A History of Mathematics From Mesopotamia to Modernity

Islam,Neglect andDiscovery 127

G

b

H

a

c

e

fd

Fig. 7Al-Khw ̄arizm ̄i’s second picture.

AB F E

CD G

Fig. 8The figure for Th ̄abit’s proof. Compare Fig. 4 (Euclid II.6). ABCD (the way of writing it seems odd, but it is necessary for the

statements to work) is the ‘squares’ of the problem, and rectangle DE (or BEGD) is the ‘roots’. Their sum is the ‘numbers’, and is

known AF=FE.

Appendix B. Th ̄abit ibn Qurra

The first type is this: square and roots equal to numbers. The way of solving it with the help of
the sixth proposition of the second book of Euclid’s elements is as follows. We take for the square a
squareABCD, and letBEbe a multiplicity of units which measures a line, equal to the given number
of roots. [So in the above example,BEis ten units.] We draw the rectangleDE[see Fig. 8]. Then it
is clear that the root isAB, and the square isABCD. In the domain of arithmetic and numbers, it is
equal to the product ofABwith a unit which measures a line. In this way, the product ofABwith
a unit which measures a line is equal to the root in the domain of arithmetic and numbers. ButBE
is such a number, equal to the given number of roots. And so the product ofABwithBEis equal
to the roots of the problem in the domain of arithmetic and numbers. But the product ofABwith
BEis the rectangleDE,asABis equal toBD. So the rectangleDEis itself equal to the roots of the
problem. And so the whole rectangleCEis equal to the square and the roots.
[The point of the repetitions seems to be that Th ̄abit is carefully reminding the reader that we
are working in a framework where numbers can be represented by lines, as they are in Euclids
arithmetic books; or by areas, if we make rectangles out of such lines, as happens in book X.
He has now drawn a figure equal to (square and roots) which, unlike al-Khw ̄arizm ̄i’s figure, is a
single rectangle.]
But the square and the roots are equal to a known number. So the rectangleCEis known, and it
is equal to the product ofAEwithAB,asABis equal toAC. So the product ofEAwithABis known
and the lineBEis known, as the number of its units is known.
In this way, the question leads to a known geometrical problem, namely: the lineBEis known,
it is produced toAB, and the product ofEAwithABis known. But in the sixth proposition of
the second book of the Elements it is shown that if the lineBEis divided in half at the pointF,