OF ARITHMETIC, BOOK 10

On the equations of simple chapters. Chapter III.

The evaluation evaluation: estimatio, referring to the numerical solution or value of the unknown of the thing is the quantity in which we experience the truth of the matters proposed in the chapter and the question. An example is when someone says: I made two parts out of 10, and I multiplied each of them by itself, and the difference of the products was 60. Since we do not know what quantity might be larger or smaller, we will set the smaller to be the unknown thing, which we call the position position: positio, the standard term for the unknown variable $x$; therefore, the larger part will be the remainder of 10, namely 10 minus 1 position. Then we follow what is proposed, and we multiply the parts by themselves: the square of the smaller becomes 1 square,

and [the square] of the larger is 1 square plus 100 minus 20 positions. Add that which is "minus" to the other part; it will become 1 square plus 100 on one side, and 1 square plus 20 positions [on the other]. The difference between these was 60 by hypothesis; we will therefore add 60 to the smaller part, and then they become equal: 1 square plus 100, and 1 square plus 20 positions plus 60. We cast out 1 square and 60 from each side; there will remain, therefore, 20 positions equal to 40. For if equals are taken away from equals, those which remain are equal This is a reference to Euclid's Common Notions.. By dividing 40 by 20 (the number of positions), the result will be 2, the evaluation of the position. In this 2, therefore, we experience the truth of the proposed question: for if its square, which is 4, is cast out from 64 (the square of 8, the remainder of 10 minus 2), the proposed Number 60 is left. It is also true of 2 what is proposed in the chapter: namely, that its square, which is 4, along with 100, is equal to the square of the position (which is again 4) and 20 positions (which are 40) and 60 joined together.