Latin-into-Hebrew:
Texts and Studies
Volume Two: Texts in Contexts
Edited by
Alexander Fidora
Harvey J. Hames
Yossef Schwartz
LEIDEN • BOSTON
2013
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
CONTENTS
LATIN-INTO-HEBREW: VOLUMES ONE AND TWO
CONTENTS OF PRESENT VOLUME
Latin-into-Hebrew: Introducing a Neglected Chapter in European
Cultural History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alexander Fidora, Resianne Fontaine, Gad Freudenthal, Harvey
J. Hames, and Yossef Schwartz
1
Introduction to this Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Alexander Fidora, Harvey J. Hames, and Yossef Schwartz
part i
textual analyses
1. The Medieval Hebrew Translations of Dominicus Gundissalinus. . . . . 19
Yossef Schwartz
2. Le Livre des causes du latin à l’hébreu: textes, problèmes, réception
Jean-Pierre Rothschild
47
3. Abraham Shalom’s Hebrew Translation of a Latin Treatise on
Meteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Resianne Fontaine
4. The Quaestio de unitate universalis Translated into Hebrew:
Vincent Ferrer, Petrus Nigri and ʿEli Habillo—A Textual
Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Alexander Fidora and Mauro Zonta
5. Ramon Llull’s Ars brevis Translated into Hebrew: Problems of
Terminology and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Harvey J. Hames
6. Latin into Hebrew (and Back): Flavius Mithridates and his Latin
Translations from Judah Romano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Saverio Campanini
7. Mordekhai Finzi’s Translation of Maestro Dardi’s Italian Algebra . . . . 195
Roy Wagner
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contents
part ii
hebrew text editions
8. Dominicus Gundissalinus: Sefer ha-nefeš (Tractatus de anima) . . . . . . 225
Yossef Schwartz (ed.)
9. Dominicus Gundissalinus (Wrongly Attributed to Boethius):
Maamar ha-eḥad ve-ha-aḥdut (De unitate et uno) . . . . . . . . . . . . . . . . . . 281
Yossef Schwartz (ed.)
10. Les traductions hébraïques du Livre des causes latin, édition
synoptique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Jean-Pierre Rothschild (ed.)
11. Judah Romano’s Hebrew Translation from Albert, De anima III . . . . . 369
Carsten L. Wilke (ed.)
12. Mordekhai Finzi’s Translation of Maestro Dardi’s Italian Algebra, a
Partial Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Roy Wagner (ed.)
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Index of Modern Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Index of Ancient and Medieval Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Index of Ancient and Medieval Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
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vii
CONTENTS OF VOLUME ONE: STUDIES
ED. BY
RESIANNE FONTAINE AND GAD FREUDENTHAL
In Memoriam: Francesca Yardenit Albertini (1974–2011) . . . . . . . . . . . . . . . .
Latin-into-Hebrew: Introducing a Neglected Chapter in European
Cultural History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alexander Fidora, Resianne Fontaine, Gad Freudenthal,
Harvey J. Hames, and Yossef Schwartz
1
9
Introduction to this Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Resianne Fontaine and Gad Freudenthal
part i
latin-into-hebrew:
the linguistic conditions of its possibility
1. Latin into Hebrew—Twice Over! Presenting Latin Scholastic
Medicine to a Jewish Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Susan Einbinder and Michael McVaugh
2. Latin in Hebrew Letters: The Transliteration/Transcription/
Translation of a Compendium of Arnaldus de Villa Nova’s
Speculum medicinae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Cyril Aslanov
3. Latin-into-Hebrew in the Making: Bilingual Documents in Facing
Columns and Their Possible Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Gad Freudenthal
4. From Latin into Hebrew through the Romance Vernaculars: The
Creation of an Interlanguage Written in Hebrew Characters . . . . . . . 69
Cyril Aslanov
5. La pratique du latin chez les médecins juifs et néophytes de
Provence médiévale (XIVe–XVIe siècles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Danièle Iancu-Agou
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part ii
latin-into-hebrew: the medical connection
6. The Father of the Latin-into-Hebrew Translations: “Doeg the
Edomite,” the Twelfth-Century Repentant Convert. . . . . . . . . . . . . . . . . 105
Gad Freudenthal
7. Transmitting Medicine across Religions: Jean of Avignon’s Hebrew
Translation of the Lilium medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Naama Cohen-Hanegbi
Appendix: Jean of Avignon’s Introduction to his Translation of
Lilium medicine, an Annotated Critical Edition and
Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Naama Cohen-Hanegbi and Uri Melammed
8. The Three Magi and Other Christian Motifs in Medieval Hebrew
Medical Incantations: A Study in the Limits of Faithful
Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Katelyn Mesler
part iii
latin-into-hebrew:
the philosophical-scientific
and literary-moral contexts
9. An Anonymous Hebrew Translation of a Latin Treatise on
Meteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Resianne Fontaine
10. Albert the Naturalist in Judah Romano’s Hebrew Translations . . . . . . 245
Carsten L. Wilke
11. Thomas Aquinas’s Summa theologiae in Hebrew: A New Finding . . . . 275
Tamás Visi
12. The Aragonese Circle of “Jewish Scholastics” and Its Possible
Relationship to Local Christian Scholarship: An Overview of
Historical Data and Some General Questions . . . . . . . . . . . . . . . . . . . . . . 295
Mauro Zonta
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13. “Would that My Words Were Inscribed”: Berechiah ha-Naqdan’s
Mišlei šuʿalim and European Fable Traditions . . . . . . . . . . . . . . . . . . . . . . 309
Tovi Bibring
part iv
latin-into-hebrew: the religious context
14. Latin into Hebrew and the Medieval Jewish-Christian Debate . . . . . . 333
Daniel J. Lasker
15. Citations latines de la tradition chrétienne dans la littérature
hébraïque de controverse avec le christianisme (XIIe–XVe s.) . . . . . . . 349
Philippe Bobichon
part v
latin-into-hebrew: final reflections
16. Traductions refaites et traductions révisées . . . . . . . . . . . . . . . . . . . . . . . . . 391
Jean-Pierre Rothschild
17. Nation and Translation: Steinschneider’s Hebräische
Übersetzungen and the End of Jewish Cultural Nationalism . . . . . . . . 421
Irene E. Zwiep
18. Postface: Cultural Transfer between Latin and Hebrew in the
Middle Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Charles Burnett
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Index of Modern Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Index of Ancient and Medieval Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Index of Ancient and Medieval Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Index of Subjects and Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
chapter seven
MORDEKHAI FINZI’S TRANSLATION
OF MAESTRO DARDI’S ITALIAN ALGEBRA
Roy Wagner
1. The Manuscript: Finzi’s Autograph Translation
The edition presented here includes the bulk of Mordekhai Finzi’s fifteenthcentury translation of Maestro Dardi’s fourteenth-century Italian algebra.1
The text does not explicitly state that it is an autograph, but a comparison
with the handwriting in Finzi’s other known autographs2 makes it evident
that it is indeed in Finzi’s own hand. According to the date given in the
manuscript, it was begun in Mantua in 1473, later than any other known
Finzi autograph (with one possible undated exception), and close to his
death in 1475.
Not much is known about Finzi’s biography.3 Documents show that he
was the owner of a 200 manuscript library (which was lost to creditors in the
1450s). A survey of his scientific work was made by Tzvi Langermann.4 Finzi
was a prolific copyist of astronomical and other scientific works, and left
behind some original contributions as well.
According to Tony Lévy,5 the known Hebrew algebraic corpus predating
the sixteenth century includes: two twelfth century works where geometric
1 The translation is analyzed in Tony Lévy, “L’ algèbre arabe dans les textes hébraïques (II).
Dans L’Italie des XVe et XVIe siècles, sources arabes et sources vernaculaires”, Arabic Sciences
and Philosophy 17 (2007): 81–107.
2 For a list of Mordekhai Finzi’s manuscripts see Giancarlo Lacerenza, “A Rediscovered
Autograph Manuscript by Mordekay Finzi”, Aleph: Historical Studies in Science and Judaism 3
(2003): 301–325.
3 Scattered information is available in Shlomo Simonsohn, History of the Jews in the Duchy
of Mantua (Hebrew) (Jerusalem: Kiryat Sefer, 1962) and in Vittore Colorni, “Genealogia della
famiglia Finzi. Le prime generazioni”, in: Vittore Colorni, Judaica minora. Saggi sulla storia
dell’ebraismo italiano dall’antichità all’età moderna (Milan: Giuffré, 1983), 329–341.
4 Y. Tzvi Langermann, “The Scientific Writings of Mordekhai Finzi”, Italia 7 (1988): 7–44.
5 Lévy, “L’algèbre arabe (II)”; Tony Lévy, “L’algèbre arabe dans les textes hébraïques (I). Un
ouvrage inédit d’ Isaac ben Salomon al-Aḥdab (XIVe siècle)”, Arabic Sciences and Philosophy
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problems are solved by what may be reconstructed as implicit algebra (Abraham bar Ḥiyya’s treatise, translated into Latin as Liber Embadorum, and
Sefer ha-Middot, attributed to Abraham ibn Ezra, which was also translated into Latin); an undated algebra in the tradition of al-Khwārizmī that
may have something to do with Abraham ibn Ezra; Moïse ibn Tibbon’s
thirteenth-century translation of al-Ḥaṣṣār’s arithmetic; Isaac ben Salomon
ben al-Aḥdab’s fourteenth-century translation of Ibn al-Banna’s arithmetic;6
Simon Moṭoṭ’s fifteenth-century algebra derived from an Italian mathematical culture; Mordekhai Finzi’s fifteenth-century translations of the algebras
of Abū Kāmil,7 Dardi and another Italian source which is yet to be analyzed
(Mantua, Biblioteca comunale, Ms. Ebr. 17, fols. 128v–130v); and one more
unanalyzed manuscript (Paris, BnF, Ms. Hébr. 1081, fols. 62v–67r).
Most of our evidence concerning the interest in Hebrew algebra is ex
silentio: there are hardly any references and few surviving copies. It is hard
to guess whether this reflects scholarly access to Arabic/Latin/vernacular
sources or more simply a lack of interest. I am also not aware of any evidence
showing Hebrew algebra reflecting back on vernacular algebra.
2. The Contents of Dardi’s Algebra
Dardi’s algebra opens with an elaborate treatise on the arithmetic of radicals. On top of the standard rules for multiplying, adding, subtracting and
dividing monomials and binomials involving numbers and square roots, it
also briefly deals with higher roots, and includes tour-de-force divisions of
numbers by three- and four-term sums of roots (see Appendix A for a list of
calculations).
Next Dardi introduces the six basic equations of algebra and the rules
for solving them: things8 equal numbers, squares equal numbers, squares
equal things, squares and things equal numbers, squares and numbers equal
things, and squares equal numbers and things. The fact that the fifth case
may involve multiple solutions is discussed, but not the possibility of its having no solutions. The presentation fits squarely within the Arabic tradition
13 (2003): 269–301; Tony Lévy, “A Newly-Discovered Partial Hebrew Version of al-Khwarizmi’s
Algebra”, Aleph 2 (2002): 225–234.
6 Ilana Wartenberg is preparing a critical edition of this text, based on her Ph.D. dissertation.
7 Martin Levey, The Algebra of Abū Kāmil: Kitāb fi al-Jābr waʾl Muqābala in a Commentary
by Mordecai Finzi (Madison, WI: University of Wisconsin Press, 1966).
8 The unknown “thing” is the predecessor of the modern x, and the “square” is the
predecessor of x2.
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as imported into Latin and vernacular Europe. Dardi then brings geometric
proofs of the rules for solving the last three (compound) cases derived from
the tradition of al-Khwārizmī and Abū Kāmil.
Only after the rules for solving the standard equations are introduced
does Dardi present the terms of algebra: numbers, the unknown thing (cosa/
cose, (‫)דבר)ים‬, the square of the unknown thing (çenso/i, ‫י‬/‫)צינסו‬, the cube of
the unknown thing (cubo/i, (‫ )מעוקב)ים‬and the fourth power of the unknown
(çenso/i di çenso/i, ‫י‬/‫י מצינסו‬/‫)צינסו‬. A brief discussion of adding, multiplying
and dividing these terms is included as well.
Then each of the six rules is demonstrated by a few examples (the Arizona
manuscript, which is the source of Finzi’s translation, is unique in providing
more than one example for rules 1–4 and 6; rule 5, which may involve
multiple solutions, is also accompanied by more examples in the Arizona
manuscript than in any of the other manuscripts). Most examples are purely
arithmetical (e.g., break 10 into two numbers that obey some arithmetical
condition), but there are three recreational/economic problems that are
more typical of vernacular Italian algebra (see Appendix B for a list of
problems and their solutions).
But Dardi’s most prominent contribution is a systematic list of 192 additional rules for solving equations that involve terms of higher power and
irrational coefficients. All of these examples, except four, are reducible to
equations of the anachronistic forms axn = b and ax2n+bxn = c. The four exceptional problems are cubic and quartic equations that cannot be reduced
to the above forms, but the solutions that Dardi brings are “special”, since
they hold only if specific restrictions apply to the coefficients. Dardi’s text
explicitly acknowledges the fact that the rules provided hold only for special cases, but does not make explicit the conditions under which they
apply.
Finzi’s translation ends with rule number 51 (the back of the folio on
which rule number 51 appears is empty, suggesting that the manuscript
was not continued elsewhere), and does not reach any of the four special
equations. Finzi either abandoned the project or died before completing it.
The partial edition brought here ends with the six standard rules.
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3. The Provenance and Special Features of Dardi’s Algebra
Finzi’s translation derives from a surviving early fifteenth-century manuscript copy of Dardi’s 1344 algebra held at Arizona State University in Tempe.9 The opening statement of this manuscript is our only source for identifying the author and date of composition. The two copies of Finzi’s autograph
are our only sources for identifying Dardi as a Pisan scholar.10 Dardi’s algebra has three other surviving copies,11 and two modern editions: Raffaella
Franci’s edition based on the Siena manuscript12 and Warren van Egmond’s
unpublished critical edition based on the Arizona manuscript,13 which will
hopefully be published soon. Portions of the text survive in other manuscripts as well.14
The received theory on the origin of Italian vernacular algebra used
to cite Fibonacci as its Italian harbinger. But Høyrup15 argued convincingly that fourteenth-century Italian writers of vernacular mathematical
treatises (“abbacists”) are not likely to have relied directly on Fibonacci.
Høyrup hypothesizes that algebra migrated into Italy from Arabic Spain
through a Catalan-Provençal mathematical culture, which may have also
been Fibonacci’s source. A comparative analysis of early Italian algebras led
Høyrup to the conclusion that Dardi’s treatise represents a branch of this
transfer of knowledge that is separate from the main branch entering Italy
through the algebra of Jacopo da Firenze.
Trying to assess the distance of Dardi’s algebra from Arabic sources is
tricky. The use of the term “drama” as synonymous with number is clearly
Arabic, but is scattered unevenly across the text. Occasionally there is talk
of the square and its root, rather than the thing and the square (e.g. on
fol. 15v of the autograph: “one square and ten things, or, say, ten of its roots”);
9 A description of the manuscript is available in Barnabas Hughes, “An Early 15th-Century
Algebra Codex: A Description”, Historia Mathematica 14 (1987): 167–172.
10 The Arizona manuscript leaves a blank space where the Hebrew copies write Pisa; the
corresponding opening folia were detached from Finzi’s autograph.
11 See Warren van Egmond, “The Algebra of Maestro Dardi of Pisa”, Historia Mathematica
10 (1983): 399–421.
12 Maestro Dardi, Aliabraa Argibra, edited with an introduction by Raffella Franci (Siena:
Murst Presso il Dipartimento di Matematica Roberto Magari dell’Università di Siena, 2001).
13 Warren van Egmond, Transcription and Edition of the Arizona Dardi Manuscript (unpublished, 2002).
14 van Egmond, “The Algebra of Maestro Dardi”, p. 419; van Egmond, The Arizona Dardi
Manuscript, I9–I10; Dardi, Aliabraa Argibra, 21–26.
15 Jens Høyrup, Jacopo da Firenze’s Tractatus Algorismi and Early Italian Abbacus Culture
(Basel: Birkhäuser, 2007), 169–176.
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such formulation is rare in Italian algebras and attests to closer contact
with an Arabic source, where the pair māl-jidhr (square-root) is sometimes
preferred over shayʾ-māl (thing-square).16 As Jens Høyrup notes, while all of
Dardi’s geometric proofs go back to Arabic sources, the first few diagrams
are lettered in a way that does not fit with this tradition. To that I can add
that the arguments, while parallel to those of the Arabs, omit all references
to Euclid, and depend on elementary cut-and-paste arguments instead.
Nevertheless, the last couple of diagrams are lettered in line with Arabic
practice. Moreover, in the last diagram, the letters describing the main
square are, instead of the standard ABCD, the odd looking ABGD. This
fits the order of the Arabic (and Greek) alphabet. This variety attests to a
combination of sources and inspirations, some closer and some farther away
from Arabic origins.
Dardi’s algebra is not only closer to scholarly Arabic algebras in terms of
the above textual residues, but also in terms of organization and reasoning.
It is by far the most elaborate and systematically organized Italian algebra; it
prefers abstract arithmetical problems over recreational-commercial ones;
it illustrates and sometimes even motivates rules with verifiable examples
(using roots of square numbers so that techniques for working with radicals can be verified with integers);17 it occasionally discusses and compares
different variations of rules for the same problem; it occasionally makes
explicit notes concerning additive commutativity and other forms of invariance under the order of arithmetic operations; it includes a systematic discussion of converting algebraic equations into other equations and occasionally discusses the impact of algebraic modeling (the choice of element
to be represented by the unknown thing) on the resulting equation (opening
the way for the study of transformations of equations, which is a prerequisite for solving higher equations); and it occasionally flirts with equations
where a sum of algebraic terms equals nothing.
All these factors made Dardi’s algebra, which was by no means well
known or widely distributed in Italy, appealing for a scholar immersed in
the Hebrew-Arabic tradition such as Mordekhai Finzi. Rather than one of
the popular commercial-recreational algebras of famous fifteenth-century
abbacists, Finzi chose to translate this obscure old algebra that retains much
more of the scent of higher Arabic learning.
16 Jeffrey A. Oaks and Haitham M. Alkhateeb, “Māl, Enunciations, and the Prehistory of
Arabic Algebra”, Historia Mathematica 32 (2005): 400–425.
17 Despite Høyrup’s claim to the contrary, some such verifications are explicit in Dardi’s
work. But these explicit verifications are indeed rare.
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4. The Arizona Manuscript is the Source of Dardi’s
Translation; The Hebrew Notes on the Arizona Manuscript
Might Not Belong to Mordekhai Finzi
I am quite certain that the Arizona manuscript is the very source of Finzi’s
translation. Finzi’s translation makes marginal notes of most of the Arizona
manuscript’s folio transitions with the correct numbering. Moreover, there
are errors that indicate that Finzi’s source is indeed the Arizona manuscript
(e.g. reading a word wrongly in a way that fits the handwriting in the Arizona
manuscript; going from the end of a line in the Arizona manuscript back to
the beginning of the same line, erasing the repeated words, and continuing
the translation).18
Moreover, the translation is almost always literal. With the exception
of some titles and conclusion lines that summarize the calculations above
them, Finzi hardly omits anything from and hardly adds anything to the
Arizona manuscript. Sentences that Finzi abridged or expanded are rather
sparsely scattered across the text. In fact, Finzi copied quite a few obvious
errors and awkward formulations that appear in the Arizona manuscript—
errors and formulation that are unlikely to have survived two copyists. Even
when Finzi restructures the sentences of the Arizona manuscript, there are
traces of the Arizona text in the form of deletions and insertion.19
One of the alleged proofs that the Arizona manuscript was Finzi’s source
is more than a hundred Hebrew glosses scattered across it, mostly
translating adjacent problems and calculations. Barnabas Hughes asked
whether these notes had been written by Mordekhai Finzi himself.20 Tony
Lévy answered the question in the affirmative.21 I would like to re-open the
question.
The Hebrew marginal glosses on the Arizona manuscript22 differ from
Finzi’s autograph in vocabulary, spelling and handwriting. For example,
the marginal glosses consistently use the verb “lerabot” for multiplication,
whereas Finzi uses “likhpol” and “lehakot” in his autograph. In fact, the difference in terminology extends not only to mathematical terms: interest
compounded annually (far capo d’anno) is described as ‫לעשות ראש משנה‬
‫ לשנה‬in the autograph and as ‫ לשים הריוח לקרן בסוף שנה‬in the marginal
18
19
20
21
22
In this edition this is recorded in footnotes marked by: !
Many footnotes in the edition attest to this phenomenon. They are marked by: @
Hughes, “An Early 15th-Century Algebra”, 172.
Lévy, “L’ algèbre arabe (II)”.
These marginal glosses are documented in this edition in footnotes marked by: ×
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glosses. The marginal glosses most often spell root (‫ )שורש‬with the letter
‫ו‬, while Finzi’s autograph consistently spells the same word without it. The
letters ‫ ק‬and ‫ ת‬are both written as a single connected line in the marginal
glosses but are broken into two disconnected lines in the autograph (even
the very few connected instances of ‫ ק‬in the autograph do not extend their
tails as far as those in the marginal glosses). The letter ‫ מ‬has a completely
open base in the autograph, but only a partially open base in the marginal
notes.
None of this conclusively disproves Lévy’s claim that Finzi is the author
of the marginal glosses. Indeed, I do not contest the fact that Finzi translated directly from the Arizona manuscript, so Finzi definitely had access
to the manuscript. Moreover, a long time may have passed between the
writing of the glosses on the Arizona manuscript and the creation of the
autograph translation in 1473. In the meantime, Finzi copied from many different sources, and his vocabulary and spelling may have been influenced
accordingly (indeed, Finzi does make a few uses of ‫ לרבות‬in his translation of
Abū Kāmil’s algebra, but not with the preposition ‫כנגד‬, which is used consistently in the Arizona manuscript). One’s handwriting can change over time,
and one may write differently when one writes personal marginal notes as
opposed to a neat book. A conclusive response to Hughes’ question would
therefore require an overall vocabulary, spelling and handwriting analysis
across all of Finzi’s autographs, whereas the analysis that I conducted was
brief and cursory. However, given all of the above, I do not think we have
enough evidence to support the attribution of the Arizona manuscript notes
to Mordekhai Finzi.23
5. The Copies of Finzi’s Translation
We know of no references to Finzi’s translation in subsequent literature,
which means that it probably did not attract too much interest. There are,
however, two known copies of Finzi’s translation of Dardi’s algebra included
in Paris, BnF, Ms. Héb. 1033 and 1029. The manuscript number 1033 is in
23
Lévy (“L’ algèbre arabe [II]”, 102ff.) mentions the reference in fols. 93r and 195v of the two
copies of Finzi’s autograph (Paris, BnF, Ms. Héb. 1033 and 1029 respectively) to fol. 121 as the
place where the discussion of the four special equations begins, which indeed fits the Arizona
manuscript (the reference in Lévy’s paper was accidentally confused, but Lévy clarified his
intention in a private communication). This is indeed more evidence to the fact that Finzi
translated from the Arizona manuscript, but not to the origin of the Arizona manuscript
marginal glosses. Lévy also consulted Mr. Garel from the BnF concerning the handwriting,
but did not record Mr. Garel’s arguments supporting his thesis.
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all likelihood a first generation descendent of Finzi’s autograph. Indeed, it
faithfully copies not only Finzi’s text with hardly any omissions and interpolations, but even copies some of the autograph’s deleted words, end of line
marks and other oddities.24 I allowed myself to use this copy to reconstruct
some of Finzi’s marginal notes that were trimmed when Finzi’s autograph
was fitted to the size of the current codex.
As for the second copy (numbered 1029), it looks neater, but is in fact of
poorer quality in terms of omissions and errors. Moreover, it leaves blank
spaces for all but one of the numbers written in Arabic figures, and omits
all arithmetical and geometric diagrams. It probably does not descend from
the first copy (numbered 1033), because it does not reproduce some of its
errors and amendments (although in principle practically all the errors and
amendments of the first copy can be reconstructed back to the version of
the autograph). I could not find good evidence to support or disprove the
hypothesis that this second copy was a first generation copy.
6. The Vocabulary of Finzi’s Translation
Finzi’s translation is highly literal. Moreover, the manuscript contains several erased words that indicate a word-by-word translation, which had to be
corrected after the first few words had already been translated, as otherwise
it would make no sense in Hebrew. It seems that Finzi did not always even
read through the whole sentence before translating it.25
This does not mean, however, that the correspondence of Italian and
Hebrew terms is one-to-one. For example, the Italian verb produre (to take
the product of two numbers), which appears mostly in the first few folios of
the manuscript, is consistently translated by the verb ‫להכות‬. The verb moltiplichare, on the other hand, is translated inconsistently by both ‫ לכפול‬and
‫להכות‬.
I include here a thematized glossary of technical terms. It is by no means
exhaustive. It is also not sensitive to which forms (nominal, verbal, active/
passive, etc.) are preferred or excluded, and to the Italian spelling, which is
highly variable. Most of the comments below are based on Sarfatti’s documentation of Hebrew mathematical terminology26 and on advice from
24
In this edition this is attested in footnotes marked by: &
Footnotes in this edition that support this conjecture can be found among those
marked by: @
26 Gad B. Sarfatti, Mathematical Terminology in Hebrew Scientific Literature of the Middle
Ages (Hebrew) (Jerusalem: Magnes Press, 1968).
25
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Naomi Aradi from the Premodern Philosophic and Scientific Hebrew Terminology (Peshat) project.
6.1. Arithmetical Terms (Excluding Roots)
produre
moltiplichare
partire
azonzere
l’aditione
somma
trare, de(s)batere, abatere
san
men
piu
contrarie (in piu/men context)
(de)sbatere (cancel piu by men)
‫להכות‬
‫ להכות‬,‫לכפול‬
‫לחלק‬
‫ להוסיף‬,‫ לקבץ‬,‫לחבר‬
‫הדיצי״אוני‬
‫כלל‬
‫ להפיל‬,‫ לגרוע‬,‫להוציא‬
‫שלם‬
‫ מעט‬,‫ גורע‬,‫ מפחת‬,‫ נפחת‬,(!‫ פוחות‬:2‫ פוחת )ע‬,‫פחות‬
‫ רב‬,‫ מוסיף‬,‫יותר‬
‫מ)ת(נגד‬
‫ להכשיל‬,‫להמעיט‬
Finzi’s terms here are well precedented, but eclectic. ‫ לכפול‬in the sense of
“to multiply” (rather than to double), for example, was a standard term
for ibn Ezra, but later discarded in favor of ‫ להכות‬by most Arab-to-Hebrew
translators. The word aditione is used only once in the Italian text, and may
have not been understood by Finzi. An analysis of the translations for piu
and men will follow in the next section of this introduction. The plurality
of translations of men has to do with the tension created by the gradual
transition of men from the role of an arithmetic operation to that of (to put
it anachronistically) a minus sign.
6.2. Root Terms
radice quara
radice chuba
radice discreta
radice indiscreta, sorda, muta
continua
radice de zonto
cum gionto, conzunta
radice de trato
voce
son
clapi
‫שרש מרובע‬
‫שרש מעוקב‬
‫ מדובר‬,‫שרש גלוי‬
‫ לא יאוזן במספר גלוי‬,‫ לא ידובר בו‬,‫ אלם‬,‫ חרש‬,‫שרש נעלם‬
‫תמידי‬
‫ שרש חבור‬,‫ שרש מחבור‬,‫ שרש נוסף‬,‫שרש מנוסף‬
‫נוסף‬
‫ שרש ממוצא‬,‫שרש מהוצאת‬
‫קול‬
‫נגון‬
‫ קשרים‬,‫קלאפי‬
The terms here are again an eclectic mix going back to bar Ḥiyya, Ibn
Ezra, translators from the Arabic, and literal translation of Italian terms.
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Note that the term continua for an irrational number (opposite to the term
discreta), used only once in the Arizona manuscript, is translated literally as
a temporal term, rather than as a spatial term. Radice di zonto refers to a root
of the sum of a number and a root (or several roots). This is an idiosyncratic
term which Finzi translated on an ad-hoc basis. Cum gionto and conzunta,
when referring to terms under the sign of a radice di zonto, are the term to
which something is added and the added term respectively. The translation
misses this distinction. Radice de trato is a root of a difference between a
number and a root (the term ‫ שרש ממוצא‬appears only in the part of the
translation that is not covered in this edition). Voce and Son express a single
arithmetic term (say, the root of a number), and demonstrate the strong
oral aspect of abbacist mathematical practice. Finzi translates these terms
literally. Clapi are the different terms in a sum of numbers and roots.27 This
term is first transliterated and then translated as ‫( קשרים‬knots).
6.3. Algebraic Terms
cosa
çenso
cubo
çenso di çenso
drama
redure, produre
schixare
question
adequation
adequa
restaurare
desfare
parte
regola del 3
2 positioni
‫דבר‬
‫צינסו‬
‫מעוקב‬
(‫צינסו מצינסו )צינסו דצינסו‬
‫דראמא‬
‫ להביא‬,‫להשיב‬
‫ ביצוע‬,‫סקיזארי‬
‫שאלה‬
‫ השואה‬,‫ תקון‬,‫שאלה‬
‫תשוה‬
‫לשלם‬
‫להשחית‬
‫חלק‬
‫כלל הג׳‬
‫שתי ההנחות‬
Finzi did not have a substantial tradition to draw on for algebraic terms. The
two Hebrew algebraic texts closest to him (Moṭoṭ’s and Aḥdab’s, the latter
possibly unknown to Finzi) do not correlate well with Finzi’s translation of
Dardi. There is, indeed, a better correlation between terms in Finzi’s translations of Dardi’s and Abū Kāmil’s algebras (of which the latter may have
depended on a combination of an Arabic source, a Spanish version and an
27 Clap means speck or spot in Catalan, which makes sense if clapi is to mean a small distinct unit of mathematical text. I could not find clapi or similar words in Italian dictionaries.
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earlier Hebrew translation), but there is no attempt at full consolidation.
Most translations of algebraic terms are simply literal. Cosa and cubo, which
have an obvious literal meaning (thing and cube), were replaced by Hebrew
terms. Çenso (an Italian variant of the Latin census, which translates māl, the
Arabic term for possession—but whose etymology was unknown to most
Italian abbacists) was simply transliterated by Finzi in his Dardi translation.
In his Abū Kāmil translation, however, Finzi used a transliteration of the
Spanish algo along with the Hebrew ‫ריבוע‬. The monetary unit drama (an
Italian transliteration of the Arabic dirham), used as equivalent to number,
was again transliterated into Hebrew—but transliterated differently in the
translations of Dardi and of Abū Kāmil (an Arabicized ‫ דירהם‬vs. an Italianized ‫)דראמא‬. Redure was used for normalizing equations (rescaling to make
the leading coefficient equal one), but also for turning a number into a root
and for other conversions. Finzi gave this term a non technical translation
by common Hebrew verbs. Schixare, the equivalent of modern cancellation
of terms above and below a fraction line, was first transliterated and then
translated literally as a sort of cutting.
The term adequation (equation) posed a challenge to Finzi. He first transliterated it, then attempted to translate it as ‫תקון‬, and subsequently misunderstood it as synonymous with question (he even interpolated a statement
to that effect on fol. 15r). Eventually Finzi realized that the meaning corresponds to something we could anachronistically term “equation,” and came
up with the term ‫השואה‬, which is precedented in al-Aḥdab’s translation. The
operations of algebra (al-jabr and al-muqabala, respectively: adding to both
sides of an equation a term subtracted on one side, and subtracting from
both sides of an equation terms that are added on both sides), whose names
Finzi considered and reconsidered in his translation of Abū Kāmil,28 are not
distinctly thematized in Dardi’s algebra, and Finzi’s translation reflects this
attitude (see fol. 18v for the casual and brief introduction of these operations through an example). A single appearance of restaurare and desfare to
designate al-jabr are translated literally, and the more common imperative
adequa, which is used for both algebraic operations, is translated as ‫תשוה‬.
The translation of parte (side of an equation, but also part of a number)
retains its original ambiguity (at one point Finzi uses the less ambiguous
term ‫צד‬, literally side, but then erases it). Finally, the rule of three and a single
reference to the algebraic method of double positioning (extrapolating the
correct solution of linear equations in two unknowns by a weighted average
28
Lévy (“L’ algèbre arabe [II]”, 98). He eventually suggested ‫ כיוון‬and ‫הפקדה‬.
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of two wrong guesses) are also translated literally, rather than by any of the
forms in the literature (e.g. Mordekhi Komṭino’s ‫ שלושת הצורות‬or al-Aḥdab’s
‫)מאזנים‬. We see here Finzi’s strong inclination towards literal translations
over an attempt at constructing a unified professional lexicon.
6.4. Geometric Terms
figura
designado
ampleça
longeça
groseça
superficie
corporale
ladi
per equale distantia
de cantoni directi
‫צורה‬
‫מצויר‬
‫ מרחב‬,‫רוחב‬
‫ מאורך‬,‫אורך‬
‫עובי‬
‫שטח‬
‫גשמי‬
‫צלעות‬
‫נכחי‬
‫נצב הזויות‬
Finzi’s geometric terms are almost perfectly in accord with those standardized in Ibn Tibbon’s addendum to The Guide for the Perplexed. The use of
‫ עובי‬for the third dimension is somewhat deviant. It is notable that Finzi
consistently uses ‫ צורה‬in his Dardi translation as opposed to ‫ תמונה‬in his
translation from Abū Kāmil (‫ צורה‬is also al-Aḥdab’s preferred translation).
6.5. Economic Terms
soldo
lira
dinari
rasone
far capo d’anno
‫דינר‬
‫ליטרה‬
‫מעות‬
(‫חשבון )ריבית‬
‫לעשות ראש משנה לשנה‬
The translation of coin names is somewhat odd. Soldo is translated as ‫דינר‬,
and so dinari awkwardly turns into ‫מעות‬.
6.6. Logical and Metamathematical Terms
Finally, the following terms used by Finzi are common medieval scholarly
terms. I include them here for completeness.
regola
comun
praticha
prova
provare
‫ כלל‬,‫סדר‬
‫כללי‬
‫הרגל‬
‫ ראיה‬,‫מופת‬
‫ לנסות‬,‫להביא מופת‬
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demostrare
certamente
amaistramento
rasone
exemplo
proprietade
natura
eser
207
‫להראות‬
‫מבואר‬
‫התחכמות‬
?‫ אמות‬,‫חשבונות‬
‫ משל‬,‫דמיון‬
‫עצמות‬
‫טבע‬
‫מהות‬
7. The Role of the Translator
It is clear that Finzi adopts the attitude seeking to “translate word by word
without additions or omission”.29 But it is also undeniable that a translation
is always an interpretation. I will quickly review here some ways in which
Finzi’s translation did some interpretive and transformative work.
7.1. Formal Abbreviations
The first point is Finzi’s commitment to verbal forms, renouncing all formal
abbreviations. Dardi’s manuscript includes various interesting notations
that facilitate its processing. Roots are designated by a special contraction of
Rx, the algebraic terms are marked by C and Ç (Cosa and Çenso respectively),
and expressions such as 10 Cose are sometimes abbreviated as 10/C (the
fraction is displayed vertically in the manuscript, as in numerical fraction
notations). An expression such as “8 C ee 180 C uguale a 900/n et a 9/Ç” is
easier to parse and recognize as an operative unit than the equivalent “8
things and 180 things are equal to 900 numbers and 9 squares”.30 But Finzi
chooses to ignore all these notational inventions and to adhere to a classical
verbal presentation.
Indeed, Finzi and his two copyists make many abbreviations, mostly
truncating off suffixes of words, including sometimes words that designate
algebraic unknowns. But these random abbreviations have no systematic
29 From Judah ibn Tibbon’s introduction to his translation of Ḥovot Halevavot, quoted in
Sarfatti, Mathematical Terminology in Hebrew, 170.
30 Some contemporary research on the cognitive work exported onto mathematical signs
can clarify this point. See, for example, David Landy, Colin Allen and Carlos Zednik, “A
Perceptual Account of Symbolic Reasoning”, paper presented at The Hebrew University’s
Institute of Advanced Studies Workshop: Philosophy and the Brain: Computation, Realization,
Representation, May 17, 2011.
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aspect, and do not serve to help identify where an equation begins and
ends. This is, perhaps, one of the reasons why it was so difficult for Finzi
to understand Dardi’s concept of adequation.
It should be noted that even manuscripts with no formal abbreviations were sometimes accompanied by a practice based on abbreviations
(Høyrup brings a beautiful example of an Arabic folio, written in strict verbal style, but whose margins are densely filled with formal abbreviations of
the main text in another hand).31 Finzi’s decision not to include any formal
abbreviations reflects a commitment to strict separation of scholarly written presentations from practical teaching, where formal abbreviations are
bound to appear.
7.2. Decimal Numbers
A similar attitude applies to number terms. Here too Finzi avoided Arabic
figures, and stuck to Hebrew letters. Again, this makes it more difficult to
parse the text and tell where a mathematical expression begins or ends.
Finzi only used decimal numbers when he reached 6 digit numbers, indicating what he considered as the limit of viability of the Hebrew number
system. At one point (fol. 30v, outside the scope of this edition) Finzi uses
vertical fractions with Arabic figures as enumerator and denominator.
Again, Hebrew also had a system of abbreviations. One could indicate
numbers by their names as well as by a semi-decimal system based on
letters (‫ מאה עשרים ואחד‬vs. ‫)קכ״א‬. But a comparison of variations in Finzi’s
manuscript and the two copies highlights the fact that these notations are
arbitrary and export no cognitive work to the written signs.
7.3. The Transformation of the Men (Minus)
The next aspect where Finzi’s translation works as an interpretation is in the
context of the term men. Men is the term used to indicate the operation of
subtraction, the opposite of e (and). But when one discusses operations on
such binomials as “root of 5 men 2”, one has to refer to the 2, ending up with
such expressions as “the 2, which is men”. This becomes even more urgent
when one has to explain in such calculations as “root of 5 men 2 times root
of 5 men 2”, which of the four terms of the product should be added and
31 Jens Høyrup, “Hesitating Progress—The Slow Development Toward Algebraic Symbolization in Abbacus and Related Manuscripts, c. 1300 to c. 1550”, in: Albrecht Heeffer, Maarten
van Dyck (eds.), Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics
(London: College Publications, 2010), 3–56 (on p. 10).
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which subtracted. Slowly but surely, men is no longer just a conjunction or
binary operation; it gradually takes on the role of an adjective, modifying
the value of the number to which it is attached and qualifying its kind.32 It is
no surprise then, that the last equation considered in Dardi’s text before the
four special cases is the breakthrough “5 things and 15 equal nothing”, where
the thing is calculated as 15/5 = 3, but this 3 is to be “debited”.33
Finzi is aware of these nuances. When men starts taking the role of an
adjective, Finzi moves from the conjunction ‫ פחות‬to the adjective ‫פוחת‬. But
the transition is subtle, and Finzi’s translation is accordingly confused. One
can find either term used in the opposite role, and an expression such as “15
and root of 100 men root 91 and men root of 36” has the first men translated
as ‫ פחות‬and the second as ‫פוחת‬.34 As the text goes on, Finzi adds to the list of
translations of men the terms ‫ נפחת‬and ‫( מפחת‬other adjectival/passive forms
of the same root) as well as ‫ גורע‬and ‫מעט‬, based on an alternative translation
of subtraction and on a literal translation of men as “less”, respectively. The
different translations are sometimes used for different men terms in the
same sentence playing identical grammatical and logical roles.
This situation is so confusing for the second (and mathematically less
able) copyist, that on several occasions he writes ‫ פחות‬and then adds a mark
indicating that the ‫ ח‬and ‫ ו‬should be inverted, and on a couple of occasions
even ends up with the hybrid monster ‫פוחות‬.
7.4. Conceptions of Numbers
The last point where translation and copying made a difference concerns
the conceptualization of root and Cosa-Çenso signs (I will use Cosa to indicate both latter terms). Here the impact is by no means unique to Finzi’s
intervention, but extends to practically all the players in the abbacist manuscript scene.
As I discuss in my work on abbacist algebra, root and Cosa signs carry
several operational meanings in terms of mathematical practice.35
32 Some perspectives on this transition in the abbacus context can be found in Roy Wagner, “The Natures of Numbers in and around Bombelli’s ‘L’algebra’”, Archive for the History
of Exact Sciences 64/5: 485–523 and Albrech Heeffer, “On the Nature and Origin of Algebraic
Symbolism”, in: Bart van Kerkhove (ed.), New Perspectives on Mathematical Practices. Essays
in Philosophy and History of Mathematics (Singapore: World Scientific Publishing, 2009), 1–27.
33 Fol. 121r of the Arizona manuscript.
34 Fol. 4r of Finzi’s autograph.
35 Wagner, “The Natures of Numbers”.
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1. Root and Cosa signs are sometimes practiced as operators modifying
the numbers they apply to. For example, in “root of 4” the square root
sign turns the number 4 into 2, and in “5 Cose” the Cose sign turns the
“5” into “5 times the value of the unknown”.
2. The same signs can also be practiced as denominators qualifying the
species of the number involved. For example “root of 5” turns “5” into
a different kind of number, belonging to the species of roots (the
ambiguity that allows to see “root of 4” both as “2” and as belonging to
the species of roots is explicitly drawn upon by Dardi in examples that
explain how to deal with radicals). Similarly, “5 Cose” can qualify the
“5” as 5 of the species of the Cosa, in the sense that “2 pounds” qualifies
the “2” as 2 units of weight.
3. Finally, root and Cosa signs can be used as indexicals pointing to
the adjacent number. When an abbacist talks about “the root” and
points to the expression “root of 5”, he may sometimes be talking
about the number 5 itself, and the same goes for Cosa (carrying the
seemingly redundant sign along with the number can indicate where
the number comes from, and why it is used at that point). This can
work in the opposite direction as well. Just as one can say “2” referring
to “2 pounds”, so, given the right context, “5” can mean “root of 5” or
“5 Cose”. This indexical role is closely related to the denominative role.
It is because root and Cosa signs mark the species of a number, that
numbers can be identified by reference to their species and vice versa.
These distinctions should not be taken lightly. The practice of signs as operators (case 1 above) homogenizes different kinds of mathematical expressions in a uniform realm of numbers, because it allows transforming one
kind into another. On the other hand, the practice of signs as denominators (case 2 above) divides different mathematical expressions into distinct
species, and draws boundaries on permissible mathematical manipulations
(reminiscent of those set by the Greek classics against conflating arithmetical and geometric magnitudes, or geometric magnitudes of different dimensions). As I argue in my paper quoted above, both attitudes were productive
in the evolution of abbacist knowledge. The practice of signs as indexicals
(case 3 above), I believe, testifies to the strong oral aspect of abbacist practice, where the possible confusion would be prevented by pointing and by
the pragmatics of dialogue. As shown in my paper, indexicals also helped
blur the line between different uses of arithmetical terms, and thus led to a
more homogeneous conception of arithmetical terms.
Dardi is well aware of these nuances. The practice of the root sign as
turning one number into another number, as well as its practice as changing
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the species (or, in Dardi’s language, “nature”) of numbers, are both applied
explicitly. Indeed, the entire logic of Dardi’s treatise depends on dealing
systematically with different species of numbers. Different rules apply to
calculations with roots and with integers, and different solution recipes
apply to equations with root and with integer coefficients. This division
to different species is precisely how Dardi could turn 6 basic equations
into 194 different cases (not including the “special” equations). But, on the
other hand, when Dardi reduces complex equations into simpler ones (such
reductions conclude many of his discussions of equations that fall outside
this edition), he explicitly states that one could treat root coefficients simply
as numbers specifying the quantity of unknowns.36
As someone who masters both approaches, Dardi is exceptionally careful,
compared to other abbacists, to separate the different uses of root and Cosa
signs. When these signs are supposed to index the number adjacent to them,
Dardi’s text usually refers to the “number called root” and “quantity of the
Cose” to avoid any possible confusion. Moreover, he makes such statements
as: “the root of this 4/9 comes to be the quotient of dividing root of 4 by root
of 9; the essence of the root of the sound or voice [term] 4/9 is 2/3”.37 We
have here a nice distinction between the formal aspect of an arithmetic term
(voice) used to express different species of numbers, and its essence—the
number it represents.
When comparing the Arizona manuscript of Dardi’s algebra to the Siena
manuscript and to Finzi’s translation, we see that Finzi is usually more conservative and respectful of such distinctions.38 However, even the Arizona
manuscript makes regular use of the plural form Cose to refer to the coefficient of the Cosa term and the singular form Cosa to refer to its value—a
distinction that can be easily lost in translation and in copying. Indeed,
while Finzi rarely errs in this respect, his second copyist blunders quite
often. Compound this with the indexical aspect of root and Cosa signs (3
above), where “5” could mean “root of 5” and vice versa, and Dardi’s delicate
conceptual structure ends up very difficult to pass on to subsequent readers.
36
E.g. equations 20–22 in either the Arizona, Siena or Finzi manuscripts.
Fol. 10r in Finzi’s autograph, 19r in the Arizona manuscript. My translation simplifies the
original ambiguous syntax. Next to this statement the manuscripts add that the ratio of two
numbers is the same as that of “voices of their roots”, meaning that, formally, dividing roots is
like dividing their numbers, although the values of the numbers and the roots are obviously
not in the same proportion.
38 A careful reading of cases 19–22 in the Arizona and Siena manuscripts will provide many
examples of everything discussed in this section.
37
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My edition marks by # some of the footnotes that indicate confusions,
corrections and variations in Finzi’s treatment of root and Cosa terms with
respect to those of the Arizona manuscript. Building on these variations, the
copyists further erode Dardi’s subtle distinctions. This nuanced and seemingly insignificant process of minute variations, which takes place, so to
speak, at the “unconscious” level of mathematics, is, I think, rather fateful.
The fact that the textual subtlety required in order to hold on to Dardi’s
distinctions is not viable in a mathematical culture of amateurs and semiprofessionals and does not travel well through written texts, contributed
towards a more homogenized early modern conception of number, where
distinctions between species of numbers were gradually losing their foothold.
Acknowledgment
I would like to thank Warren van Egmond, Naomi Aradi and Tony Lévy for
valuable discussion. I would also like to thank the first two for access to
unpublished materials. This work is supported by Israel Science Foundation
(ISF) project: “Studies in the History of Medieval Mathematics in Hebrew
and Judeo-Arabic”. I would like to thank the head of the project, Ruth
Glasner, for her support and guidance.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
finzi’s translation of maestro dardi’s italian algebra
213
Appendix A
List of Examples with Roots
Raffaella Franci’s edition of Dardi’s algebra (based on the Siena manuscript)
brings a symbolic list of Dardi’s rules for arithmetic with radicals. I chose to
complement her list with the actual numerical examples. Indeed, a list of
numerical examples can help track down genealogies of sources; but there
are additional reasons behind my choice.
First, this record allows us to be a little more sensitive to the order of
operations, which symbolic formulas tend to suppress, but which is an
important issue in abbacist rules, which are presented as recipes. Note,
however, that products may be calculated in a different order than that in
which they are subsequently summed, and so the calculations below may
not reflect precisely the way Dardi ordered things.
Second, symbolic notation gives a false image of the nature of some products. When we see, for example,
(Ra−b)×(Ra−b) = a−R(4×a×b×b)+b×b,
we see three terms on the right hand side. An abbacist, however, will consider a+b×b as a single term (an integer), and so the product would be
conceived as a binomial. These distinctions matter, as they are constitutive
of the way mathematical expressions are classified and of the analogies that
can be drawn.
Third, writing:
(a−Rb)−(c−Rd) = (a−c)+(Rd−Rb)
is completely oblivious to the negativity or positivity of the two subtractions
on the right hand side, which may change the form of this difference into,
say
(Rd−Rb)−(c−a).
This makes little difference to us, but a significant difference to an abbacist
who does not acknowledge negative terms.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
214
roy wagner
Notation
×
R
R3
multiplication
square root
is cubic root
R4 × R9 = R63
(R6) × 3 = R(6 × 3 × 3) = R54
R5 × (4 + R7) = R5 × R(4 × 4) + R5 × R7 = R80 + R35
R3 × (6 − R8) = R3 × R(6 × 6) − R3 × R8 = R108 − R24
(3 + R5) × (3 + R5) = (3 × 3 + R5 × R5) + R(4 × 3 × 3 × 5) = 14+R180
= (3×3+R5×R5) + R[(3+3)×(3+3)]×R5
(3 + R4) × (5 + R9) = 3 × 5 + 3 × R9 + 5 × R4 + R4 × R9 = 15+R81+R100+R36 = 40
(3 − R5) × (4 − R7) = 3 × 4 + R5 × R7 − 3 × R7 − 4 × R5 = 12+R35−R63−R80
(3 + R5) × (4 + R7) = 3 × 4 + (3 × R7 + 4 × R5) + R5 × R7 = 12+R63+R80+R35
= (3×4+3×R7) + (4×R5+R5×R7)
8 × 8 = (10 − 2) × (10 − 2) = 10 × 10 + 2 × 2 − 2 × 10 × 2 = 64
(3 − R5) × (3 − R5) = (3 × 3 + R5 × R5) − R[(3 + 3) × (3 + 3)]×R5 = 14−R180
= (3×3+R5×R5) − 2×3×R5
(5 + R3) × (5 − R3) = 5 × 5 + 5 × R3 − 5 × R3 − R3 × R3 = 25−3 = 22
(3 + R4) × (5 − R9) = 3 × 5 + 5 × R4 − 3 × R9 − R4 × R9 = 15+R100−R81−R36
= 15+10−9−6 = 10
R8 × (R8 − 2) = R8 × R8 − 2 × R8 = 8 − R32
(R8 − 2) × (R10 − 3) = R8 × R10 + 2 × 3 − 3 × R8 − 2 × R10 = R80+6−R72−R40
= 6+R80−R72−R40
(R12 − 2) × (R12 − 2) = (R12 × R12 + 2 × 2) − (2 + 2) × R12 = 16−R192
= (R12×R12+2×2) − 2×2×R12
(R15 − 2) × (R12 + 2) = R15 × R12 + 2 × R15 − 2 × R12 − 2×2 = R180+R60−R48−4
(R8 + 2) × (R8 − 2) = R8 × R8 + 2 × R8 − 2 × R8 − 2 × 2 = 8−4 = 4
R5 × (R7 + R10) = R5 × R7 + R5 × R10 = R35 + R50
R5 × (R12 − R8) = R5 × R12 + R5 × R8 = R60 + R40
(R5 + R7) × (R10 + R15) = R5 × R10 + R5 × R15 + R7 × R10 + R7×R15
= R50 + R75 + R70 + R105
(R5 + R7) × (R5 + R7) = R5 × R5 + 2 × R5 × R7 + R7 × R7 = 12+R35
(R5 + R7) × (R10 − R6) = R5 × R10 + R7 × R10 − R5 × R6 − R7×R6
= R50 + R70 − R30 − R42
(R10 + R7) × (R10 − R7) = R10 × R10 − R7 × R7 = 10 − 7 = 3
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
finzi’s translation of maestro dardi’s italian algebra
215
(R12 − R7) × (R15 − R10) = R12 × R15 + R7 × R10 − R12 × R10 − R7×R15
= R180 + R70 − R120 − R105
(R12 − R7) × (R12 − R7) = R12 × R12 + R7 × R7 − 2 × R12 × R7 = 12 + 7 + 2×R84
= 19 +R336
3× R4 = R9 × R4 = R36 = 6
3× R38 = R327 × R38 = R3216 = 6
R4 × R38 = R3R64 × R3R64 = R3R4096
R38 × RR16 = R3RR4096 × R3RR4096 = R3RR16777216
[1/2 + R(1/4 + R12)] × [1/2 + R(1/4 + R12)] = 1/4 + 2R(1/16 + R3/4) + (1/4 + R12)
= 1/2 + R12 + R(1/4 + R12)
R3 + R12 = R[2 × (R3 × 12) + 3 + 12] = R(12 + 15) = R27 = R[R(4×3×12)+3+12]
R6 + R7 = R[R(4 × 6 × 7) + 6 + 7] = R(R168 + 13)
R12 − R3 = R[3 + 12 − 2 × (R3 × 12)] = R(15 − 12) = R3 = R[3+12−R(4×3×12)]
R7 − R6 = R[6 + 7 − 2 × (R6 × 7)] = R(13 − R168)
(4+ R12) + (5 + R3) = (4 + 5) + (R12 + R3) = 9 + R27
(4+ R3) + (R12 − 3) = (R12 + R3) + (4 − 3) = 1 + R27
(4− R3) + (R12 − 2) = (4 − 2) + (R12 − R3) = 2 + (R12 − R3) = 2+R3
19− (10 − R12) = R12 + 19 − 10 = 9 + R12
16− (8 + R50) = (16 − 8) − R50 = 8 − R50
10− (24 − R250) = R250 + 10 − 24 = R250 − (24 − 10) = R250−14
(13 − R20) − (6 − R5) = (13 − 6) − (R20 − R5) = 7 − R5
R3 + R6 + R12 + R24 = (R3 + R12) + (R6 + R24) = R27 + R54
R4
R9
= R(4/9) = 2/3
4
R9
= R16 = R(17/9) = 11/3
R9
8
(3+R4)
= 8×(3−R4) = (24−R256) = 44/5 − R256
(3×3−4)
5
R25
= 44/5 − R(106/25)
(5+ R16)
3
20
(4− R9)
= 5/3 + R16 = 5/3 + R(17/9) = 12/3 + 11/3 = 3
R9
=
19
(2+R16)
20×(4+R9)
(4−R9)×(4+R9)
=
19
(R16+2)
=
= (80+R3600) = 113/7 + R(7323/49)
7
19×(R16−2)
(R16+2)×(R16−2)
= (R5776−38)
12
= R(401/9) − 31/6
…
(2+R4)
= …
2×R4
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
216
(19+R25)
(5+R9)
roy wagner
= (19+R25)×(5−R9)
=
(5+R9)×(5−R9)
(95+R625−R3249−R225)
16
= 515/16 + R(2113/256) − R(12177/256) − R(225/256) = 3
36
(R4+R9 +R16)39
=
=
=
=
36×(R4+R9−R16)
(R4+R9+R16)×(R4+R9−R16)
36×(R4+R9−R16)
(R144−3)
36×(R4+R9−R16)×(R144+3)
(R144−3) ×(R144+3)
36×(R4+R9−R16)×(R144+3)
135
70
(R4+R9 +R16+R25)
=
=
=
=
=
70
(R4+R9 +R16+R25)
=
=
=
=
=
=
70×(R4+R9−R16−R25)
(R4+R9 +R16+R25)×(R4+R9−R16−R25)
70×(R4 +R9−R16−R25)
(28+R1600−R144)
70×(R4 +R9−R16−R25)×(28+R1600+R144)
(28+R1600−R144)×(28+R1600+R144)
70×(R4 +R9−R16−R25)×(28+R1600−R144)
(2240+R5017600)
70×(R4 +R9−R16−R25)×(28+R1600−R144)
4480
70
[R(R1600+41) +R(R144+13)]
70×[R(R1600+41)−R(R144 + 13)]
[R(R1600+41) +R(R144+13)]×[R(R1600+ 41) − R(R144 + 13)]
70×[R(R1600 +41)−R(R144+13)]
(28+R1600−R144)
70×[R(R1600 +41)+R(R144+13)]×(28+R1600 + R144)
(28+R1600−R144)×(28+R1600+R144)
70×[R(R1600 +41)+R(R144+13)]×(28+R1600 + R144)
(2240+R5017600)
70×[R(R1600 +41)+R(R144+13)]×(28+R1600 + R144)
4480
39 In this and the following examples the products expand into as many as 12-term sums.
I omitted these expansions, but the manuscript calculates them in full.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
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217
Appendix B
Arithmetic of Algebraic Terms and Problems Solved
Again, I preferred to give numerical examples rather than general rules. The
first six examples demonstrate the six standard rules. Next there is a list of
examples of products of algebraic terms. Finally, there is a list of word problems. I reduced the word problems into symbolic terms, noted the choice
of algebraic unknown (Cosa) for deriving the equation, and recorded the
equation, solution, and reconstruction of the original unknown terms. With
three exceptions (noted below), all question are posed in purely arithmetic
terms.
Notation:
C
Ç
Ch
ÇÇ
D
⇒
Cosa (thing)
Çenso (square)
Chubo (third power)
Çenso di Çenso (fourth power)
Drama (equivalent of pure number)
derivation of equation or solution from another equation
Examples for the Six Cases:
3C = 12 ⇒ C = 12/3 = 4
2Ç = 32 ⇒ Ç = 32/2 = 16 ⇒ C = 4
2Ç = 6C ⇒ C = 6/2 = 3
2Ç+20C = 78 ⇒ Ç+10C = 39 ⇒ C = R(10/2 × 10/2 +39)−(10/2) = 3
3Ç+63 = 30C ⇒ Ç+21 = 10C ⇒ C = 10/2 +R(10/2 × 10/2 +21) = 7 or
C = 10/2 −R(10/2 × 10/2 +21) = 3
3C+4 = Ç ⇒ 3C+4 = Ç40 ⇒ C = R(3/2 × 3/2 +4)+ 3/2 = 4
Examples of Multiplication of Algebraic Terms:
3C×4D = 12C
4Ç×3D = 12Ç
2Ch×5D = 10Ch
5ÇÇ×3D = 15ÇÇ
(2+3C)×(2+3C) = 2×2 + 2×3C + 2×3C + 3C×3C = 4+12C+9Ç
40
The division by the leading coefficient 1 is explicitly recorded in the manuscript.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
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roy wagner
Problems That Reduce to the First Case:
A+B = 10; A > B; A×A−B×B = 50
B = C; A = 10−C
(10−C)×(10− C)−C×C = 100−20C ⇒ 50 = 20C ⇒ C = 21/2
B = 21/2; A = 71/2
A+B = 6; 56A+67B = 37041
A = C; B = 6−C
56C+67(6−C) = 370 ⇒ 32 = 11C ⇒ C = 211/12
A = 211/12; B = 31/12
/3 A
1
/8
1
=5
A=C
1
/3 C = 5× 1/8 ⇒ C = 17/8
A = 17/8
3A+B = 32; 6A+3B = 8042
A = C; B = 32−3C
6C+96−9C = 80 ⇒ 16 = 3C ⇒ C = 51/3
A = 51/3; B = 16
A+B = 10; B = 5
A
A = C; B = 10−C
(10−C)
= 5 ⇒ 6C = 10 ⇒ C = 12/3
C
2
A = 1 /3; B = 81/3
Problems That Reduce to the Second Case:
A+B = 10; (A−B)×(A−B) = 201/4
A = C+5; B = 5−C
2C×2C = 201/4 ⇒ Ç = (20 / ) = 51/16 ⇒ C = R(51/16) = 21/4
4
A = 71/4; B = 2¾
1
4
A×A+ 1/2A× 1/2A = 10
A=C
C×C+ 1/2C× 1/2C = 10 ⇒ Ç = 8 ⇒ C = R8
A = R8
A×¾A = 40
A=C
41
42
This is a commercial problem about buying two kinds of fabric.
This is a commercial problem about buying two kinds of objects.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
finzi’s translation of maestro dardi’s italian algebra
C×¾C = 40 ⇒ Ç =
40
(3/4)
219
= 531/3 ⇒ C = R(531/3)
A = R(531/3)
(A− 1/3A− 1/4A)×(A− 1/3A− 1/4A) = 12
A=C
5
/12C× 5/12C = 40 ⇒ Ç = 40 = 693/25 ⇒ C = R(693/25)
(25/144)
A = R(693/25)
Problems That Reduce to the Third Case:
A× 2/3A = 3A
A=C
2
/3C×C = 3C ⇒ C =
3
/3
= 41/2
2
A = 41/2
A = 2/3B;
A = 2C;
5C = 6Ç
A = 12/3;
A×B = A+B
B = 3C
⇒ C = 5/6
B = 21/2
/2A× 1/2A = 20A
A=C
20C = 1/4Ç ⇒ C =
1
20
(1/4)
= 80
A = 80
A×A = A
100
A=C
1
/100 C = Ç ⇒ C = ( / ) = 1/100
1
A = 1/100
1
100
Problems That Reduce to the Fourth Case:
A+B = 10; A < B; A×A = 1/4B× 1/4B
A = C; B = 10−C
C×C = 1/4(10−C)× 1/4(10−C) ⇒ Ç +11/3C = 62/3
C = R[1/2(4/3)× 1/2(4/3)+62/3]− 1/2(4/3) = 2
A = 2; B = 8
A×A+8×A = 33
A=C
C×C+8×C = 33 ⇒ Ç+8×C = 33
C = R(1/28× 1/28+33)− 1/28 = 3
A=3
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roy wagner
/3A× 1/3A + A = 12
A=C
1
/3C× 1/3C + C = 12 ⇒ Ç +9C = 108
C = R(1/29× 1/29+108)− 1/29 = R(1281/4)−41/2
A = R(1281/4)−41/2
1
20 lire lent at A dinari per month compounded annually equals 30 lire after 2 years.
12 dinari = soldo, 20 soldi = lira; A dinari per month therefore equals 1/20A lire per
year.
A=C
20 + C + 1/20C×(20+C) = 30 ⇒ 40C+Ç = 200
C = R(1/240× 1/240+200)− 1/240 = R(600)−20
A = R600−20
Problems That Reduce to the Fifth Case:
A+B = 10; (A−B)×(A−B) = 201/4
A = C; B = 10−C
(2C−10)×(2C−10) = 201/4 ⇒ 1915/16 +Ç = 10C43
C = 1/210+R(1/210× 1/210−1915/16) = 71/4
The other solution is irrelevant because it would render 2C−10 meaningless.
A = 71/4; B = 21/4
A+B = 10; A < B; (B−A)×(B−A)−A×A = 32
A = C; B = 10−C
(2C−10)×(2C−10)−C×C = 32 ⇒ Ç+222/3 = (131/3)C
C = 1/213−R(1/213× 1/213−222/3) = 62/3 −R(217/9)
A = 62/3 −R(217/9); B = 31/3 +R(217/9)
The solution is independent of modelling the difference between A and B as 10−2C
or as 2C−10 (there is no note in the manuscript that the choice 2C−10 made above
produces a negative term). The other solution is irrelevant because A is posited to
be smaller than B.
A+B = 10; A×B = 21
A = C; B = 10−C
C×(10−C) = 21 ⇒ 10C = Ç+21
43 The reduction to a standard equation is done in two ways. In the first the equation is
normalized and than balanced, in the second the order is reversed (in accordance with the
standard rule). In the second case the manuscript goes through the equation: 79 ¾ + 4Ç 40C = nothing.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7
finzi’s translation of maestro dardi’s italian algebra
221
C = 1/210+R(1/210× 1/210−21) = 5+R4 or
C = 1/210−R(1/210× 1/210−21) = 5−R4
A+B = 10; 3A = R8×B
A = C; B = 10−C
R8×(10−C) = R(800−160C+8Ç)
800−160C+8Ç = 3C44 ⇒ 100+Ç = 203/8C
C = R[1/2(203/8)× 1/2(203/8)−100]+ 1/2(203/8) = R(3201/216)+103/16
A = R(3201/216)+103/16; B = 103/16 −R(3201/216)45
Correction of the above:
800−160C+8Ç = 9Ç ⇒ 800 = Ç+160C
C = R(1/2160× 1/2160+800)− 1/2160 = R7200−80
A = R7200−80; B = 90−R7200
Same problem, exchanging the algebraic modeling:
B = C; A = 10−C
900−180C+9Ç = 8Ç ⇒ 900+Ç = 180C
C = 1/2180−R(1/2180× 1/2180−900) = 90−R7200
A = 90−R7200; B = R7200−80
A+B = 16; A×B = 48
A = C; B = 16−C
C×(16−C) = 48 ⇒ 16C = 48+Ç
C = 1/216−R(1/216× 1/216−48) = 8−R16 or
C = 1/216+R(1/216× 1/216−48) = 8+R16
A = 4 or 12; B = 12 or 4
Problems That Reduce to the Sixth Case:
10×A+12 = A×A
A=C
10×C+12 = C×C ⇒ 10C+12 = Ç
C = R(1/210× 1/210+12)+ 1/210 = R37+5
A = R37+5
(A− 1/3A− 1/4A)×(A− 1/3A− 1/4A) = A+1
A=C
5
/12C× 5/12C = C+1 ⇒ Ç = (519/25)C+519/25
C = R[1/2(519/25)× 1/2 (519/25)+519/25]+ 1/2(519/25) = R(1434/625)+222/25
A = R(1434/625)+222/25
44
45
This is an error, and is corrected below.
The calculation of B is wrong with respect to the given value of A.
© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7