336 11. POST-EUCLIDEAN GEOMETRYFIGURE 10. Al-Kuhi's angle trisection.
Al-Haytham also attempted to prove the parallel postulate. According to
Gray (1989, p. 45), the argument given by al-Haytham in his Commentary on
the Premises to Euclid's Book The Elements, and later in his Book on the Res-
olution of Doubts was based on the idea of translating a line perpendicular to a
given line in such a way that it always remains perpendicular. The idea is that the
endpoint of the line must trace a straight line parallel to the directing line.
4.5. Omar Khayyam. In his paper "Discussion of difficulties in Euclid" (Amir-
Moez, 1959), the Persian mathematician Omar Khayyam (1048-1131) raised a
number of questions about al-Haytham's argument. He asked how a line could move
while remaining perpendicular to a given line, and more generally, how geometry
and motion could be connected. Even admitting that Euclid allowed a line to
be generated by a moving point and a surface by a moving line, he pointed out
that al-Haytham was requiring something more in demanding that one line remain
perpendicular to another at each instant during its motion.^16
Having refuted al-Haytham's proof, Omar Khayyam himself attempted a proof
(Amir-Moez, 1959) based on a proposition that he claimed Aristotle had proved: If
two lines converge, they will (eventually) intersect. This claim raises an interesting
question, since as we have seen, Aristotle did not accept the arguments given by
scholars in Plato's Academy to prove that parallel lines exist. Given his disbelief
in a completed infinity, he probably would have liked an argument proving that
(^16) Omar Khayyam's objection is right on target from the point of view of modern physics. If the
special theory of relativity is correct, no sense can be attached to the statement that two events
occurring at different places are simultaneous. One observer may find them so, while another does
not agree. The same objection applies to Thabit ibn-Qurra's argument, which assumes a rigid
body. In special relativity rigid bodies do not exist. What al-Haytham did was to ignore all points
from the moving solid except those lying along a certain line. The relation between motion and
geometry lies at the heart of relativity theory.