The Medieval Science of Weights (Scientia de Ponderibus): Treatises Ascribed to Euclid, Archimedes Thabit ibn Qurra, Jordanus de Nemore and Blasius of Parma

224 Pages·1952·8.59 MB·English

by  Ernest A. Moody

THEMedieval Science
o f Weights
(Scientia de Ponderibus)
Treatises Ascribed to Euclid, Archimedes
Thabit ibn Qurra, Jordanus de Nemore
and Blasius of Parma
EDITED
With Introductions, English Translations, and
j?rneSf M oodj
Notes, by
M arshall Clagett
MADISON
THE UNIVERSITY OF WISCONSIN PRESS
1952
PREFAC E
When Ernst Mach, in 1883, published the first edition of his
monumental work, Die Mechanik in ihrer Entwickelung histor-
Copyright, 1952, by
isch-kritisch dargestellt, he made no mention of any contribu­
tion to mechanics between the time of Archimedes and that of
THE REGENTS OF THE UNIVERSITY OF WISCONSIN Leonardo da Vinci. The corrections and additions which steadily
expanded the Notes to Mach’s work, in its six subsequent edi­
tions, bear witness to the increase of knowledge concerning the
Copyright, Canada, 1952 scientific tradition of the Middle Ages, which has taken place
Distributed in Canada by during the past sixty years. That the philosophical works of Ar­
Burns & MacEachern, Toronto istotle were studied in minute detail by the scholastic teachers
of the thirteenth and fourteenth centuries, has long been recog­
nized; but that there also existed, in the medieval period, a tra­
dition of mathematical physics with historical links to the late
Alexandrian school of mechanicians, and to Euclid and Archi­
medes, is less generally known.
The influence of the Alexandrian tradition, and of the Archi­
medean method of demonstrating theorems in mechanics by ge­
ometrical proofs from a few explicit postulates of physical
character, is clearly exhibited in the various treatises “on
weights” associated in medieval manuscripts with the name of
Jordanus de Nemore. Sources of Greek origin, transmitted and
elaborated by Arab intermediaries, are indicated for these
treatises. But the thirteenth century author, or authors, of the
writings ascribed to Jordanus were no mere copyists or com­
pilers, and they made important additions to the materials which
they inherited.
The modern rediscovery of medieval statics, and of the in­
terest and significance of the writings attributed to Jordanus de
Nemore, is primarily due to Giovanni Vailati and Pierre Duhem,
and to the latter more than the former. Vailati’s brief estimate
of the importance of the treatises entitled De ponderibus and
attributed to Jordanus was included in a paper on the origins of
the concept of virtual work appearing in the Acte of the Acca-
demia of Turin, Volume 23, session of 1897. Vailati pointed out
that not only was the concept of virtual work being used in the
Jordanus treatises but that a directional analysis of force was
being considered for the first time with the consequent correct
analysis of the inclined plane problem. But Vailati’s analysis
was brief and not widely available so that it remained for Du­
hem to make a more thoroughgoing investigation of medieval
Printed in the United States of America statics. His work, Les Origines de la Statique, whose first vol­
Cushing-Malloy, Inc.—Ann Arbor, Michigan ume was published in 1905, contained a much more complete
v
PREFACE PREFACE
study of the Jordanus treatises on weights, and of their Greek tions to the volume, so that it now appears to be an evenly col­
and Arab antecedents. Duhem held that the thirteenth century laborative work.
author, or authors, of the Jordanus writings discovered and util­ Our indebtedness to the pioneer research of Pierre Duhem in
ized the principles of composition of forces, and of virtual dis­ this literature of the scientia de ponderibus is manifest on al­
placements, thereby anticipating by some three hundred and fifty most every page of our work. We wish likewise to acknowledge
years the achievements hitherto credited to Galileo, Descartes, the very kind cooperation of the many libraries who made their
and Bernoulli. Duhem's claims have seemed extreme to many manuscripts, or photographic reproductions of them, available
historians of science, and have been vigorously challenged. Yet to us; the names of these libraries appear in our list of Sigla of
the prerequisite for any adequate analysis of the contributions Manuscripts and Early Editions. We wish also to acknowledge
of Jordanus and his medieval associates, or for sound appraisal our indebtedness to Professor H. Lamar Crosby, of Hollins Col­
of the merits of Duhem’s thesis, has so far been lacking. The lege, for lending photographic reproductions of several manu­
texts of Jordanus and of the other auctores de ponderibus have scripts of Bradwardine*s Tractatus de proportionibus velocita­
either not been edited at all, or have been available only in rare tum in motibus, of which he has since prepared a full edition
and badly edited early editions or in inadequate editions pub­ and translation to be published shortly. Further thanks are due
lished in learned periodicals not easily accessible. None of to the John Simon Guggenheim Memorial Foundation for fellow­
these texts, moreover, has been translated into English, so that ships granted to Marshall Clagett which allowed him to collect
their content has not been accessible at all to the many compe­ photographs of many of the manuscripts used in establishing the
tent students of the sciences who do notread Latin.lt is to meet texts included in this volume.
this need that our editions and translations of the corpus of me­ There are some divergences which remain in the manner in
dieval treatises on weights have been prepared. which the two editors have cited variants, but this should not
While all the works contained in this volume have been edited prove of any difficulty to the reader since in each case the methr-
directly from manuscript sources, our editions make no claim od employed is clearly apparent. Apology is due the reader for
to be “critical” in the strict sense. They are based, in general, removing the Variant Readings and Critical Notes to the end of
on a few manuscript versions which have seemed to represent the volume. This was done to facilitate the electrotyping. It is
the most authentic manuscript traditions, with emendations on nevertheless hoped that the reader will consult the critical dis­
occasion from additional manuscripts or from the earlier edi­ cussions as he reads the various texts. The system of page ref­
tions. Variant readings are given only to the extent that they erences for the Notes and Variants should make a cumbersome
have some bearing on the meaning or interpretation of the texts. task as simple as possible.
The aim has been to provide intelligible texts authenticated by For the difficult job of typing we owe thanks to Mrs. Margaret
at least one (though in most cases, two or three) early and re­ Hundt of Madison, Wisconsin.
liable manuscript source. Some passages have remained ob­ It is hoped that these texts and their translations will be of
scure, despite all efforts at emendation. But for the most part value to students of the history of science, and serve in some
the texts, as established, are mathematically coherent and rea­ measure to remedy the pitiful shortage of modern editions, and
sonably clear in their meaning and intent. particularly of English translations, of medieval scientific writ­
The text of the Liber de ratione ponderis of Jordanus de Ne­ ings in this field.
more was originally edited as a project in a seminar for grad­
uate students at Columbia University, given under Ernest Moody’s E.A.M.
direction in the Spring Session of 1949. The members of this M.C.
seminar, who assisted Moody in collating two of the manuscripts
used as basis of the edition, were Messrs. Raymond Clements,
Arthur Ditzel, and Jason Lewis Saunders. This initial interest
in the material led to the editing of the other Jordanus treatises
and some of their antecedents of the Greek tradition. At this
point Marshall Clagett offered to contribute an edition and
translation of Thabit ibn Qurra’s Liber karastonis, with an in­
troduction and notes. Subsequently he made additional contribu­
vi
Vll
TABLE OF CONTENTS
GENERAL INTRODUCTION 1
TEXTS
f LIBER EUCLIDIS DE PONDEROSO ET LEVI
ET COMPARATIONE CORPORUM AD INVICEM 21
II LIBER ARCHIMEDIS DE INSIDENTIBUS IN
HUMIDUM (LIBER ARCHIMENIDIS DE
PONDERIBUS) 33
HI LIBER DE CANONIO 55
IV LIBER KARAS TONIS 77
V ELEMENTA JORDANI SUPER DEMONSTRA­
TIONEM PONDERUM 119
VI LIBER JORDANI DE PONDERIBUS 143
VII LIBER JORDANI DE NEMORE DE RATIONE
PONDERIS, 167
VIH TRACTATUS BLASH DE PARMA DE
PONDERIBUS 229
APPENDICES
I A FRAGMENT OF THE ROMAN BALANCE
ATTRIBUTED TO EUCLID 281
H THOMAS BRADWARDINE’S DISCUSSION OF
PROPOSITION ONE OF THE LIBER DE
PONDERIBUS 285
HI A FOURTEENTH-CENTURY COMMENTARY
ON PROPOSITION ONE OF THE LIBER DE
PONDERIBUS 293
IV A VARIANT FORM OF PROPOSITION EIGHT
OF THE ELEMENTA JORDANI 306
IX
TABLE OF CONTENTS
VARIANT READINGS 313
SIGLA OF MANUSCRIPTS AND EDITIONS 315
CRITICAL NOTES 345
BIBLIOGRAPHY 431
GENERAL INTRODUCTION
INDEX 435
Part I
by
MARSHALL CLAGETT
Part II
by
ERNEST A. MOODY
x
PART ONE
Natural science or physics in the medieval Aristotelian clas­
sification of the sciences was a science very broad in scope.^
It treated natural bodies insofar as they undergo movement or
suffer changes. It sought to elucidate the fundamental charac­
ter of matter and form in nature and to clarify the role played
by the four types of causation which Aristotle believed were
operating in nature. If particular emphasis was placed on the
final or purposeful cause throughout the Middle Ages, it is
worth noting that toward the end of the Middle Ages there was
an evident tendency to consider immediate efficient causes as
more knowable. This was particularly true in the mechanics of
local motion.
Since it dealt with the general principles of movement and
change in nature, physics was described as a “general" science.
It included as parts a number of “special" sciences where the
principles were illustrated in some particular kind of body.
Thus alchemy as a special part of physics studied certain chem­
ical changes, some real and some chimerical, that were believed
to take place in metals and other earthly bodies. Optics, astron­
omy and statics were sometimes included as part of physics.
While the general framework for natural science was supplied
by the Aristotelian tradition, the special sciences that were a
part of physics gained greatly from the Hellenistic mathematical
tradition of Euclid, Archimedes, and Ptolemy, a significant num­
ber of whose works were translated into Latin in the twelfth and
thirteenth centuries. This was true of the sciences of optics,
astronomy, and statics'. But it was in the science of statics that
the Hellenistic tradition bore greatest fruit in the thirteenth
century.
The science of statics, known in the Middle Ages as the scien­
tia de ponderibus (science of weights), was the subject of a series
of treatises “on weights," some of which were Latin translations
of earlier works from the Arabic or Greek and some of which
were original products of Latin authors. The presentation of the
texts, translations, and evaluation of the most significant of
these treatises is the object of our volume. For these treatises
constitute an important chapter in the growth of statical inves­
tigation.
The most important of the medieval statical tracts are those
which are attributed to Jordanus de Nemore, or more simply,
Jordanus. These are the texts numbered V-VII in this volume.
And of these tracts the treatise De ratione ponderis (On the
3
GENERAL INTRODUCTION
GENERAL INTRODUCTION
Nature of Weight) is, as we shall see, by far the most important.
increases proportionally as its distance from the fulcrum be­
Jordanus is sometimes identified with Jordanus of Saxony, who
cause as it increases its distance from the fulcrum it would
served as Master General of the Dominican Order from 1222.
have, if set in motion, a greater velocity. That is to say, it
But there is considerable doubt of this and we can only say that
would continually traverse a longer arc in the same time.5
he flourished about this time. For a fuller discussion of his life
Since, then, the effective weight is increased by the amount that
the reader can consult the special introduction to the text of his
the velocity is greater, hence when the actual weights are in­
Elementa de ponderibus. At any rate, Jordanus, whoever he may
versely proportional to their “velocities" (i.e., to the arcs trav­
have been, was a mathematician of great skill and considerable
ersed in the same time), the effective weights will be the same.
originality, as both his mathematical and physical works reveal.
But since in this circumstance the velocities (or arcs) are di­
So far as statics is concerned, he reworked the material which
rectly proportional to the arm lengths, the weights will also be
he inherited ultimately from the Hellenistic mechanical tradi­
inversely proportional to the arm lengths. This dynamic argu­
tion, and in doing so he continued and deepened the union of Ar­
mentis more clearly presented in one of the Arabic works com­
istotelian dynamics with Archimedean mathematical statics.
posed by Thabit ibn Qurra and called the Liber karastonis. (See
In Part Two of this Introduction and in the special introduc­
text IV, and its notes.) The essential fact is that the principle
tions to the various texts we have gone into the details of the
of virtual velocities, in a germinal form, at least, is being used
role played by the earlier statical treatises in the development
to account for a fundamental law of statics. The form of the
of statics in general and the Jordanus treatises in particular.
principle as understood by the Pseudo-Aristotle and Thabit alike
At this point we should like only to characterize briefly the two
would be something like this: In any mechanical system reduc­
main traditions in antique statics: the Archimedean and Aris­
ible to a balance or lever, the ratio of the moving force to the
totelian.^
force of the thing moved is as the inverse ratio of their veloc­
The debt we owe to Archimedes in the foundation of statics is
ities (understood in the sense of their simultaneous areal dis-
well known. His proof of the general law of the lever, which
.placements.)^
asserts the inverse proportionality of the weights or forces and
Now a better form of the principle of virtual velocities or
the lever arms on which they are suspended or act, was an in­
displacements would be, say, that the ratio of these forces var­
fluential one in the history of statics. It was a proof that was
ies inversely as their vertical, rectilinear displacements. And
essentially “static," ^ for it appealed to the symmetry in the geo­
it was in this sense that the medieval mechanician Jordanus
metrical representation of the equilibrium of a lever with equal
understood the principle. One of his commentators of the four­
arms supporting equal weights, and equally important for the
teenth century has put the principle in essentially this form:
proof was the symmetrical determination of the center of grav­
What suffices to lift a weight w through a vertical distance h
ity of two or more equal magnitudes, whose centers of gravity
will suffice to lift a weight k*w through a vertical distance h/k
lie on the same line. Archimedean statics depended, above all,
and it will suffice to lift a weight w/k through the vertical dis­
on the determinations of centers of gravity. Demonstrations
tance k*h. (See General Introduction, part two, section III, and
were wholly geometrical in character, theorems being inferred
Appendix III.)
from postulates or axioms in the Euclidian manner.
This expression could then be transformed into the concept of
Less generally known are the somewhat earlier contributions
virtual work, i.e., the concept of the equality of potential work
to statics of Aristotle and his successors. His was a more “dy­
input to work output in a system in equilibrium, once the con­
namic” approach, which, however, lacked the elegance and math­
cept of work had come to be defined as the product of a force
ematical rigor of that of Archimedes. This dynamic approach
and the distance through which it acts.
finds illustration in the general law of the lever as expressed
Jordanus, then, in the thirteenth century used the better form
in a work entitled Mechanical Problems and attributed to Aris­
of the principle of virtual displacements in terms of vertical
totle (but probably by a later author). The law of the lever is
displacements to demonstrate the general law of the lever in
accounted for in this treatise by the fact that “a longer radius
both the cases of the straight lever (see proposition E.8 or
moves more quickly than a shorter one under pressure of an
Rl.06)and the bent lever (see Rl.08), and also to give an elegant
equal weight.The account is not by any means completely
proof of the inclined plane problem (see Rl.10). At the same
clear, but the substance of his argument is probably as follows.
time, in his proposition regarding the bent lever (Rl.08) Jordanus
The effective weight of any given weight on a balance or lever
clearly recognized that it is the horizontal distance from the
4
5
GENERAL INTRODUCTION GENERAL INTRODUCTION
weight on the end of the bent lever arm to the vertical line run­ of a lever came the general principle of virtual displacements,
ning through the fulcrum that must be employed to determine which was later further refined as the principle of virtual work.
the effective power for movement of the suspended weight. Thus In the second place, the study of medieval statics illustrates
Jordanus seems to have had a deeper insight into the factors the significant achievements that could be and were made when
determining the effective force of weights in a lever or balance the abstractions and generalizations which served as principles
system than did the mechanicians who preceded him. were given even the most elementary mathematical form, and
Finally, we should note that Jordanus employs as a fundament­ further when the logical implications following from the first
al notion a principle which he calls “gravity according to posi­ principles were themselves developed in the language of quan­
tion” (gravitas secundum situm). This principle essentially held tity. For example, from his initial concept of positional gravity
that the effective weight (or force) along a potential trajectory mathematically expressed—a brilliant intuition of component for­
inclined to the vertical is measured by the vertical component ces—Jordanus proceeded by the use of the principle of virtual
of that potential trajectory. In Part Two of the General In­ displacements and the theorems of plane geometry to deduce
troduction and in the introduction and notes to the Elementa de correctly a general proposition relating interconnected weights
ponderibus, we have discussed Jordanus’ incorrect and correct on oppositely inclined planes to the lengths of those planes. Sim­
usage of this principle. When used correctly it was equivalent ilarly the neat geometrical extension of his first principles led
to the modern formulation F = w • sin a where F is the force in him to his correct theorem regarding the bent lever. Lastly, our
the direction of the inclined plane, w is the free weight of the study of medieval statics reveals the great importance for sci­
weight on the plane, and is the angle of inclination of the plane. entific development of the fact that natural science was an inte­
In summary, then, the works of Jordanus demand careful at­ gral and connected part of the general arts program. As we have
tention in the history of statics, for they seem to utilize the basic said earlier, the originality and success of Jordanus’ efforts in
principle of work to prove theorems of statics, foreshadowing i statics stemmed in part from the union of a philosophical ap-
the methods of modern mechanics; they reveal, particularly in proa<*lL.(that of Aristotle and his successors) with a moreT rlg^-
the analysis of the bent lever of Rl.08, a deeper insight into orous mathematical tradition (that of Euclid and Archimedes).
what is later called static moment; and they give what is essen­ A student of the arts faculty of a medieval university would al­
tially a “resolution” of forces in determining the effective com­ most certainly come in contact with both of these traditions in
ponent of natural gravity in a constraint system. (See Rl.004, the course of his study. The junction, then, of the philosophical
Rl.005, R1.09 and Rl.lO.) A discussion of these and other contri­ and mathematical traditions in statics was but one illustration
butions of Jordanus has been given in the second part of the of the more general interplay between the two traditions. Most
General Introduction and in the notes to the texts themselves. of us who have investigated the origins of Western science ack­
The treatises attributed to Jordanus and the various other nowledge this interplay by affirming that the principal fore-
earlier texts were copied, elaborated, and commented upon in bearers of modern science were in fact the twin traditions of
the fourteenth and fifteenth centuries. The best of these works, Greek philosophy and mathematics.
the Liber de ratione ponderis, which contains all of the basic
ideas we have attributed above to Jordanus, was published in
1565, and thus was available in print to the early modern mech­
anicians interested in statics. How widely it was read and used
is a matter of dispute, but that it played some part in the rise PART TWO
of modern statics can hardly be doubted.
Before taking up our texts in greater detail, we might note
finally how the history of this one branch of physics in an early I
period illustrates some of the truisms regarding the general
development of science that occasionally escape attention. First The science of mechanics is almost wholly a modern crea­
it illustrates the success which emerged when the ordinary tion, but like most human achievements it has roots and antece­
fruits of human experience are analytically abstracted and gen­ dents stretching far into the past. The ancientGreeks, unrivalled
eralized to form the first principles of a science. Thus from an in pure mathematics and in philosophical physics, made only
analytic intuition of what is gained and what is lost in the use slight excursions into the field of mechanics. Archimedes, pri­
6 7
GENERAL INTRODUCTION GENERAL INTRODUCTION
mariLy a mathematician, applied geometrical demonstration to dean principles from more general dynamical foundations drawn
the problem of equilibrium and thereby became the founder of ,j from Aristotle. This led to methods of establishing the general
the science of statics. Aristotle, primarily a philosopher, of­ | lever principle on the more powerful principle of work, or of
fered somewhat incautious and vague formulations of quantita­ / virtual displacements, so that statics became integrated with
tive relations between distance, time, motive power and resis­ dynamics as in modern times. While Duhem's claims may have
tance, involved in forced motions and in the free fall of heavy been exaggerated, as some recent scholars have contended,^
bodies, and to this extent he was the father of the science of dy­ the texts of the mediaeval scientia de ponderibus, which are
namics. But what was required, for the achievement of a genu­ edited in this volume, reveal a most interesting interpenetration
ine and successful science of mechanics, was a treatment of the between the mathematical tradition of Archimedes and Euclid,
fertile subject matter of the Aristotelian tradition by men of and the dynamical tradition of Aristotelian physics.
mathematical skill working with something of the rigor and clar­ The corpus of writings constituting the “science of weights,"
ity which belonged to the tradition of Archimedes. In Galileo found in numerous manuscripts of thirteenth, fourteenth and fif­
and the other seventeenth century founders of modern physics, teenth century origin, consists of three types of treatise. First,
such a combination was achieved in brilliant manner. But the there are works and fragments translated from the Greek or
task had been attempted for three hundred years prior to Gal­ Arabic. usually ascribed to Euclid or to Archimedes, which show
ileo, and much of his path had been prepared for him by the Ar­ signs of an origin within the later Alexandrian period of Greek
istotelian scholastic tradition against which he revolted.^ science. These, include the De ponderoso et levi ascribed to
The mediaeval contributions to mechanics have been brought Euclid, the De insidentibus in humidum ascribed to Archimedes,
to light only in recent years, and primarily through the histor­ a Liber de canonio which generally occurs without indication of
ical studies of Pierre Duhem.^ The most striking of these con­ authorship, and a Liber Karastonis edited by the ninth century
tributions occurred in the fourteenth century, at Paris and at mathematician Thabit ibn Qurra. To be associated withThabit’s
Oxford. The theory of impetus, developed by Jean Buridan and Liber Karastonis is a fragment attributed to Euclid, published
his Parisian disciples, yielded an analysis of projectile motions in our Appendix I, which may possibly be the original Causae
and of gravitational acceleration which has marked analogies Karastonis of which Thabit’s work is avowedly a revision or
with the Galilean and Newtonian dynamics. Even earlier, at elaboration.
Oxford, the disciples of Thomas Bradwardine carried out an The second group consists of the treatises De ponderibus
analysis of the kinematic aspects of accelerated movement, in­ ascribed in most cases to the thirteenth century mathematician
volving the concept of instantaneous velocity, which led to the Jordanus de Nemore. These appear to have been written by
statement and proof of the fundamental kinematic law relating Christian scholastic teachers in the Latin west, presumably Tn
distances to times in uniformly accelerated motion. These four­ the earlier part of the thirteenth century. Two of the treatises
teenth century contributions, originated and conceived within may with reasonable certainty be ascribed to Jordanus de Ne­
the general framework of the Aristotelian physics, constituted more: the Elementa super demonstrationem ponderum, which
a mechanics in the modern sense of the word, and this mechan­ we distinguish as Version “E,” and the Liber de ratione pond­
ics, studied and taught in the universities of northern Italy eris (Ve rsion “R"), The third treatise in this group, however,
throughout the fourteenth and fifteenth centuries, and known in is almost certainly not by Jordanus, and it does not show any
those days as “the doctrine of the moderns,” constituted a half­ influence by the writings of Jordanus. It usually bears the title
way stage between the physics of Aristotle and that of the sev­ Liber Jordani de ponderibus, or simply Liber ponderum, and we
enteenth century.^ have designated it in our collection as Version “P.” Duhem called
The mediaeval contribution to statics, found in a group of this version a “peripatetic transformation” of the Elementa of
treatises “on weights” (De ponderibus), was made in the thir­ Jordanus; but since it shows no trace of influence from this
teenth century. The significance of the mediaeval “science of work, the characterization seems unjustified, and it is more
weights” was revealed by Duhem in the first volume of his work proper to treat it as an independent .writing.
Les Origines de la Statique, published in 1905. As he pointed out, The third group of works belonging to the mediaeval scientia
the mediaeval treatment of the problem of equilibrium, inspired de ponderibus consists of a group of “commentaries” or revised
by a group of anonymous and pseudonymous fragments of late versions of the treatises comprising the second group, mostly
Greek origin, was essentially an attempt to derive the Archime- ^fourteenth century authors. In all cases, these versions show
8 9
GENERAL INTRODUCTION
GENERAL INTRODUCTION
definite dependence on the authentic treatises of Jordanus de
and from this same manuscript Curtze edited the Liber Euclidis
Nemore, either incorporating his texts verbatim, or reworking
de ponderoso et levi, publishing it in 1900 in Bibliotheca Mathe­
them with elaborations. Two of the most common of these later
matica. Finally an edition of Thabit ibn Qurra’s Liber Karastonis,
versions are represented, in part, by texts edited in our Appen­
by F. Buchner, was published in the Sitzungsberichte der Phys-
dix III and Appendix IV. The first of these is a commentary on
ikalisch-medizinischen Sozietat in Erlangen, in 1922, with a very
the Liber ponderum (Version “P’’)by a fourteenth century writer,
interesting analytical introduction.^ The general inaccessibility
which Ftetrus Apianus published as “another commentary" in his
of the learned publications in which these editions are found, as
edition of the Liber de ponderibus printed at Nuremberg in 1533;
well as the fact that they are based on single manuscripts of the
from this we have edited the commentary on Proposition One.
works edited, justifies the inclusion of editions of these treatises
The second of these later versions is the one which occurs in
in our present collection, based on more adequate manuscript
Codex Vaticanus Latinus 2975, under the title Liber Euclidis de
material.
ponderibus, from which we have edited the commentary on Prop­
II
osition Eight, dealing with the general lever principle. These
Problems of statics, with which the mediaeval treatises “on
fourteenth century versions are of historical and theoretical in­
weights" are concerned, were dealt with in ancient times by
terest, since they reveal the continuing influence of the work of
Archimedes of Syracuse (ca. 287-212 B.C.) and by the author of
Jordanus in the fourteenth century, and the debates which took
the Mechanical Problems attributed to Aristotle. Although Ar­
place over the interpretation of hiis theorems.
chimedes’ authentic writings were translated from Greek into
The Tractatus de ponderibus of Blasius of Parma, written at
Latin by William of Moerbeke, in 1269, the treatises associated
the very end of the fourteenth century, does not properly belong
with Jordanus de Necnflfe show no direct acquaintance with the
to the group of De ponderibus treatises which we have described,
genuine Archimexlean..w-orLs, probably because they were written
since it is not normally found with them in the manuscripts. We
before the latter were translated.^* The indirect influence of
have edited it in full, however, because it represents a utiliza­
Archimedes is discernible, nevertheless, in several of the anon­
tion of the content of the mediaeval “science of weights" by one
ymous and pseudonymous writings which entered into the corpus
of the men who, at the beginning of the fifteenth century, con­
of mediaeval treatises on weights. Of these we shall have more
tributed to popularizing the scientific tradition of Oxford and
to say presently.
- ■ »—•— -|%<ll fV Mi | , , , -
Paris northern Italy. The treatise of Blasius fuses together,
The Mechanical Problems ascribed to Aristotle, but presum­
in an original ordering, most of the content of the Jordanus
ably written by a member of his school (possibly Strato of Lamp­
treatises along with the theorems of hydrostatics found in the
sacus) in the first half of the third century B.C., seems to have
De insidentibus in humidum; it is, however, decidedly inferior
been the ultimate source of a number of the theorems in the
to its sources in point of critical handling of the subject matter.
Liber de ratione ponderis of Jordanus de Nemore. Yet we have
Of the eight treatises which are here edited in complete form,
no evidence to show that this pseudo-Aristotelian work was ac­
three have never before been printed. These are the Liber de can-
cessible to the Latin Middle Ages either in direct translation or
onio, the Elementa super demonstrationem ponderum of Jorda­
even in paraphrases by Arab writers. The presence of these
nus de Nemore (Version “E"), and the Tractatus de ponderibus of
tfieo rems in Jordanus’ work, however, gives reason to suspect
Blasius of Parma. The Liber de ponderibus (Version “P") was
that a mediaeval version or paraphrase of the Mechanical Prob­
edited by Petrus Apianus and printed atNuremberg in 1533, along
lems did exist, even though no manuscript of it has as yet been
with the fourteenth century commentary whose first theorem we
discovered.
have edited in our Appendix III. The Liber de ratione ponderis of
Writings of Greek origin dealing with problems of statics,
Jordanus de Nemore (Version “R") was also printed in an early
whichpassed into the mediaeval corpus of treatises on weights,
edition, along with the De insidentibus in humidum attributed to
consisted of the work known as the De canonio, apparently trans-
Archimedes; these were printed by Curtius Trojanus at Venice,
lated directly from the Greek in the thirteenth century, and some
in 1565, from a manuscript which had belonged to Niccolo Tar-
fragments ascribed to Euclid which were probably inherited by
taglia. The texts of both of these sixteenth century editions are
way of Arabic versions. The De canonio consisted of four theo­
full of errors, and copies of them are scarce and inaccessible.
rems dealing with the method of constructing the Roman Bal­
The De insidentibus in humidum was also edited by M. Curtze,
ance, or balance of unequal arms; it assumes, as already proved
from a Dresden manuscript, in Bibliotheca Mathematics (1896);
by Archimedes and others, the general lever principle, and the
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