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Masters, questions and challenges in the abacus schools

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Abstract

The mathematical scenario in Italy during the Late Middle Ages and the Renaissance is mainly dominated by the treatises on the abacus, which developed together with the abacus schools. In that context, between approximately the last thirty years of the fourteenth century and the first twenty years of the sixteenth century, the manuscript and printed tradition tell us of queries and challenges, barely known or totally unknown, in which the protagonists were abacus masters. We report in this work on the most significant examples and draw out interesting cues for thoughts and remarks of a scientific, historical and biographical nature. Five treatises, written in the fifteenth and sixteenth century, have been the main source of inspiration for this article: the Trattato di praticha d’arismetricha and the Tractato di praticha di geometria included in the codices Palat. 573 and Palat. 577 (c. 1460) kept in the Biblioteca Nazionale of Florence and written by an anonymous Florentine disciple of the abacist Domenico di Agostino Vaiaio; another Trattato di praticha d’arismetrica written by Benedetto di Antonio da Firenze in 1463 and included in the codex L.IV.21 kept in the Biblioteca Comunale of Siena; the Tractatus mathematicus ad discipulos perusinos written by Luca Pacioli between 1477 and 1480, manuscript Vat. Lat. 3129 of the Biblioteca Apostolica Vaticana; and Francesco Galigai’s Summa de arithmetica, published in Florence in 1521.

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Notes

  1. See for instance Bortolotti 1927, 169–170; Arrighi 1970, 2004, 225–236; Galuzzi et al. 2012, 36–40, 44–68.

  2. From the vast bibliography on the subject cf. Bortolotti 1927, 171–180; Masotti 1962; Gabrieli 1986; Toscano 2009.

  3. On abacus schools and abacus mathematics, cf. Franci 1998; Ulivi 2002a, b, 2008.

  4. Transcribed and published by Calzoni and Cavazzoni: cf. Pacioli 1996.

  5. Arrighi 1965, 397–398; Arrighi 2004, 156–157.

  6. “Maestro Giovanni di Bartolo began teaching around 1390. He was young, just like his teacher Antonio when he died. After his teacher’s demise, although he was only 19, some of his and Antonio’s friends helped him and convinced him to open the same school. Since he was young, although he was erudite and owned many books, he was little known to the other teachers who, moved by the envy that reigns among those who perform the same activity, and most of all among those who are teaching at present, after discussing among themselves in which way they could discourage him from his intent, decided to proceed as follows. So, believing that, due to his age, Giovanni was not expert enough, each of them selected the best students from his school: about 25 from Maestro Michele’s school, expert in various subjects, and almost as many from the school where Maestro Luca taught, although very little, and most of all his teacher, Maestro Biagio, taught. They said to each of them: it came to our knowledge that a young disciple of Maestro Antonio’s has reopened the school that he had when he was alive, and since we think that among you there is someone better than him, today you shall go follow his lessons, showing him your ability, so that he will decide to do something else. Obedient to such orders, those disciples went to the school of Maestro Giovanni who was surprised but immediately understood the reason for so many students. However, calling them one by one, he explained to all of them the subject they requested, and he did it so well that in that short time they felt they had learned much more than they had so far from the other teachers; and many of them ended up by bad-mouthing their first teachers, since they understood the deep envy they felt for him”.

  7. For more detailed information on the aforesaid schools and masters, see Ulivi 2004a, 2013.

  8. Arrighi 1967a, 401–402; Arrighi 2004, 165–166.

  9. “I understand that to you it seems a great fact that cubi, censi and cose can be equalised to the number, since in the past everybody thought it impossible; I am telling you that every equalisation can be defined and, if the procedure wasn’t very long and difficult, I would explain it to you in this letter; however, I am sending you what you have requested”.

  10. In the fourteenth century, four treatises were known reporting formulas for the solution of third-degree equations that were completely wrong: Paolo Gherardi’s Libro di ragioni, written in Montpellier (1328: BNF, Magl. XI, 87), the anonymous Trattato dell’alcibra amuchabile (c. 1365: BRF, Ricc. 2263), the treatise Aritmetica e geometria by Gilio da Siena (1384: BCS, L.IX.28), the anonymous Libro merchatantesche (1398: BCP, 2QqE13). The Aliabraa argibra, written in Pisa by some Maestro Dardi in 1344, found in different codices, and the aforementioned Libro merchatantesche propose formulas for complete third- and fourth-degree equations that are exact, but only within specific problems. The same formulas can be found in many texts of the fifteenth century. On this subject, see Franci 2013.

  11. “The cube root of 84, decreased by 5 cose, is 4; because 5 cose are 20 that, subtracted from 84, gives 64, whose cubic root is 4”.

  12. Arrighi 1967c, 774; Arrighi 2004, 208.

  13. “Ten problems to Giovanni de’ Bicci de’ Medici that said Giovanni sent to him in the past to pass them on to some expert mathematicians, who at that time had schools in this city”.

  14. “No less Lionardo Pisano clearly demonstrates that square numbers have certain properties, by which one can immediately find the solutions to the questions asked on them”.

  15. “Find a square number such that, whether you add or subtract a number, remains a square number”.

  16. See Franci 1984.

  17. Cf. Ermini 1971, 185; Ulivi 2002b, 125–126.

  18. Two more questions included in Palat. 573 and in L.IV.21 concern abacus masters from out of Florence and their relations with Florentine abacus schools. The two problems, both solved algebraically, have no particular mathematical significance; therefore, we will provide only some notes of a historical nature. The manuscript Palatino’s problem was posed by Filippo de’ Folli da Pisa to Maestro Giovanni di Bartolo in 1399 (Arrighi 1967a, 404; Arrighi 2004, 166–167). This problem is the only problem we know by Maestro Filippo. The Folli, from an illustrious family from Pisa, taught at least in 1398–1399, cooperating with his junior Iacopo di Maestro Tommaso, one of the five abacists active in Pisa in the fourteenth century who belonged to the lineage of the Bonagi dell’Abaco, perhaps the same of Fibonacci’s (Ulivi 2011, 258–271). The question of the manuscript L.IV.21 was sent to Florence by a master from L’Aquila in 1445, even if it is included by Benedetto da Firenze in the “chasi exenplari alla regola dell’algebra di Maestro Biagio”; said Biagio, called “il vecchio”, had died around 1340, after being master and collaborator of the renowned Paolo dell’Abaco, probably in the “Bottega di Santa Trinita” (Arrighi 1965, 377; Arrighi 2004, 136; M\(^{\circ }\) Biagio 1983, 116–119). As of today, the problem of the anonymous master from L’Aquila is the only statement that provides evidence of the existence of abacus schools in the capital city of Abruzzo, at least in the mid-fifteenth century.

  19. Bertanza and Dalla Santa 1907; Ulivi 2002b, 131–132; Ulivi 2008, 407.

  20. BNF, Palat. 573, cc. 138r-138v; Arrighi 1967a, 400 and 421; Arrighi 2004, 165 and 181.

  21. “Astrologists believe that the Earth has a circumference of 20400. Where two men start from the same point, and one goes east, while the other goes west. And on the first day, one travels one mile, the second day 2 miles, and the third day 3 miles, and so every day the first man has travelled one mile more than the previous day. And the second man travels one mile on the first day, 8 miles on the second, and so every day he travels the next cube number, that is 27 miles on the third day and 64 miles on the fourth. The question is: in how many days will they meet? Marco Trevigiano sent this question to Florence in 1372. And, although there are various methods, I follow this one ...”.

  22. See, for example, Loria 1982, 280; Gavagna 1999, 302–304; Gavagna 2007, 126–128; Giusti 2007, 172–173.

  23. Summa, I: distinctio secunda, tractatus quintus, 38r and 44r.

  24. “Do as I say; assume that they meet in a number of days equal to 1co.; hence, in such time, the first would have travelled \(\frac{1}{2}ce\). and \(\frac{1}{2}co\). And in that time, the second would have travelled \(\frac{1}{4}ce.ce.\) and \(\frac{1}{2} cu.\) and \(\frac{1}{4}ce.\) ...”.

  25. On the symbolism used by Pacioli, see also the next question.

  26. The German mathematician Michael Stifel, in his Arithmetica integra of 1544 (306r-307v), re-examining the problem from Cardano’s work, solved it algebraically, like Pacioli and Tartaglia, but with 44310 instead of 20400 so as to obtain the solution \(x = 20\). Soon after, observing that \({P}_2 (x)\) is the square of \({P}_1 ( x )\), without developing the two expressions, solved the \({P}_1 (x)+{P}_2 (x)=44310\) by changing the variable \(( {x^{2}+x} )/2=y\), thus avoiding the fourth-degree equation and finding x through two second-degree equations.

  27. Calvesi 1992, 18–19; Calvesi 2000, 40.

  28. Cf. Berra 2005, 125–133, 342. Oddly, it is precisely Berra, in producing these documents and only three more notary deeds where some Gabriele Aratori is mentioned, who believes “invece evidente” that the Gabriele mentioned by Cardano “non fosse in alcun modo parente del nonno materno del Caravaggio perché il suo nome non compare mai, neppure marginalmente, associato a quello dei familiari del pittore”.

  29. The first, gone missing, dates back to 1470 and was dedicated to the children of the merchant Antonio Rompiasi, of whom Luca was the tutor during his first stay in Venice.

  30. BAV, Vat. Lat. 3129, 359v-360r; Cavazzoni 1998, 594–595; Derenzini 1998, 187, 189. Note that the passage is crossed out with a stroke of the pen.

  31. “Find 3 proportional numbers so that the square of the third is equal to the sum of the squares of the other two, and if you multiply the first number by the second the result is 10; I am asking what the value of each number is. Mind you, this is a good question sent to me on the 4th April 1480 from Florence by Maestro Giovanni Sodi at the hand of Giovan Giacomo, goldsmith in Perugia, and we gave him a very exact answer, and to other questions too, and then we sent him, in our turn, other questions that so far have not been answered, etc.”.

  32. Leonardo Pisano 1857, 1862, I, 447–448.

  33. Actually, in the manuscript, the author proceeds in a rather erratic way, placing them as exponents or above the relevant coefficient.

  34. Summa, I: distinctio sexta, tractatus sextus, 93r.

  35. Piero della Francesca 2012, I, Testo e note, XIX, L-LI. We notice that in the vernacular version of Piero della Francesca’s Libellus quinque corporum regularium, which is part of the Divina proportione, published in 1509, Pacioli again used a symbolic notation limited to the first, second and fourth power, but different from both that of the Tractatus and that used by Piero in the original text: cf. Piero della Francesca 1995, I, Testo e note, XXXII and XXXIV. At the end of the second part of the Summa, in the “Particularis tractatus circa corpora regularia et ordinaria”, although taken from the geometrical part of Piero’s Trattato d’abaco, Luca obviously follows the symbolism that is used in his entire text.

  36. The “Capitoli d’arcibra” occupy the folios 124r-127r of the manuscript and are preceded by a treatise by Matteo di Niccolò Cerretani, written with a different handwriting: cf. Van Egmond 1980, 78.

  37. BMLF, Ash. 353, 124v.

  38. On Raffaello Canacci, Giovanni dei Sodi and Francesco Galigai cf. Ulivi 2004b, 125–134, 162–166; Ulivi 2012.

  39. Summa de arithmetica, 71r-71v. Cf. Franci and Toti Rigatelli 1985, 68.

  40. Summa, I: distinctio sexta, tractatus sextus, 86v-89v.

  41. Summa de arithmetica, 22r, 25r-27r.

  42. “When you wish to divide 11 in three continued proportional parts, find each part. This is the question that Maestro Agnolo dal Carmine posed to me, without telling me in which proportion he wanted it; I solved it in the double proportion”.

  43. Leonardo Pisano 1857, 1862, I, 181; Giusti 2002, 91.

  44. Cf. Pacioli 1996, 384–385 and 393. On the problems of chapter “Divisioni e partimenti di numeri” cf. Heeffer 2010.

  45. “Divide 14 in three continuously proportional parts so that, by multiplying each of them by the sum of the other two, the summed up products give 112; I am asking said quantities”. In the second formulation the numbers 14 and 112 are replaced by 20 and 160 respectively.

  46. Mazzinghi 1967, 15–16; Franci 1988, 245.

  47. Summa, I, 89v and 91r.

  48. BNF, Palat. 573, 477r-477v; Arrighi 1967b, 22–23.

  49. “Find 4 continuously proportional quantities so that the sum of the first and fourth is 18, and the sum of the second and third is 12; I am asking how much will each be”.

  50. Calandri 1982, 32–33; Høyrup 2007, 116–118, 327–328.

  51. Summa, I, 87v; Franci 1990, 25; Høyrup 2007, 117–119.

  52. Summa, I, 96v.

  53. BNF, Magl. XI.120, 5v-6r.

  54. Cf. for instance Richa 1989, 10, 92; BNF, Magl. XXV.398, 95.

  55. His name does not appear in the Baptismal Records of the Opera di Santa Maria del Fiore in Florence.

  56. On Angelo Catastini and his family cf. Negri 1973, 42; Neri di Bicci 1976, ad vocem; Giovannini and Vitolo 1981, 88, 96–98; Verde 1973–2010, 6, ad vocem. ASF, Catasto 66 (year 1427), 179v; Catasto 652 (year 1447), 19r-19v; Catasto 793 (year 1458), 297r-298r.

  57. ASF, Corporazioni religiose soppresse dal governo francese 113, 20, 152r-152v.

  58. Ulivi 2002a, 196–198.

Abbreviations

ASF:

Archivio di Stato, Firenze

BAV:

Biblioteca Apostolica Vaticana

BCP:

Biblioteca Comunale, Palermo

BCS:

Biblioteca Comunale, Siena

BMLF:

Biblioteca Medicea Laurenziana, Firenze

BNF:

Biblioteca Nazionale, Firenze

BRF:

Biblioteca Riccardiana, Firenze

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  1. Dipartimento di Matematica e Informatica U. Dini, Viale Morgagni 67/a, 50134, Florence, Italy

    Elisabetta Ulivi

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  1. Elisabetta Ulivi
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Correspondence to Elisabetta Ulivi.

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Communicated by: Menso Folkerts.

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Ulivi, E. Masters, questions and challenges in the abacus schools. Arch. Hist. Exact Sci. 69, 651–670 (2015). https://doi.org/10.1007/s00407-015-0160-1

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