Humanism

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Far more sophisticated textbooks on mathematics than the Tagliente family produced were available in print throughout the long sixteenth century. Some authors concerned themselves exclusively with bookkeeping practice. Others aimed at a more systematic arithmetic course. At the upper end, there were also mathematics-theory treatises, which will not for the most part concern us here, except insofar as they served as sources for the textbook-level works. (19)

We know that an extensive, multi-year course in commercial mathematics was available to urban students all over Northern Italy, and that it was fairly standardized by the mid-fifteenth century if not also before. Tagliente's books were too sketchy to have fit this course, but other textbooks did. The very first surviving printed book for teaching commercial math is the so-called Treviso Arithmetic of 1478. It is an altogether more substantial book than those of Tagliente and at the date it probably was intended as a teacher's manual. The Treviso book presented a basic, overall treatment of what we know was the standard arithmetic course in reckoning schools. Although primarily concerned with teaching practical skills, it covered all the basic arithmetic concepts and functions. It does not seem to have been reprinted. (20)

More influential by far than the Treviso Arithmetic was the nearly contemporary course book of Pietro Borghi, usually titled simply Book on Commercial Math (Libro de abacho). It saw at least seventeen editions between the first known one in 1484 and the end of the sixteenth century. Early editions advertise Borghi's ambition to provide a business-school course, "Here begins the noble work On Arithmetic, in which all things concerning business practice are treated." Later versions are more specific about the elementary character of the work, by way of distinguishing it from more substantial books. In 1517, for example, Melchiore Sessa and his partner Pietro Ravani advertised the Borghi with a bit of doggerel: "To the guy who really likes his maths / (first step on higher learning's paths): / Before you cross that brook, / you better read this book." (21)

Contrast this simple, down-market advertising for Borghi's elementary math book with that attached to the De arithmetica of Filippo Calandri (dates unknown), which appeared at Florence in 1492. Although Calandri covered virtually the same material and claimed a similar audience of "artisans and merchants," he and his printer took a much more traditionalizing, humanistic approach to the look of the book. There is no title page, but on the page facing the preface there appears a majestic, seated teacher in a classroom with students. The scene is labeled, "Pythagoras, Inventor of Arithmetic." The book then begins with a salutation that embedded the title of the work in a supposed network of patronage: "A small work of Filippo Calandri On Arithmetic, [dedicated] to Giuliano di Lorenzo de' Medici." The preface goes on to claim preeminence in commerce for Florence and to claim that the author went to a great deal of trouble to make sure the book was printed in "the Florentine style." The exact meaning of lo stile fiorentino is unclear, but it may very well have to do with the elegant dress of the book by comparison to the homely look of earlier printed arithmetics. Calandri is also given to dressing up his exposition with classical and patristic citations. Thus, he cites Lionardo Pisano Fibonacci (1170-1250), the usual source, on the subject of Arabic numerals, but he also adduces St. Jerome and Juvenal to the matter. (22)

The increasingly popularizing dress of Borghi's book reflected a change in the usages of theorists, math teachers, and business people alike between 1470 and 1520. In exactly this period most authors finally left Roman numerals behind in favor of Hindu-Arabic ones. Well into the sixteenth century, however, textbook authors recognized that the new system of notation was less familiar to many readers and required some new habits -- strict adherence to columns, for example. Printing was key to the rapid progress of mathematics education. It was easier to teach notation systematically when every student had a nearly identical printed textbook than when the class worked from handwritten examples. By the third decade of the sixteenth century, this idea that each student in a class had an identical (or at least closely similar) textbook was taken for granted. (23)

The increasing availability of printed texts also meant that the kinds of math books available multiplied. Two main types are easy to distinguish by style and level of detail. Primarily theoretical works like that of Luca Pacioli framed their discussions academically with humanist commonplaces about the value of mathematics. Their authors dwelt at length on definitions of terms and concepts and put readers through long series of problems. Some problems required working with enormous numbers, offered, it would seem, primarily to provide practice in computing that would reinforce theoretical concepts. Similarly, the theory books included many methods for solving the same problems. These were offered both as methods of computing and also as "proofs" for checking the accuracy of complicated computations. By contrast, more practically-oriented textbooks like Borghi's Libro de abacho simplified the definitions or omitted them entirely; offered fewer problems; never required computing with numbers larger than those commonly encountered in business; and gave a single method of computing and annotating each problem with just one or two kinds of "proof." (24) In addition to these real, substantial textbooks at various levels, there was also a form of abaco book that was just a prompt or aide-memoire. A Perugia Abbaco of 1573 is typical. Its eight leaves contain tables for multiplication, division, and currency exchange and, on the last leaf, eight simple narrative problems. Booklets like this might have been given to very elementary students while their teachers used a more formal textbook.

Like other early arithmetics for the reckoning schools, Borghi's presented the elementary mathematical functions first, but not in the order to which we are now accustomed. He began with multiplication, and then described division, addition, and subtraction. This order is so regular in early Italian books of abaco that it must have been the usual way of teaching the functions in reckoning schools at least until the fifteen twenties, even though the theoretical writers strongly recommended starting with addition. (25) Borghi's treatment of multiplication was typical. He presented thorough multiplication tables and then demonstrated their application with practical problems of increasing difficulty. The layout of the page assumed that the student would be referring back to the multiplication tables (or that they had been memorized) for each step of a problem. Pacioli had demonstrated no fewer than eight ways of visualizing a given multiplication problem on paper, because he expected his readers to comprehend all the various ways of arriving at a solution. Borghi just wanted his students to get to an answer and then move on to the next problem. (26)

NOTES

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• (19) It is important not to confuse the abaco or reckoning-school course and its textbooks with the sophisticated abaco treatises or with humanist mathematics; for the distinctions, see especially Van Egmond 1986, 59-66. Futher on the elementary course, Black 2007, 52-54, 162-163.
• (20) The Treviso book, printed in folio, is about three times the length of the quarto Luminario or the octavo Thesoro. A facsimile edition is Abbaco 1995; Swetz 1987 is a full translation. See also Smith 1908, 3-7. That it was published in the same place and year as the first known Babuino cannot be a coincidence; they were part of the same commercial course. On the standard course: Arrighi 1968; Goldthwaite 1972; Van Egmond 1976; Lucchi 2000; on Florence, Witt 1995, 93, 111-113.
• (21) Smith 1908, 16-22. Compare Borghi 1517: Chi de arte matematiche ha piacere / che tengon di certeza el primo grado / avanti che di quelli tenti el vado / vogli la presente opera vedere. This advertisement would translate more literally, "Whoever would take pleasure in mathematics / (which holds the first step to certain knowledge), / before he crosses the river into higher maths], / he'll want to read this book." It is repeated in the Bindoni edition of 1540.
• (22) Calandri 1491, 2v-3r. A reprint in similar typographic style appeared in 1518. See also Jaffe 1999, 33-35.
• (23) Hobart and Schiffman 1998, 120-123.
• (24) Jackson 1906, 45-46, 61-65, 183-184; Davis 1960, 29-31; Antinori 1994, 13-16; Ciocci 2003, 177-189.
• (25) Jackson 1906, 94-95.
• (26) Jackson 1906, 61-64, 184; Ciocci 2003, 190-193. Cataneo 1559, 7v still strenuously commended memorizing the multiplication tables, as if others might have thought it was not needed.