Mordekhai Finzi's Translation of Maestro Dardi's Italian Algebra

Latin-into-Hebrew: Texts and Studies

Volume Two: Texts in Contexts

Edited by Alexander Fidora Harvey J. Hames Yossef Schwartz

CONTENTS OF PRESENT VOLUME

Latin-into-Hebrew: Introducing a Neglected Chapter in European Cultural History ...... 1 Alexander Fidora, Resianne Fontaine, Gad Freudenthal, Harvey J. Hames, and Yossef Schwartz 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 47 Jean-Pierre Rothschild 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

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

CONTENTS OF VOLUME ONE: STUDIES ED. BY RESIANNE FONTAINE AND GAD FREUDENTHAL

In Memoriam: Francesca Yardenit Albertini (1974–2011) ...... 1

Latin-into-Hebrew: Introducing a Neglected Chapter in European Cultural History ...... 9 Alexander Fidora, Resianne Fontaine, Gad Freudenthal, Harvey J. Hames, and Yossef Schwartz 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

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

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

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 fifteenth- century 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, HistoryoftheJewsintheDuchy 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

© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7 196 roy wagner problems are solved by what may be reconstructed as implicit algebra (Abra- ham 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 radi- cals. 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 hav- ing 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. disser- tation. 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.

© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7 finzi’s translation of maestro dardi’s italian algebra 197 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/ the cube of ,( וסניצ /י ,the square of the unknown thing (çenso/i ,( רבד ( םי) ,cose and the fourth power of the unknown ( בקועמ ( םי) ,the unknown thing (cubo/i A brief discussion of adding, multiplying .( וסניצ / וסניצמי /י ,çenso/i di çenso/i) 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 addi- tional 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 excep- tional 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.

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 Tem- pe.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 convinc- ingly 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.

© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7 finzi’s translation of maestro dardi’s italian algebra 199 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 radi- cals 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 invari- ance 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 prerequi- site 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.

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 הנשמשארתושעל compounded annually (far capo d’anno) is described as in the marginal הנשףוסבןרקלחוירהםישל in the autograph and as הנשל

18 In this edition this is recorded in footnotes marked by: ! 19 Many footnotes in the edition attest to this phenomenon. They are marked by: @ 20 Hughes, “An Early 15th-Century Algebra”, 172. 21 Lévy, “L’algèbre arabe (II)”. 22 These marginal glosses are documented in this edition in footnotes marked by: ×

© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7 finzi’s translation of maestro dardi’s italian algebra 201 with the letter ( שרוש ) glosses. The marginal glosses most often spell root while Finzi’s autograph consistently spells the same word without it. The ,ו are both written as a single connected line in the marginal ת and ק letters glosses but are broken into two disconnected lines in the autograph (even in the autograph do not extend their ק the very few connected instances of has a completely מ tails as far as those in the marginal glosses). The letter 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 in his translation of תוברל accordingly (indeed, Finzi does make a few uses of -which is used consis , דגנכ Abū Kāmil’s algebra, but not with the preposition tently 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.

© 2013 Koninklijke Brill NV ISBN 978-90-04-22932-7 202 roy wagner all likelihood a first generation descendent of Finzi’s autograph. Indeed, it faithfully copies not only Finzi’s text with hardly any omissions and interpo- lations, 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 verb molti . תוכהל the manuscript, is consistently translated by the verb and לופכל plichare, on the other hand, is translated inconsistently by both . תוכהל 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 doc- umentation of Hebrew mathematical terminology26 and on advice from

24 In this edition this is attested in footnotes marked by: & 25 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).

Naomi Aradi from the Premodern Philosophic and Scientific Hebrew Ter- minology (Peshat) project.

6.1. Arithmetical Terms (Excluding Roots) להכות produre לכפול, להכות moltiplichare לחלק partire לחבר, לקבץ, להוסיף azonzere הדיצי״אוני l’aditione כלל somma להוציא, לגרוע, להפיל trare, de(s)batere, abatere שלם san פחות, פוחת (ע2: פוחות!), נפחת, מפחת, גורע, מעט men יותר, מוסיף, רב piu מ(ת)נגד (contrarie (in piu/men context להמעיט, להכשיל (de)sbatere (cancel piu by men) in the sense of לכפול .Finzi’s terms here are well precedented, but eclectic “to multiply” (rather than to double), for example, was a standard term by most Arab-to-Hebrew להכות for ibn Ezra, but later discarded in favor of 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 שרש מרובע radice quara שרש מעוקב radice chuba שרש גלוי, מדובר radice discreta שרש נעלם, חרש, אלם, לא ידובר בו, לא יאוזן במספר גלוי radice indiscreta, sorda, muta תמידי continua שרש מנוסף, שרש נוסף, שרש מחבור, שרש חבור radice de zonto נוסף cum gionto, conzunta שרש מהוצאת, שרש ממוצא radice de trato קול voce נגון son קלאפי, קשרים clapi

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.

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 appears only in the part of the אצוממשרש number and a root (the term 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 .(knots) םירשק term is first transliterated and then translated as

6.3. Algebraic Terms רבד cosa וסניצ çenso בקועמ cubo וסניצמוסניצ ( וסניצדוסניצ ) çenso di çenso אמארד drama בישהל , איבהל redure, produre יראזיקס , עוציב schixare הלאש question הלאש , ןוקת , האושה adequation הושת adequa םלשל restaurare תיחשהל desfare קלח parte ׳גהללכ regola del 3 תוחנההיתש positioni 2

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 dis- tinct unit of mathematical text. I could not find clapi or similar words in Italian dictionaries.