To make sense of almost any human endeavor examine the reward system – follow the money. Health care, politics, college sports, you name it – the knowledge of who is getting what to do what, explains a lot. However prospects for increased wealth are not a major incentive in the area of modern mathematics. Money or for that matter fame are not major drivers of mathematical discovery, witness Grigori Perelman.
Yet consider Europe’s adoption of the Hindu-Arabic numeration system and its algorithms in the 1300’s and 1400’s as described in the book, Capitalism & Arithmetic, The New Math of the 15th Century by Frank J. Swetz. At the center of this volume is an English translation of the first printed arithmetic book – The Treviso Arithmetic written in 1478 in the Venetian vernacular. Surrounding the arithmetic text lies an explanation in modern terms of the various algorithms but more importantly a description of the cultural milieu of Venice in the 15th century. Treviso is a town just north of Venice, Italy that lay on several trade routes, was an industrial center, and a nexus for Venetian merchants’ country homes. Venice at the time was the ascendant commercial center of Europe lying as it lies between Europe and Asia with access to the Mediterranean. Thus Shakespeare’s Merchant of Venice. An immense volume of trade and all its concomitant transactions required among other things accurate money-changing and division of profits not to mention good accounting methods. This created a demand for people (boys and men) who could calculate rapidly with no mistakes and driven by a need for efficiency, Venetian merchants adopted the faster and more documentable Hindu-Arabic calculation system. Europeans were soon flocking to Venice to learn commercial arithmetic. Although other parts of Europe had schools for mathematics, the math was taught too theoretically. In Venice the students were generally 12-16 year old boys of middle-class background. After they left the school, they would apprentice to a merchant or join their merchant fathers in the business. Accounting skills apparently were learned on the job.
These examples from the text indicated why the rise of capitalism created a demand for Hindu-Arabic mathematics.
If 1000 pounds and one-fifth of cinnamon are worth 130 ducats and one-quarter, what are 14616 pounds, 9 ounces, 5 sazi and one-third worth? p. 123
Three men, Tomasso, Domenego, and Nicolo, entered into partnership. Tomaso put in 760 ducats on the first day of January, 1472, and on the first day of April took out 200 ducats. Domenego put in 616 ducats on the first day of February, 1472, and on the first day of June took out 96 ducats. Nicolo put in 892 ducats on the first day of February, 1472, and on the first day of March took out 252 ducats. And on the first day of January, 1475, they found they had gained 3168 ducats, 13 grossi and one-half. Required is the share of each, so no one shall be cheated. p. 139
A merchant has 40 marks of silver containing 6 ounces and one-half fineness per mark. He has 56 marks of another kind containing 5 ounces of fineness per mark. He wishes to make these into coin containing 4 ounces and one-half of fine silver per mark. Required is to know the amount in the mixture and the amount of brass added. p. 155
The Treviso Arithmetic ends with an assortment of problems similar to elementary algebra “word problems” – work, distance, and mixture problems and also calendar problems. For accuracy all work was checked using casting out nines. Various algorithms for the elementary operations are given. Good algorithms saved paper – expensive in those days. For instance in subtraction instead of borrowing from the next digit in the minuend, one is added to the next digit of the subtrahend.
The major value of Capitalism & Arithmetic, The New Math of the 15th Century is the sense it gives of the vitality of early European capitalism. It explains by inference the architectural richness of Venice as we think of it today.