Abstract
“The present contribution is focused on a corpus of ten or so Franco-Occitan arithmetics, the production of which is centered on the fifteenth century in a defined geographical area around urban centers such as Paris, Lyons, Marseilles, Nice and Pamiers. Close to nothing is known about the teaching situations underlying their production: this problem has frequently been addressed by the historians who studied the corpus and for whom the very existence of these works emanates from a need to educate merchants. From a situation where information external to the documentation is incomplete if not non-existent, it is then interesting to re-examine the relationship between “practical arithmetics” and teaching, based on a comparative study mainly focused on the treatises and paying attention to the greatest number of clues or traces. The purpose of this study therefore consists in returning to the rather classic historical approach of highlighting the diversity of production, in order to refocus the hypotheses on the diversity of the readership and suggest new ones as to the underlying context of production and conception.”
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Notes
- 1.
This is the definition which the author (Beaujouan 1988, 73–88) uses to compare a group of 12 manuscripts that make up the corpus of practical arithmetics and are essentially those we will use. In this article we will not retain Bnf. lat. 7287, nor Suma de la art de arismetica by Francesc de Santcliment nor Bnf. lat. 7352. These manuscripts are used but will not be studied specifically as they open towards particular cultures (printing) or are clearly far from the others, like (Bibl. Nat. lat. 7287) which does not offer commercial development.
- 2.
See (Spiesser 2003a, 32–50).
- 3.
These are not textbooks of commercial practices in the way “Pratica della mercatura” is, they differ in this way from those presented by (Jeannin 2002).
- 4.
With regard to these arithmetics, Warren Van Egmond highlights that they make up the first level of mathematical literacy in the late Middle Ages (Egmond 2001).
- 5.
- 6.
Elisabetta Ulivi has been able to identify 76 abacus masters in the municipality of Florence between 1283 and 1500. She has also shown that the archives allow the masters’ families and their networks of relations to be reconstructed, see (Ulivi 2002b).
- 7.
- 8.
He is the author of two works (Ulivi 2002b), which were quite successful, Trattato di praticha d’arismetrica in 1463 (of which 3 copies remain and which Elisabetta Ulivi describes as an encyclopedia) and Trattato d’abacho of which 18 copies exist –including two from the sixteenth century– and these are considered as conceived for teaching in these Bottegga. The investigation led by Elisabetta Ulivi on the environment surrounding the Florentine abacus schools has given us a fairly precise image of not only the teaching, but also the professional “milieu” of the teachers.
- 9.
On the presentation of these rules, see (Benoit 1989a).
- 10.
The introduction of (Van Egmond 1980) gives a typology of works that reflect this diversity.
- 11.
It is evident that they do not arrive on virgin ground, cultural life continued to be inhabited by the treatises produced in vernacular language from previous centuries. Treatises on algorism translated into vernacular language have existed since the thirteenth century. The works and collections of problems even circulated in Italian. Bartoli’s dissertation kept in Avignon’s municipal library was published in (Sesiano 1984a; Bozzolo and Ornato 1983, 116–118). Bozzolo and Ornato (1983, 366) offers a diagram reviewing medieval book production.
- 12.
- 13.
For the composition of the corpus, see the table in Appendix 2.
- 14.
In this article we will take the meaning of this expression as given by Anne-Marie Chartier Chap. 2 in this volume.
- 15.
This is the case for men such as Barthélémy de Romans and Jean Adam, although we have some sources on the latter linked to missions given to him by the king. But for a man like Jean Certain, who produced a beautiful book in Marseilles, we still do not know anything more today. The approach induced from the sources themselves is the subject of new methodological and epistemological reflection among historians, a summary of which can be read in (Chastang 2008) and (Mackenzie 1991, 39–41).
- 16.
Henri Pirenne has shown that historical evidence attesting a certain merchant culture is quite abundant (Pirenne 1929). Paul Benoit and more recently Maryvonne Spiesser (2003b) have investigated the training of French merchants by studying the mathematical content, rules and problems in these works and, through comparison with the Italian model, have tried to recreate the teaching framework.
- 17.
Ginzburg (1980).
- 18.
This is one meaning of the typological work carried out by Guy Beaujouan, who investigated the manuscript traditions and suggests geographical differences between the works. This work remains the reference for classifications that can be made for these works (Beaujouan 1988).
- 19.
(Van Egmond 1980) is the most precise reference on the subject today. For a broader perspective on a European level see the International Catalogue of Mediaeval Scientific Manuscripts, http://jordanus.ign.uni-muenchen.de [consulted: 07/08/2011]. This on-line catalogue was created under the direction of Menso Folkerts.
- 20.
She offers a four-point classification: “complete” algorisms, collections of various problems, between collections of problems and “complete” algorisms and “encyclopedic” treatises (Spiesser 2003b, 62–63).
- 21.
The question of the composition of the works also quite naturally raises that of the compilation. It can be observed on the scale of the whole manuscript in the succession of the treatises, as well as the practical arithmetic text.
- 22.
“hic liber ad me pertinet quy vocoo Criberiua Paullus” (fol. 5v) and “Cribreres” (fol. 4v) can be read, and some comments like “veryrat est bon compagnion” (fol. 30r). To this we must add trial operations on the first and last pages.
- 23.
The 17 illuminations are all consultable on-line at the Liber floridus website: http://liberfloridus.cines.fr/textes/biblio_fr.html. [consulted 21/06/11]. Only the drawings have not been reproduced.
- 24.
Tournai 86: 210 mm × 70 mm; BnF. fr. 1339: 270 mm × 205 mm; Méd. Nantes 456: 286 mm × 215 mm.
- 25.
In BnF. fr. 1339 and Méd. Nantes 456 watermarks are identifiable but without being an exact, certain match, in (Briquet 1907).
- 26.
A description of this manuscript exists in (Faider and Van Sint 1950).
- 27.
Some geometrical figures have been cut out from the manuscript.
- 28.
Constructed on the model of the algorism inherited from Sacrobosco, the Algorismus linealis presents numeration and basic operations using a table with counters. This treatise can be found in BnF. fr. 1339 and Méd. Nantes 456, to which can be added the French part of Plut. 29 cod.43 at the Laurentian Library, containing an arithmetic using counters and a geometry. One can see reproductions of the manuscript and a description on the Laurentian Library website at http://teca.bmlonline.it/TecaRicerca/ (consulted 21/02/13). Manuscript BnF. fr. 1339 presents, in addition, a Greek algorism and a version of the Traité des usages de l’astrolabe by Jean Fusoris. For the treatise on fol. 80 v – 83 r, whose incipit is “Then come numbers that are made by certain figures that some call Greek algorisms” (“S’ensuite apres nombre par certaines figures que aulcuns appellent algorisme grec…”) see (King 2001; Sesiano 1985).
- 29.
The annotations born by Méd. de Nantes 456 date from the sixteenth century but it is not certain that they are from Fra Giovanni Giocondo’s hand, see (Tura 1999). Adolfo Tura also gave an edition of the geometry known as Fusoris’ with variants of the Méd. Nantes 456 and BNF. fr. 1339 texts in (Tura 2008, 119–171).
- 30.
Another excerpt from the work of Barthélemy de Romans is known in the practical arithmetic discovered by Alessandro Vitale-Brovarone (Vitale-Brovarone 1990, 29–32). This manuscript has the shelfmark Turin, Archivio di Stato, Archivio Biscaretti, Mazzo 29, n. 3. Furthermore we know that Barthélemy’s work had a decisive influence on that of Nicolas Chuquet who had himself been partly motivated at the beginning of the sixteenth century by L’arismeticque then newly written by Etienne de La Roche.
- 31.
The very precise codicological study by (Spiesser and Féry-Hue 2007) allows us to observe the relationships between the different treatises in mss. Cesena S.XXVI.6.
- 32.
The second, more theoretical treatise in Barthélemy de Roman’s work is re-employed in manuscript Bnf. lat. 7381. These are fragments involved in another manuscript situation, indicating a diffusion of these works. One could also invoke a manuscript in Latin, Bibl. de. Dijon 447, which explores rules identical to those in the manuscripts in a different way: the rules of three, of association, and of false position.
- 33.
- 34.
A study of these texts to establish criteria for comparison might be helped by the list of variables compiled by Marie-Anne Polo de Beaulieu and Jacques Berlioz on Jean Gobi for comparing a corpus collection of Exempla. See (Polo de Beaulieu 1999) and (Berlioz and Polo de Beaulieu 2001). It was necessary to modify their list to these introductory texts, we have retained nine points: length of the passages (number of occurrences); designation of the work; presentation of the arithmetic; reason for the production of the treatise, planned use for the work; designation of the author; religious authority; announcement of the plan. This set of criteria allows us to examine the differences between these arithmetics through a limited number of indications. The context of this article does not allow development of these elements, nor discussion of the value and the quality of these indicators. They were used, however, to carry out the present work.
- 35.
- 36.
The following is a breakdown of the number of occurrences (forms) by manuscript: Bnf. lat. 7352: 160(87); Compendy de Pamiers: 161(124); Bnf. fr. 2050: 104(63); Bnf. fr. 1339: 341(143); Traicté de la praticque: 199(107); Compendy de la praticque 374(165); Triparty en la science des nombres: 84(57); Méd. Nantes 456: 187(101); Kadran aux marchans: 571(292); BSG. 3143 978(445); Compendio de lo abaco: 168(113); Petit cadrans: 92(63).
- 37.
It should be emplasized that it is not certain that all these treatises are from Barthelemy (Spiesser M. and Féry-Hue, 2007).
- 38.
This identity is not uniformly accepted when you consider that Jacques Legrand in his Archiloge Sophie (c. 1415) distinguishes quite clearly between algorism and arithmetic.
- 39.
“cest science qui touche fait de comptes est elle utile et bien seant a toutes manieres de gents”.
- 40.
“grand fatigation et rompement de teste”.
- 41.
He wrote in Compendy de la praticque: “to give greater clarity and understanding to those who have or would have this compendy [meaning, the practical treatise] and would have the current work”) fol. 149r.
- 42.
Compendy de la praticque des nombres: “I have made the collection of all the rules of this compendy for the honor, respect and the singular love of the noble Guille de Claro, of Aix en Provence.” (“pour honneur et reverence et singulier amour de noble homme Guille de Claro, habitant de la cite d’Aix en Provence, ay fait la suyvant recollection de toutes les regles de celluy compendy”). Some traces of this man can be found in the archives (Spiesser and Féry-Hue 2007). He also appears in a “confirmation of the statutes and ordinances, on the subject of auditors of accounts in the charter and the regulations for recourse to their services.” (“confirmation des statuts et ordonnances, sur le faict des auditeurs des comptes tutelaires, & reglement sur le recours d’iceux”) given by King René in ‘The collection of antiquities and privileges in the town of Bourges and several other capital towns of the kingdom’, Paris, 1621, p.440.
- 43.
“a l’entendement de ceulx qui ont, ou qui auroient celluy compendy et auroient le precedent ouvrage”
- 44.
“Car le maistre reverend conditeur du livre estant au lieu de Carcassonne enseignant la science d’algorisme ou arismetique fut par aulcuns des escoliers pry de leur faire aulcun brief traictié qui leur donnast clere congnoissance des nombres proporcionalz, esquelz gisent et sont toutes raisons, fist ce petit traicti lequel donne competant cognoissance de leurs proporcions, tant comme sont necessaires a avoir clere congnoissance des raison qui est la tierce partie generale du compendy quil avoit fait aultres foiz de la pratique des nombres”. fol. 269v.
- 45.
L’Huillier (1976).
- 46.
“maistres expertz en cest art.”
- 47.
The work follows a progression similar to practical arithmetics by showing calculations with counters; it develops the rule of three and the rule of association, which it adapts to fiscal problems.
- 48.
This work attracted the attention of Thorndike, see (Thorndike 1959).
- 49.
One could continue this line of thought. Indeed, Jean Adam’s book is littered with pejorative observations on those who do wrong, those who cheat, an aspect which reminds (Mattéoni 2007).
- 50.
Lapeyre and Scheurer (1978, 3–4).
- 51.
Nicole Tilhart was appointed by letter on May 29 1470 to the office of clerc extraordinaire de la chambre des comptes. In 1476 he was commissioned to the distribution on ordinary and extraordinary finances of the Dauphiné, Valentinois and the Diois. He was promoted to General of Langue d’Oïl in 1477 (Lassalmonie 2002, 356–357).
- 52.
The difficulties of the administration of this province were highlighted recently in (Lassalmonie 2002, 359, 513).
- 53.
A goldsmith called Jean Adam appeared in “Découverte d’un trésor du XVIe siècle à Montélimar”, Bulletin d’archéologie et de statistique de la Drôme, 1887, 16 p.
- 54.
There are no studies on the liking of these men for books on science, but there are traces of purchases. Thus manuscript Bnf. fr. 2021 (dating from the thirteenth century and containing an ecclesiastical date calculator, a geometry and an algorism) bears on page 154, “This is the book of Pierre Cautet, procurer in parliament who bought it at the royal palace in Paris in one thousand iiii and fifty”. This man is mentioned among the lawyers to the court of treasury, see (Dupont-Ferrier 1936).
- 55.
The references he cites are classic, he re-uses the etymologies of Isidore de Seville and Boethius. In 1520, Etienne de La Roche used exactly the same citation.
- 56.
“merreaulx en ensuivent les plus pres que poussible est l’ordre et la forme du gect et calcul gallican sellon le stille de gecter en la chambre desdit comptes du Roy tres Chretien a Paris”, fol. 3r.
- 57.
“ce dit present traicté n’est fait que pour gens de finances de comptes et de marchandise pour savoir gecter sellon les regles de l’art d’arismetique” (fol. 4r). He notes that he will compose with “God’s help, another treatise that deals with the progression for extracting square and cube roots (…) that cannot be practiced with tokens. This is intended for land surveyors and astronomers, so will not be mentioned here.” (“l’aide de Dieu, ung aultre traicté a part c’est assavoir de progression et extraction de radices et quarreures cubic (…) qui bonnement ne se peuvent praticquer par gectoners et aussi se sera pour les geometriencs et estronomiens et ne sera icy faicte aultre mencion”).
- 58.
It has the highest number of occurrences (578) along with BSG. 3143. See the comparative presentation Table 1 in the annex. On the presentation of the Kadran aux marchans, see (Benoit 1989b).
- 59.
“faire aucune chose qui soit aggreable et plaisante”.
- 60.
“te prie mon desire amy que le vueil les revisiter, considerer et entendre”.
- 61.
The request for correction of mistakes and errors is a cliché which is found in a large number of works, such as Jean Adam’s: “[I] beg most humbly that those who discover any omissions please correct them and excuse the errors” (“supplie tres humblement a tous que le verront qui leur plese croiser les obmissions et beguinement corriger et supplier les faultes”). It is also found also in the texts of various types, see (Bourgain 2001).
- 62.
Kadran aux marchans, fol. 1r, “[I] started to set about collecting several other [texts]” (“me suis mys à faire et rassembler de pluseurs autres”).
- 63.
Ibid. “Et pourtant le veulx appeller le kadran aux marchans car tout ainsi que le kadran est guide conducteur et chemin a toutes manieres de gens pour congnoistre la limitacion du temps et du jour, ainsi ce present traicté sera guyde, enseignement et declaracion a tous marchans de bien savoir compter pour justement prandre et donner en vendant et achatant a chascun son loyal droit.”
- 64.
“Et pour ce que la congnoissance de ceste partie est ou doit estre aulx generaulx maistres de monnoyes, changeurs et recevers ou autres gens de finances.” Kadran aux marchans, fol. 67v.
- 65.
Kadran aux marchans, fol. 10r-22v, and for a description of the algorithms we find correspondences in (Van Egmond 2001).
- 66.
This idea is advanced by (Sesiano 1984b) who identified a problem dating from 1460. It is perfectly possible that the work printed in 1492 compiles earlier collections. One of the two master printers, Nicolo Benedeti appears to have published several mathematical works, as well as participating, much later, in Juan de Ortega’s publication in Barcelona. Information on the printers can be read in (Polain 1970).
- 67.
Compendio de lo abaco, fol. 4r: “this treatise will be as concise as is possible for me, for the citizens of Nice are shrewd and have knowledge of all things, and especially the said art” (“losquals tracteray sub brevibus tant coma a mi sera possible, per los citadins de la Citatat de Nisa son subtile et speculatieus en ogni causa et specialment de las dichas arts”).
- 68.
François Baby (1982) proposed Jean Toloza as author of Compendy de l’art d’algorisme, this person was appointed to be teacher by the municipality in 1423.
- 69.
“capitain governador et defendor de la universitat et communa de la ciutat de Pamias”.
- 70.
At the end of the thirteenth century, Boniface VIII endowed Pamiers a studium generale. While the university has never existed, the trustees of the town have always paid great attention to the management of their school. There are still concerns in 1439 over whether the papal bull allowed the teaching of all the sciences. (Lahondès 1883; Baby 1982).
- 71.
“captain governor and defender of the university and the municipality of the city of Pamiers” (“capitain governador et defendedor de la universitat et communa dela ciutat de Pamias.” fol. 16r.
- 72.
Verger (1997).
- 73.
“l’intencion de l’acteur est de en treter cy apres celon ce que il en a peu comprandre en l’art d’arismetique qui continuellement ce nomme chiffre ou agolisme.” BnF. fr. 2050. fol. 1r.
- 74.
A very similar sentence is found in Méd. Nantes 456 produced in the last quarter of the century. “This art will be addressed later according to what the author was able to understand and learn for both written whole numbers and fractions and using an abacus” (“En quel art sera traictié cy apres selon que l’acteur en a peu comprendre et savoir tant de numbre entier que rompu et tant par la plume que par le giet”).
- 75.
For example, he performs a compound interest calculation in which he manipulates a number of 16 figures.
- 76.
- 77.
“Siec se la segonda partita daquesta compendi que parla de nombre ro” fol. 50r.
- 78.
“Lo tertz capitol qui ensenha de multiplicar” fol. 24r.
- 79.
“Com deves ajustar” fol. 19. In the case of Traicté de la praticque and Kadran aux marchans the quality of the copy accentuates this structure.
- 80.
During the development of these problems the Kadran aux marchans qualifies the latter as “request” and proposes a living relationship centered around a question/response dialectic, evoking dispute in a problem. Kadran aux marchans, fol. 40r: “And these questions are difficult to understand, as it may be that the applicant wishes for a solution using the rule of three, and the defendant, or the respondent answers in another way, but it is enough to answer” (“Et tieulx questions sont difficiles à entendre car par aventure peult estre que le demandeur la demandra par la reigle de 3 et le deffendeur ou le respondeur respondra en autre maniere mais il soufist de responder”).
- 81.
Möhren (2006, 98–99).
- 82.
Only Méd. Nantes 456 bears comments from a later reader aiming at attaching the text to other elements of a more humanistic mathematics culture.
- 83.
Benoit (1985).
- 84.
Spiesser (2003b, 188).
- 85.
In this algorithm for multiplication, almost no intermediate result is noted. And it is difficult to implement when the numbers become quite large.
- 86.
Lamassé (2012).
- 87.
That is: “if this is worth this, what is the value of that?”.
- 88.
That is: “multiply and divide”.
- 89.
Ibid.
- 90.
In modern terms, this is a rule of linear interpolation.
- 91.
This approach is particularly well developed in his section on the rule of three, where he presents a series under a general rule and then applications with currency, with the following numerical expressions:
4
7
12
4 + ½
7
12
4 + ½
7 + \( \frac{2}{3} \)
12
4 + ½
7 + \( \frac{2}{3} \)
13 + \( \frac{3}{4} \)
Bnf. fr. 1339 and the Compendio de lo abaco operate in a similar way with other values.
- 92.
Lamassé (2007).
- 93.
It is essentially the content of the manuscript of Cesena, i.e. the Traicté de la praticque, the Compendy en la science des nombres and a third part containing geometry. The Speculative that belongs to the same book does not appear to have had the same success.
- 94.
Chastang (2008, 269).
- 95.
Ibid.
- 96.
Paul Benoit was the first to note that Nicolas Chuquet raised the problem of the relationship of an interest problem with the commercial reality to which it was supposed to correspond. Refusing the solution to this problem provided by other authors, he writes: “thus such calculations(/problems) are without value and I think that there are no partnerships of this kind between merchants “ (ainsi telles raisons sont nulles et croy que entre marchand nulles telles compagnies se font)”. Benoit (1988).
- 97.
For a more precise study on these copies, see (Lamassé 2007).
- 98.
In folios 41r – 43r et 116v – 122r respectively.
- 99.
In folios 279r – 285r et 285r – 288v.
- 100.
The contribution of Alexei Volkov Chap. 10 in this volume proposes a similar reflection by the analysis of lists of problems obtained by the variation of one parameter.
- 101.
Indeed, he reserves a section to interest problems.
- 102.
Triparty en la science des nombres, fol. 280v: “For some the calculation is finished. For such calculations to be valid, one must assume that it is the principal or the capital that earns, and not what [the capital and the interest] earns. However, this is not really so, as the earnings and what the earnings earn for the merchant can be used and increase proportionally like the principal day by day, month by month and year by year” (“Et est faicte la raison selon le stile et opinion d’aulcun. Et a ce que telz comptes puissent estre valables il convient presupposer que le principal ou le chatel tant seulement gangne et non le gaing. Et pour tant qu’il n’est pas ainsi car le gaing et la gaing du gaing faiz en marchandise peuvent profiter et gangner proporcionellement comme le principal de jour en jour de moys en moys et de an en an par quoy il peult entrevenir plus grant gain.”).
- 103.
Traicté de la praticque, fol. 117r: “Note that in this calculation, as for those that follow, it must be understood that the only earnings come from the principal, and not from what the principal earns, because otherwise these calculations could not be done”. (“Et note que en ceste raison et aussi es autres cy apres ensuivant l’on doit entendre que le principal gaigne et non pas le gaing car autrement telles raisons ne se pourroient pas bien faire.”).
- 104.
For example in the Kadran aux marchans: “fair and just right” (“loyal droit”).
- 105.
One can compare the partnership with the contractual problems presented in (Wolff 1954).
- 106.
This name is given to numbers expressed with monetary subdivisions. Jean Certain gives an example of these arithmetical complexities in subtraction exercises, Kadran aux marchans, fol. 6-9r.
- 107.
These manuscripts are presented in (Bompaire 2006). All these money changer manuscripts present problem sets that are quite general in their presentation, as evidenced in mss. Bnf. fr. 1514.
- 108.
This is about a bank placement by a man in favor of his daughter who has just been born, BnF. fr. 2050, fol. 93v.
- 109.
“les questions deputables de la reigle de troys”.
- 110.
Jens Høyrup has tried to write a history of this type of problem (Høyrup 2000, 64–67).
- 111.
Which he justifies in this way: “In the rule of three there are some issues that are more for discussion than for immediate use. To know how to answer these questions I have given you several examples, though not as many as I would like wished, because I have so much desire to keep to the point that I have left them for the end of the book, even if reason demands that they are organized in order [of the rule]. Nevertheless, as I have read that happiness always comes first and is necessary (fol.39r) although I do not put a useful example in the order of the rule through oversight or ignorance, I do put it at the end with the others.” (“En la reigle de troys se font pluseurs questions qui sont plus desputables que proufitables. Et pour savoir respondre à tieulx questions vous en mectray si apres pluseurs et non mye toutes celles que je vouldroye bien car j’ay si grant desir de parfournir les choses principalles que je laisseray partie des accidentalles pour la fin de ce livre, bien que raison feroit que elles fussent par ordre. Neantmoins j’ay ouy dire que le bonnure seroit tousiours premier au plus necessaires et aussi (fol. 39 r ) si je laisse quelque chose neccessaire dire pour leur ordre pour oubly ou non savance je auray matiere de les acompaigner avecques les autres alla fin”).
- 112.
“ilz se font pluseurs manieres de compaignies avecques convenances ou paches expres qui seroient difficilles à juger”, fol. 45r.
- 113.
“[A]utre raisons de compaignies y a qui sont plus disputables que prouffitables”, fol. 46v.
- 114.
Nicolas Chuquet’s remark on the discrepancy between real commerce and the solution proposed for calculating interest points to the shortcomings of arithmetics that circulated at this time.
- 115.
Genthilhomme (2001).
- 116.
In a sense quite close to that used by Kuhn in the expression “normal science”.
- 117.
The latter are particularly well shown in (Labarthe 2005). In a section on false position, BnF. fr. 2050 teaches the calculation of compound interest taking as an example the opening of an account on the birth of a daughter. Without doubt its author shows his command of arithmetic by this, which contributes to his prestige, but it also reveals practices that do not use the usual formalization.
- 118.
In a section on false position, BnF. fr. 2050 teaches the calculation of compound interest taking as an example the opening of an account on the birth of a daughter. Certainly it shows by the same his command of arithmetic, which contributes to his prestige, but it also reveals practices that do not use the usual formalization.
- 119.
A complete description is found in (Spiesser and Féry-Hue 2007).
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Appendices
Appendices
1.1 Appendix 1
Relationship densities for problems shared by the texts (The value “x” is a link indicator explained in Lamassé 2007)
1.2 Appendix 2: Brief Profile of the Corpus
Brief description and comparison of the manuscripts
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Lamassé, S. (2014). Relationships Between French “Practical Arithmetics” and Teaching?. In: Bernard, A., Proust, C. (eds) Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives. Boston Studies in the Philosophy and History of Science, vol 301. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5122-4_6
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