In 1479, a few months before his eleventh birthday, Niccolò Machiavelli left the school where he had learned to read and write and went to study with a teacher named Piero Maria. The future author of The Prince spent the next twenty-two months mastering Hindu Arabic numerals, arithmetic techniques, and a dizzying array of currency and measurement conversions. Mostly he made word problems like these:
If 8 Braccia fabric is worth 11 guilders, what is 97 Braccia worth?
20 braccia cloth is worth 3 lire and 42 pounds of pepper is worth 5 lire. How much pepper is equivalent to 50 Braccia fabric?
One type of problem reflected the currency shortage of the era. Goods that would be sold for a price in coins cost a premium if the buyer pays in other goods. (These problems assume familiarity with trade conventions, and therefore confuse the modern reader.)
Two men want to exchange wool for cloth, one has wool and the other has cloth. A fabric canna is worth 5 lire and is bartered for 6 lire. A hundredweight of wool is worth 32 lire. What should it be bartered for?
Two men want to swap wool and cloth. A fabric canna is worth 6 lire and 8 lire in barter. The hundredweight of wool is worth 25 lire and is bartered at such a price that the man with the material realizes that he has earned 10 percent. At what price was the hundredweight of wool bartered?
Others were brain teasers that were disguised with seemingly realistic details.
A merchant was across the sea with his companion and wanted to travel by sea. He came to the port to leave and found a ship on which he put a load of 20 sacks of wool and the other a load of 24 sacks. The ship began its voyage and set out to sea.
The captain of the ship then said, “You have to pay me the freight charges for this wool.” And the merchants said: “We have no money, we take each of us a sack of wool and sell it, pay ourselves and give us the surplus.” The master sold the sacks and paid himself, and returned to the dealer who had 20 sacks and 8 lire and the dealer who had 24 sacks and 6 lire. Tell me how much each sack was sold for and how much freight was billed to each of the two dealers?
Along with their famous humanistic arts and letters, the trading cities of early modern Italy promoted a new form of education: schools known as Botteghe d’abaco. The term literally means “abacus workshops,” but the instruction had nothing to do with counting beads or accounting for boards. On the contrary, a maestro d’abaco, also known as an abakist or abbachist, taught students to do math with pen and paper instead of moving the counters on a board.
The schools took their misleading name from the Liber Abbaci, or Book of Calculation, published in 1202 by the great mathematician Leonardo of Pisa, better known as Fibonacci. The young Leonardo was raised in North Africa by his father, who represented Pisan merchants in the customs house of Bugia (now Béjaïa, Algeria). He learned to calculate with the nine Hindu digits and the Arabic zero. He was addicted.
After Fibonacci improved his math skills on his trip across the Mediterranean, he eventually returned to Pisa. There he published the book in which the number system used today was enthusiastically introduced.
Fibonacci’s novel pen-and-paper accounting methods were ideal for Italian textile retailers who wrote a lot of letters and needed permanent business records. Starting in Florence in the early fourteenth century, specialized teachers began teaching the new system and producing manuals in the local language. Consistent sellers, the books also served as textbooks for children, as a reference work for dealers and, with their tricky puzzles, as leisure material.
Future merchants and craftsmen usually completed apprenticeships and work in the classrooms of the abacists. But even in Machiavelli, destined for higher education and a career in statecraft and letters, basic training in business mathematics was common. In a society based on commerce, cultural literacy included computation.
It was while teaching generations of children how to convert hundreds of weights of wool into braccia of cloth or how to allocate the profits of a business venture to its disparate investors, the abacists invented the multiplication and division techniques we still use today. They made small but important advances in algebra, a technical college that was despised for being too commercial, and developed solutions to common practical problems. In addition, they mainly advised on construction projects. They were the first Europeans to make a living solely from math.
In his seminal 1976 study of nearly 200 abacus manuscripts and books, the historian of mathematics Warren Van Egmond emphasizes their practicality – a significant departure from the classical view of mathematics inherited from the Greeks as a study of abstract logic and the ideal shapes. The abacus books treat math as useful.
“When you study arithmetic,” he writes, “you learn how to calculate prices, calculate interest, and calculate profits; if you study geometry, you learn how to measure buildings and calculate areas and distances; if you study astronomy.” is to learn how to create a calendar or determine holidays. “Most of the price problems concern textiles.
Compared to school geometry, the abacus manuscripts are actually down-to-earth with their problems when exchanging material for pepper. But they do not despise abstraction. Rather, they associated an abstract expression with the physical world. The transition from physical counters to pen numbers is indeed a move towards abstraction. Symbols on one side represent sacks made of silver or cloth bolts and the relationships between them.
Students learn to ask the question: how do I express this practical problem in numbers and unknowns? How can I better identify the patterns of the world – the flow of money in and out of a business, the relative values of fabrics, fibers, and dyes, the pros and cons of bartering over cash – by converting them to math? The abacists taught their students that mathematics can model the real world. It doesn’t exist in a separate area. It is useful knowledge.