Archimedes in the Middle Ages. III. The Fate of the Medieval Archimedes. Parts i+ii

305 Pages·1978·8.26 MB·English

by  Marshall Clagett

Archimedes in the Middle Ages
VOLUME THREE
Memoirs of the
AMERICAN PHILOSOPHICAL SOCIETY
held at Philadelphia
for Promoting Useful Knowledge
Volume 125 Part A
ARCHIMEDES
in the Middle Ages
VOLUME THREE
THE FATE OF THE MEDIEVAL ARCHIMEDES
1300 to 1565
Part I: The Moerbeke Translations of
Archimedes at Paris in the
Fourteenth Century
Part II: The Arabo-Latin and Handbook
Traditions of Archimedes in the
Fourteenth and Early Fifteenth
Centuries
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1978
Copyright © 1978 by The American Philosophical Society
Library of Congress Catalog Card Number 76-9435
International Standard Book Number 0-87169-125-6
US ISSN 0065-9738
PREFACE
With this volume, I conclude my direct investigation of Archimedean
traditions in the Middle Ages. As will be evident to the reader, special
effort has also been made to disentangle the medieval elements still
working in the Renaissance. Indeed the major part of this volume is so
directed. My objective here, as always, has been to present as fully as
possible the textual material on which the investigation is based. Many
texts or parts of texts have been edited for the first time from manuscripts
and some have been re-edited from earlier editions. I have appended
English translations to each of these texts except for the case of the
extracts from Giorgio Valla’s De expetendis et fugiendis rebus opus
(added to Part III, Chapter 2, Section V) since the material included
there is presented more fully, though in somewhat different form, in
the Praeparatio ad Archimedis opera of Francesco Maurolico that is given
as Text A in Part III, Chapter 5 and accompanied by a complete
English translation. All of the translations are my own except for that
given as Text B of Part III, Chapter 4, Section III, which is an English
translation attributed to Thomas Salusbury of Federigo Commandino’s
De Us quae vehuntur in aqua (and of Tartaglia’s Italian translation of
Book I of Archimedes’ De insidentibus aquae, the latter produced from
the Latin translation of William of Moerbeke, and presented by me in
Section II of the same chapter). In addition to these more or less
complete texts, I have also presented in the footnotes countless extracts
drawn from manuscript and printed sources. In the discursive chapters
and sections preceding the texts, or in the chapters without texts, I have
attempted to analyse the understanding of Archimedean texts displayed
by Latin authors. The hurried reader will no doubt go immediately to the
last chapter (Part III, Chapter 7) for a retrospective summary of the
medieval Archimedean material. But I hope that such a reader will be led
back to the texts themselves and their more detailed treatment in the
earlier chapters. One infelicity of the machine composition used for this
volume ought to be observed by the reader. The dot of multiplication,
which is positioned properly for magnitudes represented by capital letters,
is often too high for minuscules, thus leaving a rather strange appearance
to some of the formulas. However, this does not, I believe, produce
any mathematical ambiguity.
As in Volume Two, I have gathered the diagrams together in a
separately bound fascicle that also includes Appendixes, a Bibliography of
works cited in both Volumes Two and Three, and Indexes. I do not cite
the textual page numbers in the subheads on the pages of diagrams (as I
did in Volume Two) since the diagrams often do double and even triple
duty in the discussions, the Latin texts, and the English translations, and
this would accordingly produce a rather confusing set of numbers in each
subhead. But the diagrams are all numbered in an easily identifiable
order, with the Roman numeral standing for the Part and the succeeding
Arabic numerals for the Chapter, the Section and the ordinal position
of the diagram within the given section. For the preparation of the
Indexes I owe special thanks to my former assistant. Dr. Charles
Zuckerman. I must also thank once more my secretary, Mrs. Ann Tobias,
for preparing and mounting the finished diagrams, as well as for typing
and retyping this long and complex volume. I have already acknowledged
the manifold help I have received from others in the preface to Volume
Two. To these, I should add the names of two recent assistants, Glenn
Sterr and Peter Marshall, who helped in reading the proofs of Volume
Three. I should also acknowledge the persistent help and friendship of
my colleague Herman Goldstine, whose wide knowledge of mathematics
and its history was always at my beck and call. Finally, I must again
thank the Director and staff of the Institute for Advanced Study—my
academic home-—and the American Philosophical Society (and par­
ticularly, its editorial staff) for their help-—financial, and of other kinds—
in publishing this volume.
M.C.
VI
Contents
PAGE
Part I: The Moerbeke Translations of Archimedes at Paris
in the Fourteenth Century
Chap. 1: The Quadripartitum numerorum of Johannes
de Muris
Text: Johannis de Muris Quadripartitum numero­
rum, Book IV, Tract 1, Chapter 31
Chap. 2: TheDe arte mensurandi attributed to Johannes
de Muris 11
Chap. 3: The Problem of Proportional Means in Chapter
7 of the De arte mensurandi 19
Text: De arte mensurandi, Chapter 7, Proposi­
tion 16 24
Chap. 4: Johannes de Muris’ Version of Proposition
III of Archimedes’ On the Measurement of the Circle 31
Text: De arte mensurandi. Chapter 8, Proposition 1 36
Chap. 5: The Hybrid Circuli quadratura of 1340 45
Text: Circuli quadratura 52
Chap. 6: The Use of the Moerbeke Translations of
Archimedes in Chapter 10 of the De arte mensurandi 89
Text: De arte mensurandi. Chapter 10 92
Chap. 7: The Use of the Moerbeke Translations of
Archimedes by Nicole Oresme and Henry of Hesse 125
Chap. 8: An Archimedean-Type Proof of the Law of
the Lever 145
Text: Anonymi demonstratio 151
Part II: The Arabo-Latin and Handbook Traditions of Archimedes
in the Fourteenth and Early Fifteenth Centuries
Chap. 1: The Glasgow Collection of Quadrature Tracts 157
Text; The Glasgow Version of the Liber Archi-
menidis de quadratura circuli 162
Chap. 2: Philippus Elephantis’ Knowledge of Archimedes 185
Text: Philippi Elephantis Mathematica, Con­
sideratio 3, Pars 9, Propositiones 18, 21-25, 28 194
Vll
PAGE
Chap. 3: The Handbook Tradition of Archimedes
Sect. I: Archimedes and the Tradition of the
Agrimensores 199
Sect. II: Archimedes and the Medieval Handbooks
of Geometry in the Twelfth and Thirteenth
Centuries 206
Sect III: Archimedes and the Handbook Tradition in
the Fourteenth and Early Fifteenth Centuries 220
Chap. 4: Giovanni Fontana and the Medieval Archimedes
Sect. I: Fontana’s Life and Works 239
Sect. II: Fontana and Archimedes 259
Text: Tractatus Johannis Fontane de trigono
balistario, Bk II, Chapter 24 270
Part III: The Medieval Archimedes in the Renaissance, 1450-1565
Chap. 1: The Medieval Archimedes Toward the Middle
of the Fifteenth Century
Sect. I: Nicholas of Cusa 297
Sect. II: Leon Battista Alberti 316
Chap. 2: The Medieval Archimedes in the Second Half
of the Fifteenth Century
Sect. I: Jacobus Cremonensis 321
Sect. II: Regiomontanus 342
Sect. Ill: Piero della Francesca 383
Sect. IV: Luca Pacioli 416
Sect. V: Giorgio Valla 461
Text: Giorgio Valla, De expetendis et fugiendis
rebus opus, Bk. XI, Chap. 8; Bk. XIII, “De
duobus cubis ad unum redactis” 467
Chap. 3: Leonardo da Vinci and the Medieval Archimedes 477
Chap. 4: The Fate of William of Moerbeke’s Transla­
tions of Archimedes in the Sixteenth Century
Sect. I: Andreas Coner, the Corrector of Moerbeke’s
Archimedes 525
Sect. II: Niccolo Tartaglia and the Moerbeke Transla­
tions of Archimedes 538
Text: Niccolo Tartaglia, Ragionamenti, Rag. I
De insidentibus aquae 592
vni
PAGE
Sect. Ill: Federigo Commandino’s Version of William
of Moerbeke’s Translation of On Floating Bodies 607
Text A: Archimedis De Us quae vehuntur in aqua 636
Text B: Thomas Ssilmbury, Archimedes, His Tract
De Incidentibus Hurnido 682
Chap. 5; Francesco Maurolico and the Medieval
Archimedes
Sect. I; Life and Principal Works of Francesco
Maurolico 749
Sect. II: Some General Aspects of Maurolico’s
Concern "with Archimedes 771
Sect. Ill: Maurolico’s Use of Medieval Archimedean
Texts 788
Text A: The Praeparatio ad Archimedis opera of
Franceso Maurolico 813
Text B: The Archimedis De circuli dimensione
libellus of Francesco Maurolico 873
Text C: The Archimedis Liber de sphaera et cylin­
dro of Francesco Maurolico 908
Text D: The Archimedis Quadratura parabolae of
Francesco Maurolico 1023
Chap. 6: Archimedean Problems in the First Half of the
Sixteenth Century: The Influence of the Medieval
Traditions
Sect. I: Spiral Lines 1055
Sect. II: The Crown Problem 1066
Text A: Archimedis de incidentibus in humidis
cum Francisci de Mello commentariis 1086
Text B: Pierre Forcadel, Le Livre d’Archimede
des pois and Une piece du Livre d’Euclide intitule
du leger et du pesant 1098
A Commentary on Texts A and B 1146
Text C: Propositions from the Liber de ponderibus
Archimenidis translated by Niccolh Tartaglia. 1159
Sect. Ill: The Problem of Proportional Means 1163
Sect. IV: The Quadrature of the Circle and Kindred
Problems 1179
Chap. 7: The Medieval Traditions of Archimedes in
the Renaissarjce: A Retrospective Summary 1225
IX
PAGE
Part IV: Appendixes, Bibliography, Diagrams, and Indexes
Appendixes
I. Corrections and Additions to Volume One
Sect. 1: Short Corrections and Additions 1249
Sect. 2; A Variant Form of the Naples Version of
the De mensura circuli 1257
Sect. 3: Extracts from Savasorda’s Liber emba­
dorum and Leonardo Fibonacci’s Practica
geometrie 1265
Sect. 4: The De ponderibus Archimenidis 1286
II. The Quadrature by Lunes in the Later Middle Ages 1312
Text A: Quadratura circuli per lunulas, Ver­
sion III 1318
Text B: Johannes de Muris, De arte mensu­
randi, Chap. 6, Propositions 26-29 1321
Text C: The De lunularum quadratura at­
tributed to Leon Battista Alberti 1326
III. Biographical Accounts of Archimedes in the Middle
Ages 1329
IV. Some Specimens of Renaissance Translations of
Archimedes
Sect. 1: Jacobus Cremonensis 1342
Sect. 2: Antonius de Albertis 1357
Sect. 3: Federigo Commandino 1366
Sect. 4: The Florentine Archimedes 1368
Bibliography 1393
Diagrams 1413
Indexes for Volumes Two and Three
I. Scientific and Mathematical Terms 1519
II. Manuscripts Cited 1557
III. Names and Works 1563