Jacopo da Firenze's Tractatus Algorismi and Early Italian Culture

by

Jens Høyrup

Medieval Italy was unique in Western Europe in having schools to teach ten year old (or thereabouts) boys the basics of merchant mathematics. These `abbaco' schools were first recorded in Bologna in 1265, and by 1280 there were state-supported schools in many of the larger city-states of Northern Italy. Jacopo da Firenze's Tractatus Algorismi is one of the first texts for these schools that we have. Høyrup provides a sketch of these schools, their texts, students, and teachers in pages 27-40, and has an excellent analysis of the transmission of algebra from Islamic sources to the Italian abbaco schools in pages 147-189.

The most interesting part of his discussion of the abbaco schools is his proposed re-interpretation of our understanding of the provenance of these abbaco schools. The traditional view of the abbaco schools is that Fibonacci, having learned algebra and merchant mathematics from his travels thoughout the Mediterranean and from his contacts with Islamic scholars, wrote the Liber Abaci in 1202, and that this book provided a template for merchant mathematics. As the Italian communes of the thirteenth century became more merchant-centered, these city-states saw the value in providing schools that taught their youths the mathematics needed for commerce. Fibonacci's text provided an organizing principle, and so the abbaco schools were born. Høyrup provides a different narrative. He separates three phenomena from the traditional narrative: the mathematics itself, the texts, and the educational institutions. The mathematics of the abbaco schools is the mathematics of a widespread merchant culture that Høyrup describes as extending from Western Europe through Africa and Islamic lands to India, and perhaps to China. The Islamic tools of formal algebra are included in this mathematics, although they seem to be not necessary for the everyday merchant. The texts of the abbaco schools are not, says Høyrup, related to Fibonacci's Liber Abaci . Finally, Høyrup seems to suggest that the abbaco schools, as educational institutions focused on mathematics, were probably modelled on Islamic institutions that had flourished in the recently-conquered Catalonia.

Fibonacci is an interesting person in all of this. He was a scholar at the court in Sicily of Emperor Frederick II in the early thirteenth century (at least through 1225), and he retired to Pisa, where he died around 1250. He is a generation or two removed from the first abbaco teachers. His books, like the Liber Abaci , were written in Latin, and presented to the Emperor. Høyrup suggests that Fibonacci was a cultural hero to those early abbaco teachers--he was a mathematician who gained fame in Southern Italy with the Emperor, and who then returned home to Pisa to finish out his life. Perhaps some of these early abbaco teachers knew him personally. But how much influence did Fibonacci's writings, as opposed to his reputation, have on these teachers? Fibonacci wrote in Latin for the Emperor's court, while the abbaco teachers wrote in the vernacular for merchants. There are questions of language and of class here, as well as a more immediate questions: how many copies of the Liber Abaci were made? and were these copies accessible to abbaco teachers? Høyrup suggests that Fibonacci's texts had little influence on the abbaco schools. He notes that Jacopo da Firenze's Tractatus Algorismi shares very few problems in common with the Liber Abaci , and that Jacopo uses very few Latinisms in his Tuscan text. He concludes, from this as well as from other considerations, that Jacopo drew on a mathematical tradition that was not the Latin tradition represented by Fibonacci.

So where does Høyrup locate the tradition from which Jacopo drew? He argues that Jacopo left Florence to go to Montpellier, to draw upon a Provençal tradition that originally came from Catalan, and before that from the Maghreb. By tracing the history of the presentation of the six cases of quadratic equations and other aspects of the Jacopo's text (like the lack of both Arab expressions and Latin expressions), he excludes Jacopo working directly with either Arabic sources or with Latin translations of Arabic sources. He concludes that Catalonia was the only region of Europe that had sustained an interest in algebra and (at that time) spoke a Romance language that Jacopo could follow. He further argues that Jacopo's text was the only one in circulation that early, and that most of the abbaco tradition follows from Jacopo, and hence from this Provençal/Catalonian tradition. I find Høyrup's argument separating the Liber Abaci from the foundation of the abbaco schools convincing; his argument about the influence of Jacopo seems less convincing. If the abbaco schools started by 1265, and were flourishing by 1280, then there should have been, in Northern Italy, both a mathematical tradition and a textual tradition. Unfortunately, Jacopo's text seems to be the earliest we have; even so, Høyrup's suggestion that Jacopo brought new mathematics back to Italy seems somewhat unsubstantiated. Still, Høyrup presents a convincing argument that Northern Italian algebra does not stem from Fibonacci alone (if at all), but included influences from Iberia and Provençe.

The translation of the Tractatus Algorismi seems to me to be very well-done. He tries to be as literal as possible, and presents the translation in two columns, with the original in the left hand column (including the original diagrams, as redone by Høyrup! an excellent touch) and his translation in the right hand column. The Tractatus Algorismi is a fascinating text, opening with Jacopo's presentation of mathematics as one of the highest gifts of God. Høyrup (pages 45-147) discusses the contents of the text, which start with the numerals and move to multiplication and division, arithmetic of fractions, the Rule of Three, interest and merchant partnership problems, a little geometry, and finishing with algebra and a list of the value of the various coins in circulation in the Mediterranean around 1307. One fascinating aspect of these abbaco texts is the counter-factual problem:

Further we shall say thus, if 5 times 5 would make 26, how much would 7 times 7 make at this same rate.

Høyrup provides details on how he translated the text, and how he came to order the three surviving copies (the Vatican copy, the Florence copy, and the Milan copy) of Jacopo's text from earliest to latest. His main translation is of the Vatican text, and he provides the text (but does not translate) of the Milan copy--the text of the Florence copy has already been transcribed.

I found Høyrup's book to be fascinating and well-researched, and I would recommend it to anyone interested in medieval European mathematics.

482 pages in English, published in hardcover by Birkhäuser Basel 2007.

List Price: 139 dollars.

ISBN-10: 3764383909
ISBN-13: 978-3764383909