A History of Mathematics From Mesopotamia to Modernity

Islam,Neglect andDiscovery 113

A

KL

EGF

HM

CBD

Fig. 4The diagram for Euclid proposition II.6. The line AB is bisected at C(AC=CB), and BD is added. If now AK=BD, then ‘the

rectangle AD by DB’ means the area of the rectangle ADMK; and this, together with the square on CB (which equals the square

LHEG) is said to be equal (in area) to the square CEFD on CD. The proof is fairly obvious.

the chance of finding either translation in a library is slim.^8 However, it is a very interesting

document. Th ̄abit was one of the groups of Greek translators, and much of his prolific work

expanded on Greek texts, commenting or dealing with problems which they raised. Here he uses

his knowledge to draw on Euclid’s proposition II.6 (for the case above described) and prove—in

some sense—that the formula is the right one. Unfortunately, unlike al-Khw ̄arizm ̄i, he does not

have an easy style, at least here.

Proposition II.6, in its particular form, says:

Let the straight line AB be bisected at the point C, and let a straight line BD be added to it in a straight line (see Fig. 4)

I say that the rectangle AD by DB together with the square on CB equals the square on CD.

Those who believe that the results of book II should be interpreted as a form of algebra interpret
this by saying: call AB ‘a’ and BD ‘b’; then BC=a/2, and CD=b+(a/ 2 ); the proposition says that:

(a+b)b+

(a

2

) 2

=

(

b+

a

2

) 2

(1)

It is now on the whole thought unhistorical to claim that Euclid was thinking in such terms (see
the remarks on this in Chapter 2). However, there is evidence that the Islamic translators of Euclid
at some stage did come to use some sort of algebraic translation—after all, they now had algebra
to help them. In the early tenth century the philosopher al-Far ̄ab ̄i wrote that the rational numbers
correspond to the rational quantities, and the irrational numbers to the irrational quantities (cited
Youschkevitch 1976, p. 169). The distinction between numbers and lengths, which sometimes
seemed so important to the Greeks, was being eroded, and in commentaries by Arabic writers on
Euclids books V and X (which exercised them greatly) we can find many similar examples. Th ̄abit
does say that he is investigating the case ‘square and roots equal to numbers’; but it is typical of his
more hurried and more abstract approach that he gives no numbers as examples. You can find his
argument in Appendix B. The end-result (of the extract) is that the root which we are looking for
is ‘known’—in classical geometric terms, it can be constructed.

8. Luckey’s German version, with the original text and his discussion, is in Sezgin (ed.) 1999, pp. 195–216.