Pen-and-Paper Arithmetic Is Useful When You're Selling Textiles

Somebody had to invent those techniques you learned in elementary school.


In 1479, a few months shy of his eleventh birthday, Niccolò Machiavelli left the school where he'd 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, arithmetical techniques, and a dizzying assortment of currency and measurement conversions. Mostly he did word problems like these:

If 8 braccia of cloth are worth 11 florins, what are 97 braccia worth?

20 braccia of cloth are worth 3 lire and 42 pounds of pepper are worth 5 lire. How much pepper is equal to 50 braccia of cloth?

One type of problem reflected the era's shortage of currency. Goods that would sell for one price in coins cost a premium if the buyer paid with other goods. (These problems assume familiarity with trading conventions and therefore present ambiguities to the modern reader.)

Two men want to barter wool for cloth, that is, one has wool and the other has cloth. A canna of cloth is worth 5 lire and in barter it is offered at 6 lire. A hundredweight of wool is worth 32 lire. For what should it be offered in barter?

Two men want to barter wool and cloth. A canna of cloth is worth 6 lire and in barter it is valued at 8 lire. The hundredweight of wool is worth 25 lire and in barter it is offered at such a price that the man with the cloth finds he has earned 10 percent. At what price was the hundredweight of wool offered in barter?

Others were brain teasers dressed up in ostensibly realistic detail.

A merchant was across the sea with his companion and wanted to journey by sea. He came to the port in order to depart and found a ship on which he placed a load of 20 sacks of wool and the other brought a load of 24 sacks. The ship began its voyage and put to sea.

The master of the ship then said: "You must pay me the freight charge for this wool." And the merchants said: "We don't have any money, but take a sack of wool from each of us and sell it and pay yourself and give us back the surplus." The master sold the sacks and paid himself and returned to the merchant who had 20 sacks 8 lire and to the merchant who had 24 sacks 6 lire. Tell me how much each sack sold for and how much freightage was charged to each of the two merchants?

Along with their famed humanist arts and letters, the mercantile cities of early modern Italy fostered a new form of education: schools known as botteghe d'abaco. The phrase literally means "abacus workshops," but the instruction had nothing to do with counting beads or reckoning boards. To the contrary, a maestro d'abaco, also known as an abacist or abbachista, taught students to calculate with a pen and paper instead of moving 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. Brought up in North Africa by his father, who represented Pisan merchants in the customs house at Bugia (now Béjaïa, Algeria), the young Leonardo learned how to calculate using the nine Hindu digits and the Arabic zero. He was hooked.

After honing his mathematical skill as he traveled throughout the Mediterranean, Fibonacci eventually returned to Pisa. There he published the book that enthusiastically introduced the number system we use today.

Fibonacci's novel methods of pen-and-paper reckoning were ideal for Italian textile merchants, who wrote lots of letters and needed permanent account records. Beginning in Florence in the early fourteenth century, specialized teachers began teaching the new system and producing handbooks in the vernacular. Consistent sellers, the books served simultaneously as children's textbooks, merchants' reference tools, and, with their brain-teasing puzzles, recreational materials.

From the abacists' classrooms, future merchants and artisans typically graduated to apprenticeships and work. But a grounding in commercial math was also common for those like Machiavelli, who were destined for higher education and a career of statesmanship and letters. In a society based on trade, cultural literacy included calculation.

As they drilled generations of children on how to convert hundredweights of wool into braccia of cloth or to allocate the profits from a business venture to its unequal investors, the abacists invented the multiplication and division techniques we still use today. They made small but important advances in algebra, a subject universities scorned as too mercantile, and devised solutions to common practical problems. On the side, they did consulting, mostly for construction projects. They were the first Europeans to make a living entirely from math.

In his seminal 1976 study of nearly 200 abacus manuscripts and books, historian of mathematics Warren Van Egmond emphasizes their practicality—a significant departure from the classical view of mathematics, inherited from the Greeks, as the study of abstract logic and ideal forms. The abacus books treat math as useful.

"When they study arith­metic," he writes, "it is to learn how to figure prices, compute interest, and calculate profits; when they study geometry it is to learn how to measure buildings and calculate areas and distances; when they study astronomy it is to learn how to make a calendar or determine holidays." Most of the price problems, he observes, concern textiles.

Compared to scholastic geometry, the abacus manuscripts, with their problems about trading cloth for pepper, are indeed down to earth. But they don't scorn abstraction. Rather, they wed abstract expression to the physical world. The transition from physical counters to pen-and-ink numerals is in fact a movement toward abstraction. Symbols on a page represent bags of silver or bolts of cloth and the relationships between them.

Students learn to ask the question, How do I express this practical problem in numbers and unknowns? How do I better identify the world's patterns—the flow of money in and out of a business, the relative values of cloth, fiber, and dyes, the advantages and disadvantages of barter over cash—by turning them into math? Mathematics, the abacists taught their pupils, can model the real world. It does not exist in a separate realm. It is useful knowledge.

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  1. I can see that I’m going to have to get this book.

    1. That’s why she’s here 🙂

  2. “(These problems assume familiarity with trading conventions and therefore present ambiguities to the modern reader.)”

    Your “modern reader” must have gotten a lot fewer story problems in math class than I did. These look pretty ordinary to me.

    1. Yes, the “brain teaser” is a straight forward “2 equations in 2 unknowns” problem. The other problems are just ratio (or proportion) problems.

      Though the discipline of algebra was hundreds of years old in Machiavelli’s time, these problems can be solved without algebra. Techniques for solving these types of problems are probably thousands of years old.

      Using algebra makes solving these types of problems a lot easier. The easier a solution to a problem is the less likely a mistake will be made. Fewer mistakes are very valuable to a merchant.

      1. Oh, I’m sure the classes then were very useful. I’m just saying this is not that different from modern “story problems” in any middle school math class. Very like the ones my 12 year old son is seeing.

        In fact, I might get him this book for Christmas, he’d probably like it.

        1. I agree! These are problems I give to my students. Of course, I dress them up in modern clothes, not in Renaissance trappings.

    2. I think that comment refers to the example in the next paragraph – “A canna of cloth is worth 5 lire and in barter it is offered at 6 lire.” The interpretation of that sentence is not immediately obvious. A modern reader would assume that a unit of cloth is worth the same 5 lire whether for direct sale or in barter. Even after thinking about it at length, I’m struggling to figure out why the barter value in the example is higher than for direct sale value.

      1. Hypothesis – the in-barter value was higher because, assuming you really want the thing being bartered for, you would be able to avoid the transaction costs of making a separate purchase. That contradicts most modern experience where a) transaction costs are relatively low and b) the in-barter value is lower because the thing you are getting is less fungible than money.

        Regardless of whether my hypothesis is true, clearly there’s some sort of trade convention that must be understood for the problem to make sense. Sure, you could grind the numbers without knowing that convention but you won’t really understand the answer.

        1. I think you have this backwards. I think when the author says ““A canna of cloth is worth 5 lire and in barter it is offered at 6 lire,” it means that the price of the cloth will be higher if in barter than if the buyer is paying cash (coin). Makes sense.

          1. You’re selling cloth, and you want lire. If the person buying it offers you wine in return, instead of lire, you ask for 6 lire worth of wine instead of 5 lira currency, because you’ll incur some expense and inconvenience converting the wine to lire.

            1. Of course, then you have things like triangular trade, which really make it interesting.

  3. This book is, I think, in the same vein as Mark Kurlansky’s wonderful books Cod and Salt. I enjoyed them quite a lot. Perhaps the author should have simply called this Cloth.

  4. “The Fabric of Civilization: How Textiles Made the World”

    Beer, not textiles.

    1. No, the beer didn’t make the world, they were too busy playing with the cantaloupe and Roman the buffalo.

  5. “They made small but important advances in algebra, a subject universities scorned as too mercantile”

    Huh, some things never change

  6. An interesting fact about Libra Abaci is that while it was published in 1202, it wasn’t until 2002 that the first complete English translation appeared. 800 years!

    From wikipedia:

    “The first appearance of the manuscript was in 1202. No copies of this version are known. A revised version of Liber Abaci, dedicated to Michael Scot, appeared in 1227 CE.[7][8] There are at least nineteen manuscripts extant containing parts of this text.[9] There are three complete versions of this manuscript from the thirteenth and fourteenth centuries.[10] There are a further nine incomplete copies known between the thirteenth and fifteenth centuries, and there may be more not yet identified.[10] [9]

    There was no known printed version of Liber Abaci until Boncompagni’s Italian translation of 1857. [9] The first complete English translation was Sigler’s text of 2002.[9]”

  7. All this is quite interesting, but again I wonder why trade in textiles is different from trade in other goods in this respect.

    Merchants need arithmetic, whatever they deal in. “How many of this size container of olive oil is the same as one of those,” “What’s the wheat-to-wine exchange ratio?” etc.

    1. You could probably write a similar book about several other articles of trade, but textiles do have the characteristic of being non-perishable and having a very wide market.

      Salt might be a reasonable comparison, but, excluding today’s trade in specialty salts, salt is pretty much interchangeable, while fabrics are diverse.

      1. Sure, but the non-perishability doesn’t seem all that relevant to the need for arithmetic. You need that for trade in any kind of goods.

        What I’m wondering is whether there particular aspects of textile trade that make it special in that regard.

        1. Non-perishability may well be relevant in that due to transport speeds, trade in perishable goods was generally local, while non-perishable goods were transported and traded over long distances.

          The long distance trade complicates tracking inventory and revenues and who is buying what and how much.

          Non-perishable goods are also more likely to be sold on “credit”.

        2. What makes textiles different from jewels, spices and tea is that Virginia Postrel isn’t selling a book about jewels, spices and tea.

  8. Trump did not succeed (entirely) in implementing a “Muslim ban” on immigration, but Prof. Postrel has successfully banned Muslims from her history of mathematics and fabrics.

    1. That’s an insulting and worthless comment. Dr. Postrel hasn’t written a history of mathematics, and she has mentioned the origins of the math involved in fabric trade appropriately.

      Why don’t you peddle your vitriol elsewhere.

      1. No, she throws around terms like “algebra” but only mentions the contributions of Europeans. Before that she jumped around from Assyria to Greece to Africa. She makes much of a 1203 Italian publication which a couple of centuries before Arabic (ahem) numerals were widely used by Europeans. In fact we owe more to Arabs than to anyone else for the development of mathematics and our numeric system.

        1. The wiki History of Algebra page seems to make the same mistake, skipping around from Babylon to Ancient Egypt to Greece to China to India to the Islamic world to Europe.

          Fun trivia from that article: ‘Rhetorical algebra’ was used until pretty late in the game, i.e. “equations are written in full sentences. For example, the rhetorical form of x + 1 = 2 is “The thing plus one equals two” or possibly “The thing plus 1 equals 2″. Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.”

          That sounds pretty tedious…

        2. That’s baloney. You should brush up on your history. FYI, “arabic” numerals were invented in India, along with zero.

          The “muslim” contributions are minor in comparison to those of India, Greece, Babylonia,

          BTW, regarding “muslim” contributions, most of the important pillars of math were established before the birth of Mohammed.

        3. “…she throws around terms like “algebra” but only mentions the contributions of Europeans. ”

          She used the word “algebra” exactly once in the post, in this phrase:

          “They made small but important advances in algebra” (referring to the abacists.

          What’s your problem with that?

  9. Reading this fascinating post reminded me of the general topic of the pragmatic aspect of measurement [see for example ].

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