126 A History ofMathematics
A simple number is any number which may be pronounced by itself without reference to root
or square.
A number belonging to one of these classes may be equal to a number of another class; you may
say, for instance,‘squares are equal to roots’, or ‘squares are equal to numbers’, or ‘roots are equal
to numbers’.
[Al-Khw ̄arizm ̄i then deals with examples of these cases before continuing as follows.]
I found that these three kinds: namely, roots, squares, and numbers, may be combined together,
and thus three compound species arise; that is, ‘squares and roots equal to numbers’; ‘squares and
numbers equal to roots’; ‘roots and numbers equal to squares’.
Roots and squares are equal to numbers: for instance, ‘one square, and ten roots of the same,
amount to thirty-nine dirhems’; that is to say, what must be the square which, when increased by
ten of its own roots, amounts to thirty-nine? The solution is this: you halve the number of the roots,
which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add
this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from
it half the number of the roots, which is five; the remainder is three. This is the root of the square
which you sought for; the square itself is nine.
[...]
[Geometrical demonstration]
We have said enough so far as numbers are concerned, about the six types of equation. Now,
however, it is necessary that we should demonstrate geometrically the truth of the same proposi-
tions which we have explained in numbers. Therefore our first proposition is this, that a square and
ten roots equal thirty-nine units.
The proof is that we construct a square of unknown sides, and let this figure represent the square
which, together with its roots, you wish to find. Let the square, then, beab[Fig. 3.] of which any
side represents one root. Since then ten roots were proposed with the square, we take a fourth
part of the number ten and apply to each side of the square an area of equidistant sides, of which
the length should be the same as the length of the square first described and the breadth two and
a half, which is a fourth part of ten. Therefore, four areas of equidistant sides are applied to the
square,ab. Of each of these the length is the length of one root of the squareaband also the
breadth of each is two and a half, as we have said. These now are the areasc,d,e,f. Therefore, it
follows from what we have said that there will be four areas having sides of unequal length, which
also are regarded as unknown. The size of the areas in each of the four corners, which is found
by multiplying two and a half by two and a half, completes that which is lacking in the larger or
whole area. Whence it is we complete the drawing of the larger area by the addition of the four
products, each two and a half by two and a half; the whole of this multiplication gives twenty-five
(Fig. 7).
And now it is evident that the first square figure, which represents the square of the unknown,
and the four surrounding areas make thirty-nine. When we add twenty-five to this, that is, the four
smaller squares which indeed are placed at the four angles of the squareab, the drawing of the
larger square, calledGH, is completed. Whence also the sum total of this is sixty-four, of which
eight is the root, and by this is designated one side of the completed figure. Therefore when we
subtract from eight twice the fourth part of ten, which is placed at the extremities of the larger
squareGH, there will remain but three. Five being subtracted from eight, three necessarily remains,
which is equal to one side of the first squareab.