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Latin-into-Hebrew:Texts and Studies

Volume Two: Texts in Contexts

Edited by

Alexander FidoraHarvey J. HamesYossef Schwartz

LEIDEN • BOSTON2013


CONTENTSLATIN-INTO-HEBREW: VOLUMES ONE AND TWO

CONTENTS OF PRESENT VOLUME

Latin-into-Hebrew: Introducing a Neglected Chapter in EuropeanCultural History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Alexander Fidora, Resianne Fontaine, Gad Freudenthal, HarveyJ. Hames, and Yossef Schwartz

Introduction to this Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Alexander Fidora, Harvey J. Hames, and Yossef Schwartz

part i

textual analyses

1. The Medieval Hebrew Translations of Dominicus Gundissalinus. . . . . 19Yossef Schwartz

2. Le Livre des causes du latin à l’hébreu: textes, problèmes, réception 47Jean-Pierre Rothschild

3. Abraham Shalom’s Hebrew Translation of a Latin Treatise onMeteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Resianne Fontaine

4. The Quaestio de unitate universalis Translated into Hebrew:Vincent Ferrer, Petrus Nigri and ʿEli Habillo—A TextualComparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Alexander Fidora andMauro Zonta

5. Ramon Llull’s Ars brevis Translated into Hebrew: Problems ofTerminology and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Harvey J. Hames

6. Latin into Hebrew (and Back): Flavius Mithridates and his LatinTranslations from Judah Romano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Saverio Campanini

7. Mordekhai Finzi’s Translation of Maestro Dardi’s Italian Algebra . . . . 195RoyWagner


vi contents

part ii

hebrew text editions

8. Dominicus Gundissalinus: Sefer ha-nefeš (Tractatus de anima) . . . . . . 225Yossef Schwartz (ed.)

9. Dominicus Gundissalinus (Wrongly Attributed to Boethius):Maamar ha-eḥad ve-ha-aḥdut (De unitate et uno) . . . . . . . . . . . . . . . . . . 281Yossef Schwartz (ed.)

10. Les traductions hébraïques du Livre des causes latin, éditionsynoptique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Jean-Pierre Rothschild (ed.)

11. Judah Romano’s Hebrew Translation from Albert, De anima III . . . . . 369Carsten L. Wilke (ed.)

12. Mordekhai Finzi’s Translation of Maestro Dardi’s Italian Algebra, aPartial Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437RoyWagner (ed.)

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503Index of Modern Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507Index of Ancient and Medieval Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509Index of Ancient and Medieval Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513


contents vii

CONTENTS OF VOLUME ONE: STUDIESED. BY

RESIANNE FONTAINE AND GAD FREUDENTHAL

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

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

Introduction to this Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Resianne Fontaine and Gad Freudenthal

part i

latin-into-hebrew:the linguistic conditions of its possibility

1. Latin into Hebrew—Twice Over! Presenting Latin ScholasticMedicine to a Jewish Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Susan Einbinder andMichael McVaugh

2. Latin in Hebrew Letters: The Transliteration/Transcription/Translation of a Compendium of Arnaldus de Villa Nova’sSpeculummedicinae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Cyril Aslanov

3. Latin-into-Hebrew in the Making: Bilingual Documents in FacingColumns and Their Possible Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Gad Freudenthal

4. From Latin into Hebrew through the Romance Vernaculars: TheCreation of an Interlanguage Written in Hebrew Characters . . . . . . . 69Cyril Aslanov

5. La pratique du latin chez les médecins juifs et néophytes deProvence médiévale (XIVe–XVIe siècles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Danièle Iancu-Agou


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part ii

latin-into-hebrew: the medical connection

6. The Father of the Latin-into-Hebrew Translations: “Doeg theEdomite,” the Twelfth-Century Repentant Convert . . . . . . . . . . . . . . . . . 105Gad Freudenthal

7. Transmitting Medicine across Religions: Jean of Avignon’s HebrewTranslation of the Liliummedicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Naama Cohen-Hanegbi

Appendix: Jean of Avignon’s Introduction to his Translation ofLiliummedicine, an Annotated Critical Edition andTranslation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Naama Cohen-Hanegbi and Uri Melammed

8. The Three Magi and Other Christian Motifs in Medieval HebrewMedical Incantations: A Study in the Limits of FaithfulTranslation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Katelyn Mesler

part iii

latin-into-hebrew:the philosophical-scientificand literary-moral contexts

9. An Anonymous Hebrew Translation of a Latin Treatise onMeteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Resianne Fontaine

10. Albert the Naturalist in Judah Romano’s Hebrew Translations . . . . . . 245Carsten L. Wilke

11. Thomas Aquinas’s Summa theologiae in Hebrew: A New Finding . . . . 275Tamás Visi

12. The Aragonese Circle of “Jewish Scholastics” and Its PossibleRelationship to Local Christian Scholarship: An Overview ofHistorical Data and Some General Questions . . . . . . . . . . . . . . . . . . . . . . 295Mauro Zonta


contents ix

13. “Would that MyWords Were Inscribed”: Berechiah ha-Naqdan’sMišlei šuʿalim and European Fable Traditions . . . . . . . . . . . . . . . . . . . . . . 309Tovi Bibring

part iv

latin-into-hebrew: the religious context

14. Latin into Hebrew and the Medieval Jewish-Christian Debate . . . . . . 333Daniel J. Lasker

15. Citations latines de la tradition chrétienne dans la littératurehébraïque de controverse avec le christianisme (XIIe–XVe s.) . . . . . . . 349Philippe Bobichon

part v

latin-into-hebrew: final reflections

16. Traductions refaites et traductions révisées . . . . . . . . . . . . . . . . . . . . . . . . . 391Jean-Pierre Rothschild

17. Nation and Translation: Steinschneider’s HebräischeÜbersetzungen and the End of Jewish Cultural Nationalism . . . . . . . . 421Irene E. Zwiep

18. Postface: Cultural Transfer between Latin and Hebrew in theMiddle Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447Charles Burnett

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455Index of Modern Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461Index of Ancient and Medieval Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463Index of Ancient and Medieval Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469Index of Subjects and Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473


chapter seven

MORDEKHAI FINZI’S TRANSLATIONOFMAESTRO DARDI’S ITALIAN ALGEBRA

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1. The Manuscript: Finzi’s Autograph Translation

The edition presented here includes the bulk ofMordekhai Finzi’s fifteenth-century translation of Maestro Dardi’s fourteenth-century Italian algebra.1The text does not explicitly state that it is an autograph, but a comparisonwith the handwriting in Finzi’s other known autographs2 makes it evidentthat it is indeed in Finzi’s own hand. According to the date given in themanuscript, it was begun in Mantua in 1473, later than any other knownFinzi autograph (with one possible undated exception), and close to hisdeath in 1475.

Not much is known about Finzi’s biography.3 Documents show that hewas the owner of a 200manuscript library (whichwas lost to creditors in the1450s). A survey of his scientific work was made by Tzvi Langermann.4 Finziwas a prolific copyist of astronomical and other scientific works, and leftbehind some original contributions as well.

According to Tony Lévy,5 the known Hebrew algebraic corpus predatingthe sixteenth century includes: two twelfth century works where geometric

1 The translation is analyzed inTonyLévy, “L’algèbre arabedans les texteshébraïques (II).Dans L’ Italie des XVe et XVIe siècles, sources arabes et sources vernaculaires”, Arabic Sciencesand Philosophy 17 (2007): 81–107.

2 For a list of Mordekhai Finzi’s manuscripts see Giancarlo Lacerenza, “A RediscoveredAutographManuscript byMordekay Finzi”, Aleph: Historical Studies in Science and Judaism 3(2003): 301–325.

3 Scattered information is available in ShlomoSimonsohn,History of the Jews in theDuchyof Mantua (Hebrew) (Jerusalem: Kiryat Sefer, 1962) and in Vittore Colorni, “Genealogia dellafamiglia Finzi. Le prime generazioni”, in: Vittore Colorni, Judaica minora. Saggi sulla storiadell’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)”; TonyLévy, “L’algèbre arabedans les texteshébraïques (I).Un

ouvrage inédit d’ Isaac ben Salomon al-Aḥdab (XIVe siècle)”, Arabic Sciences and Philosophy


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problemsare solvedbywhatmaybe reconstructed as implicit algebra (Abra-ham bar Ḥiyya’s treatise, translated into Latin as Liber Embadorum, andSefer ha-Middot, attributed to Abraham ibn Ezra, which was also trans-lated into Latin); an undated algebra in the tradition of al-Khwārizmī thatmay have something to do with Abraham ibn Ezra; Moïse ibn Tibbon’sthirteenth-century translation of al-Ḥaṣṣār’s arithmetic; Isaac ben Salomonben al-Aḥdab’s fourteenth-century translation of Ibn al-Banna’s arithmetic;6SimonMoṭoṭ’s fifteenth-century algebra derived from an Italianmathemat-ical culture; Mordekhai Finzi’s fifteenth-century translations of the algebrasof 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 moreunanalyzed manuscript (Paris, BnF, Ms. Hébr. 1081, fols. 62v–67r).

Most of our evidence concerning the interest in Hebrew algebra is exsilentio: there are hardly any references and few surviving copies. It is hardto guess whether this reflects scholarly access to Arabic/Latin/vernacularsources ormore simply a lack of interest. I amalso not aware of any evidenceshowing 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 anddividing monomials and binomials involving numbers and square roots, italso briefly deals with higher roots, and includes tour-de-force divisions ofnumbers by three- and four-term sums of roots (see Appendix A for a list ofcalculations).

Next Dardi introduces the six basic equations of algebra and the rulesfor solving them: things8 equal numbers, squares equal numbers, squaresequal things, squares and things equal numbers, squares andnumbers equalthings, and squares equal numbers and things. The fact that the fifth casemay involvemultiple 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’sAlgebra”, 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 AbūKāmil: Kitāb fi al-Jābr waʾlMuqābala in a Commentaryby Mordecai Finzi (Madison, WI: University of Wisconsin Press, 1966).

8 The unknown “thing” is the predecessor of the modern x, and the “square” is thepredecessor of x2.


finzi’s translation of maestro dardi’s italian algebra 197

as imported into Latin and vernacular Europe. Dardi then brings geometricproofs of the rules for solving the last three (compound) cases derived fromthe tradition of al-Khwārizmī and Abū Kāmil.

Only after the rules for solving the standard equations are introduceddoesDardi present the terms of algebra: numbers, the unknown thing (cosa/cose, )םי(רבד ), the square of the unknown thing (çenso/i, י/וסניצ ), the cube ofthe unknown thing (cubo/i, )םי(בקועמ ) and the fourth power of the unknown(çenso/i di çenso/i, י/וסניצמי/וסניצ ). A brief discussion of adding, multiplyingand dividing these terms is included as well.

Then eachof the six rules is demonstratedby a fewexamples (theArizonamanuscript, which is the source of Finzi’s translation, is unique in providingmore than one example for rules 1–4 and 6; rule 5, which may involvemultiple solutions, is also accompanied by more examples in the Arizonamanuscript than in any of the othermanuscripts).Most examples are purelyarithmetical (e.g., break 10 into two numbers that obey some arithmeticalcondition), but there are three recreational/economic problems that aremore typical of vernacular Italian algebra (see Appendix B for a list ofproblems 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 andirrational coefficients. All of these examples, except four, are reducible toequationsof the anachronistic formsaxn=bandax2n+bxn=c. The four excep-tional problems are cubic and quartic equations that cannot be reducedto the above forms, but the solutions that Dardi brings are “special”, sincethey hold only if specific restrictions apply to the coefficients. Dardi’s textexplicitly acknowledges the fact that the rules provided hold only for spe-cial cases, but does not make explicit the conditions under which theyapply.

Finzi’s translation ends with rule number 51 (the back of the folio onwhich rule number 51 appears is empty, suggesting that the manuscriptwas not continued elsewhere), and does not reach any of the four specialequations. 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 manu-script copy of Dardi’s 1344 algebra held at Arizona State University in Tem-pe.9Theopening statementof thismanuscript is our only source for identify-ing the author and date of composition. The two copies of Finzi’s autographare our only sources for identifying Dardi as a Pisan scholar.10 Dardi’s alge-bra has three other surviving copies,11 and two modern editions: RaffaellaFranci’s edition based on the Siena manuscript12 and Warren van Egmond’sunpublished critical edition based on the Arizona manuscript,13 which willhopefully be published soon. Portions of the text survive in other manu-scripts as well.14

The received theory on the origin of Italian vernacular algebra usedto cite Fibonacci as its Italian harbinger. But Høyrup15 argued convinc-ingly that fourteenth-century Italian writers of vernacular mathematicaltreatises (“abbacists”) are not likely to have relied directly on Fibonacci.Høyrup hypothesizes that algebra migrated into Italy from Arabic Spainthrough a Catalan-Provençal mathematical culture, which may have alsobeen Fibonacci’s source. A comparative analysis of early Italian algebras ledHøyrup to the conclusion that Dardi’s treatise represents a branch of thistransfer of knowledge that is separate from the main branch entering Italythrough the algebra of Jacopo da Firenze.

Trying to assess the distance of Dardi’s algebra from Arabic sources istricky. The use of the term “drama” as synonymous with number is clearlyArabic, but is scattered unevenly across the text. Occasionally there is talkof the square and its root, rather than the thing and the square (e.g. onfol. 15v of the autograph: “one square and ten things, or, say, ten of its roots”);

9 Adescriptionof themanuscript is available inBarnabasHughes, “AnEarly 15th-CenturyAlgebra Codex: A Description”, Historia Mathematica 14 (1987): 167–172.

10 The Arizona manuscript leaves a blank space where the Hebrew copies write Pisa; thecorresponding opening folia were detached from Finzi’s autograph.

11 SeeWarren van Egmond, “The Algebra of Maestro Dardi of Pisa”,HistoriaMathematica10 (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 (un-published, 2002).

14 van Egmond, “The Algebra of Maestro Dardi”, p. 419; van Egmond, The Arizona DardiManuscript, 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.



such formulation is rare in Italian algebras and attests to closer contactwith an Arabic source, where the pairmāl-jidhr (square-root) is sometimespreferred over shayʾ-māl (thing-square).16 As Jens Høyrup notes, while all ofDardi’s geometric proofs go back to Arabic sources, the first few diagramsare lettered in a way that does not fit with this tradition. To that I can addthat the arguments, while parallel to those of the Arabs, omit all referencesto Euclid, and depend on elementary cut-and-paste arguments instead.Nevertheless, the last couple of diagrams are lettered in line with Arabicpractice. Moreover, in the last diagram, the letters describing the mainsquare are, instead of the standard ABCD, the odd looking ABGD. Thisfits the order of the Arabic (and Greek) alphabet. This variety attests to acombinationof sources and inspirations, somecloser and some farther awayfrom Arabic origins.

Dardi’s algebra is not only closer to scholarly Arabic algebras in terms ofthe above textual residues, but also in terms of organization and reasoning.It is by far themost elaborate and systematically organized Italian algebra; itprefers 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 comparesdifferent variations of rules for the same problem; it occasionally makesexplicit notes concerning additive commutativity and other forms of invari-ance under the order of arithmetic operations; it includes a systematic dis-cussion of converting algebraic equations into other equations and occa-sionally discusses the impact of algebraic modeling (the choice of elementto be representedby theunknown thing) on the resulting equation (openingthe way for the study of transformations of equations, which is a prerequi-site for solving higher equations); and it occasionally flirts with equationswhere a sum of algebraic terms equals nothing.

All these factors made Dardi’s algebra, which was by no means wellknown or widely distributed in Italy, appealing for a scholar immersed inthe Hebrew-Arabic tradition such as Mordekhai Finzi. Rather than one ofthe popular commercial-recreational algebras of famous fifteenth-centuryabbacists, Finzi chose to translate this obscure old algebra that retainsmuchmore of the scent of higher Arabic learning.

16 Jeffrey A. Oaks and Haitham M. Alkhateeb, “Māl, Enunciations, and the Prehistory ofArabic Algebra”, Historia Mathematica 32 (2005): 400–425.

17 Despite Høyrup’s claim to the contrary, some such verifications are explicit in Dardi’swork. But these explicit verifications are indeed rare.


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4. The Arizona Manuscript is the Source of Dardi’sTranslation; 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’stranslation. Finzi’s translationmakes marginal notes of most of the Arizonamanuscript’s folio transitions with the correct numbering. Moreover, thereare errors that indicate that Finzi’s source is indeed the Arizonamanuscript(e.g. reading awordwrongly in away that fits the handwriting in theArizonamanuscript; going from the end of a line in the Arizona manuscript back tothe beginning of the same line, erasing the repeated words, and continuingthe translation).18

Moreover, the translation is almost always literal. With the exceptionof some titles and conclusion lines that summarize the calculations abovethem, Finzi hardly omits anything from and hardly adds anything to theArizona manuscript. Sentences that Finzi abridged or expanded are rathersparsely scattered across the text. In fact, Finzi copied quite a few obviouserrors and awkward formulations that appear in the Arizona manuscript—errors and formulation that are unlikely to have survived two copyists. Evenwhen Finzi restructures the sentences of the Arizona manuscript, there aretraces of the Arizona text in the form of deletions and insertion.19

One of the alleged proofs that the Arizona manuscript was Finzi’s sourceis more than a hundred Hebrew glosses scattered across it, mostlytranslating adjacent problems and calculations. Barnabas Hughes askedwhether these notes had been written by Mordekhai Finzi himself.20 TonyLévy answered the question in the affirmative.21 I would like to re-open thequestion.

The Hebrew marginal glosses on the Arizona manuscript22 differ fromFinzi’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 dif-ference in terminology extends not only to mathematical terms: interestcompounded annually (far capo d’anno) is described as הנשמשארתושעל

הנשל in the autograph and as הנשףוסבןרקלחוירהםישל in the marginal

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: ×



glosses. The marginal glosses most often spell root ( שרוש ) with the letter,ו while Finzi’s autograph consistently spells the same word without it. Theletters ק and ת are both written as a single connected line in the marginalglosses but are broken into two disconnected lines in the autograph (eventhe very few connected instances of ק in the autograph do not extend theirtails as far as those in the marginal glosses). The letter מ has a completelyopen base in the autograph, but only a partially open base in the marginalnotes.

None of this conclusively disproves Lévy’s claim that Finzi is the authorof the marginal glosses. Indeed, I do not contest the fact that Finzi trans-lated directly from the Arizona manuscript, so Finzi definitely had accessto the manuscript. Moreover, a long time may have passed between thewriting of the glosses on the Arizona manuscript and the creation of theautograph translation in 1473. In themeantime, Finzi copied frommany dif-ferent sources, and his vocabulary and spelling may have been influencedaccordingly (indeed, Finzi doesmake a fewuses of תוברל in his translation ofAbūKāmil’s algebra, but not with the preposition דגנכ , which is used consis-tently in the Arizonamanuscript). One’s handwriting can change over time,and one may write differently when one writes personal marginal notes asopposed to a neat book. A conclusive response to Hughes’ question wouldtherefore require an overall vocabulary, spelling and handwriting analysisacross all of Finzi’s autographs, whereas the analysis that I conducted wasbrief and cursory. However, given all of the above, I do not think we haveenough evidence to support the attribution of theArizonamanuscript notesto 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 includedin 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 twocopies of Finzi’s autograph (Paris, BnF, Ms. Héb. 1033 and 1029 respectively) to fol. 121 as theplacewhere the discussion of the four special equations begins,which indeed fits theArizonamanuscript (the reference in Lévy’s paper was accidentally confused, but Lévy clarified hisintention in a private communication). This is indeed more evidence to the fact that Finzitranslated from the Arizona manuscript, but not to the origin of the Arizona manuscriptmarginal 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, itfaithfully 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 linemarks and other oddities.24 I allowed myself to use this copy to reconstructsome of Finzi’s marginal notes that were trimmed when Finzi’s autographwas fitted to the size of the current codex.

As for the second copy (numbered 1029), it looks neater, but is in fact ofpoorer quality in terms of omissions and errors. Moreover, it leaves blankspaces for all but one of the numbers written in Arabic figures, and omitsall arithmetical and geometric diagrams. It probably does not descend fromthe first copy (numbered 1033), because it does not reproduce some of itserrors and amendments (although in principle practically all the errors andamendments of the first copy can be reconstructed back to the version ofthe autograph). I could not find good evidence to support or disprove thehypothesis 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 sev-eral erased words that indicate a word-by-word translation, which had to becorrected after the first fewwords had already been translated, as otherwiseit would make no sense in Hebrew. It seems that Finzi did not always evenread through the whole sentence before translating it.25

This does not mean, however, that the correspondence of Italian andHebrew terms is one-to-one. For example, the Italian verb produre (to takethe product of two numbers), which appears mostly in the first few folios ofthe manuscript, is consistently translated by the verb תוכהל . The verbmolti-plichare, on the other hand, is translated inconsistently by both לופכל and

תוכהל .I include here a thematized glossary of technical terms. It is by nomeans

exhaustive. It is also not sensitive to which forms (nominal, verbal, active/passive, etc.) are preferred or excluded, and to the Italian spelling, which ishighly 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)

produre תוכהלmoltiplichare תוכהל,לופכלpartire קלחלazonzere ףיסוהל,ץבקל,רבחלl’ aditione ינוא״יצידהsomma ללכtrare, de(s)batere, abatere ליפהל,עורגל,איצוהלsan םלשmen טעמ,ערוג,תחפמ,תחפנ,)!תוחופ:2ע(תחופ,תוחפpiu בר,ףיסומ,רתויcontrarie (in piu/men context) דגנ)ת(מ(de)sbatere (cancel piu by men) לישכהל,טיעמהל

Finzi’s terms here are well precedented, but eclectic. לופכל in the sense of“to multiply” (rather than to double), for example, was a standard termfor ibn Ezra, but later discarded in favor of תוכהל by most Arab-to-Hebrewtranslators. The word aditione is used only once in the Italian text, and mayhave not been understood by Finzi. An analysis of the translations for piuand men will follow in the next section of this introduction. The pluralityof translations of men has to do with the tension created by the gradualtransition of men from the role of an arithmetic operation to that of (to putit anachronistically) a minus sign.

6.2. Root Terms

radice quara עבורמשרשradice chuba בקועמשרשradice discreta רבודמ,יולגשרשradice indiscreta, sorda, muta יולגרפסמבןזואיאל,וברבודיאל,םלא,שרח,םלענשרשcontinua ידימתradice de zonto רובחשרש,רובחמשרש,ףסונשרש,ףסונמשרשcum gionto, conzunta ףסונradice de trato אצוממשרש,תאצוהמשרשvoce לוקson ןוגנclapi םירשק,יפאלק

The terms here are again an eclectic mix going back to bar Ḥiyya, IbnEzra, 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 termdiscreta), used only once in the Arizonamanuscript, is translated literally asa temporal term, rather than as a spatial term.Radice di zonto refers to a rootof the sum of a number and a root (or several roots). This is an idiosyncraticterm 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 towhich something is added and the added term respectively. The translationmisses this distinction. Radice de trato is a root of a difference between anumber and a root (the term אצוממשרש appears only in the part of thetranslation that is not covered in this edition). Voce and Son express a singlearithmetic term (say, the root of a number), and demonstrate the strongoral aspect of abbacist mathematical practice. Finzi translates these termsliterally. Clapi are the different terms in a sum of numbers and roots.27 Thisterm is first transliterated and then translated as םירשק (knots).

6.3. Algebraic Terms

cosa רבדçenso וסניצcubo בקועמçenso di çenso )וסניצדוסניצ(וסניצמוסניצdrama אמארדredure, produre איבהל,בישהלschixare עוציב,יראזיקסquestion הלאשadequation האושה,ןוקת,הלאשadequa הושתrestaurare םלשלdesfare תיחשהלparte קלחregola del 3 ׳גהללכ2 positioni תוחנההיתש

Finzi did not have a substantial tradition to draw on for algebraic terms. Thetwo Hebrew algebraic texts closest to him (Moṭoṭ’s and Aḥdab’s, the latterpossibly unknown to Finzi) do not correlate well with Finzi’s translation ofDardi. There is, indeed, a better correlation between terms in Finzi’s trans-lations of Dardi’s and Abū Kāmil’s algebras (of which the latter may havedepended on a combination of an Arabic source, a Spanish version and an

27 Clapmeans 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.



earlier Hebrew translation), but there is no attempt at full consolidation.Most translations of algebraic terms are simply literal. Cosa and cubo, whichhave an obvious literal meaning (thing and cube), were replaced by Hebrewterms.Çenso (an Italian variant of the Latin census, which translatesmāl, theArabic term for possession—but whose etymology was unknown to mostItalian abbacists) was simply transliterated by Finzi in his Dardi translation.In his Abū Kāmil translation, however, Finzi used a transliteration of theSpanish algo along with the Hebrew עוביר . The monetary unit drama (anItalian transliteration of the Arabic dirham), used as equivalent to number,was again transliterated into Hebrew—but transliterated differently in thetranslations of Dardi and of Abū Kāmil (an Arabicized םהריד vs. an Italian-ized אמארד ). Redurewas used for normalizing equations (rescaling to makethe leading coefficient equal one), but also for turning a number into a rootand for other conversions. Finzi gave this term a non technical translationby common Hebrew verbs. Schixare, the equivalent of modern cancellationof terms above and below a fraction line, was first transliterated and thentranslated literally as a sort of cutting.

The term adequation (equation) posed a challenge to Finzi. He first trans-literated it, then attempted to translate it as ןוקת , and subsequently misun-derstood it as synonymous with question (he even interpolated a statementto that effect on fol. 15r). Eventually Finzi realized that the meaning corre-sponds to something we could anachronistically term “equation,” and cameupwith the term האושה , which is precedented in al-Aḥdab’s translation. Theoperations of algebra (al-jabr and al-muqabala, respectively: adding to bothsides of an equation a term subtracted on one side, and subtracting fromboth sides of an equation terms that are added on both sides), whose namesFinzi considered and reconsidered in his translation of Abū Kāmil,28 are notdistinctly thematized in Dardi’s algebra, and Finzi’s translation reflects thisattitude (see fol. 18v for the casual and brief introduction of these opera-tions through an example). A single appearance of restaurare and desfare todesignate al-jabr are translated literally, and the more common imperativeadequa, which is used for both algebraic operations, is translated as הושת .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 ambiguousterm דצ , literally side, but thenerases it). Finally, the rule of three anda singlereference to the algebraic method of double positioning (extrapolating thecorrect solution of linear equations in two unknowns by a weighted average

28 Lévy (“L’algèbre arabe [II]”, 98). He eventually suggested ןוויכ and הדקפה .


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of two wrong guesses) are also translated literally, rather than by any of theforms in the literature (e.g. Mordekhi Komṭino’s תורוצהתשולש or al-Aḥdab’s

םינזאמ ). We see here Finzi’s strong inclination towards literal translationsover an attempt at constructing a unified professional lexicon.

6.4. Geometric Terms

figura הרוצdesignado ריוצמampleça בחרמ,בחורlongeça ךרואמ,ךרואgroseça יבועsuperficie חטשcorporale ימשגladi תועלצper equale distantia יחכנde cantoni directi תויוזהבצנ

Finzi’s geometric terms are almost perfectly in accord with those standard-ized in Ibn Tibbon’s addendum to The Guide for the Perplexed. The use of

יבוע for the third dimension is somewhat deviant. It is notable that Finziconsistently uses הרוצ in his Dardi translation as opposed to הנומת in histranslation from Abū Kāmil ( הרוצ is also al-Aḥdab’s preferred translation).

6.5. Economic Terms

soldo רנידlira הרטילdinari תועמrasone )תיביר(ןובשחfar capo d’anno הנשלהנשמשארתושעל

The translation of coin names is somewhat odd. Soldo is translated as רניד ,and so dinari awkwardly turns into תועמ .

6.6. Logical andMetamathematical Terms

Finally, the following terms used by Finzi are common medieval scholarlyterms. I include them here for completeness.

regola ללכ,רדסcomun יללכpraticha לגרהprova היאר,תפומprovare תוסנל,תפומאיבהל



demostrare תוארהלcertamente ראובמamaistramento תומכחתהrasone ?תומא,תונובשחexemplo לשמ,ןוימדproprietade תומצעnatura עבטeser תוהמ

7. The Role of the Translator

It is clear that Finzi adopts the attitude seeking to “translate word by wordwithout additions or omission”.29 But it is also undeniable that a translationis always an interpretation. I will quickly review here some ways in whichFinzi’s translation did some interpretive and transformative work.

7.1. Formal Abbreviations

The first point is Finzi’s commitment to verbal forms, renouncing all formalabbreviations. Dardi’s manuscript includes various interesting notationsthat facilitate its processing. Roots are designated by a special contraction ofRx, the algebraic terms aremarkedbyCandÇ (Cosa andÇenso respectively),and expressions such as 10 Cose are sometimes abbreviated as 10/C (thefraction is displayed vertically in the manuscript, as in numerical fractionnotations). An expression such as “8 C ee 180 C uguale a 900/n et a 9/Ç” iseasier to parse and recognize as an operative unit than the equivalent “8things and 180 things are equal to 900 numbers and 9 squares”.30 But Finzichooses to ignore all these notational inventions and to adhere to a classicalverbal presentation.

Indeed, Finzi and his two copyists make many abbreviations, mostlytruncating off suffixes of words, including sometimes words that designatealgebraic unknowns. But these random abbreviations have no systematic

29 From Judah ibn Tibbon’s introduction to his translation of Ḥovot Halevavot, quoted inSarfatti,Mathematical Terminology in Hebrew, 170.

30 Some contemporary research on the cognitive work exported ontomathematical signscan clarify this point. See, for example, David Landy, Colin Allen and Carlos Zednik, “APerceptual Account of Symbolic Reasoning”, paper presented at The Hebrew University’sInstitute of Advanced StudiesWorkshop: 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 andends. This is, perhaps, one of the reasons why it was so difficult for Finzito understand Dardi’s concept of adequation.

It should be noted that even manuscripts with no formal abbrevia-tions were sometimes accompanied by a practice based on abbreviations(Høyrup brings a beautiful example of an Arabic folio, written in strict ver-bal style, but whose margins are densely filled with formal abbreviations ofthe main text in another hand).31 Finzi’s decision not to include any formalabbreviations reflects a commitment to strict separation of scholarly writ-ten presentations from practical teaching, where formal abbreviations arebound to appear.

7.2. Decimal Numbers

A similar attitude applies to number terms. Here too Finzi avoided Arabicfigures, and stuck to Hebrew letters. Again, this makes it more difficult toparse the text and tell where a mathematical expression begins or ends.Finzi only used decimal numbers when he reached 6 digit numbers, indi-cating what he considered as the limit of viability of the Hebrew numbersystem. At one point (fol. 30v, outside the scope of this edition) Finzi usesvertical fractions with Arabic figures as enumerator and denominator.

Again, Hebrew also had a system of abbreviations. One could indicatenumbers by their names as well as by a semi-decimal system based onletters ( דחאוםירשעהאמ vs. א״כק ). But a comparison of variations in Finzi’smanuscript and the two copies highlights the fact that these notations arearbitrary and export no cognitive work to the written signs.

7.3. The Transformation of theMen (Minus)

The next aspectwhere Finzi’s translationworks as an interpretation is in thecontext of the term men. Men is the term used to indicate the operation ofsubtraction, the opposite of e (and). But when one discusses operations onsuch binomials as “root of 5men 2”, one has to refer to the 2, ending up withsuch expressions as “the 2, which is men”. This becomes even more urgentwhen one has to explain in such calculations as “root of 5 men 2 times rootof 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 Symbol-ization in Abbacus and RelatedManuscripts, c. 1300 to c. 1550”, in: Albrecht Heeffer, Maartenvan Dyck (eds.), Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics(London: College Publications, 2010), 3–56 (on p. 10).



which subtracted. Slowly but surely, men is no longer just a conjunction orbinary operation; it gradually takes on the role of an adjective, modifyingthe value of the number to which it is attached and qualifying its kind.32 It isno surprise then, that the last equation considered in Dardi’s text before thefour special cases is the breakthrough “5 things and 15 equal nothing”, wherethe 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 anadjective, Finzi moves from the conjunction תוחפ to the adjective תחופ . Butthe transition is subtle, and Finzi’s translation is accordingly confused. Onecan find either term used in the opposite role, and an expression such as “15and root of 100 men root 91 and men root of 36” has the first men translatedas תוחפ and the second as תחופ .34 As the text goes on, Finzi adds to the list oftranslations ofmen the terms תחפנ and תחפמ (other adjectival/passive formsof the same root) as well as ערוג and טעמ , based on an alternative translationof subtraction and on a literal translation of men as “less”, respectively. Thedifferent translations are sometimes used for different men terms in thesame sentence playing identical grammatical and logical roles.

This situation is so confusing for the second (and mathematically lessable) copyist, that on several occasions hewrites תוחפ and then adds amarkindicating that the ח and ו should be inverted, and on a couple of occasionseven ends up with the hybrid monster תוחופ .

7.4. Conceptions of Numbers

The last point where translation and copying made a difference concernsthe conceptualization of root and Cosa-Çenso signs (I will use Cosa to indi-cate both latter terms). Here the impact is by no means unique to Finzi’sintervention, but extends to practically all the players in the abbacist manu-script scene.

As I discuss in my work on abbacist algebra, root and Cosa signs carryseveral operational meanings in terms of mathematical practice.35

32 Some perspectives on this transition in the abbacus context can be found in Roy Wag-ner, “The Natures of Numbers in and around Bombelli’s ‘L’algebra’ ”, Archive for the Historyof Exact Sciences 64/5: 485–523 and Albrech Heeffer, “On the Nature and Origin of AlgebraicSymbolism”, in: Bart van Kerkhove (ed.), New Perspectives on Mathematical Practices. EssaysinPhilosophyandHistory ofMathematics (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 modifyingthe numbers they apply to. For example, in “root of 4” the square rootsign 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 thespecies of the number involved. For example “root of 5” turns “5” intoa different kind of number, belonging to the species of roots (theambiguity that allows to see “root of 4” both as “2” and as belonging tothe species of roots is explicitly drawn upon by Dardi in examples thatexplain 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” qualifiesthe “2” as 2 units of weight.

3. Finally, root and Cosa signs can be used as indexicals pointing tothe adjacent number. When an abbacist talks about “the root” andpoints to the expression “root of 5”, he may sometimes be talkingabout the number 5 itself, and the same goes for Cosa (carrying theseemingly redundant sign along with the number can indicate wherethe number comes from, and why it is used at that point). This canwork in the opposite direction as well. Just as one can say “2” referringto “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, thatnumbers can be identified by reference to their species and vice versa.

These distinctions should not be taken lightly. The practice of signs as oper-ators (case 1 above) homogenizes different kinds of mathematical expres-sions in a uniform realm of numbers, because it allows transforming onekind into another. On the other hand, the practice of signs as denomina-tors (case 2 above) divides different mathematical expressions into distinctspecies, and draws boundaries on permissiblemathematical manipulations(reminiscent of those set by the Greek classics against conflating arithmeti-cal and geometricmagnitudes, or geometricmagnitudes of different dimen-sions). As I argue inmy paper quoted above, both attitudes were productivein the evolution of abbacist knowledge. The practice of signs as indexicals(case 3 above), I believe, testifies to the strong oral aspect of abbacist prac-tice, where the possible confusion would be prevented by pointing and bythe pragmatics of dialogue. As shown in my paper, indexicals also helpedblur the line between different uses of arithmetical terms, and thus led to amore homogeneous conception of arithmetical terms.

Dardi is well aware of these nuances. The practice of the root sign asturning one number into another number, as well as its practice as changing



the species (or, in Dardi’s language, “nature”) of numbers, are both appliedexplicitly. Indeed, the entire logic of Dardi’s treatise depends on dealingsystematically with different species of numbers. Different rules apply tocalculations with roots and with integers, and different solution recipesapply to equations with root and with integer coefficients. This divisionto different species is precisely how Dardi could turn 6 basic equationsinto 194 different cases (not including the “special” equations). But, on theother hand, whenDardi reduces complex equations into simpler ones (suchreductions conclude many of his discussions of equations that fall outsidethis edition), he explicitly states that one could treat root coefficients simplyas numbers specifying the quantity of unknowns.36

As someonewhomasters both approaches,Dardi is exceptionally careful,compared to other abbacists, to separate the different uses of root and Cosasigns.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 theCose” to avoid any possible confusion. Moreover, he makes such statementsas: “the root of this 4/9 comes to be the quotient of dividing root of 4 by rootof 9; the essence of the root of the sound or voice [term] 4/9 is 2/3”.37 Wehavehere anice distinctionbetween the formal aspect of an arithmetic term(voice) used to express different species of numbers, and its essence—thenumber it represents.

When comparing the Arizona manuscript of Dardi’s algebra to the Sienamanuscript and to Finzi’s translation, we see that Finzi is usually more con-servative and respectful of such distinctions.38 However, even the Arizonamanuscript makes regular use of the plural form Cose to refer to the coeffi-cient of the Cosa term and the singular form Cosa to refer to its value—adistinction that can be easily lost in translation and in copying. Indeed,while Finzi rarely errs in this respect, his second copyist blunders quiteoften. Compound this with the indexical aspect of root and Cosa signs (3above), where “5” couldmean “root of 5” and vice versa, and Dardi’s delicateconceptual 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.37 Fol. 10r in Finzi’s autograph, 19r in the Arizonamanuscript.My translation simplifies the

original ambiguous syntax. Next to this statement the manuscripts add that the ratio of twonumbers is the same as that of “voices of their roots”, meaning that, formally, dividing roots islike dividing their numbers, although the values of the numbers and the roots are obviouslynot in the same proportion.

38 Acareful readingof cases 19–22 in theArizonaandSienamanuscriptswill providemanyexamples of everything discussed in this section.


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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 withrespect to those of theArizonamanuscript. Building on these variations, thecopyists further erode Dardi’s subtle distinctions. This nuanced and seem-ingly insignificant process of minute variations, which takes place, so tospeak, 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’sdistinctions is not viable in a mathematical culture of amateurs and semi-professionals and does not travel well through written texts, contributedtowards a more homogenized early modern conception of number, wheredistinctions between species of numbers were gradually losing their foot-hold.

Acknowledgment

I would like to thank Warren van Egmond, Naomi Aradi and Tony Lévy forvaluable discussion. I would also like to thank the first two for access tounpublishedmaterials. This work is supported by Israel Science Foundation(ISF) project: “Studies in the History of Medieval Mathematics in Hebrewand Judeo-Arabic”. I would like to thank the head of the project, RuthGlasner, for her support and guidance.



Appendix A

List of Examples with Roots

Raffaella Franci’s edition of Dardi’s algebra (based on the Sienamanuscript)brings a symbolic list of Dardi’s rules for arithmetic with radicals. I chose tocomplement her list with the actual numerical examples. Indeed, a list ofnumerical examples can help track down genealogies of sources; but thereare additional reasons behind my choice.

First, this record allows us to be a little more sensitive to the order ofoperations, which symbolic formulas tend to suppress, but which is animportant issue in abbacist rules, which are presented as recipes. Note,however, that products may be calculated in a different order than that inwhich they are subsequently summed, and so the calculations below maynot reflect precisely the way Dardi ordered things.

Second, symbolic notation gives a false image of the nature of some prod-ucts. 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 con-sider a+b×b as a single term (an integer), and so the product would beconceived as a binomial. These distinctions matter, as they are constitutiveof the waymathematical expressions are classified and of the analogies thatcan be drawn.

Third, writing:

(a−Rb)−(c−Rd) = (a−c)+(Rd−Rb)

is completely oblivious to the negativity or positivity of the two subtractionson 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 abbacistwho does not acknowledge negative terms.


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Notation

× multiplicationR square rootR3 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



(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

R4R9

= R(4/9) = 2/3

4R9

= R16R9

= R(17/9) = 11/3

8(3+R4)

= 8×(3−R4)(3×3−4)

= (24−R256)5

= 44/5−R256R25

= 44/5−R(106/25)

(5+R16)3

= 5/3+R16R9

= 5/3+R(17/9) = 12/3+11/3 = 3

20(4−R9)

= 20×(4+R9)(4−R9)×(4+R9)

= (80+R3600)7

= 113/7+R(7323/49)

19(2+R16)

= 19(R16+2)

= 19×(R16−2)(R16+2)×(R16−2)

= (R5776−38)12

= R(401/9)−31/6…

(2+R4)= …

2×R4


216 roy wagner

(19+R25)(5+R9)

= (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)(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(R4+R9+R16+R25)

= 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.



Appendix B

Arithmetic of Algebraic Terms and Problems Solved

Again, I preferred to give numerical examples rather than general rules. Thefirst six examples demonstrate the six standard rules. Next there is a list ofexamples of products of algebraic terms. Finally, there is a list of word prob-lems. I reduced the word problems into symbolic terms, noted the choiceof algebraic unknown (Cosa) for deriving the equation, and recorded theequation, solution, and reconstruction of the original unknown terms.Withthree exceptions (noted below), all question are posed in purely arithmeticterms.

Notation:

C Cosa (thing)Ç Çenso (square)Ch Chubo (third power)ÇÇ Çenso di Çenso (fourth power)D 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 orC = 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 = 12C4Ç×3D = 12Ç2Ch×5D = 10Ch5ÇÇ×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.


218 roy wagner

Problems That Reduce to the First Case:

A+B = 10; A > B; A×A−B×B = 50B = C; A = 10−C(10−C)×(10−C)−C×C = 100−20C ⇒ 50 = 20C ⇒ C = 21/2B = 21/2; A = 71/2

A+B = 6; 56A+67B = 37041A = C; B = 6−C56C+67(6−C) = 370 ⇒ 32 = 11C ⇒ C = 211/12A = 211/12; B = 31/12

1/3 A1/8

= 5

A = C1/3 C = 5× 1/8 ⇒ C = 17/8A = 17/8

3A+B = 32; 6A+3B = 8042A = C; B = 32−3C6C+96−9C = 80 ⇒ 16 = 3C ⇒ C = 51/3A = 51/3; B = 16

A+B = 10; BA= 5

A = C; B = 10−C(10−C)

C= 5 ⇒ 6C = 10 ⇒ C = 12/3

A = 12/3; B = 81/3

Problems That Reduce to the Second Case:

A+B = 10; (A−B)×(A−B) = 201/4A = C+5; B = 5−C2C×2C = 201/4 ⇒ Ç = (20

1/4)4

= 51/16 ⇒ C = R(51/16) = 21/4A = 71/4; B = 2¾

A×A+ 1/2A× 1/2A = 10A = CC×C+ 1/2C× 1/2C = 10 ⇒ Ç = 8 ⇒ C = R8A = R8

A×¾A = 40A = C

41 This is a commercial problem about buying two kinds of fabric.42 This is a commercial problem about buying two kinds of objects.



C×¾C = 40 ⇒ Ç = 40(3/4)

= 531/3 ⇒ C = R(531/3)

A = R(531/3)

(A− 1/3A− 1/4A)×(A− 1/3A− 1/4A) = 12A = C5/12C× 5/12C = 40 ⇒ Ç = 40

(25/144)= 693/25 ⇒ C = R(693/25)

A = R(693/25)

Problems That Reduce to the Third Case:

A× 2/3A = 3AA = C2/3C×C = 3C ⇒ C = 3

2/3= 41/2

A = 41/2

A = 2/3B; A×B = A+BA = 2C; B = 3C5C = 6Ç ⇒ C = 5/6A = 12/3; B = 21/2

1/2A× 1/2A = 20AA = C20C = 1/4Ç ⇒ C = 20

(1/4)= 80

A = 80

A×A = A100

A = C1/100 C = Ç ⇒ C = (

1/100)1

= 1/100A = 1/100

Problems That Reduce to the Fourth Case:

A+B = 10; A < B; A×A = 1/4B× 1/4BA = C; B = 10−CC×C = 1/4(10−C)× 1/4(10−C) ⇒ Ç+11/3C = 62/3C = R[1/2(4/3)× 1/2(4/3)+62/3]− 1/2(4/3) = 2A = 2; B = 8

A×A+8×A = 33A = CC×C+8×C = 33 ⇒ Ç+8×C = 33C = R(1/28× 1/28+33)− 1/28 = 3A = 3


220 roy wagner

1/3A× 1/3A + A = 12A = C1/3C× 1/3C + C = 12 ⇒ Ç+9C = 108C = R(1/29× 1/29+108)− 1/29 = R(1281/4)−41/2A = R(1281/4)−41/2

20 lire lent at A dinari permonth compounded annually equals 30 lire after 2 years.12 dinari = soldo, 20 soldi = lira; A dinari per month therefore equals 1/20A lire peryear.

A = C20 + C + 1/20C×(20+C) = 30 ⇒ 40C+Ç = 200C = R(1/240× 1/240+200)− 1/240 = R(600)−20A = R600−20

Problems That Reduce to the Fifth Case:

A+B = 10; (A−B)×(A−B) = 201/4A = C; B = 10−C(2C−10)×(2C−10) = 201/4 ⇒ 1915/16+Ç = 10C43C = 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 = 21/4A+B = 10; A < B; (B−A)×(B−A)−A×A = 32A = C; B = 10−C(2C−10)×(2C−10)−C×C = 32 ⇒ Ç+222/3 = (131/3)CC = 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 ofmodelling the difference betweenA and B as 10−2Cor as 2C−10 (there is no note in themanuscript that the choice 2C−10made aboveproduces a negative term). The other solution is irrelevant because A is posited tobe smaller than B.

A+B = 10; A×B = 21A = C; B = 10−CC×(10−C) = 21 ⇒ 10C = Ç+21

43 The reduction to a standard equation is done in two ways. In the first the equation isnormalized and than balanced, in the second the order is reversed (in accordance with thestandard rule). In the second case the manuscript goes through the equation: 79 ¾ + 4Ç -40C = nothing.



C = 1/210+R(1/210× 1/210−21) = 5+R4 orC = 1/210−R(1/210× 1/210−21) = 5−R4

A+B = 10; 3A = R8×BA = C; B = 10−CR8×(10−C) = R(800−160C+8Ç)800−160C+8Ç = 3C44 ⇒ 100+Ç = 203/8CC = R[1/2(203/8)× 1/2(203/8)−100]+ 1/2(203/8) = R(3201/216)+103/16A = R(3201/216)+103/16; B = 103/16−R(3201/216)45

Correction of the above:

800−160C+8Ç = 9Ç ⇒ 800 = Ç+160CC = R(1/2160× 1/2160+800)− 1/2160 = R7200−80A = R7200−80; B = 90−R7200

Same problem, exchanging the algebraic modeling:

B = C; A = 10−C900−180C+9Ç = 8Ç ⇒ 900+Ç = 180CC = 1/2180−R(1/2180× 1/2180−900) = 90−R7200A = 90−R7200; B = R7200−80

A+B = 16; A×B = 48A = C; B = 16−CC×(16−C) = 48 ⇒ 16C = 48+ÇC = 1/216−R(1/216× 1/216−48) = 8−R16 orC = 1/216+R(1/216× 1/216−48) = 8+R16A = 4 or 12; B = 12 or 4

Problems That Reduce to the Sixth Case:

10×A+12 = A×AA = C10×C+12 = C×C ⇒ 10C+12 = ÇC = R(1/210× 1/210+12)+ 1/210 = R37+5A = R37+5

(A− 1/3A− 1/4A)×(A− 1/3A− 1/4A) = A+1A = C5/12C× 5/12C = C+1 ⇒ Ç = (519/25)C+519/25C = R[1/2(519/25)× 1/2 (519/25)+519/25]+ 1/2(519/25) = R(1434/625)+222/25A = R(1434/625)+222/25

44 This is an error, and is corrected below.45 The calculation of B is wrong with respect to the given value of A.

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