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Friday, November 15, 2013

Capitalism & Arithmetic: The New Math of the 15th century

by Jamie Lutton

I came across a book about the early history of Arabic numbers in the West, a subject I had not thought that much about before.
The evolution of Arabic numerals

The book is Capitalism & Arithmetic: The New Math of the 15th century, annotations and notes by Frank J. Weitz, translation by David Eugen Smith.

 It is the annotated text of Treviso Arithmetic, a Northern Italian math textbook published in 1478, written by an anonymous math institutor. a very basic math book designed for young students who were not familiar with Arabic numerals.

In the West, educated people knew about Arabic numbers for 500 years before this book was written, but it took that long before a  climate developed for their general adaption. That environment was the world of the Venetian and other Northern European merchants, and the need for very accurate record keeping in business transactions, such as currency conversion and figuring interest. This added to the explosion of printed books after 1460, and the interest in and desire to be able to compute with Arabic numbers..

This book is an Incunabula, or "cradle book," one of the first printed books from the early days of printing, just a generation after Fust and Gutenberg.    Frank J. Weitz published it, with annotations and a brief background  in the introduction and the afterword of this text. He illustrated the book profusely with with amazing woodblocks from the era.

The push to adopt Arabic numerals came from commerce. This part of Italy was where merchants made fortunes in trade of cloth, dyes, spices, and other goods.

Only with Arabic numbers was double-entry book accounting possible.   Also how to derive interest on a loan, and how to convert different currencies into each other, so that money from one country could be converted to another, which is vital for commerce across  borders.  Loaning money was also how banking houses made great fortunes, as money was loaned to buy goods overseas.

This ancient textbook uses story problems, very similar to ones in modern 21st century math books in public schools. For example, on page 138:

Three merchants have invested their money in a partnership, which to make the problem clearer I shall mention them by name. The first was named Piero, the second Polo, and the third Zuanne. Piero put in 112 ducats,Polo put in 200 ducats, Zuanne 142 ducats. At the end of a certain time they found that they had gained 360 ducats. Required is to find how much falls to each man so that no man shall be is cheated. 

 Sound familiar? This anonymous teacher's book from 1478 resembles current textbooks closely. this method of teaching math has been handed down unchanged for hundreds of years to teach arithmetic, subtraction, multiplication and division, plus fractions as in the example word problem.

There were adequate competing computation systems in place, using Roman numerals, before this, but  quick computations were difficult. roman numerals do not have a place holder for zero, a new concept in Arabic numerals.

This was the most revolutionary concept of all.

We can see the use of Roman numerals still, in dating books and public buildings and documents.

Also,  the use of the abacus, which persisted in the Far East up to my lifetime. This text  illustrated very well how those competing systems worked, such as 'counting tables' using the abacus, and even what they looked like, using illustrations from the time. There is even an illustration showing a completion between the two system, with the Goddess Arithmetic presiding over it.

What I found fascinating was the beginning of the text, in teaching students what a number is, from scratch.  Children in the modern era are taught numbers from when they are infants, the concept of what a number is is absorbed early. This book demonstrates how to teach what numbers are philosophically,
I can see why Frank Sweitz and Eugene Smith were both fascinated by this ancient textbook. 
For fans of medieval history, who have run out of exciting books to read, put this book on your list. Or those who wish to read about the history of math. This book would make an invaluable text to those who want to make math more exciting when teaching adults, or children in an advanced class of math.

If you don't understand math, you can be cheated very easily by the unscrupulous you employ, either to invest your money, or to do your books, or even the bank you deal with for a mortgage.


  1. Most of the arithmetic books contains story-line to serve the problem and this is kind of good approach as it motivates the reader to solve the problem and improve his skills in maths. Thank you Jamie for telling the name of the book. I'll read it.

    HCBL Co-Operative Bank

  2. Can anyone tell me the origin of the wood cut illustrations?
    They are amazingly appropriate, so much so I wondered if they were skilful fakes... I do hope not!

  3. It's from Margarita Philosophica: