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.
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
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
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?
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