Medieval Math Problems

All students will remember having to do math word problems – they existed in the Middle Ages too!


The Liber mahameleth, which has just been translated into English, was one of the largest books on mathematics written in the Middle Ages. It was created in the 12th century by a John of Seville, a Christian living in al-Andalus. He wrote the 23 chapters in Latin, but much of it was based on Arabic mathematical knowledge. After covering various theories of arithmetic, such as how to divide fractions or multiply integers, the Liber mahameleth moves on to deal with math problems related to daily life and doing business: how to share profits, hire workers, buy and sell goods. Here are a few of the math problems the text gives, along with the solutions it provides:


1. A messenger is sent to a town and advances daily by twenty miles. In how many days will another messenger, sent five days later and advancing daily by thirty miles, overtake him?

You will do the following. Take the difference between twenty and thirty, namely ten, as the principal. Next, multiply five by twenty and divide the product by the principal; this will give ten, and so many days did the second messenger walk. The first has been on his way that many days plus five, which is fifteen, when the second messenger met him.

2. A messenger, sent from one town to another at a distance of four hundred miles, advances daily by twenty miles. Another messenger is sent fifteen days later and is to overtake him at the entry of the town so that they walk in at the same time. How many miles must he cover daily?


You will do the following. Multiply fifteen by twenty; this produces three hundred. Subtract it from four hundred; this leaves a hundred. Divide it by twenty; this gives five. Dividing four hundred by this gives eighty, and so many miles does the second need to cover daily so as to overtake the first after five days.

3. A ship moves from one place to another at a distance of three hundred miles. It sails daily twenty miles but is driven back five miles daily by the wind. In how many days will it reach this place?

You will do the following. Add five to twenty; this makes twenty-five. Next, subtract five from three hundred; this leaves two hundred and ninety-five. Divide it by the difference between five, by which it goes back, and twenty, which it sails, that is, by fifteen; this gives nineteen and a remainder of ten. Add this ten to five miles, thus making fifteen. Denominate it from twenty-five; this gives three fifths of a day. Adding these three fifths of a day to nineteen will make nineteen and three fifths of a day, and in so many days will it reach the intended place.


4. A snake comes out of a hole daily by a third of its length and returns daily by a fourth of its length. In how many days will it be out completely?

You will do the following. Multiply the denominators of a third and a fourth, thus producing twelve. Add its third and its fourth, which are three and four; this gives seven, which you keep in mind. Next, subtract three from twelve; this leaves nine. Divide it by the difference between three and four, which is one; this gives nine. Next, denominate three from seven; this gives three sevenths. Adding it to nine makes nine and three sevenths, and in so many days and fraction of a day will it be out completely.

The Liber Mahameleth: A 12th-century Mathematical Treatise, has been edited and translated by Jacques Sesiano, and is published in three volumes by Springer. Click here to read more about the book.



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