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In this manner, whether the squares be many or few, (i e. multiplied or divided by any number), they are reduced to a single square; and the same is done with the roots, which are their equivalents; that is to say, they are reduced in the same proportion as the squares.
As to the case in which squares are equal to numbers; for instance, you say, “a square is equal to nine;”[1] then this is a square, and its root is three. Or “five squares are equal to eighty;”[2] then one square is equal to one-fifth of eighty, which is sixteen. Or “the half of the square is equal to eighteen;”[3] then the square is thirty-six, and its root is six.
Thus, all squares, multiples, and sub-multiples of them, are reduced to a single square. If there be only part of a square, you add thereto, until there is a whole square; you do the same with the equivalent in numbers.
As to the case in which roots are equal to members; for instance, “one root equals three in number;”[4] then the root is three, and its square nine. (5) Or “four roots are equal to twenty;”[5] then one root is equal to five, and the square to be formed of it is twenty-five. Or “half the root is equal to ten;”[6] then the
