F
M
G
Z
H
K
E
A
B
Figure 5.4 Ab¯u K¯amil’s geometric proof that
√ √ √ 9· 4 = 9·4 = 6.
Considered from our modern point of view, the last part of the argument is simple and straightforward. The square EMFZ is 4 and the square EKAB is 9. Hence, if we read proportions as equality of fractions, what Ab¯u K¯amil has shown is that 4/ZHKE = ZHKE/9, or, in other √ words, that the rectangle ZHKE represents the geometric mean of 4 and 9, which is 9·4 or simply 6, as we wanted to show. But this is precisely what Ab¯u K¯amil does not have the tools to do. Ratios are not for him fractions, and proportions are not equalities of fractions. The proportions were not even written in the simple symbolic language that I have used here, but in a completely rhetorical manner: “The ratio of surface EMFZ and surface ZHKE is the same as the ratio of surface ZHKE and surface EKAB.” So, the only tools he had for handling the proportions and deducing the result in a rigorous way (and this is what he wanted to do) were those afforded by the theories of proportion appearing in Euclid’s Elements. Now, in principle, there are two propositions in the Elements that could be used to handle this case, namely, the following: VI.17: If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines are proportional. VII.19: If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
In modern symbols, they can be formulated as follows: VI.17: A:E :: E:D ⇔ AD = E2 . VII.19: A:B :: C:D ⇔ AD = BC (and hence A:E :: E:D ⇔ AD = E2 ).
Although both symbolic expressions would seem to provide the step that was necessary for Ab¯u K¯amil to complete his proof, both propositions (as originally conceived)
104 | NUMBERS IN MEDIEVAL ISLAM