317. In order to give a certain rule for this operation, we must consider attentively the square of the root a + b, which is a2 + 2ab + b2, in order that we may reciprocally find the root of a given square.

318. We must consider therefore, first, that as the square, a2 + 2ab + b2, is composed of several terms, it is certain that the root also will comprise more than one term; and that if we write the terms of the square in such a manner, that the powers of one of the letters, as a, may go on continually diminishing, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is a2, it is certain that the first term of the root is a.

319. Having therefore found the first term of the root, that is to say, a, we must consider the rest of the square, namely, 2ab + b2, to see if we can derive from it the second part of the root, which is b. Now, this remainder, 2ab + b2, may be represented by the product, (2a + b)b; wherefore the remainder having two factors, (2a + b), and b, it is evident that we shall find the latter, b, which is the second part of the root, by dividing the remainder, 2ab + b2, by 2a + b.