Islam,Neglect andDiscovery 107
have been made. However, al-Uql ̄idis ̄i does take the trouble to explain his rules where he thinks
it necessary. Why repeat ‘Is, is not, is’ to know where to start in root extraction? Why double
the extracted root before shifting? These questions are answered in book III chapter 6 ‘Queries on
Roots’. The book is in no way an advanced theorem–proof Greek text—but it makes no pretence
to being that. It is a completely practical text on how to do arithmetic with Indian numbers, and
the shadowy al-Uql ̄idis ̄i understands exactly what is required of such a book. We do not even know
whether his text was popular—no other writers refer to it, and it seems to have survived by chance.
There is a great contrast in the comprehensive geometric text written about the same time in
Egypt by ab ̄u-l-Waf ̄a al-Buzj ̄an ̄i. EntitledKit ̄ab f ̄im ̄a yah.t ̄aju ilayhi al-s.ani’min a‘m ̄al al-handasah(The
sufficient book on geometric constructions necessary for the artisan), this has to date only been
published in Arabic and Russian (Ab ̄u-l-Waf ̄a 1966, 1979). It is therefore not a text easily available
to the reader; but it has been considered important by Youschkevitch and Høyrup (who used the
Russian version) and Berggren (who used an extract translated by Woepcke in the 1850s). We have
done our best with the Russian text.
Ab ̄u-l-Waf ̄a was at the other end of the scale from al-Uql ̄idis ̄i; a court mathematician and
astronomer working in Baghdad who wrote (lost) commentaries on the classical works of Euclid
and Diophantus and numerous other works on mathematics, astronomy, and other sciences. That
he thought it useful to devote time to writing textbooks for artisans is the more significant. As ibn
Khald ̄un says, in the passage which immediately precedes the story of Euclid as geometer (which
we quote in Chapter 3):
In view of its origin, carpentry needs a great deal of geometry of all kinds. It requires either a general or a specialized
knowledge of proportion and measurement, in order to bring the forms (of things) from potentiality to actuality in
the proper manner, and for the knowledge of proportions one must have recourse to the geometrician. (Ibn Khald ̄un
1958, II, p. 365)
However, while the world of calculators who might have used al-Uql ̄idis ̄i’s book is fairly easy
to imagine from his text, the artisans who needed the ‘Book on geometric constructions’ seem
more enigmatic. It is clear that ab ̄u-l-Waf ̄a had in mind an actual audience, but he wished to raise
the level:
[M]ethods and problems of Greek geometry...and Ab ̄u-l-Waf ̄a’s own mathematical ingenuity are used to improve
upon practitioners’ methods, but...the practitioners’ perspective is also kept in mind as a corrective to otherworldly
theorizing.
Interesting passages include Chapter 1, on the instruments of construction; and 10.i and 10.xiii, which discuss the
failures of the artisans as well as the shortcomings of the (too theoretical) geometers. (Høyrup 1994, pp. 103, 312)
Indeed, the quote which opens this section is just such a criticism of geometers. As an example
of ab ̄u-l-Waf ̄a’s method, here is the very classical construction of a regular pentagon (Fig. 2).
If someone asks how to construct on the lineABa regular pentagon, then we raise from pointBa perpendicularBC
[toAB] equal to the lineAB. We divideABin half at the pointD, we describe withDas centre and radiusDCthe arcCE,
and we extend the lineABto the pointE. Then we draw arcs with each of the pointsA,Bas centres and with radius
equal toAE. They meet at the pointG. We join the linesAGandBG. We have the triangleABG, which is the triangle
of the pentagon. (Abu-l-Waf ̄a 1966, p. 71–2)
From this point on, the construction is easy (see Chapter 2, Appendix B); AGB is an isosceles
triangle whose base angles are 72◦, and the isosceles triangles BFG and AHG which complete
the pentagon have their short sides equal to AB. There is, as Høyrup remarks, no proof; and the