Elisabetta Ulivi's biography of Benedetto da Firenze, a fifteenth century master of the abacus (maestro d'abaco or abacisto), is a welcome contribution to the literature on the abacisti and the arte d'abaco in fifteenth-century Florence and in late medieval and renaissance Italy more generally. Before the publication of Ulivi's work, very little was known of Benedetto. He was called in to take part in renovations of the Palazzo della Signoria as an official measurer in 1475 and Leonardo da Vinci mentioned him in a list of notable persons of the quattrocento in his Codice Atlantico. He is also known as the author of an arithmetic tract dated from 1463 (La reghola de algebra amuchabale, ed. L. Salamone, Siena 1982), but other references to Benedetto were less certain and most of the details of his life remained obscure. Ulivi's biography has now redressed the balance. In the course of her research, she scoured the State Archives of Florence (l'Archivio di Stato di Firenze) and turned up 150 documents on Benedetto and his family, only three of which, those concerning Benedetto's work during the renovations of the Palazzo della Signoria, had evidently been cited previously. Her research has enabled her to construct a detailed history of Benedetto and to attribute to him arithmetic tracts formerly of uncertain attribution or mistakenly attributed to other abacisti.
The book is divided into two parts. The first part, which occupies a less than a third of the book, contains the main text, which functions, in effect, as an introduction to the two appendices that follow in the second part of the book. The text is made up of a brief introduction and six short chapters. The first chapter begins with a discussion of the libro d'abaco, the sort of mathematic tract typical of the later middle ages and early renaissance. Ulivi states that about three hundred libri d'abaco have survived to the present. Most are Italian and above all Florentine, written in the vernacular, and they are for the most part practical in character, dominated by problems of commercial mathematics. The chapter then discusses the scuole d'abaco and the masters of the abacus who taught in them. These schools were private enterprises, often run by two or more masters working together, who specialised in preparing students for a life of work as a merchant or artisan. Twenty schools and some eighty masters of the abacus are attested in attested in Florence from the later thirteenth century to the early sixteenth century. The libri d'abaco written by these masters were intended for the most part not as textbooks for students, but as guides for those who had already studied in a scuola d'abaco.
Chapters two and three trace Benedetto's family history. Benedetto was the last of eight sons of Antonio di Cristofano di Giudo (1381-1464), who made his fortune as a weaver in the silk industry, and Taddea di Domenico di Piero (1387-1470/80). Ulivi dates Benedetto's birth to the beginning of 1429, shortly after his father had purchased a new family house not far from the Piazza del Duomo in the no longer extant Piazza Padella adjacent to the house of the famous Florentine architect Filippo di Ser Brunellesco, better known as Filippo Brunelleschi. The more or less secure dating of Benedetto's birth to 1429 rules out the possibility that the Florentine might have been the 'Benedetto da Firenze' mentioned as a master of the abacus in Brescia in 1436. Benedetto married Pippa dei Tinghi in 1457, but there is no evidence to suggest that the union ever produced children. By the time that Benedetto's father had died seven years later, at least four of his brothers had also died. Benedetto was in fact the principal heir of his father's patrimony. Benedetto himself died evidently intestate in 1479, probably in November, most likely the victim of an epidemic that swept through much of Italy in the later 1470s. His wife followed him to the grave within a few months, certainly by the end of February, also intestate and probably a victim of the same epidemic.
In chapters four and five, Ulivi focuses on Benedetto's work as a master of the abacus and on the Florentine scuole d'abaco in which he practised his profession. Forty-five of the documents on Benedetto uncovered by Ulivi identify him specifically as a master of the abacus. Most of these documents date from 1448 when he evidently assumed the title to his death in 1479, but three are posthumous. From 1448 to 1451, Benedetto taught in the Scuola di Orsanmichele, which until 1448 and again from 1451 onwards was a grammar school in which Latin, logic, and rhetoric were taught. During the period from 1451 to 1469, he taught in the Scuola dei Santi Apostoli, in his own residence in the Scuola di Piazza Padella, and in the Scuola di Santa Maria della Scala. In 1468, while teaching in the latter of these schools, Benedetto was even charged with sodomy, probably by one of his students and, according to Ulivi, probably wrongly. In any event, Benedetto's career survived the accusation and he went on to teach at the Scuola di Corticina dell'abaco for the last decade of his life.
The last chapter of the first part looks at Benedetto's arithmetic tracts. Arrighi attributed to Benedetto one tract in codex L.IV.21 in the Communal Library of Siena (la Biblioteca comunale di Siena) nearly forty years ago. This tract, dated from 1463, was published as La reghola de algebra amuchabale in 1982 by Salamone, who likewise attributed it to Benedetto. Ulivi's analysis of the orthography merely strengthens the attribution of this tract, which, until now, has constituted the extent of Benedetto's known work. Ulivi has nevertheless identified Benedetto as the author of four other works, even if her new attributions perhaps leave some room for doubt. She suggests that two anonymous tracts in codices Palatino 573 and 577 in the National Library of Florence (Biblioteca nazionale di Firenze) were the work of Benedetto. The tract in Palatino 573 was written for a member of the Rucellai family shortly before 22 April 1460 by an author declaring himself to be a student of Domenico d'Agostino Vaiaio. Domenico d'Agostino, whose actual surname was Cegia, was not a master of the abacus but a wealthy furrier specialising in squirrel pelts, as the appellation 'Vaiaio' suggests, who also had an interest in mathematics. A notarial act of 1466 indeed establishes a connection between Benedetto and Domenico, but it does not in any way suggest that Benedetto was a student of Domenico. The tract in Palatino 577, according to Ulivi, was written soon after the tract in Palatino 573. She attributes it to Benedetto on the basis of orthographic evidence, namely the fact that this tract has the same opening as the tract in codex L.IV.21. Ulivi also attributes to Benedetto a tract in codex Ottoboniano Latino 3307 in the Vatican Library (Biblioteca Apostolica Vaticana), which she describes as an abridged version of the two tracts in the National Library of Florence. Finally, she argues that the Tractato d'abbacho in the codex Acquisti e Doni 154 in the Medici Library of Florence (Biblioteca Medicea Laurenziana di Firenze) published by Arrighi in 1974 and attributed to Pier Maria Calandri was also composed by Benedetto.
Ulivi's new attributions, provided they can stand up to scrutiny, raise Benedetto from relative obscurity into the upper echelons of the Florentine masters of the abacus, and for this alone, the work deserves praise. The two appendices that form the second part of the book, and constitute over two thirds of the volume, are especially noteworthy. The first appendix reproduces, whether in part or in their entirety, the 150 documents on Benedetto and his family that Ulivi found in the State Archives of Florence. The second appendix, based mainly on the secondary literature, gives a list of the known masters of the abacus in Florence from the later thirteenth century to the early sixteenth century, roughly sketches the genealogies of a few of them, and identifies the scuole d'abaco of Florence attested from the first half of the fourteenth century through the first three decades of the cinquecento. The book also has an up-to-date bibliography and two indices that cover the text and both of the appendices.
There is little question of the value of this book, but it conspicuously fails to put the Florentine abacisti in their proper historical context. On the really big questions of interest to the historian, for example the role of the Florentine abacisti in the commercial success of Florence and in the so-called "commercial revolution of the middle ages" in general, it is utterly silent. Apart from a few lines in the first chapter, she also largely ignores the broader contours of mathematics teaching in Florence, and what she does have to say on the subject is inadequately referenced. Had she devoted more attention to the wider context and given more space to mathematics teaching in Florence, her biography of Benedetto would probably appeal to a broader audience. As it is, the book will be of interest mainly to historians of mathematics. Even so, anyone interested in the place of mathematics teaching in the commercial revolution will find it useful, and future research on the abacisti of Florence will need to take it into account. The list of Florentine masters of the abacus in the second appendix provides the essential groundwork for further prosopographical research while raising the fundamental question as to how far the activities of these maestri can be pushed back. The first known Florentine masters of the abacus are attested only towards the end of the thirteenth century, but by that time Florence was already the dominant city in Tuscany and Florentines were already establishing themselves as expert administrators throughout Italy and beyond. The so-called tradizione abacistica in Florence perhaps has even earlier origins than the list implies. More research is needed. Still, Ulivi has made a positive contribution to the study of the tradizione abacistica and of the Florentine maestri d'abaco, and her diligent work in the State Archives of Florence will serve as something of a benchmark for future works on masters of the abacus.