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remainder is three. This is the root of the square, and the square itself is nine.
The proceeding will be the same if the instance be, “half of a square and five roots are equal to twenty-eight dirhems;”[1] that is to say, what must be the amount of a square, the moiety of which, when added to the equivalent of five of its roots, is equal to twenty-eight dirhems? Your first business must be to complete your square, so that it amounts to one whole square. This you effect by doubling it. Therefore double it, and double also that which is added to it, as well as what is equal to it. Then you have a square and ten roots, equal to fifty-six dirhems. Now halve the roots; the moiety is five. Multiply this by itself; the product is twenty-five. Add this to fifty-six; the sum is eighty-one. Extract the root of this; it is nine. Subtract from this the moiety of the number of roots, which is five; the remainder is four. This is the root of the square which you sought for; the square is sixteen, and half the (7) square eight.
Proceed in this manner, whenever you meet with squares and roots that are equal to simple numbers: for it will always answer.
