In What Manner the Square Root May Be Extracted from Numbers

We shall now see in what manner the square Root may be extracted from numbers. Therefore, the number whose Root we wish to extract being written down, we shall begin from the first figure placed on the right side of the said number, putting a point beneath it; which done, leaving the figure that follows, we shall put another point beneath the third, and so beneath the fifth in order, always leaving one figure between, when there are many.

Then, beginning from the last point placed on the left side, we shall find a square number that is equal to all the number found from that point backward toward the left — or that is nearest to it, provided it does not exceed it; whose Root we shall place beneath the figure at the point, and whatever remains we shall always place above this number.

Beyond this, we shall double the Root that was placed beneath the point; and what arises we shall place beneath the figure that follows immediately after that point on the right side, accommodating the figures one after another toward the left. This done, we shall see how many times the double of the Root is contained in that number which is placed above the Root and its double; and the result — which will be the Root of another square number — we shall place beneath the following point, multiplying it by the result of the doubling, and taking the product from the number placed above.

But one must observe that there remain a number equal to the square number of this Root, so that, the one being subtracted from the other, nothing remains; for then we shall have exactly the true square root of the proposed number, which will be contained among the roots of the Squares that are placed beneath the points. And if there should remain a number greater than the Square, then one could have only the irrational and surd Root, in the manner I have demonstrated elsewhere; and it will be needful to have recourse to continuous Quantity, operating in the manner that I am about to show in the second part.

And because it is very difficult to treat this matter in the universal, we shall therefore come to a particular example, that what has been said may be understood. Let us suppose, then, that we wished to extract the square Root of 1225.

I say, first, that we must place a point beneath the first figure placed on the right, which is the 5; then, leaving the second that follows, we shall make another point beneath the third — that is, beneath the 2; which done, we shall find a square number equal to, or a little less than, 12, and it will be 9, of which 3 is the Root. This we shall first accommodate beneath the point placed on the left side — that is, beneath the 2; then we shall take the 9 from 12, and 3 will remain, which we shall place above the pointed 2, joining it with the non-pointed 2, and we shall have 32. Doubling now the Root — that is, the 3 placed beneath the point — we shall have 6, which we shall accommodate beneath the non-pointed 2; and we shall see how many times it is contained in 32, and it will be 5 times, and 2 will remain over. This, then, joined with the pointed 5, gives 25; which, being equal to 25 — the square number that arises from 5, which is its Root — gives us exactly that which is sought: that is, the Root, which will be 35. We shall therefore place this second Root beneath the pointed 5; and, taking from 32 the 30 that arises from the multiplication of this Root by the double of the first, there will remain 2, which, with the pointed 5, makes 25 (as we have said); and so, taking from this the 25 that is the second square number, nothing will remain; and we shall have exactly the square root of the proposed number, which, as I have said, is 35, found beneath the points in the example set below — since 35 multiplied by itself renders exactly 1225, which is its Square.

[Editorial note: Here Zarlino’s original contains a figure showing the scratch-work of the extraction. The number 1225 is written in a row, with points beneath the 5 and the leftmost 2; the successive remainders (3, then 0) stand above and the doubled partial root (6) below; and at the foot, labelled “Radice quadrata … del proposto numero” (the square root of the proposed number), the result is set out digit by digit beneath the points — 3, 5 — that is, 35.]