Capitalism and Arithmetic: The New Math of the 15th Century

This literary gem by Frank Swetz focuses upon the first English translation of the Treviso Arithmetic of 1478, which is the earliest known example of a printed book on mathematics. The Treviso itself bore no title, and no author’s name. Its adopted title is solely derived from its place of origin — the little town of Treviso, which lay on a major trading route in Northern Italy.

Written in the common Venetian dialect of the 15^th century, the Treviso was intended for merchants who wished to learn methods of computation other than those based upon Roman numerals, abaci and counting boards. Such archaic modes of commercial arithmetic were inadequate for the book-keeping and accountancy needed to sustain the rapidly growing mercantile practices of the Venetian empire.

As with many innovative mathematical ideas, there was a long delay in the incorporation of the Hindu-Arabic numerals into everyday European mathematics. And, because nearly all scholarly ideas were presented in Latin, they were particularly inaccessible to those whose livelihoods arose from commercial activities.

Another stultifying factor was the power of the ‘reckoning masters’, who were keen to maintain their lucrative monopoly of the accountancy business, along with its use of counting boards, abaci and Roman numerals. Even 200 years later in Shakespeare’s time, much commercial arithmetic was still based upon such inefficient methods.

Let me see. Every ‘leven wether tods: every tod yields pound and odd shilling; fifteen hundred [sheep] shorn, what comes the wool to?

I cannot do’t without counters. (The Winter’s Tale).

The Treviso offered a crash course of arithmetic based upon our Hindu-Arabic system of numeration. The associated computational techniques were introduced and illustrated by variety of hypothetical trading transactions like the (translated) example below.

Three merchants 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 Zuanne142 ducats. At the end of a certain period, they had gained 563 ducats. Required is to know how much falls to each man so that no one shall be cheated.

Other examples concern calendric calculations, weights and measures, profit margins on spice, sugar, silk and other luxury goods for wealthy members of 15^th century European society. Clues to this are given by another instructional example from the Treviso:

Two merchants wish to barter. One has 1 pexo of balsam worth 150 ducats. He wishes trade this for wax at 5 ducats per hundredweight, sugar at 6 ducats a hundredweight and ginger at 8 ducats a hundredweight. He wishes the same amount of one substance as the rest of the three kinds of merchandise. Required is how much he will have of each.

Although the translated version of the Treviso Arithmetic occupies only 140 pages of Capitalism and Arithmetic, the remaining 200 pages provide commentary on the ‘new arithmetic’ and the business practices of the early Renaissance. It specifically examines the course of trade, prices, systems of money, how people dressed and what they ate. In fact, as claimed on the publisher’s blurb, Capitalism and Arithmetic provides greater understanding of the historical interaction between mathematics and the social milieu of the late medieval period

And even if readers aren’t particularly interested in the history of arithmetic and number systems, the warmly elegant prose with which Frank Swetz portrays the mathematical and social influence of the Treviso Arithmetic will make this book compulsive reading. In addition, there are lovely illustrations based upon woodcuts of medieval commercial activities. There is also an abundance of notes and literary references.

Finally, I have to say that I’m really puzzled as to why this book has remained for 25 years hidden and unread in one of my bookcases. Consequently, I can’t complain that it hasn’t previously been subject to an MAA Online review.

Peter Ruane, having left secondary school unable to cope with fractions and decimals, subsequently worked through the 1906 edition of Pendlebury’s ‘Shilling Arithmetic’ whilst employed as a stevedore on the Liverpool docks. Here is one example from that book: ‘Express £3 13s 4½d as the decimal of a £.’ A non-Pendlebury exercise requires one to work entirely within the system of Roman numerals to find the number of days in XXVII years.