Archimedes in the Middle Ages, Vol. 4: A supplement on the Medieval Latin traditions of conic sections (1150-1566). Part ii

54 Pages·1980·2.03 MB·English

by  Marshall Clagett

ARCHIMEDES
in the Middle Ages
VOLUME FOUR
Memoirs of the A SUPPLEMENT ON THE MEDIEVAL
AMERICAN PHILOSOPHICAL SOCIETY
LATIN TRADITIONS OF CONIC SECTIONS
held at Philadelphia
(1150- 1566)
for Promoting Useful Knowledge
Volume 137 Part B
Part II: Bibliography, Diagrams, and Indexes
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1980
PART II
Copyright 1980 by the American Philosophical Society
Bibliography, Diagrams,
and Indexes
Library of Congress Catalog Card Number 62-7218
International Standard Book Number 0-87169-137-X
US ISSN 0065-9738
BIBLIOGRAPHY
Alberti, L. B. Della prospettiva. Ed. of A. Bonucci in Opere volgari, Vol. 4, Florence, 1847.
Albertus Magnus. Book of Minerals. Trans, of D. Wyckof, Oxford, 1967.
---------------. Mineralium libri V. Ed. of A. Borgnet in Opera omnia, Vol. 5, Paris, 1890.
Alhazen, see Ibn al-Haytham.
Amodeo, F. “II trattato delle coniche di Francesco Maurolico,’’ Bibliotheca mathematica,
3. Folge, Vol. 9 (1908-09), pp. 123-38.
Anonymous. De duabus lineis semper approximantibus sibi invicem et nunquam concur­
rentibus. See my edition and English translation in Chap. 2 above.
--------------. De sectione optimi speculi: vocatur Mukefi. For MSS, see Chap. 5, η. 1.
--------------. Speculi almukefi compositio. See my edition and English translation in Chap. 4
above. Note my edition also of Regiomontanus’ version of this work in Chap. 5
above, where variant readings are given to A. H. Gogava’s edition of 1548 under
the title De sectione conica orthogona quae parabola dicitur (see Ptolemy below).
Anthemius of Tralles. See Huxley, G. L., below.
Apollonius of Perga. Apollonii Pergaei Conicorum libri IV. Cum commentariis R. P. Claudii
Richardi. Antwerp, 1655.
--------------.Apollonii Pergaei Conicorum libri octo et Sereni Antissensis De sectione cylindri
et coni libri duo. Ed. of E. Halley. Oxford, 1710.
---------------. Apollonii Pergaei Conicorum libri quattuor, una cum Pappi Alexandrini lem­
matibus et commentariis Eutocii Ascalonitae, Sereni Antinsensis philosophi libri duo.
Trans, and commentaries by F. Commandino. Bologna, 1566.
---------------. Apollonii Pergaei Conicorum sectionum libri V. VI. et VII. Trans, from the
Arabie of Christianus Ravius. Kilonii, 1669.
--------------. Apollonii Pergei philosophi, mathematicique excellentissimi Opera. Trans, of
G. B. Memmo. Venice, 1537.
--------------. Apollonii Pergaei Quae graece exstant cum commentariis antiquis. Ed. of
J. L. Heiberg, 2 vols., Leipzig, 1891-93. Abbreviated as Gr 1 and Gr 2.
---------------. Apollonius of Perga. The Treatise on Conic Sections. Paraphrase of T. L.
Heath. Cambridge, 1896, repr. 1961.
--------------. Conics. Arabic translation by Hilal ibn Abi Hilal al-Himsi. Ms. Oxford, Bodl.
Library, Marsh 667.
--------------. Conics (Books I—III). English transl. by R. C. Taliaferro in Great Books
of the Western World. Vol. 11, Chicago, 1952.
--------------. Elementa conica Apollonii Paergei (!) et Archimedis opera nova et breviori
methodo demonstrata a Ioanne Alphonso Borellio. Rome, 1679.
--------------. Francisci Maurolyci Messanensis Emendatio et restitutio Conicorum Apollonii
Pergaei. Messina, 1654.
--------------. Liber de pyramidibus. A fragment trans, into Latin by Gerard of Cremona, in
Heiberg’s Apollonii Pergaei Quae graece exstant, Vol. 2, pp. LXXIII-LXXX. See
also MS Paris, BN lat. 9335, 85v-86r.
--------------. See also L. M. L. Nix.
'Arafat, W. See Ibn al-Haytham.
Archimedes. Opera omnia. Ed. of J. L. Heiberg. 3 vols., Leipzig, 1910-15.
--------------. The Works of Archimedes. English trans, and paraphrase by T. L. Heath.
Cambridge, 1897; repr., New York, 1950.
--------------. See Dijksterhuis, E. J., below, and also the numerous entries in the Bibliog­
raphy appended to Vol. 3 of my Archimedes in the Middle Ages.
Barozzi, F. Admirandum illud geometricum problema tredecim modis demonstratum: Quod
docet duas lineas in eodem plano designare, quae nunquam invicem coincidant,
etiam si in infinitum protrahantur: et quanto longius producuntur, tanto sibiinvicem
propriores evadunt. Venice, 1586. See Chap. 7, Text B.
465
466 ARCHIMEDES IN THE MIDDLE AGES BIBLIOGRAPHY 467
Bauch, G. Die Anfdnge des Humcmismus in Ingolstadt, in Historische Bibliothek, Vol. 13 Fontana, Giovanni. Dicta atque notata. Notes on Alhazen’s De speculis comburentibus,
(Munich/Leipzig, 1901). which I have edited and translated as Text B of Chap. 5 above.
Boncompagni, B. “Intorno alia vita ed ai lavori di Francesco Barozzi,” Bullettino di bibliog- France. Catalogue general des manuscrits des bibliotheques publiques de France. Departe-
rafia e di storia delle scienze matematiche e fisiche, Vol. 17 (1884), pp. 795-848. ments. Vol. 5: Dijon. Paris, 1889.
--------------. See Leonardo Fibonacci. Fusoris, Jean. His version of Witelo’s Perspectiva. For the MS and extracts from it, see
Borelli, G. A. See Apollonius. above, Chap. 5, n. 9.
Boyer, Μ. N. “Barocius, Franciscus,” Dictionary of Scientific Biography, Vol. 1, New York, --------------. Libellus de seccione mukefi. See my edition and English translation as Text A
1970, p. 468. in Chap. 5 above.
British Library. A Catalogue of the Manuscripts of the Cottonian Library Deposited in the Gerard of Cremona. See Apollonius and Ibn al-Haytham.
British Museum. Vol. 1, London, 1802. Ghetaldi, M. Nonnullae propositiones de parabola. Rome, 1603.
Calcagnini, C. Opera aliquot. Basel, 1544. Gogava, A. H. See Ptolemy and Anonymous, Speculi almukefi compositio.
Camerarius, J., See Durer, A. Heath, T. L. A History of Greek Mathematics. Vol. 2, Oxford, 1921.
Campanus. See Euclid. ---------------. “The fragment of Anthemius on burning mirrors and the 'Fragmentum mathe­
Cantor, M., Vorlesungen iiber Geschichte der Mathematik. Vol. 2, 2nd ed., Leipzig, 1913. maticum Bobiense’,” Bibliotheca mathematica, 3. Folge, Vol. 7 (1906-07), pp. 225-
Cardano, G. De subtilitate libri XXI. Nuremberg, 1550. Augmented ed., Basel, 1554. 33.
Cavalieri, B. Lo specchio ustorio. Bologna, 1632. --------------. See Apollonius and Archimedes.
Clagett, M. “A Medieval Latin Translation of a Short Arabic Tract on the Hyperbola,” Heiberg, J. L. “Die Kenntnisse des Archimedes iiber die Kegelscnitte,” Zeitschrift fur
Osiris, Vol. 11 (1954), pp. 359-85. Mathematik und Physik, XXV. Jahrg. (1880), Hist.-lit. Abtheilung, pp. 41-67.
--------------. Archimedes in the Middle Ages. Vol. 1, Madison, Wise., 1964. Vols. 2-3, --------------. See Apollonius and Archimedes.
Philadelphia, 1976-78. Heiberg, J. L. and E. Wiedemann, “Eine arabische Schrift iiber die Parabel und parabo-
--------------. “The Life and Works of Giovanni Fontana,” Annali dell’Istituto e Museo di lische Hohlspiegel,” Bibliotheca mathematica, 3. Folge, Vol. 11 (1910-11), pp. 193 —
Storia della Scienza di Firenze, Anno I (1976), pp. 5-28. 208.
--------------. “The Works of Francesco Maurolico,” Physis, Anno XVI (1974), pp. 149-98. Huillard-Breholles, J. L. A. Historia diplomatica Friderici Secundi. Vol. 2, Paris, 1852.
Curtze, Μ. “II. Der Briefwechsel Regiomontan’s mit Giovanni Bianchini, Jacob von Speier Vol. 5, Paris, 1859.
und Christian Roder,” Urkunden zur Geschichte der Mathematik im Mittelalter und Huxley, G. L. Antemius of Tralles: A Study in Later Greek Geometry. Cambridge, Mass.,
der Renaissance, in Abhandlungen zur Geschichte der mathematischen Wissenschaf- 1959.
ten, XII. Heft (1902), pp. 185-336. Ibn al-Haytham. De speculis comburentibus. Latin text, analysis and German trans, in J. L.
Devreese, R. Le Fonds grec de la Bibliotheque Vaticane des origines a Paul V. Citta del Heiberg and E. Wiedemann, “Ibn al Haitams Schrift iiber parabolische Hohlspiegel,”
Vaticano, 1965. Bibliotheca mathematica, 3. Folge, Vol. 10 (1909-10), pp. 201-37. Cf. MS. Paris,
Dijksterhuis, E. J. Archimedes. Copenhagen, 1956. BN lat. 9335, 86r-88v. See also English trans, in H. J. J. Winter and W. 'Arafat,
Diodes. Burning Mirrors. The Arabic Translation of the Lost Greek Original. Ed. and transl. “Ibn al-Haitham on the Paraboloidal Focussing Mirror,” Journal of the Royal Asiatic
of G. J. Toomer. Heidelberg / New York, 1976. Society of Bengal. Science, Vol. 15 (1949), pp. 25-40.
Durer, A. Albertus Durerus Nurembergensis . . . versus e Germanica lingua in Latinam --------------. Majamit' al-rasa’il .... ibn al-Haitham (Arabic), Hyderabad, 1357 A. H.
. . . Quatuor his suarum institutionum geometricarum libris, lineas, superficies et --------------. Perspectiva. Latin trans, published by F. Risner in Opticae thesaurus Alhazeni
solida corpora tractavit, adhibitis designationibus ad eam rem accommodissimis. Arabis libri septem . . . Item Vitellonis Thuringopoloni opticae libri X. Basel, 1572.
Latin trans, of Unterweysung der Messung by J. Camerarius. Paris, 1532. I have Separate pagination for each of the works. For Latin MSS of Alhazen, see Chap. 1
cited this under the title Institutiones geometricae. above, n. 41.
--------------. Unterweysung der Messung. Nuremberg, 1525. Fasc. reprint, Zurich, 1966.
--------------. See Woepcke, F.
Efodi. See Profiat Duran.
Kepler, H. Ad Vitellionem paralipomena quibus Astronomiae pars optica traditur. Frankfurt,
Emden, A. B. A Biographical Register of the University of Oxford. Vol. 1, Oxford, 1957.
1604.
Euclid. Euclidis Megarensis . . . Elementorum geometricorum libri λ:v. Basel, 1546. One of
--------------. Nova stereometria doliorum . . . Accessit Stereometriae Archimedeae Sup­
the standard editions that contains both the Campanus and Zamberti versions.
plementum. Lincii, 1615.
--------------. Elementorum geometricorum libri tredecim ex traditione doctissimi Nasiri-
Knorr, W. R. “Archimedes’ Neusis-Constructions in Spiral Lines,” Centaurus, Vol. 22
Dini Tusini: nunc primum Arabice impressi. Rome, 1594.
(1978), pp. 77-98.
Eutocius. See Apollonius and Archimedes, where the various editions that contain these au­
thors and Eutocius’ commentaries are listed. Under Apollonius, see in particular the Krause, M. “Stambuler Handschriften islamischer Mathematiker,” Quellen und Studien
versions of Commandino, Halley, and Heiberg. Under Archimedes, see the edition zur Geschichte der Mathematik, Astronomie und Physik, Abt. B: Studien, Vol. 3
of Heiberg. Also consult the Index of Names and Works under Eutocius. (1936), pp. 437-532.
Fibonacci, Leonardo. See Leonardo. La Hire, P. de. Sectiones conicae. Paris, 1685.
Fine, O. De speculo ustorio, ignem ad propositam distantiam generante, liber unicus. Ex Leonardo Fibonacci. Scritti di Leonardo Pisano. Ed of B. Boncompagni. Vol. 2, Rome,
quo duarum linearum semper approprinquantium et nunquam concurrentium colligi­ 1862.
tur demonstratio. Paris, 1551. See corrected ed. in Chap. 7, Text A. Lindberg, D. C. A Catalogue of Medieval and Renaissance Optical Manuscripts. Toronto,
Folkerts, M. “Werner, Johann,” Dictionary of Scientific Biography, Vol. 14, New York, 1975.
1976, pp. 272-77. --------------. John Pecham and the Science of Optics.Madison, 1970.
468 ARCHIMEDES IN THE MIDDLE AGES BIBLIOGRAPHY 469
Madan, F. A Summary Catalogue of Western Manuscripts in the Bodleian Library at Oxford. Ravius, C. See Apollonius.
Vol. 4. Oxford, 1897. Regiomontanus. Additions to the Speculi almukefi compositio. See my ed. and English trans,
Maimonides, Moses. Dux seu Director dubitantium aut perplexorum. Paris, 1520. in Chap. 5 above. Text D.
--------------. The Guide of the Perplexed. Trans, of S. Pines. Chicago, 1963. --------------. See Anonymous, Speculi almukefi compositio, for Regiomontanus' version of
Margoliouth, G. Catalogue of the Hebrew and Samaritan Manuscripts in the British Museum. that work.
Part III. London, 1915. Richard, C. See Apollonius.
Maurolico, F. Opuscula mathematica. Venice, 1575. Risner, F. See Ibn al-Haytham and Witelo.
--------------. See Apollonius and Clagett, M. Rose, P. L. “A Venetian Patron and Mathematician of the Sixteenth Century: Francesco
Memmo, G. B. See Apollonius. Barozzi, 1537-1604,” Studi Veneziani, N. S., Vol. 1 (1977), pp. 119-78.
Mirot, L. “La Proces de maitre Jean Fusoris,” Memoires de la Societe de I'Histoire de Paris Sabra, A. I. “Ibn al-Haytham,” Dictionary of Scientific Biography, Vol. 6, New York,
et de TIle-de-France, Vol. 27 (1900), pp. 137-287. 1972, pp. 189-210.
Moise Provenzale. Opera nova di giometria in dimostrare come possano uscire due linee Sacerdote, G. “Le livre de l'Algebre et la probleme des asymptotes de Simon Motot,”
sopra una superficie le quali proccedeno si accostino sempre ne poseno incontrarsi Revue des etudes juives, Vol. 27 (1893), pp. 91-105; Vol. 28 (1894), 228-46; Vol. 29
mai, cosa meravigliosa e prodotta per verace e certa damo Mose d’Egitto Heb. nel (1894), 111-26.
suo Directio dubitantium a cap. 73 del primo composta per Magistro Moise Pro­ Serenus. See Apollonius, eds. of Halley and Commandino.
venzale Hebreo. Mantua, 1550. al-Sijzi. On the matter of two lines—one of which is a straight line and the other a perimeter
Motot, Simeon. See Sacerdote, G. of a hyperbolic section—that always approach each other on being extended in­
Mugler, C. Dictionnaire historique de la terminologie geometrique des Grecs. Paris, 1958. definitely but without meeting. In Arabic, MS Leiden, Bibl. der Rijksuniversiteit,
Murdoch, J. E. “Euclides Graeco-Latinus. A Hitherto Unknown Medieval Latin Translation Cod. Or. 14(6), pp. 226-31.
of the Elements Made Directly from the Greek,” Harvard Studies in Classical Phi­ Spiazzi, G. “Barozzi, Francesco,” Dizionario biografico degli Italiani, Vol. 6, Rome, 1964,
lology, Vol. 71 (1966), pp. 249-302. pp. 495-99.
Nebbia, G. “Ibn al-Haytham nel millesimo anniversario della nascita,” Physis, Anno 9 Steinschneider, M. Catalogus librorum Hebraeorum in Bibliotheca Bodleiana. 2nd ed.,
(1967), pp. 165-214. Berlin, 1931.
Nix, L. M. L., Das funfte Buch der Conica des Apollonius von Perga in der arabischen Tartaglia, N. La quinta parte del General trattato de’ numeri et misure. Venice, 1560.
Vbersetzung des Thabit ibn Corrah. Leipzig, 1889. Toomer, G. J. See Diodes.
Pappus. Pappi Alexandrini Collectionis quae supersunt. Ed. of F. Hultsch. Vol. 2, Berlin, al-Tusi. See Euclid.
1887; repr. Amsterdam, 1965. Unguru, S. “A Very Early Acquaintance with Apollonius of Perga’s Treatise on Conic
Para vicini Bagliani, A. “Witelo et la science optique a la cour pontificale de Viterbe (1277),” Sections in the Latin West,” Centaurus, Vol. 20 (1976), pp. 112-28.
Melanges de VEcole Erancaise de Rome: moyen age, temps modernes, Vol. 87 --------------. “Pappus in the Thirteenth Century in the Latin West,” Archive for the History
(1975), pp. 425-53. of Exact Sciences, Vol. 13 (1974), pp. 307-24.
Paris. Catalogus codicum manuscriptorum Bibliothecae Regiae. Pars tertia, Vol. 4, Paris, --------------. See Ibn al-Haytham.
1794. Valla, G. De expetendis et fugiendis rebus opus. Venice, 1501.
Pecham, John. See Lindberg, D. C. Vienna. Tabaluae codicum manu scriptorum praeter graecos et orientales in Bibliotheca
Peletier, J. Commentarii tres: I. De dimensione circuli. II. De contactu linearum; et de Palatina Vindobonensi. Vol. 4, Vienna, 1870.
duabus lineis in eodem plano neque parallelis, neque concurrentibus. III. De consti­ Vogel, K. “Fibonacci, Leonardo,” Dictionary of Scientific Biography. Vol. 4, New York,
tutione horoscopi. Basel, 1563. 1971, pp. 604-13.
Porta, G. B. della. Magiae naturalis sive de miraculis rerum naturalium libri Illi. Naples, Watson, A. G. “A Merton College Manuscript Reconstructed: Harley 625, Digby 178, fols.
1558. Augmented ed. with title Magiae naturalis libri XX. Naples, 1589. 1-14, 88-115; Cotton Tiberius B.IX, fols. 1-4, 225-35,” Bodleian Library Record,
Poulle, E. “Fine, Oronce,” Dictionary of Scientific Biography, Vol. 15, New York, 1978, Vol. 9 (1976), pp. 207-16.
pp. 153-57. Werner, J. In hoc opere haec continentur: Libellus Ioannis Verneri Nurembergen. super
--------------. “Fusoris, Jean,” Dictionary of Scientific Biography, Vol. 15, New York, 1978, vigintiduobus elementis conicis. Eiusdem commentarius seu paraphrastica enarratio
pp. 162-63. in undecim modos conficiendi eius problematis quod cubi duplicatio dicitur. Eiusdem
---------------. Un Constructeur d’instruments astronomiques au XVe siecle Jean Fusoris. commentatio in Dionysidori problema, quo data sphaera plano sub data secatur
Paris, 1963. ratione, alius modus idem problema conficiendi ab eodem Ioanne Vernero novissime
compertus demonstratisque. Eiusdem Ioannis, de motu octavae sphaerae, tractatus
Profiat Duran. Commentary on Chap. 73 of Bk. I of the Guide of the Perplexed (Hebrew).
duo. Eiusdem summaria enarratio theoricae motus octavae sphaerae. Nuremberg,
Sabionetta, 1553. Latin Trans, by F. Barozzi, falsely ascribed by him to Rabbi Samtou.
1522. See my ed. and English trans, of Libellus super vigintiduobus elementis conicis
See above, Chap. 7, n. 41, and Text B, Extract VIII.
in Chap. 6 above, Text.
Provenzale, Moise. See Moise.
Wickersheimer, E. Dictionnaire biographique des medecins en France au moyen age. Vol. 1,
Ptolemy. Cl. Ptolemaei Pelusiensis mathematici operis quadripartiti in Latinum sermonem
Paris, 1936.
traductio: Adiectis libris posterioribus Antonio Gogava Graviens, interprete. . . .
Wiedemann, E. See Heiberg, J. L., and E. Wiedemann, and Ibn al-Haytham.
Item, De sectione conica orthogona, quae parabola dicitur; Deque speculo ustorio,
Winter, H. J. J. See Ibn al-Haytham.
libelli duo, hactenus desiderati, restituti ab Antonio Gogava Graviensi. Lovanii,
Witelo. Vitellionis mathematici . . . Περί οπτικής, id est de natura, ratione, et projec-
1548.
470 ARCHIMEDES IN THE MIDDLE AGES
tione radiorum visus, luminum, colorum atque formarum, quam vulgo perspectivam
vocant, libri x. Nuremberg, 1535; rep. 1551. For ed. of F. Risner in 1572, see Ibn
al-Haytham.
--------------. Witelonis Perspectivae liber primus. Book I of Witelo’s Perspectiva. Ed. of S.
Unguru. Wroclaw, Krakow, 1977. A revision of Unguru’s doctoral dissertation, Uni­
versity of Wisconsin, 1970.
---------------. See Fusoris.
Woepcke, F. L’Algebre d’Omar alkhayyami. Paris, 1851.
Zamberti, B. See Euclid.
Zeuthen, H. G. Die Lehre von den Kegelschnitten im Altertum. Copenhagen, 1886.
Zinner, E. Leben und Wirken des Job. Muller von Konigsberg gennant Regiomontanus.
2nd ed., Osnabruck, 1968.
Diagrams
DIAGRAMS 473
472 ARCHIMEDES IN THE MIDDLE AGES
Figures 1.5-1.7
Figures 1.1-1.4
curves here and in Figs. 1.8 and 1.9 are
circular arcs in the MSS.
Fig. 1.5
c
Fig. 1.3 Fig. 1.4
Fig. 1.6 Fig. 1.7
474 ARCHIMEDES IN THE MIDDLE AGES
DIAGRAMS 475
Figure 1.8
Figure 1.9
As given in the edition As in the Sloane MS
h
As in the Sloane MS. The same figure appears
in the Royal College MS [with letters that
differ given here in brackets]. In the Royal
College MS letter n is missing, as is the
hyperbolic branch tp. The latter MS adds a
third line through point z that lies between
kzl and ozr, a line designated as Izm, thus
making two points l. No doubt this added line
was mistakenly taken from Fig. 1.8.
Fig. 1.8
Fig. 1.9
476 ARCHIMEDES IN THE MIDDLE AGES
DIAGRAMS 477
Figures 2.1-2.4 Figures 2.5-3.2
D
d
Only the base circle appears in MS P of
In MS P of Version B the letters z and h are Version B. In MS O the line dh was not drawn
interchanged, no doubt so that the figure as a diameter and diameter beg (here added as
might serve also for Proposition 2. In this and a broken line) is missing.
the rest of the figures I have drawn the base
Fig. 2.3
circles in perspective. In the manuscripts
they are drawn as circles in the plane of
triangle abg.
Fig. 2.1 This and the succeeding diagrams through
Fig. 2.5 Fig. 3.8 have been reproduced from Clagett,
Archimedes in the Middle Ages, Vol. 2, where
details concerning their appearance in the
autograph of Moerbeke’s translations are
given.
Fig. 3.1
T K M H
In MS O straight lines dh and he were not In MS O the letters m and n are falsely
drawn parallel to lines zb and zg respectively, interchanged, as are letters s and o. In MS P
as they should have been. In MS P of Version of Version B the letters are correctly placed
B lines bzg and dhe are drawn as straight but the whole orientation is reversed. In MS
lines, i.e. as diameters, and the figure serves O the base diameter bg is misdrawn as a bent Fig. 3.2
for both Propositions 1 and 2. I have added line. I have added the broken line that
the broken lines. represents part of the hyperbolic curve.
Fig. 2.2 Fig. 2.4
DIAGRAMS 479
478 ARCHIMEDES IN THE MIDDLE AGES
Figures 3.6-3.10
Figures 3.3-3.5
Λ
K
N
Fig. 3.4
z
As given by Risner; the MSS (as presented by
9. K Unguru) do not include the base circle but
only its diameter. The letter D appears as E
B in the MSS. This letter is not mentioned in
the text. Note that all of the letters in the
diagrams and text, both in the MSS and
Risner’s edition, are minuscules but I have
S2
followed Unguru in using majuscules.
Fig. 3.9
z
Fig. 3.7
A
Fig. 3.3
As taken from the manuscript with all letters
and lines removed except those referred to
in note 6 above. ABG should be a parabola As given by Risner; again the MSS do not
but appears as a semicircle in the manuscript. include the base circle.
Fig. 3.8 Fig. 3.10
Fig. 3.5