- GEOMETRY IN THE MUSLIM WORLD 333
distinction between magnitude and number, they actually operated with quadratic
and quartic irrationals as if they were numbers.
4.1. The parallel postulate. The Islamic mathematicians continued the later
Hellenistic speculation on Euclid's parallel postulate. According to Sabra (1969),
this topic came into Islamic mathematics through a commentary by Simplicius on
Book 1 of the Elements, whose Greek original is lost, although an Arabic transla-
tion exists. In fact, Sabra found a manuscript that contains Simplicius' attempted
proof. The reworking of this topic by Islamic mathematicians consisted of a crit-
icism of Simplicius' argument followed by attempts to repair its defects. Gray
(1989, pp. 42-54) presents a number of these arguments, beginning with the ninth-
century mathematician al-Gauhari. Al-Gauhari attempted to show that two lines
constructed so as to be parallel, as in Proposition 27 of Book 1 of the Elements
must also be equidistant at all points. If he had succeeded, he would indeed have
proved the parallel postulate.
4.2. Thabit ibn-Qurra. Thabit ibn-Qurra (826-901), whose revision of the Ara-
bic translation of Euclid became a standard in the Muslim world, also joined the
debate over the parallel postulate. According to Gray (1989, pp. 43-44), he con-
sidered a solid body moving without rotating so that one of its points Ñ traverses a
straight line. He claimed that the other points in the body would also move along
straight lines, and obviously, they would remain equidistant from the line generated
by the point P. By regarding these lines as completed loci, he avoided a certain
objection that could be made to a later argument of ibn al-Haytham, discussed
below. Thabit ibn-Qurra's work on this problem was ground-breaking in a number
of ways, anticipating much that is usually credited to the eighteenth-century math-
ematicians Lambert and Saccheri. He proved, for example, that if a quadrilateral
has two equal adjacent angles, and the sides not common to these two angles are
equal, then the other two angles are also equal to each other. In the case when
the equal angles are right angles, such a figure is called—unjustly, we may say—a
Saccheri quadrilateral, after Giovanni Saccheri (1667-1733), who like Thabit ibn-
Qurra, developed it in an attempt to prove the parallel postulate. Gray prefers
to call it a Thabit quadrilateral, and we shall use this name. Thabit ibn-Qurra's
proof amounted to the claim that a perpendicular drawn from one leg of such a
quadrilateral to the opposite leg would also be perpendicular to the leg from which
it was drawn. Such a figure, a quadrilateral having three right angles, or half of a
Thabit quadrilateral, is now called—again, unjustly—a Lambert quadrilateral, af-
ter Johann Heinrich Lambert (1728-1777), who used it for the same purpose. We
should probably call it a semi-Thabit quadrilateral. Thabit's claim is that either
type of Thabit quadrilateral is in fact a rectangle. If this conclusion is granted, it
follows by consideration of the diagonals of a rectangle that the sum of the acute
angles in a right triangle is a right angle, and this fact makes Thabit's proof of the
parallel postulate work.
The argument of Thabit ibn-Qurra, according to Gray, is illustrated in Fig. 8.^13
Given three lines /, m, and ç such that I is perpendicular to ç at Ε and m intersects
it at A, making an acute angle, let W be any point on m above ç and draw a
perpendicular WZ from W to n. If Ε is between A and Z, then I must intersect m
by virtue of what is now called Pasch's theorem. That much of the argument would
(^13) We are supplementing the figure and adding steps to the argument for the sake of clarity.