Old and New Arithmetic

"Three merchants have invested their money in a partnership, whom to make the problem clearer I will mention by name. The first was called Piero, the second Polo, and the third Zuanne. Piero put in 112 ducats, Polo 200 ducats, and Zuanne 142 ducats. At the end of a certain period they found that they had gained 563 ducats. Required is to know how much falls to each man so that no one shall be cheated."

This problem appears in a mathematics text 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 town 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 15th 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 accompany it. Together, 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," mathematics historian Frank J. Swetz wrote in Capitalism and Arithmetic: The New Math of the 15th Century (Open Court, 1987), 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," Swetz continued. "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 an 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 commented.

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.

Pages from the Treviso Arithmetic (1478), the earliest known example of a printed book on arithmetic.

Problems involving currency, for example, could get quite complicated. 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.

What about Piero, Polo, and Zuanne? The author provides a model solution to the problem of dividing their profits: "In this and all problems of partnership, you set down all of the shares one after the after and find their sum, which becomes your divisor."

Piero put in 112 ducats

Polo put in 200 ducats

Zuanne put in 142 ducats

The sum 454 (Divisor)

You then apply the relevant algorithm to compute that Piero's share comes to 138 ducats. 21 grossi, 11 pizoli and 190/454; Polo's share to 248 ducats, 0 grossi, 13 pizoli and 242/454; and Zuanne's share to 176 ducats, 2 grossi, 2 pizoli and 22/454. Of course, you then check your answer.

"To prove the case for all three, that no one has been cheated, take the sum of the profits of all three," the author instructs. Since these amount to exactly 563 ducats, which is the total given, no one has been cheated."

Despite the author's best intentions, 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.

Originally posted August 5, 1996