|Ivars Peterson's MathTrek|
August 5, 1996
This problem appears in a mathematics textbook known as the Treviso Arithmetic. The original book, written in a Venetian dialect, had no formal title, and its author is unknown. Treviso is the northern Italian city where the book originated in 1478.
Intended for self study and aimed at a broad audience not necessarily versed in Latin, this volume had a very practical bent. Venice, along with its country outpost Treviso, was a major trade center during the fifteenth century, and the book's language, examples, and problems reflected a wide range of commercial concerns.
The book also introduced a "new math," promoting the use of the Hindu-Arabic numeral system and the pen-and-ink computational algorithms that accompanied this notation. They were well-suited to the bookkeeping essential for burgeoning worldwide enterprises and clearly superior to Roman numerals and the abacus for handling daily business dealings.
"As the activities of the merchant profession moved from the limited scope of the itinerant peddler to the entrepreneurship of the international commercial house, preparation for entry into the business world became more prolonged and rigorous," writes mathematics historian Frank J. Swetz in Capitalism and Arithmetic, which includes an English translation of the Treviso Arithmetic.
"A merchant had to be literate, if not in several languages, at least in his own; therefore, boys aspiring to the merchant profession attended the basic grammar schools," he continues. "Then, upon securing a fundamental literacy and numeracy they advanced onward at ages 11 to 12 to a special secondary school to study commercial arithmetic."
The Treviso is the earliest known printed mathematics book in Europe, appearing even before a printed edition of Euclid's Elements. The fact that a book devoted to commercial arithmetic was printed before Euclid "tells much about the real mathematics climate of this time," Swetz comments.
The Treviso Arithmetic begins on a personal, modest note. "I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic," the anonymous author noted. "Therefore, being impelled by my affection for them, and by the value of the subject, I have to the best of my small ability undertaken to satisfy them in some slight degree."
Presumably a teacher, the author then set the stage by echoing words that go back to Aristotle. "All things which have existed since the beginning of time have owed their origin to number." He went on to discuss the five fundamental operations: numeration, addition, subtraction, multiplication, and division.
The rest of the book goes from algorithm to worked example to word problem and solution, step by step cycling to increasingly difficult tasks and combinations of operations.
Problems involving currency, for example, could get quite complicated in those days. For example, merely to subtract the sum of 2820 lire, 4 soldi, 3 grossi, and 27 pizoli from the sum of 8433 lire, 4 soldi, 3 grossi, 27 pizoli, you would need to know that 20 soldi = 1 lira, 12 grossi = 1 soldo, and 32 pizoli = 1 grosso and proceed accordingly. For many other problems, it's useful to know that 24 grossi = 1 ducat.
Investment problems sometimes involved calendar calculations, another type of reckoning replete with quirks. And here's a type of problem that ought to sound familiar: "If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?" Some things never seem to change!
The book concludes with a point-by-point summary of the key facts and formulas that a diligent student ought to remember and use.
Despite the author's best intentions, however, his textbook apparently was not a popular success. Only one edition was ever published. Perhaps it wasn't commercial enough. Perhaps it failed to do justice to the true complexity of the financial transactions, accounting practices, and mercantile activities typical of that period.
Nonetheless, exploring the byways of practical mathematics through the pages of the Treviso Arithmetic is both illuminating and great fun. It provides a fascinating glimpse of arithmetic as it was taught and used centuries ago.
Copyright © 1996 by Ivars Peterson.
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