Mathematics in the Early Italian Renaissance

 Angelo Mazzocco, Spanish and Italian
Mark Peterson, Mathematics

Index | Excerpt of Piero | Precis | Trattato d'Abaco | Article on Piero

Excerpt from Trattato d'Abaco by Piero della Francesca ca 1470?

The Rule of the Three Things says you should multiply the thing which the man wants to know by that which is dissimilar, and divide the result by the other; and that which comes out is of the nature of that which is dissimilar, and always the divisor is similar to the thing which the man wants to know.

Example: 7 loaves of bread are worth 9 Libre, what will 5 loaves be?

Do it like this: multiply the quantity which you want to know by the value of 7 loaves of bread, that is 9 Libre, i.e. 5 times 9 makes 45, divide by 7, and what comes out is 6 Libre, remainder 3 Libre; making soldi, they are 60 soldi, divide by 7 and you get 8 soldi remainder 4 soldi; making denari they are 48, divide by 7 you get 6 denari and 6/7. Thus 5 loaves of bread by this reasoning are worth 6 Libre, 8 soldi, and 6 6/7 denari.

The 9 Libre are worth 7 loaves of bread, what are 10 Libre worth?

Multiply the 7 loaves of bread by the 10 Libre, getting 70, divide by the 9 Libre and you get 7 7/9 loaves. Thus you will have for 10 Libre 7 7/9 loaves of bread.

3 1/3 loaves of bread cost 15 Libre, 2 soldi, 3 denari. What will 10 loaves cost?

Do it like this: Multiply 10 by 15 Libre, 2 soldi, 3 denari, getting 151 Libre, 2 soldi, 6 denari; which quantity is to be divided by 3 1/3 loaves of bread. Make them whole numbers, you will have 10 loaves of bread and 453 Libre, 7 soldi, 6 denari; divide first the Libre, which are 453, by 10 you get 45 Libre remainder 3 Libre; making soldi they are
60, and 7 makes 67 soldi, divide by 10 you get 6 soldi remainder 7 soldi; making denari they are 84, and the 6 which there are already make 90, divide by 10 you get 9 denari. Putting it all together you will have 45 Libre, 6 soldi, 9 denari, so much are 10 loaves of bread worth according to this reasoning.

A pound of silk is worth 5 Libre, 3 soldi, what will 8 ounces be? You should multiply 8 ounces by 5 Libre, 3 soldi, get 41 Libre, 4 soldi; to divide by 1 pound would not be right, you should convert to ounces, which for one pound is 12. Thus divide 41 Libre by 12, you get 3 Libre remainder 5 Libre, which are 100 soldi and the 4 soldi which you already have make 104; divide by 12 you get 8 soldi remainder 8; getting denari there are 96, divide by 12 you get 8. And putting it all together: there are 3 Libre, 8 soldi, 8 denari; so much are 8 ounces of silk worth.

A thousandweight of things costs me 87 Libre, I ask how much will 3456 pounds cost? (Exercise for the reader!)


4 companions enter into a partnership; the first enters in the month of January and invests 100 Libre, the second enters in April and invests 200 Libre, the third enters in July and invests 300 Libre, and the fourth enters in October and invests 400 Libre; and they stay together until the next January. They have earned 1000 Libre, I ask how much each one takes for himself?

Do it like this: Suppose first each one earns 2 denari per Libra per month for the time they have been together. The first, who invested 100 Libre, has been in the company for one year, at 2 denari per Libra per month, 100 Libre earn 10 Libre. Now the second, who has been in the company 9 months and invested 200 Libre, at 2 denari per Libra per month, gets 15 Libre. And the third, who has been in the company 6 months, 300 Libre at 2 denari per month per Libra gets 15. The fourth, who has been 3 months, at 2 denari per month, 400 gets 10 Libre. Now do like this:
suppose there are four in the company; the first gets 10 Libre, the second gets 15 Libre, the third gets 15 Libre, the fourth 10 Libre; put together this makes 50, which is the divisor. They have earned 1000, see what each one takes. Multiply 10 by 1000, get 10000, divide by 50 you get 200; so much does the first one take. For the second, multiply 15 by 1000, get 15000, divide by 50 you get 300; so much does the second one take. For the third, multiply 15 by 1000, get 15000, divide by 50, you get 300; so much does the third one take. Multiply 10 by 1000, get 10000, divide by 50 you get 200: so much does the fourth one take. The first takes 200, the third 300, the second 300, the fourth 200.

Three make a company, the first invests 58, the second 87, the third I don't know. They have earned 368, the first takes 86. What will the second one take, what did the third invest, and how much does he take?
(For the reader.)


And now I intend to say a few necessary things about algebra, which treats of fractions and integers and of roots and squares, or of simple numbers. When numbers are multiplied by themselves, then those numbers are called roots, and the products which come out are called squares. And when the numbers are considered neither roots or squares, then they are called simple numbers. Thus according to this definition all numbers are sometimes roots, or squares, or simple numbers. And, of these, algebra gives 6 rules, three simple and three composite. The three simple ones
are when in arithmetical or geometrical questions you have the thing or the root equal to the number, or the squares equal to the things, or the square equal to the number. For when the things are equal to the number, you should divide the number by the things, and what comes out is the thing. And when the squares are equal to the things, you should divide the things by the squares, and what comes out is the thing. And when the squares are equal to the number, you should divide the number by the squares, and the root of that which comes out is the thing. And the composite ones are when the squares and the things are equal to the numbers, and when the squares and the numbers are equal to the things, and when the square equals the things and the number. When the squares and the things are equal to the number you want to compute a square, you halve the things and multiply by itself, and what you get add to the number; and
the root of the sum minus the half of the things is the thing. When the squares and the number are equal to the things, you want to compute a square, and halve the things and multiply by itself and subtract the number; and the radical of the remainder minus the half of the things is the thing. And when the squares are equal to the things and the number,
you should compute a square, and halve the things, multiply with itself and add the number; and the root of the sum plus the half of the things is the thing.

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