilamowitz-Moellendorff, U. 16;

74n23 , III, 115, 245n83 Wilson, N.G. 531n28 Witelo 229 Woepcke, F. 224(nnI9,31), 294, 307,

593n57 Wright, J.K. 688nl49

Ya'akov b. Makhir 6,442,450, 451nl4 Youschkevitch, A.P. 750n86

852 Textual Studies in Ancient and Medieval Geometry

De ysoperimetris 595, 614(nn45,57), 680n46, 689, 741n1, 743n18; adapta­tions 686n132, 687n147, 688(nn154,157), 741n1, 810, 816n3; and John of Tynemouth 676; citation of Archimedes' circle theorem 607

al-Zarqali 687n149 Zenodorus 124n91, 169n3, 227,

379-381, 386-387, 398n29, 400(nn55-56), 402, 405, 502,

510(nn43-44), 690-691, 710, 723, 728, 730, 734-737, 741n7, 811-812; date 398n31, 742n14, 748n79, 751n92; On /soperimetric Figures, lost tract (Z*) 697-698, 701-702, 704, 710, 715-716, 724-728, 732, 734; relation to extant accounts 724-728; on koilogonion 747(nn60,62)

Zerachiah b. Yitshaq Ha-Sefardi 751n90 Zeuthen, H.G. 238, 240nlO, 245n83,

747n58 Ziegler, K. 242n34, 243n37, 796n38