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A survey of our present
knowledge of the medieval translations of Euclid's Elements
into Latin would clarify the relations of many texts and
would distinguish them from later reworkings of the texts.
First let us distinguish between translations of Euclid's
Elements into Latin directly from the
Greek and translations from the Arabic (which in turn had
come from the Greek).
We begin with the direct translations.
The most important was that of Boethius, which has been
made in about the year 500. Only parts of this translation
are extant in four different fragments dating from the eighth
to the eleventh centuries. [Note 1: See M. Folkerts,
The Importance of the Pseudo-Boethian Geometria
During the Middle Ages, in: Boethius and the Liberal
Arts. A Collection of Essays, edited by Michael
Masi (Bern/Frankfurt/Las Vegas: Peter Lang, 1981), pp.187-209.]
These fragments are:
-
in the so-called "third"
recension of Cassiodorus' Institutiones [Note 2: Extant
are: I def. 1-12.14.13.15-23, post. 1-5, ax. 1.3.2.7; II
def. 2; V def. 1-8.11.9.10.13.12.14-16.18.17.]
-
in manuscripts
of the Corpus agrimensorum [Note 3: Extant
are: I def. 1-12.14.13.15-23, post. 1-5, ax. 1.3.2.7, prop.
1-3 with proofs.]
-
in the so-called Geometry
I attributed to Boethius [Note 4: Extant are:
I def. 1-12.14.13.15-23, post. 1-5, ax. 1.3.2.7; II def.
1.2, prop. 1; III def. 1-6.8-11; IV def. 1.2, prop. 1; III
def. 6.8; I prop. 2-4.6-8.(9).10-18.21.23.26-28.31-37.39-41.43.42.44-48;
II prop. 1.3-6.9-12.14; III prop. 3.7 beginning. 22 end.
27.30-33; IV prop. 1-4.6.8.12.11; III prop. 7 end. 9.12.10.13.14.16.18.19.24.22
beginning (all propositions without proofs).]
-
in the
so-called Geometry II attributed to
Boethius. [Note 5: Extant are: I def. 1-12.14.13.15-23,
post. 1-5, ax. 1.3.2.7; II def. 1.2; III def. 1-6.8-11;
IV def. 1.2; I prop. 1-8.(9.)10-41.43.42.44-48; II prop.
1.3-6.9-12.14; III prop. 3.7 beginning. 22.27.30-33; IV
prop. 1-4.6.8.12.11 (all propositions without proofs); further,
I prop. 1-3 with proofs.]
The first three texts
seem to have originated in Corbie in the eighth century;
the fourth text was compiled in Lorraine in the first half
of the eleventh century. I have tried to reconstruct the
original translation from these fragments. [Note 6: Menso
Folkerts, "Boethius" Geometrie II, ein mathematisches
Lehrbuch des Mittelalters (Wiesbaden: Franz Steiner,
1970), esp. pp.173-217.] In the last few years little
that is new has appeared on this subject, except for the
discovery of ms Madrid, BN 9088, which contains Geometry
I. Perhaps one should also mention that the
twelfth-century Liber Ysagogarum
Alchorismi has been found to contain axioms and
enunciations which may belong to the Boethius tradition
but are not among the extant Boethius fragments. [Note 7: See
the unpublished dissertation of Bruce G. Dickey, Adelard
of Bath: An Examination Based on Heretofore Unexamined Manuscripts
(Ph.D. dissertation, University of Toronto, 1982).]
There
are two other anonymous Latin Euclid fragments from the
fifth and ninth centuries respectively. The older of them
is a palimpsest now in the Biblioteca Capitolare of Verona,
ms XL (38). From the original text there are three double
folios with parts of Euclid's Elements,
books XI-XIII. [Note 8: There are parts of the propositions
XI 24-25; XII 2-3.8; XIII, 2-3.7. The books XII and XIII
are marked as XIIII and XV.] Geymonat, who edited the
text, [Note 9: Euclidis Latine facti fragmenta
Veronensia, ed. M. Geymonat (Milano, Varese: Istituto
Editoriale Cisalpino, 1964).] thinks that the writing
can be dated to the last years of the fifth century and
that we have here part of the original Boethius' translat
ion.
The second fragment was written at the beginning of the
ninth century in a north-east French scriptorium; according
to Bernhard Bischoff, it may be connected with the Palatine
library of Charlemagne. [Note 10: B. Bischoff, in: Mittelalterliche
Studien, Bd.3 (Stuttgart: Anton Hiersemann, 1981),
pp.14.158.] This fragment, now in the University Library
at Munich, [Note 11: Ms. 2o 757. Besides older editions
and comments by M. Curtze and A. A. Björnbo, there is
a modern edition by M. Geymonat, Nuovi frammenti della geometria
'boeziana' in un codice del IX secolo?, in: Scriptorium,
XXI (1967) 3-16.] contains Euclid, I 37 to 38 and II,
8 to 9. The translator obviously did not know the mathematical
contents nor did he master the Latin grammar: he transliterated
the mathematical terms as Greek expressions having similar
letters and took Greek letters indicating the endpoints
of segments as number symbols. Therefore, it seems very
unlikely that this unique fragment has anything to do with
the Boethian tradition.
Another direct translation -- this
time fully extant -- was made in South Italy or Sicily in
the twelfth century. There are translations of the Almagest,
the minor writings of Euclid (Data,
Optics, Catoptrics)
and the Elementatio physica of Proclos
of the same provenance and date. J. E. Murdoch has made
a thorough analysis of this translation of the Elements, [Note 12: J.
E. Murdoch, Euclides Graeco-Latinus. A Hitherto Unknown
Medieval Latin Translation of the Elements
Made Directly from the Greek, in: Harvard Studies
in Classical Philology, LXXI (1966) pp.249-302.]
and H. L. L. Busard has edited the text. [Note 13: H.
L. L.Busard, The Mediaeval Latin Translation of
Euclid's Elements Made Directly from the Greek
(Stuttgart: Franz Steiner, 1987).] Unfortunately, we
cannot say exactly how the manuscript exemplar used by the
translator may be related to the extant Greek manuscripts.
It should perhaps be added that, despite Heiberg's analysis
of the manuscripts available to him, [Note 14: See the
Prolegomena critica in Euclidis
Elementa. Edidit I.L.Heiberg. Vol.V (Lipsiae:
B.G.Teubner, 1888), pp.XXIII-CXIII.] the transmission
of the Greek text of the Elements is
itself unclear. There were of course later translations
direct from the Greek by humanist scholars like Zamberto,
but these do not concern us now. In any case there has
been very little research on them.
Far more interesting
than the early translations from the Greek are the twelfth-century
translations into Latin from the Arabic. Unfortunately
the transmission is extremely complicated and is still partly
unknown -- though in the last few years H. L. L. Busard
and others have thrown some light on the situation. In
addition, our understanding of the Arabic translations of
Euclid by Hajjaj and Ishaq-Thabit has been deepened.
I should like to make some remarks on some of these results.
The
systematic examination of the Arabic-Latin tradition of
Euclid's Elements began in 1953 with
Marshall Clagett's fundamental article. [Note 15: M. Clagett,
The Medieval Latin Translations From the Arabic of the Elements
of Euclid, With Special Emphasis on the Versions of Adelard
of Bath, in: Isis XLIV (1953) 16-42.]
For the first time three different texts were distinguished:
version I, "a translation"; version II, "an abridgement";
version III, "an editio". From an examination of the most
important manuscripts Clagett was able to characterize these
three versions. He also tried to find the relations between
the three versions and to decide which were genuine translations
from the Arabic and what were the Arabic originals of these
translations. He further gave a general description of
the other Euclid translations from Arabic, namely those
of Gerard of Cremona and Hermann of Carinthia, and also
of medieval reworkings of the text. After this article
was written it was the accepted theory that there are three
Adelard versions of Euclid and a reworking by Campanus.
Later, J. E. Murdoch showed that this gives too simple
a picture and pointed to a number of "mélanges" and reworkings. [Note 16: J.
E. Murdoch, The Medieval Euclid: Salient Aspects of the
Translations of the Elements by Adelard
of Bath and Campanus of Novara, in XIIe
Congrès International d'Histoire des Sciences, Colloques
(= Revue de synthèse, 3e
série, nos 49-52; Paris 1968, pp.67-94.] Of all these
versions and redactions only the Campanus reworking was
in print before 1967 -- and that in the fifteenth and sixteenth
centuries. Since this time H. L. L. Busard has published
editions of the following translations: Hermann of Carinthia
(1967-1977), [Note 17: H. L. L. Busard, The
Translation of the Elements of Euclid from the Arabic into
Latin by Hermann of Carinthia(?): books
I-VI, Janus, LIV (1967) 1-140, and published
separately (Leiden: E. J. Brill, 1968); books VII-IX, Janus,
LIX (1972) 125-187; books VII-XII (Amsterdam: Mathematisch
Centrum, 1977).]
Gerard of Cremona (1983), [Note 18: H.
L. L. Busard, The Latin translation of the Arabic
version of Euclid's Elements commonly
ascribed to Gerard of Cremona (Leiden: New Rhine
Publishers, 1983).] Adelard I (1983). [Note 19: H. L.
L. Busard, The First Latin Translation of Euclid's
Elements Commonly Ascribed to Adelard of Bath
(Toronto: Pontifical Institute of Mediaeval Studies, 1983).]
A critical edition of the Adelard II version, which has
an important place in the transmission of the text, has been being
prepared by Busard and myself.
(H. L. L. Busard, M. Folkerts (eds.), Robert of Chester's (?) Redaction of Euclid's Elements, the so-called Adelard II Version. 2 vols. Basel / Boston / Berlin: Birkhäuser Verlag, 1992.)
In the introductions to
his editions Busard has collected the available information
about the authors and their texts, and in an article he
gave information on further, hitherto unknown, reworkings
of the text. [Note 20: H. L. L. Busard, Some Early Adaptations
of Euclid's Elements and the Use of
its Latin Translations, in: Mathemata. Festschrift
für Helmuth Gericke, ed. M. Folkerts and U.
Lindgren (Stuttgart: Franz Steiner, 1985), pp.129-164.]
Yet other results about the connections between the Arabic
and Latin versions were published in 1985 by P. Kunitzsch. [Note 21: Paul
Kunitzsch, Findings in Some Texts of Euclid's Elements
(Mediaeval Transmission, Arabo-Latin), in: Mathemata.
Festschrift für Helmuth Gericke, ed. M. Folkerts
and U. Lindgren (Stuttgart: Franz Steiner, 1985), pp.115-128.]
These questions were also treated by R. Lorch and M. Folkerts
in the Adelard Colloquium held at the Warburg Institute
in 1984 and again by C. Burnett, M. Folkerts and R. Lorch
in an Addendum published in the proceedings
of this conference. [Note 22: R. Lorch, Some Remarks
on the Arabic-Latin Euclid, in: Adelard
of Bath. An English Scientist and Arabist of the Early Twelfth
Century, ed. by Charles Burnett (London: The Warburg
Institute, 1987), pp. 45-54. - M. Folkerts, Adelard's Versions
of Euclid's Elements,
in: Adelard of Bath ..., pp.55-65. -
Addendum, pp.65-68.
]
Questions about
the Arabic-Latin translations of Euclid cannot be separated
from questions about the state of the Arabic text. Which
Arabic versions could the translators have had before them
in the twelfth century? It is known that the Elements
had been translated or reworked by al-Hajjaj, Ishaq b.
Hunayn and Thabit b. Qurra. We know far too little about
the characteristics of these versions. Only the Hajjaj
version is in print and that from an incomplete and contaminated
manuscript. [Note 23: Ed. by R. O. Besthorn et al., Codex
Leidensis 399,1. Euclidis Elementa ex
interpretatione al-Hadschdschadschii cum commentariis al-Narizii
(Copenhagen, 1893-1932).] The study of the Arabic tradition
has been scarcely begun. After Klamroth's excellent article
(1881) on the Arabic Euclid, [Note 24: M. Klamroth,
Über den arabischen Euklid, in: Zeitschrift
der Deutschen Morgenländischen Gesellschaft,
XXXV (1881) 270-326, 788.] the only editions of pre-thirteenth
century texts are books V and VII-IX of the Ishaq-Thabit
version. These editions are Harvard Ph.D. theses by John
Engroff and Gregg De Young. [Note 25: J. W. Engroff, The
Arabic Tradition of Euclid's Elements: Book v
(unpublished Ph.D. dissertation: Harvard University, 1980),
not seen; G. De Young, The Arithmetic Books of
Euclid's Elements in the Arabic Tradition (unpublished
Ph.D. dissertation: Harvard University, 1981). The results
are summarized in G. De Young, The Arabic Textual Traditions
of Euclid's Elements, in: Historia
Mathematica, XI (1984) 147-160.]
According
to the Fihrist of Ibn al-Nadim, Hajjaj
translated the Elements twice. [Note 26: M.
Steinschneider, Euklid bei den Arabern, in: Zeitschrift
für Mathematik und Physik, hist.-lit. Abth.,
XXXI (1886) 81-110.] Of the two translations the second
"for al-Ma'mun," he says, is better than the first "for
Harun." Hajjaj I appears to have been lost. The first
six books and a few definitions of book VII of one -- as
it seems, the second -- Hajjaj version, together with Nayrizi's
commentary, are extant in ms Leiden 399,1 -- and this is
the printed version I have just mentioned. It is possible
that we also have books XI-XIII in the Hajjaj version,
if we can believe the scribe of the Arabic manuscript København
81. The problem here is that the readings in ms København
81 is too close to the corresponding passages in the Ishaq-Thabit
manuscripts to represent an independent translation;
in fact M. Klamroth in 1881 claimed that there was no more
divergence than one would expect from normal manuscript
transmission. [Note 27: See note 24.]
But we should also note that Kunitzsch's recent investigation
revealed sufficient differences to justify the assumption
of some kind of independent transmission. [Note 28: See
note 21.] There are manuscripts in Leningrad, Teheran
and in the El Escorial which probably also carry a text
similar to København 81, but they have not yet been investigated.
Another
translation is ascribed to Ishaq b. Hunayn. No manuscript
of this translation is known. But Thabit's redaction of
this text is extant in at least 19 manuscripts. The by
far oldest manuscript is Teheran, Malik 3586, which -- according
to the subscriptions in books III, VI, and VII -- was written
in A.H.343 = A.D.954/55; the missing part at the end of
book VII is now Teheran, Danishgah 2120. The manuscripts
do not all carry the same text. At least two of them are
contaminated with Hajjaj readings. [Note 29: El Escorial
907; Leningrad, Akademia Nauk C 2145.] Further, De Young
has identified two recensions, which he calls A and B, in
books VII and VIII within the "Ishaq-Thabit" tradition.
The manuscripts are not constant in their affiliation.
P. Kunitzsch attributes one recension to Hajjaj and the
other to Ishaq-Thabit. [Note 30: See note 21, pp.116-117.]
In the thirteenth century the two texts -- al-Hajjaj and
Ishaq b. Hunayn improved by Thabit ibn Qurra -- were
available to Nasir al-Din al-Tusi. He made several remarks
about the differences between them in
his Tahrir,
of which we have several manuscripts. There is another
tahrir, also attributed
to Nasir al-Din and printed in Rome in 1594, which gives
us other information about the two translations. The thirteenth
century redactor tells us that certain theorems were omitted
by Hajjaj but are included by Ishaq-Thabit. [Note 31: See
C. Thaer, Die Euklid-Überlieferung durch at-Tusi,
in: Quellen und Studien zur Geschichte der Mathematik,
Astronomie und Physik, Abt. B: Studien, III (1936)
116-121. The propositions are: I,45; VI,12; VII,24,25 (in
Thabit's numbering; not present in Heiberg's Greek); X,27,28
(in Heiberg; X,21,22 in Thabit).]
We come now to the
four most important twelfth century Latin texts which are
assumed to be translations from the Arabic: 1) the one ascribed
to Hermann of Carinthia, 2) the one ascribed to Gerard of
Cremona, 3) and 4) the two versions ascribed to Adelard
of Bath, called Adelard I and Adelard II. The "Hermann"
translation contains only books I-XII; it was obviously
not much used and is extant only in one manuscript Paris
BN Lat. 16646. Recently Busard has found that there is
another witness to this translation, a few citations in
ms Vat. Lat. 1268; and he thinks that these reflect a more
reliable text than the Paris manuscript. [Note 32: Busard
(see note 20), pp.133-134.] It appears that Hermann followed
a Hajjaj text, because he leaves out the propositions omitted
by Hajjaj but retained in Ishaq-Thabit, [Note 33: See
note 31.] but no Arabic manuscript has thus far been found
that could have been the one which Hermann translated.
Gerard
of Cremona is well known for his extreme literalness in
translating Arabic texts. As it stands however, the "Gerard"
translation of the Elements is not always
word for word. It seems indeed that a literal translation
-- no doubt originally by Gerard -- has been reworked later
giving it a more standardized and uniform wording and a
better Latin style which is not so severely "Arabicized"
as Gerard's translations normally are -- here I base my
conclusions on Kunitzsch's results. [Note 34: See note
21, pp.119-120.] The hypothesis that the text has been
reworked finds some support in the fact that the earliest
manuscript is from the fourteenth century. A comparison
of selected passages of the Gerard text with Arabic manuscripts [Note 35: By
R.Lorch; see note 22.] show that Gerard basically follows
the Ishaq-Thabit version. The source manuscript for his
translation must, however, have included borrowings from
the Hajjaj version because, inserted in the Ishaq-Thabit
text translated by Gerard, we find isolated words or formulas
or even passages of several words which are recognizably
in the Hajjaj wording. There are some alternative proofs
"from another book," Gerard says, which may have been taken
from Hajjaj.
Now to the text that since Clagett's important
article has been called "Adelard I." From this text we
have four manuscripts which contain books I - VIII, one
for books X.36 - XV.2, one for book X.36-49, but none at
all for books IX to X.35. It is quite possible that Adelard
itself was the author of Adelard I: Clagett's ascription
of the text to Adelard of Bath is based solely on the attribution
in ms Oxford, Trinity College 47, which appears to be the
oldest. But there are other witnesses for Adelard's authorship:
the spelling el for the article (normally,
Gerard and others used al); and the
inflecting of an Arabic word according to Latin inflexion
-- both being known features of Adelard's work. [Note 36: See
Kunitzsch (note 21), p.124.] Adelard I is clearly a translation,
not a commentary. It must have been translated from the
Arabic, since many of its technical terms are transcriptions
from Arabic, some of which are not found in other versions.
Curiously, there are yet more Arabic terms in the margins
of one -- just one -- manuscript, Bruges 529. Technical
terms are not always consistently used. Thus we sometimes
find inconiunctivus for equidistans,
quadratura for ductu,
differens for residuum.
Busard supposes that the translator was constantly seeking
the right Latin expressions for the various Arabic terms
and that Adelard I is more primitive than Adelard II. [Note 37: Busard
(note 19), p.17.] Certainly there is no Greek influence
in version I -- so hypotenusa, gnomo,
parallelogrammum, ysosceles,
ortogonaliter, and so on do not appear,
though they are frequent in Adelard II.
Which Arabic translation
or mixed version was used in the translation of Adelard
I can only be decided after a detailed comparison of Latin
against Arabic. Busard has shown that there is an almost
literal agreement between Adelard I and the fragment of
a Syriac redaction of book I which has come down to us. [Note 38: Busard
(note 19), pp.18-19.] Two theorems compared in detail
by R. Lorch [Note 39: See note 22.] reveal, with one
uncomfortable exception, affinities with the Hajjaj phraseology.
But further investigations by Kunitzsch [Note 40: Kunitzsch
(note 21), p.119.] give quite contrary indications, i.e.,
that Adelard I depends on Ishaq-Thabit rather than Hajjaj.
Kunitzsch has noted that Adelard I has some sections of
a complete literalness against the Arabic, while other sections
show a high degree of "literary Latin" transformation.
Clearly, the text has been reworked.
Of the three "Adelard"
versions (I, II, and III) Adelard II was easily the best
known, as the more than 50 extant manuscripts show. Further,
Campanus used it as his base text to make what became the
standard medieval "Euclid." He and other compilers in the
thirteenth and fourteenth centuries took over the enunciations
and provided new proofs.
The most striking feature of Adelard
II is the form of the proofs: Though some proofs are given
in great detail in Adelard II, most are abbreviated. Sometimes
only an indication is given of the theorems on which the
proof depends. A remarkable characteristic of this text
is that the enunciations in the better manuscripts come
after the proofs.
The principal characteristics
of Adelard II and its relationship with the other two "Adelard"
versions and with the Campanus text have already been sufficiently
treated by Murdoch. [Note 41: See note 16.] I should
like to add here some facts which were not known to Murdoch: [Note 42: See
Folkerts (note 22).]
1) Some enunciations in Adelard II
agree literally with the Boethius excerpts that I have mentioned,
but other related enunciations have an "Arabic" formulation.
2)
There are some manuscripts of Adelard II which carry an
essentially different text. Significant variants of this
type occur especially in books VII to IX and XI to XIII
where different sets of proofs are given. It must be stated
that there was no canonical text for Adelard II in the
Middle Ages but that it was at an early stage reworked and
commented upon.
Since 1953 when Clagett's article appeared,
it was the common opinion that Adelard translated or reworked
the Elements at least twice and that
Adelard I as well as Adelard II were both from Adelard.
The main argument was that Adelard is named as author in
the Trinity College manuscript of version I and in many
manuscripts of version II. Adelard himself said in his
treatise on the astrolabe that he has translated the Elements,
and it seems to be clear that there was a translation by
him. But there are doubts that versions Adelard I and Adelard
II are both from Adelard: Adelard I and Adelard II have
little in common beyond some shared definitions and enunciations,
and we should remember here that Hermann and Adelard II
also share some definitions and enunciations. Furthermore,
the differences are great: different enunciations, different
proofs, different Arabisms. There is little ground for
believing that they are the work of the same translator
or redactor.
My work on Adelard II in collaboration with
Dr. H. L. L. Busard has brought new factors to light, and
I think we are able to give a better hypothesis about the
origin of the Adelard II text. Contrary to what has been
assumed, Adelard II appears to have originated as a collection
of Euclidean definitions, axioms, postulates, and enunciations;
but the proofs came later and piecemeal, influenced by the
translations from the Arabic that I have mentioned above.
The collection of enunciations appears on so on and not
to have been translated from Arabic but to have come from
Latin sources. We note in this context that there are only
four Arabic expressions in transliteration in this part
of the text, [Note 43:
elmuhain,
elmunharifa, alkaide
and mutekefia.] and all of them can
be found in the Hermann or the Adelard I text. Therefore
it is not necessary to assume that they were translated
from the Arabic for Adelard II.
There are some positive
reasons to show that the proofs in Adelard II were probably
not original to Euclid. When we look at the earliest manuscripts
of the text, we find that several of them contain no proofs
at all, or at most very general, short indications how one
might prove the proposition. In fact, there are about ten
manuscripts written before the end of the twelfth century,
but only two of them contain proofs beyond book VI. When
proofs are given they are of various lenghths and natures.
Further, in some manuscripts one finds the proofs before
the enunciations, in others they are beside the enunciation;
in yet others there is a proof in the regular position but
also an indication in the manner just spoken of in the
margin.
At this point it is perhaps sensible to mention
two particular manuscripts: München CLM 13021, and Paris
BN Lat. 10257. CLM 13021 was written in Prüfening near
Regensburg, probably in the sixties of the twelfth century,
and contains one of the earliest witnesses to the Adelard
II text. The first part of the text in this manuscript
is similar to the Greek-Latin tradition associated with
Boethius; but from book IV on it is Adelard II. There are
no proofs. There is another copy of the same text in ms
München CLM 23511, from the end of the twelfth century.
The next manuscript mentioned above (Paris BN Lat. 10257)
was originally in Chartres and also comes from the twelfth
century; it is especially important in our story. [Note 44: See
G. D. Goldat, The Early Medieval Traditions of
Euclid's Elements (unpublished dissertation: University
of Wisconsin, 1957).] This manuscript has not only the
Boethian excerpts in a contaminated form but also enunciations
for all fifteen books. A notable feature of this work
is the presence of what appears to be transcriptions of
Greek terms for the various irrational quantities
in book X. In book I-III the Paris manuscript shows some
resemblance to the Munich manuscript and to an Oxford manuscript
(Digby 98) which has an incomplete text. In the margin
of some manuscripts containing Adelard II there are enunciations
of ms Paris BN Lat. 10257 which are refered to by the words
alia translatio.
In the Chartres manuscript now in Paris there are very
short indications of proofs in the margin near the figures,
and these are only for parts of book I. These excepted,
there are eleven other proofs in book I which are somewhat
longer. They differ from the corresponding passages in
Adelard II, but they are arranged in the same way.
Another
manuscript from Chartres is ms 497/498, the Eptateuchon
written by Thierry of Chartres in about A.D.1140, now unfortunately
destroyed. It seems to have contained the oldest witness
to Adelard II. It comprised the most modern texts of its
time for the seven liberal arts,
including both a text in the Greek-Latin Boethius tradition,
the so-called Geometry II and also originally at least
a part of the text of Adelard II. Some folios had been
lost, but book VII-IX and the fragment of book XIV-XV contain
no proofs, and the text fully confirms what we have deduced
above from other manuscripts. The contents of this codex
leads us to suppose that perhaps someone or some group
from this part of France, maybe even Chartres itself, could
have been responsible for making the original collection
that later formed what we know as "Adelard II." As we shall
see later, it is possible to give some more evidence for
this proposal.
Let us make a first conclusion: It seems
very likely that the Adelard II text circulated in the beginning
without proofs. This is not only confirmed by the early
manuscripts Chartres 498 and München CLM 13021, but also
by the fact that -- unlike all the other Greek-Latin and
Arabic-Latin versions and the Greek and Arabic tradition
-- the porisms have been appended to the enunciations.
If there had originally been proofs, then we must assume
that the porisms would have been at the end of the proofs
and not at the end of the enunciations. The enunciations
in Adelard II as we have them are a compilation of several
texts: of the Boethius tradition - there are some striking
identities between the Adelard II and the Boethius text
-, of Adelard I which in my opinion was translated by Adelard
himself, and of Hermann of Carinthia who seems to have finished
his Euclid translation not later than about A.D.1140.
As
I said, I think it is possible to give some more details
about the compiler of the enunciations and definitions of
the text now called Adelard II. There are good reasons
to assume that this was Robert of Chester (Ketton). This
Robert lived in Spain about A.D.1141-1147, was archdeacon
of Pamplona in 1143, and lived in London about 1147-1150. [Note 45: See
C. H. Haskins, Studies in the History of Mediaeval
Science (Cambridge/Mass., 1924), esp. chapter
III and pp.120-123 in chapter VI.] Robert was a friend
of Hermann of Carinthia, and we know that Peter the Venerable,
abbot of Cluny, found Robert and Hermann in 1141 in the
region of the Ebro where they were engaged in astrological
studies. In this time Hermann translated the astronomical
tables of al-Khwarizmi and the astrological work of Albumasar,
and Robert worked with the assistance of Hermann on the
first Latin translation of the Qur'an.
Robert's translation of the algebra of al-Khwarizmi, dated
Segovia 1145, may be said to mark the beginning of European
algebra. In 1143 Hermann completed the De
essentiis and the Planisphere
which he dedicated to his teacher Thierry of Chartres.
Besides the striking coincidence of date, there are other
indications which lead us to the conclusion that the so-called
Adelard II text of the enunciations of Euclid was written
by Robert of Chester, Hermann's associate in 1141:
-
In the preface to his translation of the Iudicia
of al-Kindi Robert states that he occupied himself with
Euclid's Elements. [Note 46: Quamquam
post Euclidem, Theodosii cosmometrie libroque proportionum
libencius insudarem ...: Haskins (note 45), p.121.]
-
As I said, Robert wrote a revision of Adelard's version
of the tables of al-Khwarizmi. Its incipit
is very similar to that of the Adelard II manuscripts [Note 47: Tables
of al-Khwarizmi (Madrid BN 10016): Incipit liber
Ezeig id est chanonum Alghoarizmi per Adelardum Bathoniensem
ex arabico sumptus et per Rodbertum Cestrensem ordine digestus;
Adelard II: Incipit liber geometrie Euclidis translatus
ab Adelardo Bathoniensi de arabico in latinum.]:
only the additional phrase is missing in the Adelard II
incipit, and we can assume
that probably the meaning of the title is that the text
is based upon the translation of Adelard, i.e. Adelard I.
-
In Robert's translation of the Algebra
of al-Khwarizmi there is an appendix which summarizes the
rules for solving the six types of equations. [Note 48: It
should be noted that all known manuscripts of this text
are from the fifteenth century and that in the appendix
these manuscripts have symbols for the powers of the unknown
which were used not before the second half of the fifteenth
century.] Similarly, we find in book X of Adelard II
an introduction where the author gives the definitions of
the various irrational quantities and later summarizes
the six binomial lines and the six apotomes.
-
Since Hermann was a pupil of Thierry of Chartres and he
dedicated his Planisphere
to him, it is very plausible that Robert, Hermann's friend,
sent his work to Thierry, who inserted it in his Eptateuchon.
There
remains the problem of the proofs in Adelard II. The proofs
in the various manuscripts of Adelard II are sometimes very
different. Usually the content is the same, but the formulation
is sometimes radically different. The proofs often contain
interesting Arabic transcriptions and other traces of
Arabic origin. There are more Arabic words in the proofs
than in the text of the enunciations, but I think these
words too can be explained without the assumption that
there was a translation from the Arabic other than that
by Hermann or Adelard I. The fact that in many cases the
proofs are not in agreement with each other and that there
are abridged or additional proofs in some manuscripts seems
to indicate that the proofs were not written by only one
person. I think that originally indications of proofs
were given in the margin similar to those in ms Paris BN
Lat. 10257 and in the Adelard I manuscripts Bruges 529 and
Oxford Trinity College 47. [Note 49: As to the last two manuscripts,
see Busard (note 19), p.22, note 10.] In some of
the oldest Adelard II manuscripts [Note 50: Oxford, Trinity
College 47, and London, British Library Royal 15 A 27.]
the proofs are in the margins next to the propositions.
Later they were written mostly before the enunciations,
but there are differences between the manuscripts, too.
In a later state of development the proofs seem to be enlarged
by more than one person, and this can explain the differences
in the proofs within the manuscripts. Some obscure names,
such as Ocreatus, Eggebericus,
Lincol' Zeob' Rog' Hel'and
Reginerus, which are given in some proofs,
may indicate that these persons contributed to the text
of the proofs or have given additional proofs. But all
this must be more rigorously investigated. We may therefore
state the following:
- No Arabic manuscript has
been found from which any of the three Latin translations
attributed to Hermann, Gerard, or Adelard I could have been
made.
- None of these translations, as we have them, presents
a pure Euclid text: either the originals were themselves
reworked, or the translator took his material from several
sources. The transmission of the Elements
is thus much more complex than that of Archimedes and the
Almagest.
- It appears correct to assume
that the Adelard II text, as we have it, is not a translation
from the Arabic but a compilation of different texts, among
them the Arabic-Latin translations of Hermann and Adelard
I, and that the original collection of Adelard II was made
from them by Robert of Chester. Therefore there were not
four but only three, translations of Euclid's Elements
in the twelfth century.
The three known translations
and the Adelard II compilation were only the beginning of
an intensive activity after the Arabic texts had become
known. Between the twelfth and fifteenth centuries the
text was reworked many times. Of these the most important
are:
1) The version called Adelard III by Clagett has the
same enunciations as Adelard II, which it cites but different
and fuller proofs. Although it contains Arabic terms not
in Adelard II, it is in all probability a commentary rather
than an independent translation. It was written or compiled
probably at the end of the twelfth century; the only manuscript
which transmits the complete text, Oxford, Balliol College
257, seems to have been written in the first half of the
thirteenth century. One indication of the authorship of
Adelard III is the ambiguous ascription by Roger Bacon,
editio specialis Alardi Bathoniensis.
2)
In the middle of the thirteenth century Campanus compiled
his well-known reworking of the Elements.
He took over the enunciations in Adelard II, but the proofs
do not correspond. Campanus' method of working has been
analyzed by Murdoch. [Note 51: See note 16.] Particularly
notable is Campanus' habit of including contemporary and
near contemporary material in his reworkings. In this material
is to be found Nayrizi's commentary of the Elements
and, above all, Jordanus' Arithmetica
-- the latter is particularly notable in the definitions
of books VII and VIII. [Note 52: Busard (see note 20),
p.134. 135.]
3) Not so well known is a commentary on
the Elements that exists in five manuscripts. [Note 53: Wien
5304; Paris BN Lat. 7292; Vat. Regin. lat. 1268, fol.1-69r;
Bonn, S 73; see Busard (note 20), pp.131-132.] This
commentary also depends on Adelard II. Busard maintains [Note 54: Busard
(note 20), pp.136-142.] that the author was acquainted
with more than one version II manuscript and perhaps with
a version III text. It is particularly interesting that
the commentary ascribed to Albertus Magnus is in turn based
on this anonymous commentary. [Note 55: See P. M. J. E. Tummers,
Albertus (Magnus)' commentaar op Euclides' Elementen
der geometrie, Nijmegen 1984. ] If Albert
is really the author, in this work at least he was not very
original.
Busard mentions a number of other reworkings
in which contemporary material is mixed with the text.
The transmission is made more complicated by the presence
in many manuscripts of various translations, commentaries
and compilations and of additional proofs.
There is very
little scholarly work about the Euclid text in the West during
the fifteenth century. It is well-known that in 1482 the
Campanus version was printed in Venice -- this was, except
of editions of fragments of the Boethian tradition, the
first printed Euclid in the West --, that in 1498 Giorgio
Valla published books XIV and XV together with commentaries
and took over Euclidean material into his encyclopedic De
expetendis et fugiendis rebus (Venice, 1501), [Note 56:
See Paul Lawrence Rose, The Italian Renaissance
of Mathematics. Studies on Humanists and Mathematicians
from Petrarch to Galileo, Genève, 1975, pp.46-50.]
and that Bartolomeo Zamberti published a new Latin translation
from the Greek in 1505. [Note 57: See Rose (note 56), p.51.]
But we know from two Campanus manuscripts [Note 58: Stuttgart
HB XI 24, and Vat. Palat. lat. 1352.] that already in
1450 Pope Nicholas V, who was also responsible for the new
Archimedes translation made by Jacobus Cremonensis, assembled
a Greek Euclid text. [Note 59: On Nicholas' activities in
mathematics see Rose (note 56), pp.36-38.] Some years
later Regiomontanus tried to reconstruct the original Euclid
text with the help of Bessarion's Greek manuscripts. [Note 60:
See Rose (note 56), pp.44-46. For Euclid manuscripts in
the possession of Regiomontanus see M. Folkerts, Regiomontans
Euklidhandschriften, in: Sudhoffs Archiv
LVIII (1974) 149-164. There may be added ms Boston, Medical
Library 24, which has the same introduction as Regiomontanus'
manuscript Nürnberg Cent. VI.13.] At least the first
books of Euclid's Elements were taught
within the university curriculum of the artes
liberales. There are some manuscripts of the
fifteenth century which seem to be notes or copies of students,
but up to now there is no systematical research on them. [Note 61:
I have listed some of them in chapter 20 of the enclosed
list of manuscripts, but I assume that there are many more
texts.]
Therefore it is clear that Clagett's 1953 model
is only a simple guide to the reality. It is no surprise
that the transmission of such an important and oft-used
work as the Elements should be so very
complicated. Although in the last years many manuscripts
have been analyzed and the most important texts have been
edited or are to be edited, much more work must be done
before we shall have a Euclid in the Middle Ages.
|
.W.8.12), f.1-179v, A.D.1452 (I-XV)