Archimedes in the Middle Ages. V. Quasi-Archimedean Geometry in the Thirteenth Century . Parts i-iv

306 Pages·1984·17.94 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
Memoirs of the
Version known as the Liber de triangulis Iordani, and an Appended
AMERICAN PHILOSOPHICAL SOCIETY
Text of John Dee’s Inventa circa illam coni recti atque rectanguli sec­
held at Philadelphia tionem quae ab antiquis mathematicis Parabola appellabatur.
for Promoting Useful Knowledge
Volume 157 Part A Parts I-III. Texts and Analysis.
Part IV. Appendixes.
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1984
PREFACE
In this second supplemental volume I have presented the texts, English
translations, and analyses of three of the most interesting geometrical works
Copyright 1984 by the American Philosophical Society
of the thirteenth century: the Liber de motu of Gerard of Brussels, the Liber
for its Memoirs series, Volume 157
philotegni of Jordanus de Nemore, and the expanded and altered version of
Jordanus’ work completed by an unknown author under the title of Liber
Publication of this and other volumes has been made possible
de triangulis lordani. These texts will allow the student of medieval geometry
by a generous grant from The Institute for Advanced Study,
to see how Latin mathematicians responded to the wave of translations from
Princeton, New Jersey.
Greek and Arabic which appeared in the preceding century and particularly
how these mathematicians utilized certain Archimedean and quasi-Archi-
medean techniques. The works which were translated in the twelfth century
and proved particularly important for the authors of the works that I have
edited here included the De mensura circuli of Archimedes, the Archimedean-
type text Liber de curvis superficiebus Archimenidis (put together or translated
by Johannes de Tinemue), and the Verba filiorum of the Banü Müsá, another
Archimedean-like work. These three Archimedean works I edited in Volume
One of my study. The translations of the twelfth century also included various
versions of the Elements of Euclid and a translation by Gerard of Cremona
of a now lost version of the Liber divisionum [jigurarum] of Euclid, an
anonymous Liber de ysoperimetris based ultimately on a work of Zenodorus,
and the Liber de similibus arcubus of Ahmad ibn Yüsüf, together with nu­
merous fragments translated from the Arabic that concerned triangles and
other polygons and the theory of proportions.
I should explain why I picked these three works to edit. The Liber de motu
I first edited in a preliminary fashion in 1956. Since that time I have found
another manuscript of it and altered my interpretation and understanding
Library of Congress Catalog Card No. 62-7218 of the text, particularly as to the way in which it fits into medieval geometry.
International Standard Book Number; 0-87169-157-4 Hence I have thoroughly revised the text, extended the variant readings,
US ISSN: 0065-9738 provided an English translation, and given a lengthy analysis of the math­
ematical contents of the text. It obviously belongs with works under the
rubric of “Quasi-Archimedean Geometry” since it borrowed and widely used
the particular form of the method of exhaustion that appeared in the Liber
de curvis superficiebus. Furthermore, the author developed a method not
unlike that used by Archimedes to compare areas of figures and their motions
by comparing the corresponding line elements of these figures. This was a
brilliant tour de force that singles out this author as one of the most original
mathematicians of the Middle Ages. The second of the works, the Liber
philotegni of Jordanus, the separate existence of which has never before been
recognized, is here edited for the first time. It is clearly a masterpiece, which
treats of the areas and perimeters of triangles (and other polygons), of their
divisions, of their comparisons one to the other when inscribed and/or cir­
cumscribed in circles and when possessing equal perimeters. Only two manu­
Readers familiar with my earlier volumes will see that I have adopted the
scripts contained the last and most interesting of the propositions of this
same format and organization here. Diagrams, along with the Bibliography
work (Propositions 47-63), and one of these manuscripts has no diagrams
and the Indexes, are included in a separate fascicle, and thus are easily related
while the other is wanting a number of crucial diagrams. I mention these
to references to them in the texts. 1 have attempted to give quite complete
deficiencies only to inform the reader that the reconstruction of texts and
information on the nature of the variant forms of the diagrams from manu­
diagrams of the last part of this treatise was often a difficult task. Still it
script to manuscript, and this information is encapsulated in the legends
seems to me that my reconstruction makes both mathematical and linguistic
attached to the diagrams. In general, analysis of the three texts is included
sense and I trust that the reader will agree. I have put it forward as a quasi-
in the chapters preceding them, though particular points are often raised in
Archimedean work primarily because it adopts a kind of geometrical trig­
footnotes to the English translations of the texts. I have been especially gen­
onometry that was familiar to Aristarchus and Archimedes (as well as to
erous in giving variant readings because so often we find variant forms of
other later Greek geometers including especially the author of the Liber de
the proofs in the margins or texts of the manuscripts. Occasionally I have
ysoperimetris). It will be evident to the reader that Jordanus takes his basic
translated such variant forms of the proofs and given them in the footnotes
Proposition 5, which he learned from the Liber de ysoperimetris, and from
to the English translations of the texts.
it develops many nice theorems in a way that appears to be original with
Once more I am pleased to acknowledge the assistance I have received
him. I further believe that the main objective of the treatise, in seeking out
from librarians and scholars abroad, and I have attempted to single out each
theorems concerning inscribed and circumscribed polygons, probably arose
instance of that help in the body of the text or in the notes. Here at The
because of the Archimedean concern with such polygons found in the Liber
Institute for Advanced Study I was particularly fortunate to have the help
de curvis superficiebus and perhaps the De mensura circuli. Finally I chose
of two stalwart friends: Dr. Herman Goldstine and Professor Harold Chemiss.
the Liber de triangulis lordani (a work till now assigned to Jordanus) to edit
How often and with what patience did Dr. Goldstine listen to my efforts to
not only because it was poorly edited by Curtze on the basis of a single
reconstruct some badly preserved proof of Jordanus, and if I have been
manuscript and thus deserves a new edition but also because of its relationship
successful in these reconstructions not a little credit belongs to him. Only
to the Liber philotegni upon which it is based and which it vastly alters and
Professor Cherniss will know how important to me his linguistic guidance
because it contains a good many extra propositions translated from the Arabic
has been. I must also acknowledge the expert help of my assistant Mr. Mark
that concern some of the basic problems of Greek geometry often associated
Darby and my secretary Mrs. Ann Tobias. Mr. Darby has constantly read
with the name of Archimedes. For example it includes the trisection of an
proof, checked references, and prepared the indexes, while Mrs. Tobias has,
angle (associated with Archimedes’ name in the pseudo-Archimedean Lem­
as usual with these volumes, typed the manuscript more than once, read the
mata, a work which reduces the trisection to a neusis that can be easily
proofs, and prepared the final diagrams in exemplary fashion. Also helpful
related to the neusis that the author of Proposition IV.20 drew from the
have been the library and administrative staffs of the Institute. They have
Verba filiorum of the Banü Müsá, a neusis not unlike those found in the On
combined to make the Institute the ideal place for scholarly work. Furthermore
Spiral Lines of Archimedes), the inserting of two mean proportionals between
I would surely be remiss if I did not thank the Institute and its Director,
two given magnitudes so that all four magnitudes are in continued proportion
Harry Woolf, for the financial aid given to the publication of this volume.
(a problem whose Greek solutions are summarized by Eutocius in his Com­
Last, I thank my colleagues at the American Philosophical Society for pub­
mentary on the Sphere and the Cylinder of Archimedes), and the construction
lishing these difficult and costly volumes and the editorial staff of the Society
of a regular heptagon (a problem of which an Archimedean solution is extant
for “putting the book into light,” as Renaissance authors were wont to say.
in an Arabic work attributed to Archimedes). It is my hope, then, that this
new edition with its accompanying English translation and its extended anal­
Marshall Clagett
ysis will make clear the relationship the Liber de triangulis has to the Liber
philotegni and its contributions to medieval geometry. So much then for the
texts that make up the main subject of this volume. The reader will also
note that I have edited for the first time the Inventa of John Dee on the
parabola in Appendix II. This work belongs more properly to my Volume
Four, but during the preparation of that Volume I was unfamiliar with the
contents of Dee’s work. Then, on reading it, I realized that it was a work
that joined the medieval traditions of works on the parabola with Archimedean
propositions developed in Archimedes’ On the Quadrature of the Parabola.
Hence its inclusion as an appendix to my new volume.
Appen. Ill: Sources Related to Some Propositions in Jor­
danus’ Liber philotegni and in the Liber de triangulis
Contents lordani
A. Sources Related to Propositions 5, 6, 14, 18-23,
and 25 of the Liber philotegni and the Correspond­
PAGE ing Propositions of the Liber de triangulis lordani 577
Preface B. Sources for Propositions IV. 10, IV.12-IV.28 of
the Liber de triangulis lordani 591
Part I: The Liber de motu of Gerard of Brussels
Part V: Bibliography, Diagrams, and Indexes
Chap. 1: Gerard of Brussels and the Composition of the
Liber de motu Bibliography 607
Chap. 2: An Analysis of the Mathematical Content of the
Diagrams 613
Liber de motu 13
Chap. 3: The Text of the Liber de motu 53 Index of Latin Mathematical Terms 669
The Book on Motion of Gerard of Brussels, Latin
Index of Manuscripts Cited 701
Text 63
The English Translation 111 Sigla in Alphabetical Order 704
Index of Citations of Euclid’s Elements 705
Part II: The Liber philotegni of Jordanus de Nemore
Chap. 1: Jordanus and the Liber philotegni 145 Index of Names and Works 711
Chap. 2: The Mathematical Content of the Liber philotegni 153
Chap. 3: The Text of the Liber philotegni 187
The Book of the Philotechnist of Jordanus de Nemore,
Latin Text 196
The English Translation 258
Part III: The Liber de triangulis lordani
Chap. 1: The Liber de triangulis lordani: Its Origin and
Contents 297
Chap. 2: The Text of the Liber de triangulis lordani 333
The Book on Triangles of Jordanus, Latin Text 346
The English Translation 430
Part IV: Appendixes
Appen. I: Corrections and Short Additions to the Earlier
Volumes 481
Appen. II: The Inventa of John Dee Concerning the Pa­
rabola 489
The Latin Text 513
The English Translation 548
PART I
The Liber de motu of Gerard
of Brussels
CHAPTER 1
Gerard of Brussels and the
Composition of the Liber de motu
In Volume One of this work I briefly mentioned the use that a little-known
geometer Gerard of Brussels made of Archimedes’ On the Measurement of
the Circle and the Book on the Curved Surfaces of Archimedes attributed to
Johannes de Tinemue.* I did not at that time examine Gerard’s work, the
Liber de motu, in detail since I had already published an edition of it in
1956 and had discussed its role in the development of medieval kinematics,
both in the edition and in my The Science of Mechanics in the Middle Ages}
In the twenty-eight years since the appearance of the first edition certain
questions of the proper interpretation of the Liber de motu have arisen,^ and
furthermore another manuscript of it has been located (see MS E among
the sigla of Chapter Three below). Because of the importance of the text to
geometrical studies such as those I have treated in the first four volumes of
my Archimedes in the Middle Ages, I have decided to publish a corrected
version of the text with a fuller presentation of the variant readings, an
accompanying English translation, and a detailed analysis of the text. It is
hoped that the new text and translation will lead to a closer consideration
of the work than has hitherto been possible.
In the twentieth century attention was first called to a short fragment of
the De motu that appears in a manuscript of the Bibliothèque Nationale (BN
' M. Clagett, Archimedes in the Middle Ages, Vol. 1 (Madison, Wise., 1964), pp. 9-10.
^ For the first edition of the Liber de motu, see M. Clagett, “The Liber de motu of Gerard of
Brussels and the Origins of Kinematics in the West,” Osiris. Vol. 12 (1956), pp. 73-175.
Cf. my The Science of Mechanics in the Middle Ages (Madison, 1958; 3rd reprint, 1979),
Chap. 3.
^ V. Zubov, “Ob ‘Arkhimedovsky traditsii’ v srednie veka (Traktat Gerarda Bryusselskogo
‘Odvizhenii’),” Istoriko-matematicheskie issledovaniya, Vol. 16 (1965), pp. 235-72.1 must thank
Prof. Michael Mahoney of Princeton University for sending me a draft of his English translation
of this article, which has been of great help to me in going through Zubov’s article. While Zubov’s
article proved of considerable assistance to me in preparing my analysis of the content of the
treatise in Chapter Two below, let me say that I have found some cases in which I disagree with
Zubov and some in which he has overlooked difficulties in the text. I believe that the new analysis
solves all of the principal difficulties that appear in Gerard’s work.
ARCHIMEDES IN THE MIDDLE AGES COMPOSITION OF THE LIBER DE MOTU
lat. 8680A, 4r-5r; siglum P below) by Pierre Duhem."* Duhem knew nothing when in 1921 G. Enestrom published a short article in which he correctly
of the author of this work, nor its original title since the fragment he discovered identified Duhem’s fragment as a part of a longer work by one Master Gerard
was without indication of author or title. Furthermore, that fragment included of Brussels.^ He made this identification on the basis of three manuscripts:
only the initial postulates of Book I (which I have numbered 1-8) and Prop­ those of Berlin, Oxford, and Naples (see the manuscripts B, O, and iV described
osition 1.1. Thus the remaining twelve propositions were missing. But frag­ in Chapter Three below), though in actuality he saw only the first page of
mentary as this piece was, Duhem sensed its significance for the study of the tract in the Berlin manuscript and nothing from the other manuscripts,
medieval kinematics. He suggested in fact that the composition of the work his knowledge of manuscripts O and N coming orally from A. A. Bjombo.
inaugurated kinematic studies in the West. He correctly saw that the object Enestrom spoke of the treatise as containing three “chapters” with four, five,
of this preliminary material was to prove that the motion of a rotating line and four propositions respectively. In fact the “chapters” are specified as
or radius was made uniform by the speed of its middle point. That is, if the “books” in manuscripts B and N and remain undesignated in the other
speed of the middle point is given to all points of an equal line segment manuscripts (see the variant readings for the beginning of each book in the
moving in translation always parallel to itself, that motion of translation in text below). Since Enestrom had in hand a photograph of only the first page
the same time produces a rectangular area equal to the curvilinear area of the Berhn copy, he confined himself (like Duhem) to publishing the pos­
produced by the rotating segment. In Gerard’s terminology the rotating seg­ tulates and the enunciation of Proposition 1.1, designating both the postulates
ment is said “to be moved equally as its middle point” (see my discussion and the enunciation as “propositions.” In regard to the author he suggested
of the intentions of Gerard’s treatment in Chapter Two below). Furthermore that because two of the manuscripts (in fact, as we now know, three of them:
' ^ Duhem discovered that Thomas Bradwardine, whom we can call the founder O, B, and N) have the Flemish form “de Brussel” rather than the common
of the school of kinematics at Merton College, Oxford, cited the De motu Latin form “de Bruxella,” the author might have been Flemish. However,
in his Tractatus de proportionibus of 1328, giving our treatise the title De the recently discovered manuscript E contains in its colophon the reading
proportionalitate motuum et magnitudinum.^ Though refuting the conclusion “de bruxella” (see the variant reading for the colophon), which makes that
of Gerard’s initial proposition, Bradwardine nevertheless made the earlier suggestion somewhat more doubtful. Enestrom further suggested that Ricardus
tract one of his points of departure, as we shall see in Chapter Two below. de Uselis may be identical with Gerardus de Brussel, the praenomen being
With only the Parisian fragment at hand, Duhem was able to say little that a slip of the pen and the “Uselis” arising from “Uccle,” a town near Brussels,
was precise concerning the author or the tract’s date of composition. In fact while I suggested in my first edition of the De motu that the author of the
he was able only to establish the year 1328 as a terminus ante quem for the De sex inconvenientibus might have seen a manuscript in which the author’s
composition of the tract. But Duhem also noticed in a somewhat later four­ name was mutilated and read “. . . russelis,” on the basis of which he read
teenth-century treatise entitled De sex inconvenientibus that a view concerning the “r” as “Ricardus” and the remainder as “Uselis,” having missed a scribal
the motion of rotation similar to that expressed in the Parisian fragment was reading for double “s.”* This is a suggestion I would not press strongly.
assigned to a Ricardus de Versellys (according to one manuscript of the De In fact modem references to Gerard of Brussels’ Liber de motu go back
sex inconvenientibus) or Ricardus de Uselis (according to another manuscript much further than the accounts of Duhem and Enestrom. In the sale catalogue
of it).^ But Duhem admitted that it was impossible for him to say whether of Libri manuscripts, we find a description of the manuscript that later passed
Ricardus composed the tract on motion or merely repeated views he found to Berlin.^ The authors of this catalogue describe this codex (with the catalogue
expressed in the tract written by someone else. However, progress was made no. 665) as of the twelfth century, though Enestrom in two different places
would date the manuscript as of the fourteenth century,'® and I would prefer
a date in the thirteenth century (see Chapter Three, Sigla, MS B). The Libri
* P. Duhem, Études sur Léonard de Vinci, Vol. 3 (Paris, 1913), pp. 292-94, where Duhem,
apparently starting his numbering with the first of two unnumbered folios, gives the pagination
of the fragment as folios 6r-7r. ’ G. Enestrom, “Sur l’auteur d’un traité ‘De motu’ auquel Bradwardin a fait allusion en 1328,”
’ Ibid.; cf the edition of H. Lamar Crosby, Thomas of Bradwardine. His Tractatus de Pro­ Archivio di storia della scienza. Vol. 2 (1921-22), pp. 133-36.
portionibus. Its Significance for the Development of Mathematical Physics (Madison, 1955; 2nd * Clagett, “Gerard of Brussels,” p. 103.
pr. 1961), p. 128. Notice that Themon Judei in a question on the motion of the moon, which ® Catalogue of the Extraordinary Collection of Splendid Manuscripts, Chiefly Upon Vellum,
he determined at Erfurt in 1349, cites the De motu under the titles De proportionibus motuum in Various Languages of Europe and the East, Formed by M. Guglielmo Libri. . . . Which Will
et motorum and De proportione motuum et motorum. See H. Hugonnard-Roche, L’Oeuvre be Sold by Auction by Messrs. S. Leigh Sotheby and John Wilkinson, (London, 1859), pp. 145-
astronomique de Thémon Juif maître parisien du XIV‘ siècle (Paris, 1973), pp. 337, 345, 353- 48. Note that in publishing the comment made by Gregory that is quoted below in the text the
55. He writes with the context of Bradwardine in mind. authors of the catalogue omitted “Gerardi” from the title given in the first line of the comment,
* Duhem, Études, Vol. 3, p. 295. Duhem cites Paris, BN lat. 6559, 34r, 36r, and BN lat. 7368, as my inspection of Gregory’s comments attached to MS B reveals.
162r and 164r. Cf. MS Venice, Bibl. Naz. Marc. Lat. VIII, 19, 129r and my Science of Mechanics, Enestrom, op. cit. in n. 7, p. 135, and his “Das Bruchrechnen des Jordanus Nemorarius,”
p. 262, n. 8. Bibliotheca mathematica, 3. Folge, Vol. 14 (1913-14), p. 42.
ARCHIMEDES IN THE MIDDLE AGES COMPOSITION OF THE LIBER DE MOTU
catalogue {op. cit. in note 9, p. 147) quotes from the first part of the manuscript work often appears with the works of Jordanus de Nemore, with the Liber
a comment by the celebrated David Gregory (who prepared a kind of analytical de curvis superficiebus Archimenidis of Johannes de Tinemue, with the On
index of the manuscript): the Measurement of the Circle of Archimedes, and with the Elements of
Euclid. This suggests, as my analysis of the contents of the Liber de motu
The next is Liber Magistri [Gerardi] de Brussel de Motu. It contains three books
in the next chapter tends to confirm, that these various works were integral
in seven leaves. In the first, there are four propositions; in the second, five; in
parts of a lively geometrical activity at the end of the twelfth century and
the third, four. This was never printed that I know. It does not handle motion
in the present acceptance of the word, it only shows, that in the rotation of Lines the beginning of the thirteenth.'^ While remarking on the possible close ties
and Plain (.0 Figures about an immovable Axis whereby surfaces and solids are of Gerard’s work with the mathematical activity of Jordanus de Nemore, I
generated, there is sometimes equal motion in different generations, and sometimes hasten to admit that there is no definite evidence of which author precedes
more in one than in another. There are some initial small instances of the the other.But, as I note in my analysis of Proposition 1.4 in the next chapter,
proposition: Tantum movetur Figura quantum ejus centrum gravitatis. several propositions of Jordanus’ Liber philotegni also deal with the rela­
As we shall see in Chapter Two, this is by no means an accurate description tionships existing, mutually and separately, between inscribed and circum­
of the treatise, but it shows at least a fleeting acquaintance with the tract in scribed regular polygons, and indeed they are probably the source of Gerard’s
the seventeenth century. knowledge of such relationships.
What may we say about the author and the date of the tract? It is clear As for Gerard’s citations of the other geometrical works mentioned above,
that all but one of the extant manuscripts listed below in Chapter Three date we should note first that he cites Archimedes’ On the Measurement of the
from the thirteenth century. More explicitly we know that MS E was once Circle under the title De quadratura circuli (e.g., see Prop. I.l, line 32, and
a part of the collection of manuscripts described by Richard of Foumival in var. to line 180 in Tradition I; and Prop. II. 1, line 112). In doing so, he is
his Biblionomia,^^ a catalogue composed when he was chancellor of the no doubt referring to the second tradition of Gerard of Cremona’s translation
church at Amiens.'^ We know that he was already chancellor in 1246 at the of that work rather than to the first tradition which bore the title De mensura
time of the death of his brother Amoul, and we also know that Richard was circuli.Incidentally Archimedes’ name does not appear in the citations
no longer living in 1260.*^ Hence his Biblionomia must have been written given in the Liber de motu. Nor does an author’s name appear in the citations
before that year. Birkenmajer in his fine study of the work suggests “vers to the De curvis superficiebus. Gerard of Brussels in the text and the scribe
1250” and surely this would not be far off the true date.*'* The item in the of MS O in the margins always cite this work under the title of De piramidibus
Biblionomia reads: (see Prop. I.l, Trad. I, var. to hnes 77-78; 1.2, var. to lines 7-9, text line 11;
II.2, lines 6 and 17; II.3, hnes 103-04). When preparing my edition of the
43. Jordani de Nemore liber philothegny CCCCXVII [.' LXIIII] propositiones
Liber de curvis superficiebus, I was unable to find any extant manuscript in
continens. Item ejusdem liber de ratione ponderum, et alius de ponderum pro-
which the work bore the title De piramidibus, which is ordinarily reserved
portione. Item cujusdam ad papam de quadratura circuli. Item Gerardi de Bruxella
for a fragment of Apollonius’ Conics translated from the Arabic by Gerard
subtilitas de motu. In uno volumine cujus signum est littera D.
of Cremona.'^ However it is possible that a manuscript briefly described in
As I pointed out in my edition of the Liber de motu this also constitutes the Foumival’s Biblionomia contains the De curvis superficiebus under that title:^°
earliest datable reference to Jordanus’ longer treatise on weights, the Liber
de ratione ponderum, as well as to his Liber philotegni which I edit in Part 42. Dicti Theodosii liber de speris, ex commentario Adelardi. Item Archimenidis
Arsamithis liber de quadratura circuli. Liber de piramidibus. Liber de ysoperi­
II below. In fact, as I note under the rubric Sigla in Chapter Three, Gerard’s
metris. Item libri de speculis, de visu et de ymagine speculi. In uno volumine
cujus signum est littera D.
" N. R. Ker, Medieval Manuscripts in British Libraries, Vol. 2 (Oxford, 1977), p. 547 (concerning
I suggest that the title De piramidibus used here refers to the De curvis
Cr. 1.27, our MS£).
L. Delisle, Le Cabinet des manuscrits de la Bibliothèque Nationale, Vol. 2 (Paris, 1874), superficiebus since it is given immediately after Archimedes’ De quadratura
p. 521. circuli, and the De curvis superficiebus is often found in close proximity to
Histoire littéraire de la France, Vol. 23 (Paris, 1856), p. 717.
A. Birkenmajer, Études d’histoire des sciences et de la philosophie du moyen age (Wroclaw,
etc., 1970), p. 119. Cf. R. H. Rouse, “Manuscripts belonging to Richard de Foumival,” Revue Clagett, Archimedes, Vol. 1, passim. Vol. 3, pp. 212-13.
d’histoire des textes. Vol. 3 (1973), pp. 253-69 (whole article), and particularly pp. 255, 260. See my remarks on Jordanus’ career in Part II, Chap. 1 below.
Rouse believes that this is one of the manuscripts prepared for Foumival. He notes earlier (p. Clagett, Archimedes, Vol. 1, p. 31.
254) that the commissioned books were “contemporary with his [i.e. Foumival’s] mature years, Ibid., Vol. 4, p. 3. Notice that Zubov, op. cit. in note 2, p. 243, misidentifies the De
ca. 1225-1260.” piramidibus with Jordanus’ De triangulis when in fact it was the Liber de curvis superficiebus.
Delisle, Le Cabinet, Vol. 2, p. 526; Cf Birkenmajer, Études, p. 166. “ Delisle, Le Cabinet, Vol. 2, p. 526; Cf. Birkenmajer, Études, p. 166.
ARCHIMEDES IN THE MIDDLE AGES COMPOSITION OF THE LIBER DE MOTU
Archimedes’ work in extant manuscripts.^' How does one explain the use
Sx! S = TJ T^^"^ (2) the Physics of Aristotle, available in several translations,
2
by Gerard of Brussels of De piramidibus for a title of the De curvis super-
with its demonstration (given in quasi-geometric form) of the three cases of
ficiebusl I suppose that the most sensible answer is that Gerard read a manu­
“quicker,” “slower” and “equal” motions;^^ (3) the Elementatio physica of
script of the De curvis superficiebus in which the work was without title and
Proclus, available in a twelfth-century translation from the Greek, with its
so he decided, on the basis of the prominent position given to cones in that
repetition of the Aristotelian rules cast in strictly geometrical form;^^ and (4)
work (cf. Propositions I, IV, V, VII, and IX of the De curvis superficiebus
the Mechanica attributed to Aristotle, which was perhaps (though not surely)
in my text in Volume One), to call it De piramidibus. Not only does Gerard
translated into Latin in the twelfth century, and which contained the statement
depend specifically on Propositions I and IV of the De curvis superficiebus,
from which Gerard’s tract begins: “but of two points that which is farther
but, even more fundamentally, he uses a form of the exhaustion procedure
from the fixed center is the quicker.”^^ Having examined the possible influ-
in several propositions that is like the form used in the De curvis superficiebus,
a form which the Renaissance mathematician Francesco Maurolico called
Clagett, Science of Mechanics, pp. 166-68.
• the “easier way” and which was ultimately based on Proposition XII. 18 of
25 pp. 175-83.
Euclid’s Elements.^^ Speaking of Euclid’s Elements, we should add that Gerard H. Boese, Die mittelalterliche Übersetzung der hroixf^ffis ‘PvaiKrt des Proclus. Procli Diadochi
of Brussels’ Liber de motu displays a thorough knowledge of the Elements. Lycii Elementatio physica (Berlin, 1958), p. 34:
But unlike most medieval geometers he did not cite the work. The only “8. Inequali celeritate motorum celerius in equali tempore maius movetur.
Esto enim inequaliter motorum celerius A, tardius autem B, et moveatur A ab G super D in
specific references to the Elements are found in additions made by the scribe
ZI tempore. Quoniam ergo tardius est B, in ZI tempore non veniet ab G super D. Celerius enim
of MS O (see the variant to Prop. I.l, Trad. I, line 63, citing Prop. 1.16 of
est prius in finem veniens, tardius autem posterius. Moveatur ergo in ZI tempore in E veniens.
the Elements, and the variant to Prop. II. 1, Une 120, citing Props. VI. 17 and In eodem igitur tempore A quantitatem GD pertransit, B vero quantitatem GE; maior autem
VI. 1 of the Elements) and citations added to the text by the compositor of GD quam GE. Celerius ergo in eodem tempore maiorem quantitatem pertransit.
Tradition II of Prop. I.l (see line 46, citing VI.4; line 60, citing V.15; line 9. Si fuerint mota inequalis celeritatis, sumentur quedam tempora, plus quidem tardioris,
minus vero celerioris, in quibus celerius quidem maiorem movetur quantitatem, tardius autem
81, citing 1.29; line 85, citing 1.26; and line 93, citing 1.34). Thus the knowledge
minorem.
of Euclid shown by Gerard is never specific enough for us to determine which
Sint enim inequalis celeritatis A, B, et A quidem celerius, B vero tardius. Quoniam ergo
version he used of the several translations of Euclid that might have been celerius in eodem tempore maiorem pertransit quantitatem, in ZI tempore A quidem GD per-
available to him in the late twelfth or early thirteenth century.^^ Perhaps his transeat et B, GE. Et quoniam A in toto ZI tempore pertransit GD quantitatem, ergo GT in
readers’ knowledge of the Elements was so taken for granted by him that he minori pertransiet quam sit ZI. Sumatur ergo illud tempus minus et sit ZK. Quoniam ergo A
quidem in ZK pertransit GT, B vero in ZI pertransit GE, maior autem GT quam GE maiusque
felt no need to give exact citations. At any rate, it is clear that Gerard’s proofs
tempus ZI quam ZK, sumpta ergo sunt tempora quedam, maius quidem ZI eius quod
show proper knowledge of Euclidian propositions concerning similar triangles,
est B, minus vero ZK eius quod est A, in quibus A quidem progreditur maiorem GT, at vero
the relationships of cylinders and cones, the Euclidian theory of proportions, B minorem GE.
and other areas of Euclidian geometry. 10. Inequaliter motorum celerius in minori tempore equalem pertransit quantitatem.
In addition to the geometrical tracts that he knew and used (like the De Sint enim inequaliter mota sitque celerius A quam B moveaturque A in ZI tempore quantitatem
GD, at vero B in eodem minorem, scilicet GE. Quoniam ergo A in toto ZI quantitatem GD
quadratura circuli, the De curvis superficiebus, and the Elements) and the
progreditur, minorem GE in minori progredietur; progrediatur ergo in ZK. At vero B quantitatem
geometrical tract of Jordanus Liber philotegni that he may have known, there
GE in ZI progrediebatur, plus vero ZI tempus quam ZK. Equalem ergo quantitatem, scilicet
are certain other treatises with kinematic rules that may have influenced his GE, A quidem in minori tempore progreditur, B vero in maiori.
treatment of motion. These include the following texts: {\) On the Moved Aliter; Esto A quam B celerius et moveatur B quantitatem GE in ZI tempore; A ergo in
Sphere of Autolycus, translated by Gerard of Cremona, with its defini­ eodem tempore movetur GE vel in maiori vel in minori. Sed si in eodem, erunt equalis celeritatis;
si autem in maiori, erit tardius, positum est autem celerius. In minori ergo tempore progredietur
tion of “equal [i.e. uniform] motion” and its statement of the funda­
A quantitatem GE.”
mental kinematic proportion for two uniform but unequal motions:
I have made slight changes in punctuation in Prop. 9, writing “A, B,” instead of “A B” and
“B, GE” instead of “B GE.” Like the editor I have not given any diagrams, but the magnitudes
specified are so obvious that no diagrams are needed to follow the text.
See the MSS listed in Clagett, Archimedes, Vol. 1, pp. xix-xxvi (where MS B is mistakenly
Clagett, Science of Mechanics, pp. 182-83, and see pp. 5, 71. In the last reference (p. 71)
written “Q.150” instead of “Q.510”—see p. xxiv).
I give the one passage that may indicate the existence of a medieval translation of the Pseudo-
For a description and discussion of the “easier way” see M. Clagett, Archimedes in the AristoteUan Mechanica. It is found in the De arte venandi cum avibus of Frederick II: “Portiones
Middle Ages, Vol. 3 (Philadelphia, 1978), pp. 798-808.
circuli quas faciunt singule penne sunt de circumferentiis equidistantibus, et illa que facit portionem
Clagett, “The Medieval Latin Translations from the Arabic of the Elements of Euclid, with maioris ambitus et magis distat a corpore avis iuvat magis sublevari aut impeUi et deportari,
Special Emphasis on the Versions of Adelard of Bath,” Isis, Vol. 44 (1953), pp. 16-42; cf. J. quod dicit Aristotiles (/) in libro de ingeniis levandi pondera dicens quod magis facit levari
Murdoch, “Euclid: Transmission of the Elements,” Dictionary of Scientific Biography, Vol. 4 pondus maior circulus.” Stili one could argue that, even if there was no medieval translation of
(New York, 1971), pp. 437-59.
the Mechanica, the basic approach to statics found in the works of Jordanus and in the Liber
10 ARCHIMEDES IN THE MIDDLE AGES COMPOSITION OF THE LIBER DE MOTU 11
ences of the first two and the fourth of these works on Gerard in great detail John de Tynemuth, who, among other positions, held that of archdeacon at
in my earlier publications, I shall refrain from saying anything more about Oxford from 1215.^° But I have found nothing to connect him with the
them at this point. As for the third work, the inclusion here of the actual writing of mathematical works.
text in footnote 26 and the observation that it contained Aristotle’s rules on The fortune of Gerard of Brussel’s work has been alluded to in the beginning
velocity which I discussed in my earlier works should make any further of this chapter. The Liber de motu certainly exerted some influence on the
discussion unnecessary. kinematic concepts found in the Tractatus de proportionibus of Thomas
Though the reader will now have some sense of the works that influenced Bradwardine and in the works of his successors at Merton College, Oxford,
Gerard’s composition of the Liber de motu, he will still have no precise and at the University of Paris (e.g. on Themon Judei). Because this kinematic
information about the life of the author and where he studied and taught. material of the fourteenth century has received wide consideration since
In four of the MSS (see the variants for the title of the treatise in MSS O, Pierre Duhem first drew attention to it, I shall not go over it again, except
B, and N, and for the colophon in E) Gerard is spoken of as Magister to note a few specific citations in my discussion of the content of the Liber
Gerardus. It seems possible that this is an indication of some university de motu in the next chapter. I know of no description of this work after the
connection for Gerard. But with which university I do not know. At this fourteenth century until David Gregory’s brief estimate of it in the seventeenth
early stage of university history, Paris seems the most likely, just as it does century, although it seems possible that the French scholar Charles de Bouefles
for Jordanus, whose works appear so often in the same codices as Gerard’s {or Bovelles) read Gerard’s work in the early sixteenth century, and perhaps
De motu. Though the Liber de motu is the only extant work ascribed to Nicholas of Cusa had seen it a half century earlier.^* Furthermore, John Dee,
Gerard of Brussels, Gerard himself seems to mention another of his works the well-known mathematician and magician of the sixteenth century, singles
in Proposition II. 3 of the Liber de motu when he tells us that he has proved out its presence in MS O (Oxford, Bodleian Library, MS Auct. F.5.28, 116v-
elsewhere that the ratio of the curved surfaces of similar polygonal bodies is 125r), which he had borrowed from Ricardus Bruamus.^^
equal to the square of the ratios of their sides (see the translation below.
Prop. II.3, note 1). There is the intriguing possibility that our Gerardus is
identical with another mathematician of about the same period, Magister
Gernardus, who composed an Algorismus demonstratus (entitled in its two
parts Algorismus de integris and Algorismus de minutiis), which bears a close
relationship to the mathematical works of Jordanus.^^ But, though one manu­
script of the Algorismus seems to have Gernandus instead of Gernardus, I
conclude from Enestrom’s Ust of its manuscripts that no manuscript has
Gerardus, at least in the hand of an original scribe.^^ So, in fact, the person
of Gerard of Brussels remains as elusive as that of his more illustrious con­
temporary Jordanus (see Part II below) or as that of Johannes de Tinemue,
who played some role in the preparation or translation of the De curvis
superficiebus. In regard to the latter, we can note the existence of a canonist,
A. B. Emden, A Biographical Register of the University of Oxford to A.D. 1500, Vol. 3
karastonis of Thabit ibn Qurra (translated by Gerard of Cremona) assumes (no doubt ultimately (Oxford, 1959), p. 1923.
from the Mechanica) that a weight that is on an arm that is farther from the center of the balance Clagett, Archimedes, Vol. 3, pp. 1180, 1182 n. 4, 1187, 1192 n. 19, 1196.
moves faster than one at a position closer to the center, and it is precisely this idea assumed for M. R. James, Lists of Manuscripts Formerly Owned by Dr. John Dee (Oxford, 1921), p.
points that Gerard of Brussels uses at the beginning of his treatise. Hence, one can perhaps say 13, item 43.: “1 borowed one volume of master bruem (/) written in parchment in 4“ two ynches
that this basic idea is so well-known from the statical traditions circulating in the thirteenth thik in which are many and good bokes and Jordan de datis numeris and Gerardus Brussellensis
century that it is unnecessary to assume a translation of the Mechanica. de motu which 1 never saw elsewhere and mr bruarun’s name is written on the back.” This is
G. Sarton, Introduction to the History’ of Science, Vol. 2 (Baltimore, 1931), p. 616. See the identical with MS. C.13 (see p. 16). I have identified the manuscript with O by the reference to
edition of G. Enestrom, “Der ‘Algorismus de integris’ des Meisters Gernardus,” Bibliotheca Bruamus (see F. Maddan, H. H. E. Craster, and N. Denholm-Young, A Summary Catalogue
mathematica, 3. Folge, Vol. 13 (1912-13), pp. 289-332, and “Der ‘Algorismus de minutiis’ des of Western Manuscripts in the Bodleian Library at Oxford, Vol. 2, Part 2 (Oxford, 1937), p.
Meisters Gernardus,” ibid.. Vol. 14 (1913-14), pp. 99-149. 708. It had been tentatively and wrongly identified by James with the Libri MS 665 (my MS
Ed. cit. of Algorismus de integris in note 28, p. 290. The aberrant form “Gernandus” appears B; see the Sigla in Chapter Three below). A. G. Watson has kindly informed me that he and
in MS Oxford, Bodl. Libr., Digby 161, Ir. Enestrom says that this manuscript was formerly R. J. Roberts have also made the identification with O in their edition of John Dee’s Library
attributed to Gerard of Cremona (and indeed in a later hand on the title page we find “Gerhardus”); Catalogue {London, The Bibliographical Society, forthcoming), which includes further comments
cf. R. B. Thomson, “Jordanus de Nemore: Opera,” Mediaeval Studies, Vol. 38 (1976), p. 112. on MS O.