Here Begins the Fourteenth Chapter, On Finding Square and Cubic Roots, and on the Multiplication, Division, and Subtraction of Them, and On the Treatment of Binomials and Apotomes and their Roots

Abstract

Let me insert in this chapter on roots certain necessary results that are said to be key, and they are clearly demonstrated in Euclid’s second book; it suffices to proceed to the definitions of them according to arithmetic [1]. The first of them is with the number separated into parts; the products of the parts by the entire separated number added together are equal to the separated number squared, namely the product of the number by itself. For example, let 10 be parted into 2, 3, and 5. I say that the sum of the products of the two, the three, and the five by the 10, namely 20, 30, and 50, is equal to the product of 10 by itself, that is 100. Also if some number is separated into parts, and each part is multiplied by some other number, and all of the products are added together, they will equal the product of the separated number by the other number; for example, if 10 is separated into the abovesaid parts, and each part is multiplied by some other number, we say 12, and the products are added together, namely the 24, the 36, and the 60, then undoubtedly the resulting 120 will be equal to the product of the 10 and the 12. Also if a number is separated into any two parts, then the product of each part by itself plus double the product of one part by the other equals the square of the entire number; for example, if 12 is separated into 5 and 7, the product of 5 by itself will be 25, and the 7 by itself will be 49, and the double of 5 times 7 will be 70; these numbers added together make 144, namely the product of the entire number by itself.