Reviewed by Maurice S. Newman University of Alabama

This fascinating window into the world of mercantile capitalism at the end of the fifteenth century, together with its associated arithmetical practices, offers an unusual insight into how the burgeoning trade of the time was enhanced by the use of arabic numbers and printed arithmetic texts. This is accomplished by the somewhat unusual device of a completely translated Italian book, the “Treviso Arithmetic,” within another book that analyzes the arithmetical practices and the cultural background of the times.

The so-called Treviso Arithmetic was, as its principal claim to fame, the first printed arithmetic book. It was printed at Treviso, near Venice, in 1478 and bore no title and no author’s name. The author was most likely a reckoning master, familiar with the computational processes of the time, and writing for a group of pupils or friends about practical uses of these arithmetical concepts. Probably it was not the best book on the subject written around that period but it was the first printed book, it was written in the vernacular for popular consumption, and it was short and to the point. The book was translated by David Eugene Smith, Chairman of the Mathematics Department at Teachers College, Columbia University in 1907 but has not been previously published except for fragmentary references by Smith in articles and speeches.

Picking up where Smith left off, Swetz has published a study of early Renaissance arithmetic based on the Treviso text and has delved deeply into the mathematical and sociological significance of the contents. As Swetz says in his preface, the “study focuses on a book and its contents but, perhaps more importantly, it also concerns a time (the early Renaissance), a place (the Venetian Republic), and circumstances (the rise of mercantile capitalism and the economic beginnings of indus-trialization), and how these three aspects molded and affected the directions of human involvement with mathematics.” [xvii].
Swetz has done his research admirably. His arguments and suggestions are well supported by some twenty-one pages of extensive chapter and notes, together with a bibliography of historical references, a general bibliography, and a good index.

Chapter One sets the stage for the study of discussing the social and intellectual changes that were going on, the growth of towns and cities, the rise of the Venetian Republic, the position of Treviso astride the great overland trade routes, the develop-ment of the Italian Reckoning School, the breaking of the monopoly on intellectual knowledge that came with the printing of texts in the vernacular such as the Treviso Arithmetic, and the subsequent impetus to the rise of a successful middle class.

Chapter Two consists of a short introduction and the free translation by David Eugene Smith of the Treviso Arithmetic in its entirety. The first sixty pages of the Treviso text (pp. 40 to 100) deal with the basic operations of numeration, addition, subtraction, multiplication, and division. The unknown author was selective in his use of methods and was apparently guiding his readers into the best possible approaches to the practical problems that they would face. This part of the text is likely to be useful to students of the history of mathematics particularly as it may relate to such new areas of knowledge as the mathematical operations within computers. The general reader may prefer to skip over several pages and rely on the interpretation supplied by Swetz later in the book. The remainder of the Treviso text deals with practical problems and algorithms that can be useful in their solution. Some of these problems, particularly those involving multiple currencies, could be challenging to students at the high school level. The general reader should read through the problems so as to gain the background for the analysis and discussion in later chapters. The methods of solution may only have appeal to the dedicated historians of mathematics.

In Chapter Three, Swetz begins his interpretation of the Treviso Arithmetic in relation to the environment and times. The required use of Roman numerals was breaking down before the pressure of mercantile needs although acceptance of the arabic numbers was somewhat slow in coming. The algorithms that could be put to use in trading situations were easy to learn and could be used without elaborate equipment such as count-ing boards. The evolution of numerals over five hundred years which is presented in Table 3.1 is a most interesting exhibit.

The remainder of the chapter deals with the basic functions of addition and subtraction as they are given in the Treviso text. This is supported by reference to other texts of contemporary and earlier times to show how current usage had developed. The examples are limited to adding or subtracting two numbers and stress is laid on various methods of proof such as “casting out nines” or use of the inverse operation. A minimum number of illustrative problems is given and it would appear that the author expected the student to learn by doing. The cost of printing at that time would also restrict the author to the basic fundamental requirements of his craft.

Chapter Four deals with multiplication and Chapter Five explains division as found in the Treviso text. Each of these chapters is short and to the point. These were considered difficult to teach in a way that the student could fully understand. The concept of multiplication as repeated addition [p. 67] and division as repeated subtraction [p. 85] as used in the various methods described is insightful when we consider that this is the basic method used by an electronic computer. Again, considerable stress is placed on proving out the work by various methods and Swetz interprets the methods used against an historical background to put them in proper perspective.

Chapter Six describes the types of problems that are used in the Treviso text and categorizes them as follows:

1. The Rule of Three
2. Tare and Tret
3. Partnership
4. Barter
5. Alligation
6. Rule of Two
7. Pursuit
8. Calendar Reckoning

These types of problems were clearly of some importance in the late fifteenth century but would be solved algebraically today if they needed to be solved at all. While most people would not know when Easter or Passover will fall next year, neither do they have any urgent desire to know until those holidays come closer. The problems in connection with the determination of partnership earnings bear a marked similarity, however, to some that have appeared on CPA examinations within recent memory. Barter was obviously a complicated problem when different weights and measures were in use along with multiple currencies and problems of valuation. These still exist although the amount of barter in the commercial world today is relatively nowhere nearly as great.

The final chapter, as its title implies, gives a fascinating glimpse of fifteenth century life, trade, and applied mathematics. Swetz has done an excellent job in recreating the life and times through the analysis and investigation of the problems contained in the Treviso tome. The various commodities that are mentioned in the book are made to reveal the background of the traders, manufacturers, merchants, and eventual purchasers and have them come to life.

The scope of the trade items mentioned in the Treviso is broad, and includes “saffron, pepper, cinnamon, ginger, sugar, wheat, silver, cotton, crimson cloth, French wool, balsam, and wax” [p. 258]. While monetary problems are not too prevalent in the Treviso, Swetz uses other texts of the period to indicate the difficulties in changing from other currencies, the lack of high standards in the coinage of the period, the use of counterfeit coins and the varied weights and measures used in the burgeoning international trade. Swetz points out that Venetian money [p. 271] and Venetian weights and measures [p. 279] became the standard for international trade during that period.

The book closes with a fitting tribute to Professor David Eugene Smith inasmuch as the book probably could not have been written without his prior research interest and translation.