Archimedes in the Middle Ages. V. Quasi-Archimedean Geometry in the Thirteenth Century . Part v

59 Pages·1984·3.01 MB·English

by  Marshall Clagett

ARCHIMEDES
in the Middle Ages
VOLUME FIVE
Quasi-Archimedean Geometry in the Thirteenth Century. A Supple­
mentary Volume Comprising the Liber de Motu of Gerard of Brussels,
the Liber philotegni of Jordanus de Nemore together with its longer
Version known as the Liber de triangulis Iordani, and an Appended
Text of John Dee’s Inventa circa illam coni recti atque rectanguli sec­
tionem quae ab antiquis mathematicis Parabola appellabatur.
Part V. Bibliography, Diagrams, and Indexes
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1984
Copyright 1984 by the American Philosophical Society
for its Memoirs series, Volume 157
Publication of this and other volumes has been made possible
by a generous grant from The Institute for Advanced Study,
Princeton, New Jersey.
Memoirs of the
AMERICAN PHILOSOPHICAL SOCIETY
held at Philadelphia
for Promoting Useful Knowledge
Volume 157 Part B
Library of Congress Catalog Card No. 62-7218
International Standard Book Number: 0-87169-157-4
US ISSN: 0065-9738
PART V
Bibliography, Diagrams, and Indexes
BIBLIOGRAPHY
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269 (partial French translation).
Anaritius (=al-Nairizi). See Curtze; and Besthom and Heiberg.
Anonymous. Liber de triangulis datis. First two propositions developed from the Liber philotegni.
See Appendix III.A for the text of these propositions.
Anonymous. Liber de triangulis Iordani. I have given a new text and English translation of this
above in Part III. Also see pseudo-Jordanus for the earlier text of Curtze.
Antolin, G. Catalogo de los codices latinos de la Real Biblioteca del Escorial. Vol. 3, Madrid,
1913.
Apollonius of Perga. Apollonii Pergaei Conicarum sectionum libri V, VI, et VII. Translation
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---------------. Apollonii Pergaei Conicorum libri octo et Sereni Antissensis De sectione cylindri
et coni libri duo. Ed. of E. Halley, Oxford, 1717.
Archibald, R. C. Euclid’s Book on Divisions of Figures. Cambridge, 1915.
Archimedes. On the Construction of the Regular Heptagon. For studies of this, see Part III, Chap.
1, nn. 18-20.
---------------. Opera quae quidem extant omnia. Basel, 1544. First edition of the Greek text and
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Besthom, R. O., and J. L. Heibeig, eds. Codex Leidensis 399.1 Euclidis Elementa ex interpretatione
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Birkenmajer, A. Etudes d’histoire des sciences et de la philosophic du moyen age. Wroclaw, etc.,
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Bjombo, A. A. Die mathematischen S. Marcohandschrifien in Florenz. Ed. of G. C. Garfagnini,
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132, 193-224. This edition includes corrections and additions.
---------------. “Studien iiber Menelaos’ Spharik,” Abhandlungen zur Geschichte der mathematischen
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Bjombo, A. A., and S. Vogl. “Alkindi, Tideus und Pseudo-Euklid,” Abhandlungen zur Geschichte
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607
608 ARCHIMEDES IN THE MIDDLE AGES BIBLIOGRAPHY 609
---------------. The Translation ofthe Elements of Euclid from the Arabic into Latin by Hermann Duhem, P. “A propos du •PiXorex^m de Jordanus de Nemore,” Archiv fur die Geschichte der
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Busard, H. L. L., and P. S. van Koningsveld. “Der Liber de arcubus similibus des Ahmed ibn Emden, A. B. A Biographical Register of the University of Oxford to A. D. 1500. Vol. 3, Oxford,
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Calder, I. R. F. “John Dee: Studied as an English Neoplatonist.” Thesis, London University, 2 Enestrom, G. “Das Bruchrechnen des Jordanus Nemorarius,” Bibliotheca mathematica, 3. Folge,
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Campanus. See Busard; and Euclid. ---------------. “Der ‘Algorismus de integris’ des Meisters Gemardus,” Bibliotheca mathematica,
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Various Languages of Europe and the East, Formed by M. Guglielmo Libri. . . Which
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1976-80.
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---------------. Opera a Campano . . . tralata. Venice, 1509.
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---------------. See Moody.
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Gerard of Brussels. Liber de motu. For my new edition, see above, Part I of this volume. Also
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see Clagett.
Significance for the Development of Mathematical Physics. Madison, Wise., 1955; 2nd
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---------------. “Jordanus de Nemore,” Dictiomry of Scientific Biography. Vol. 7, New York,
plementum to the Teubner text of Euclid: Euclidis opera omnia ediderunt I. L. Heiberg
1973, pp. 171-79.
et H. Menge.)
---------------. Nicole Oresme: "Deproportionibus proportionum" and “Adpauca respicientes.”
---------------. “Ueber eine Handschrift der Konigl. offentl. Bibliothek zu Dresden,” Zeitschrift
Madison, Wise., 1966.
Jur Mathematik und Physik, Vol. 28 (1883), Historisch-literarische Abtheilung, pp. 1-
Haskins, C. H. Studies in the History of Mediaeval Science, 2nd ed., Cambridge, Mass., 1927.
13.
Heath, T. L. Aristarchus of Samos, The Ancient Copernicus. Oxford, 1913.
---------------. “Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance.
---------------. See Euclid.
I. ‘Liber embadorum’ des Savasorda in der Ubersetzung des Plato von Tivoli,” Abhand-
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wendungen, 12. Heft (1902), pp. 1-183.
Bibliotheca mathematica, 3. Folge, Vol. 10 (1909-10), pp. 201-37.
---------------. See pseudo-J ordanus.
Hermann of Carinthia. De essentiis. A Critical edition with translation and commentary by
Dee, John. De speculis comburentibus libri 5. In primis duobus: Inventa Joannis Dee Londinensis
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Histoire litteraire de la France. Vol. 23, Paris, 1856.
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BIBLIOGRAPHY 611
610 ARCHIMEDES IN THE MIDDLE AGES
Ptolemy. Almagestum. Venice, 1515.
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of Arabic Science, Vol. 3 (1979), pp. 309-86.
---------------. Medieval Mathematical Latin Writings in the University Library, Basel. Unpublished
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Roriczer, M. Geometria deutsch. See Shelby.
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Vol. 3 (1973), pp. 253-69.
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Ahmad al-Birum. Hannover, 1927.
Le Fevre: Jordani Nemorarii. . . Elementa Arithmetica cum demonstrationibus Jacobi
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Scott, E. J. L. Index to the Sloane Manuscripts in the British Museum. London, 1904.
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Sharpe, K. Sir Robert Cotton 1586-1631. History and Politics in Early Modem England. Oxford,
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Smith, D. E. Rara arithmetiea. Boston and London, 1908.
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Diagrams
614 ARCHIMEDES IN THE MIDDLE AGES DIAGRAMS 615
Liber de motu, Figure I.la
Liber de motu, Figures I.la(con.)-I.lb
DIAGRAMS SIMPLIFIED
FROM FIG. I.la
For all parts of Proposition 1.1 in Tradition
I, and for preliminary proof in Tradition II.
Tradition II. For Tradition II, for first half of Tradition II. For second half of first proof of
main proof of first first proof of second part. second part, for all second proof of second
part. part, and for corollary.
The bracketed letters refer to letters in Tradition II exclusively. A letter without brackets (1)
when unaccompanied by a letter in brackets is a letter that appears in both traditions, but (2)
when accompanied by a letter in brackets is ordinarily in Tradition I only. Exceptions are
evident in the following comments. Letter E appears on the diagrams of MSS in both traditions
but is only used in the text of Tradition II. Letter B, appearing in the diagrams of both
traditions, is used in the proof of the second part of the proposition in Tradition II, and in the
corollary in Tradition I. It is replaced by the letter [F] in the text of the corollary in Tradition
II, which letter is missing from all the diagrams in the MSS. Letters [K], [N], and [L] are found
only in MS V of Tradition II. In place of them MS P (as well as MSS OBN) has letters G, K, This figure is missing in MS V. In MS P it
and I. The letter I have designated as [AT appears immediately next to O in MS P. In MS V it contains both [K] and R, although in the text
is located farther up the radius, where I have designated it as [A], The diagram of MS P has only [K] is used. In this figure the bracketed
letter D between letters C and [£*], which last letter seems to be substituted for letter F, which letters are only in MS F, though V appears in
does not appear on the diagram of MS P, though of course it is in the text in all MSS. Letter C the text of Trad. I. The letter [RT is mentioned
is missing from the diagram in MS P, appearing only in the diagram of MS V, though it is in in the text of MS F but is missing on the
the text in both MSS P and V. In MS P, concentric circles are drawn through letters C, D, and diagram.
[ET (assuming that in MS P letter [JET is meant to mark the point where F appears in the other
diagrams). In MSS N and V, circles are drawn through C and F. Only a single circle (drawn Fig. I.lb
through F) is on the diagrams of MSS OB. Summarizing the divergent letters on the radius in
the diagram of MS V, we note the following letters (proceeding from the center to the
circumference at F): O, B, [A], [O'], C [FT, [G], and F. The diagram thus omits the letters
marked [AT, D and E on this composite diagram. The letters appropriate for each part of the
proposition in both traditions have been added to the simplified diagrams (A)-(D) which
accompany this composite diagram.
Fig. I.la
DIAGRAMS 617
616 ARCHIMEDES IN THE MIDDLE AGES
Liber de motu, Figures 13b-1.4
Liber de motu, Figures 1.2-13«
B
uses Fig. I.lb and this figure is
repeated in MSS OBN.
Fig. 1.2
The lettering is that of MS O. MSS BV are badly mixed up in their lettering. MS
N has B at the intersection of KH extended and CO. I have added prime signs
to I', O', B\ M', 71, and G because each of these letters is used twice. Points /', B1
are given in MSS ON. MS V has C and B for these points and MS B has E and
B. I have used Z' for a sign that sometimes looks like the abbreviation for
“quam” or “que” and sometimes like that for “et”. In the figure of MS N the 71
looks like M. The line of which 71 is the center is given in MS N as QE, and
perhaps also in MS B. But I have not added these letters since they do not
appear in the text. They are also missing in the diagrams of O and V.
Fig. I.3a
618 ARCHIMEDES IN THE MIDDLE AGES
DIAGRAMS 619
Liber de motu, Figures II.la-II.lb
Liber de motu, Figures II.2-II3b
Square Z appears in MS O only.
MS V omits this figure.
Fig. II.la
In the manuscripts the two
figures are drawn the same
size, though different sizes are
implied by the text. MS V
omits this figure.
Fig. II.3a
Points I, L, and M are misplaced on MSS O
and N but are accurately located on MS B.
MS V omits this figure.
Fig. II.lb
MS V omits this
figure.
Fig. II.3b