Table Of ContentPRACTICAL GEOMETRY
IN THE HIGH MIDDLE AGES
ARTIS CUIUSUBET CONSUMMATIO
Memoirs of the
AMERICAN PHILOSOPHICAL SOCIETY
AND THE
Held at Philadelphia
For Promoting Useful Knowledge
Volume 134 V PRATIKE DE GEOMETRIE
Edited with Translation and Commentary
by
STEPHEN K. VICTOR
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square • Philadelphia
1979
To Sue
and our family
Copyright © 1979 by The American Philosophical Society
Library of Congress Catalog Card Number 78-73170
International Standard Book Number 0-87169-134-5
US ISSN 0065-9738
CONTENTS
INTRODUCTION
I. The Place of Practical Geometry in the Middle Ages 1
The Nature of Practical Geometry 2
Practical Geometry in Education 31
Theory and Practice in Geometry 42
Practical Geometry and Practical Concerns 53
II. The Contents of Artis cuiuslibet consummatio and
the Pratike de geometrie 74
III. Procedures in the Editions, Translations, and
Commentary 92
Editing Artis cuiuslibet consummatio 93
Editing the Pratike de geometrie 101
Translating the Texts 104
About the Commentary 105
ARTIS CUIUSLIBET CONSUMMATIO
I. Planimetry -- Planimetria 108
II. Altimetry -- Altimetria 220
III. Volumetry -- Crassimetria 326
IV. Fractions -- De minutiis 384
THE PRATIKE DE GEOMETRIE
I. Planimetry (From ACC) -- Mesure des planeces 473
III. Altimetry and Volumetry (From ACC) -- Mesure
des hautesches et des perfondeces et des
crasse mesures 507
V. Other Geometric Material 522
VI. Mercantile Material 550
SELECTED BIBLIOGRAPHY 602
INDEX OF LATIN TECHNICAL TERMS 615
INDEX OF OLD FRENCH TECHNICAL TERMS 623
INDEX OF ASTRONOMICAL PARAMETERS 628
GENERAL INDEX 630
vii
PREFACE
Since beginning research toward a doctoral disserta
tion on practical geometry in the Middle Ages, I have changed
my views on the subject as I uncovered new evidence justi
fying a different approach. My first assumption, and that
of most of the people I have spoken to about the topic, was
that "practical geometry" must relate somehow to architecture,
surveying and city planning, to those areas, in other words,
where geometry plays a central role in the exercise of other
professions. The study of medieval buildings, fields and
towns from extant physical evidence was not a fruitful
approach for me, and I have left it to those better trained
in the methods of archeology and art history. Since I was
working as an historian of science, I chose to concentrate
on the written tradition of treatises called "practical
geometry."
The study in depth of the two treatises presented
as part of this book, the practical geometry beginning
"Artis cuiuslibet consummatio ..." (ACC) and the Old
French adaptation of it, the Pratike de geometrie, led me
to another view of the subject, namely that practical
geometry has its greatest importance as a popularization
of mathematics. The treatises on practical geometry were
a way of teaching some basic principles to those who would
not remain in school or university long enough to become
philosophers or theologians and would not necessarily
exercise a mathematical profession, but who might want,
or even need, some mathematics in their everyday lives.
The sampling of arithmetic and astronomy in ACC and of
commercial arithmetic and metrology in the Pratike argues
for the generally pedagogic, rather than scholastic,
IX
PREFACE PREFACE
purpose of the treatises. The development of a vernacular Both ACC and the Pratike are popularizing efforts, but ACC's
version of ACC is further evidence that the practical geome readers were more comfortable in Latin while those of the
tries sought their homes outside of the universities, perhaps Pratike were more at home in French. Together the treatises
in the bureaucratic and commercial milieus. Nonetheless, as give a picture of popular learning in mathematics during the
High Middle Ages.
the Introduction shows, the formalized structures of univer
sity education had an influence even on the non-scholastic The introductory essay on The Place of Practical
tradition of practical geometry. As the tradition developed, Geometry in the Middle Ages has been enlarged to include
the practical geometries acquired an increasingly theoretical discussion of fourteenth-century materials. I have more
underpinning, to the point where they are sometimes considered fully justified the stemma of manuscripts of ACC for this
works on measurement rather than simply practical geometries. edition, establishing the relationships among all known
ACC, whose author attempts to demonstrate most of manuscripts. I have also included the marginal annotations
his geometric results, belongs fairly early in the tradition made by a later medieval reader of ACC. The various indexes
of demonstrated practical geometry. By the time of its com I have prepared should help the scholar find access to the
position in the late twelfth century, a Latin Euclid was various parts of the large book this work has grown into.
widely available in the West. Thus practical geometries, The Latin and Old French indexes are meant as a resource for
with their roots in the Roman agrimensores, Boethius, pseudo- tracking down obscure technical usages. The Index of Astro
Boethius and Gerbert, were no longer the sole repositories nomical Parameters seemed useful because of the extensive
of geometric knowledge. If a practical geometry was to have astronomical material in the book. Since ACC and the Pratike
an impact, it had to offer something more than introductory cover so many topics, the editions, with their commentaries,
geometry to the potential reader. The author of ACC apparent translations and indexes should help scholars in a variety of
ly broadened his audience by increasing the scope of his work. fields, including the history of geometry, arithmetic, astro
The long sections on astronomy and fractional arithmetic nomy, metrology and commerce. I hope my work is as helpful
included in ACC indicate that he meant for his work to be to them as theirs have been to me.
a general treatment of the mathematical sciences; the A very special note of appreciation goes to John E.
treatise covers three of the four branches of the quadrivium, Murdoch, who encouraged and supervised my work when the core
music being omitted. of this book was a doctoral dissertation at Harvard and who
I have made substantial revisions and have supple continued to provide support as the work developed. Guy
mented my original doctoral dissertation study considerably Beaujouan has shared his wealth of knowledge over the course
in preparing this work for publication. I have added an of many years. Marshall Clagett has been very encouraging,
edition and discussion of the thirteenth-century adaptation offering advice and a place to work at the Institute for
of ACC, the Pratike de geometrie. Even though the Pratike Advanced Study. Working space, time and access to libraries
had been edited before, I felt a new edition from both manu have also been arranged by Henry Copeland of the College of
scripts, with translation, reference to its sources in ACC, Wooster, Everett Mendelsohn of the History of Science Depart
and explicative commentary would be useful to researchers in ment at Harvard, Charles F. Montgomery of the Yale University
the history of science and the romance languages. With the Art Gallery and Rick Beard of Yale's Center for American Art
addition of the Pratike, this work acquires a further dimension. and Material Culture.
x xi
PREFACE
The National Science Foundation supported part of my
research through a doctoral dissertation grant. A Fulbright
Fellowship permitted a year of study and research in Europe.
Sincere thanks are due to a number of scholars who have read
parts of the manuscript and saved me from many errors;
Caroline Walker Bynum, Alfred Foulet, Owen Gingerich, Otto I.
Neugebauer, Lon R. Shelby, Noel Swerdlow and Charles Zucker- THE PLACE OF PRACTICAL GEOMETRY
man have all helped in various stages of the work. IN THE MIDDLE AGES
The Publications Committee of the American Philo
sophical Society has been most understanding, suggesting The history of the medieval traditions of practical geometry
production of this book by photo-offset from typescript as is very little known; no full-length or general history of the
a cost-saving measure so that the text and variants, and subject is available. Although it is not my intention to provide
translations and notes appear on facing pages. Although such a history here, some knowledge of the place of practical •
this approach has meant considerable work for my typist and geometry in the Middle Ages is useful as a preliminary to under
me, I expect the users' convenience to justify our effort. standing the late twelfth-century practical geometry, Artis
I am very grateful to Roger Reed who did the drawings and cuiuslibet consummatio, and its thirteenth-century adaptation,
to Donna Gold who typed this complex book in a manner suit the Pratike de geometrie, both of which I have edited below.
able for publication. Information that will be useful in evaluating the significance
To my wife, Susanne F. Roberts, whom I have exas of the practical geometries Artis cuiuslibet consummatio and the
perated for too long with problems of artificial measurement, Pratike de geometrie will also be relevant to understanding the
cube root, and astronomical parameters, I owe the deepest place of any of the medieval practical geometries. The materials
debt of thanks. Her suggestions have been well-taken, her which shed light on the place of medieval practical geometry
help is always unstinting, and her sighs of relief will offer insights into several aspects of practical geometry.
echo through the house when this is done. The first concern is the nature of practical geometry. One
might ask what was the place of practical geometry among the sci
ences, and what was included in the treatises themselves. In
other words, what was practical geometry thought to be by those
who composed treatises on the subject and by those who wrote on
the classification of the sciences?
The place of practical geometry in education is another area
of investigation. Its role in educational curricula, testimony
about its pedagogical effectiveness and its inclusion in works
having another educational purpose all provide information on
the place of practical geometry within medieval educational tra
ditions .
xii 1
INTRODUCTION THE PLACE OF PRACTICAL GEOMETRY 3
A third topic to consider is the relationship between prac The first use of the distinction geometria theorica et
tical geometry and theoretical geometry. Both theoretical practica seems to be that of Hugh of St. Victor in his prac
and practical works seem to be sources of the contents of the tical geometry, the Practica geometriae.^ Dividing the whole
extant practical geometries. The question therefore arises of philosophy into theorica and practica is an ancient tradi
of the proper relationship of theory and practice. As several tion, going back to Plato and Aristotle. Hugh uses the same
of the practical geometries address themselves to the problem, distinction in his work on the classification of the sciences,
2
the relation indicates something of what the authors of the the Didascalicon. For the ancients, all of mathematics be
treatises felt their place to be. longs on the theoretical side of the dichotomy. In the Didas
A last consideration on the place of practical geometry in calicon, Hugh follows the ancients, placing geometry and all of
the Middle Ages is how it relates to practical concerns. Sev mathematics within the theoretical sciences. In the Practica
3
eral treatises hint at some connection between practical geo geometriae, Hugh divides geometry into theoretical and practical
metry and what might be called the practice of geometry. Sur branches. In other words, Hugh takes the theory-practice dyad
veying and architecture are domains where one would expect from the division of all of philosophy and applies it within one
to find practical geometry applied to practical concerns. of the branches of theoretical philosophy, thereby crossing the
boundaries of a domain to use the distinction in a new way.^
The Nature of Practical Geometry Since it is Hugh who first uses the terms geometria theorica
et practica, we might expect to see some hint of the separation
Practical geometry is defined by some of the treatises on
of the parts of geometry in the Didascalicon. It happens that
the classification of the sciences. During the twelfth and
one of the major innovations in the Didascalicon offers to the
thirteenth centuries, the nature of practical geometry is
individual arts the distinction, analogous to that of theorica-
described with increasing precision in the literature on
practica, of agere de arte and agere per artem. Hugh introduces
classification. Its place is more and more narrowly defined
these terms in an effort to delineate better the areas of the
as it is separated from other sciences. As its place is more
various arts and to limit the incursion of one art into the area
carefully circumscribed, its nature is gradually revealed.
of another.’’ Agere de arte has to do with discussing or teach
In other words, practical geometry is not defined as an ab
ing about the art; agere per artem is the application of the
stract concept, but in considering what it does and does not
principles of the art to other endeavors.
do.
How close the pair, de arte, per artem, is to the pair,
Throughout the same period and into the fourteenth century,
theorica, practica, may be established by considering any of a
the practical geometries themselves vary widely. The contents
of one particular treatise can be quite different from that
of another. What is included in the various treatises does 1Baron, 1955b: pp. 298-302.
not seem to follow a general pattern of development. The
^Ed. Buttimer, 1939. Tr. Taylor, 1961.
nature of a treatise is reflected in its parts. The number 3
Ed. Baron, 1966: pp. 1-64. Another MS of part of the treat
and names of the parts tell something of the author's inten ise is BL Digby 166, ff.6v-8r.
tion for his practical geometry. Because the same name may 4
This practice of changing the domain of a distinction occurs
apply to different contents, we should establish, to some
elsewhere in Hugh's work; see Baron, 1955a: p. 111.
extent at least, what is included in the various sections of
5Hunt, 1948: pp. 99-100.
the different practical geometries.
4 INTRODUCTION THE PLACE OF PRACTICAL GEOMETRY 5
6
number of twelfth-century glosses. An anonymous introduction parts, at least formally. The fuller differentiation, as
to astronomy, written in a blank space where a diagram was to found in the Practica geometriae, would have completed the
have been drawn, tells us treatment of geometry in the Didascalicon. Further, it would
The craftsmen (that is, the doers) of this art are two, have fit in well with distinctions found there. Why then does
one acting concerning the art (agens de arte), the other
Hugh use the distinction only in his Practica geometriae?
by means of the art (ex arte). The agens de arte is the
teacher of the art, the agens ex arte is the practition Knowing the state of geometric knowledge at Hugh's time may
er of the art.' help resolve the problem.
Since ex arte is merely a reformulation of per artem, Hugh
In the section of the Didascalicon where Hugh describes how
would probably have concurred that the agens de arte, or
he studied the quadrivium, his approach to all the quadrivial
teacher, deals with the theoretical aspect of the art and
sciences is primarily practical. Hugh's learning of geometry
that the agens ex arte, the practitioner, deals with the prac
seems to have involved measurement and comparison; no mention
Q
tical aspects of the art. Still the distinction, de arte,
is made of proving theorems. It is obvious that Hugh did not
per artem, enunciated in Hugh's Didascalicon, does not fully
read Euclid as a youth; Latin translations only of an extremely
suffice to separate theoretical from practical geometry.
small part of the Elements were available. Still there was no
Nonetheless, in his Practica geometriae, Hugh does provide a
shortage of works on geometry in circulation by about 1100;^
differentiation which is not only adequate but also has full
the two pseudo-Boethian geometries were available, as were Ger-
er implications. Theoretical geometry, Hugh tells us, deals
bert's geometry and other similar works.^ These were based pri
with measurements intellectually, finding dimensions by con marily on compilations used by the Roman agrimensores. Since
templation through reason alone; practical geometry uses cer the agrimensores were practicing surveyors, it is not surprising
tain instruments and interprets by means of other proportions.
that their treatises have a distinctly practical orientation.^
The separation given in Hugh's practical geometry yields spe
cific information about the nature of theoretical and practi 9
Ed. Buttimer, 1939: pp. 114-115: "Ipso exemplo oculis
cal geometry. However, the distinction in the practical geo subiecto, quae ampligonii, quae orthogonii, quae oxygonii
metry does not preclude the simpler and more general differ differentia esset, patenter demonstravi. Utrumne quadratum
aequilaterum duobus in se lateribus multiplicatis embadum
entiation between de arte and per artem. The pair de arte- impleret, utrobique procurrente podismo didici . . . Haec
per artem could have been used to divide geometry into two puerilia quidem fuerant, sed tamen non inutilia, neque ea
scire stomachum meum onerat." See tr. Taylor, 1961: pp. 136-137.
^In his Practica geometriae, Hugh tells us he is writing his
6Ibid., pp. 100-105. treatise "not as though to forge a new work but to collect the
scattered old ones," ed. Baron, 1966: 15.1-2: "Practicam geo
^Oxford, Corpus Christi College, MS 283, f.81v, as quoted
metric nostris tradere conatus sum, non quasi novum cudens opus
by Hunt, 1948: p. 104, nn. 1-4: "Opifices (glossed id est sed vetera colligens dissipata."
factores) huius artis sunt duo, unus agens de arte, alter ex
arte. Agens de arte est doctor ipsius artis, agens ex arte ^The second of the geometries attributed to Boethius has been
est practicus ipsius artis." edited by Folkerts, 1970. There is no complete edition of the
first pseudo-Boethian geometry. Books I and II may be found in
®Ed. Baron, 1966: 16.27-32: "Considerandum est quod omnis
Migne, 1844-1864: vol. 63, cols. 1352-1364. Books I, III, IV
geometrica disciplina aut theorica est, id est speculativa,
and parts of V are available in Blume, Lachmann and Rudorff,
aut practica, id est activa. Theorica siquidem est que spacia
1848-1852, v. I: pp. 377-412. Gerbert'.s Geometry and mariy re
et intervalla dimensionum rationabilium sola rationis specu
lated texts have been edited by Bubnov, 1899.
latione vestigat, practica vero est que quibusdam instrumentis
agitur et ex aliis alia proportionaliter coniciendo diiudicat." 12
Probably the most significant source of geometric knowledge
in the early Middle Ages is the collection of materials on sur-
THE PLACE OF PRACTICAL GEOMETRY
6 INTRODUCTION 7
14
the Didascalicon, it may have been the appearance of Adelard
The legacy of the Roman surveyors gives the works of pseudo-
of Bath's version of the Elements which caused Hugh to elaborate
Boethius and Gerbert a practical flavor as well. Thus the — pj---- °
his definition of geometry. Perhaps even Hugh's application
geometry available in about 1100 could easily be considered
of the distinction practica-theorica to geometry was a way of
practical geometry, even though the term was not in use at
differentiating the geometry available in the very early twelfth
the time. The practical nature of geometry in Hugh's descrip
century from the "new" geometry of Euclid, which certainly de
tion accurately reflects the geometric treatises we know to
served to be called theoretical geometry. Thus, what had been
have been available at the time of Hugh's childhood.
the extant contents of geometry before the recovery of Euclid
Elsewhere in the Didascalicon, Hugh gives the parts of
in a translation from the Arabic became known as practical geo
geometry as planimetria, altimetria, and cosmimetria; he de
metry. This newly named geometry had no place among the tradi
scribes geometry as a means of measuring material things like
tional sciences. Hugh of St. Victor named it but did not place
the sea (its depth), a tree (its height), the world, a ball,
13 it within his classification of the sciences.
and an egg. It is quite inaccurate to consider altimetry,
The assimilation of Arabic learning in the West has other ef
planimetry, and cosmimetry to be parts of geometry in general.
fects on the definition and elaboration of practical geometry.
Hugh seems to agree that they are part of the more restricted
Dominicus Gundissalinus, Archdeacon of Segovia, as translator
subject of practical geometry, for in the Fractica geometriae
from the Arabic was in an excellent position to know the Arabic
he gives the same triplet as its parts. Also, Hugh's descrip
traditions concerning geometry. He also was in contact with
tion of geometry as material measure fails to account for ab
the intellectual milieu of Northern France.^ It is no surprise,
stract geometry, which deals with relations besides those of
then, to find that Gundissalinus has an important influence on
measure. We are thus led to assume that at the time Hugh
the development of that part of the classification of the sci
wrote the Didascalicon, he was not sufficiently familiar with
ences that relates to practical geometry. One aspect of Gun
Euclid's Elements to have listed parts of theoretical geometry
dissalinus 's work concerns the relation of practical geometry
or even to have realized that the geometry which he knew was
to theoretical geometry; another aspect defines the place of
not all of geometry. If the Practica geometriae came after
practical geometry with respect to the practice of certain
crafts.
veying, land division, etc. deriving from the practical manu
als of the agrimensores. On the geometric traditions arising
from their works, see Cantor, 1875, and Thulin, 1911. On the
agrimensores in general, see Dilke, 1971. They both seem to date from the period 1125-1130. See
Baron, 1955a: p. 116 for the date of the Practica geometriae.
■^Ed. Buttimer, 1939: p. 33: "De geometria. Geometria Taylor, 1961, in his introduction to the PidascaTicon, p. 3,
tres habet partes, planimetriam, altimetriam, cosmimetriam. suggests a date in the late 1120's for that work”.
Planimetria planum metitur, id est, longum et latum, et ex
tenditur ante et retro, dextrorsum et sinistrorsum. Alti ^It is difficult to give a precise date for Adelard's trans
metria altum metitur et extenditur sursum et deorsum. Nam lation. In a comment in his Astrolabe, cited in Haskins, 1924:
et mare altum dicitur, id.est profundum, et arbor alta, id p. 25, Adelard tells us that the translation of the Elements
est sublimus. Cosmus mundus interpretatur, et inde dicta predates his Astrolabe. Haskins, pp. 23-29, dates the Astrolabe
est cosmimetria, id est mensura mundi. Haec metitur sphae in the period 1142-1146. It is most likely that the translation
rica, id est, globosa et rotunda, sicut est pila et ovum, came after 1126.
unde etiam a sphaera mundi propter excellentiam dicta est
cosmimetria, non quia tantum de mundi mensura agat, sed 16See Haring, 1964: pp. 271-286.
quia mundi sphaera inter omnia sphaerica dignior sit." See
tr. Taylor, 1961: p. 70.
THE PLACE OF PRACTICAL GEOMETRY 9
8 INTRODUCTION
proach gives us both knowledge and deed, once we have the the-
In the work of Gundissalinus on the classification of the
22
oretical basis. Thus, both the theoretical and the practical
sciences, the De ortu scientiarum, a distinction comparable
aspects of a science are not only mutually beneficial, but in
to that between theory and practice is found in the pair
terdependent in Gundissalinus's scheme. The interdependence
extrinsecus and intrinsecus. What must be known before be
of theory and practice is a theme that occurs frequently in
fore beginning the practice of an art is ars extrinsecus ;
the treatises on practical geometry, as we shall see below.
what must be known in order to practice the art is the ars
In assimilating Arabic knowledge and traditions to those of
intrinsecus. For Gundissalinus, the extrinsic art is very
----------- . . 19 the West, Gundissalinus was in a pivotal position for elaborat
closely related to theory; the intrinsic to practice.
ing new distinctions reflecting Arabic learning in a Western
Gundissalinus intends the distinction to apply to all the sci
context. With respect to the divisions of practical geometry,
ences; all of them have their extrinsic and intrinsic aspects.
Gundissalinus serves as a bridge between two traditions. We
Even though Gundissalinus has distinguished the extrinsic
shall see that Gundissalinus, in response to the craft tradi
and intrinsic approaches to a subject, he in no way means for
tions reflected in an Arabic classification of the sciences,
them to be independent; they are complementary rather than
20 distinguishes between two sorts of practical geometry.
separate approaches to the same subject. In fact, so close
Gundissalinus's discussion of each of the sciences is ar
is the reliance of the intrinsecus on.the extrinsecus that
ranged into several topics. Certain topics applying to geometry
Gundissalinus warns us that
as a whole have but one answer. Seven other topic headings re
It would be disgraceful for someone to exercise any
art and not know what it is, and what subject matter 21 quire two answers, one for theoretical geometry and one for
it has, and the other things that are premised of it.
practical geometry. For a few of the topics, there are two an
In addition to Gundissalinus’s plea for the necessity of an
swers for practical geometry, indicating that Gundissalinus
extrinsic or theoretical knowledge of an art as a prerequisite
recognizes two types of practical geometry. The seven headings
to its practice, he specifically emphasizes the merits of the
for which the distinction between theoretical and practical
practical approach to an art. The intrinsic or practical ap-
geometry is made are presented in the table following
17Ed. Baur, 1903.
Ibid., p. 44: "Ars enim extrinsecus non tradit actum,
set sciendam tantum; ars vero intrinsecus et actum dat et
^®The distinction seems to originate in Varro; it comes to
scienciam."
the twelfth century from Marius Victorinus's commentary on
Cicero's De inventione ; see Hunt, 1948: p. 98. 23
Page and line numbers, separated by a period, are given
after each of the answers. Unfortunately, there is a gap in
■^Ed. Baur, 1903: p. 44: "Cum autem omnis ars dividatur
the manuscript where the parts of practical geometry were to
in theoricam et practicam, quoniam vel habetur in sola cog-
have been discussed. The complete list of topics for geometry
nicione mentis--et est theorica— , vel in exercicio operis--
are given in Baur's ed., 1903: p. 102: "Circa geometriam
et est practica--: profecto ars extrinsecus pertinere videtur
quoque hec eadem inquirenda sunt: quid sit ipsa, quod genus
ad theoricam, ars vero intrinsecus ad practicam."
eius, que materia, que partes, que species, quis artifex,
quod instrumentum, quod officium, quis finis, quare sic dica
7®Ibid., p. 43: "Non autem ista sic distinguimus quasi
tur, quo ordine sit legenda."
artem extrinsecus et intrinsecus duas artes esse velimus, sed
quod hiis duobus modis una et eadem ars docetur."
^Ibid. , p. 44: "Turpe enim esset alicui, si aliquam artem
exerceret, et, quid ipsa esset et cuius generis, et quam ma
teriam haberet, et cetera que premissa sunt, ignoraret."
