  Euclid’s fifth “parallel” postulate states that if a line falls on two given lines such that the two interior angles add up to less than two right angles, then the given lines must meet on that side. This statement is equivalent to several more easily understood assertions, such as: there is exactly one parallel to a given line that passes through a given point; or, the angles of a triangle add up to two right angles. It has been known since the 19th century that there are non-Euclidean geometries that violate these properties; indeed, it is not yet known whether the space in which we live satisfies them. The parallel postulate, however, was not subject to doubt at Khayyam’s time, so it is more appropriate to think of Islamic efforts in this area as part of the tradition of improving upon Euclid rather than as the origin of non-Euclidean geometry. Khayyam’s reconstruction of Euclid is one of the better ones: he does not try to prove the parallel postulate. Rather, he replaces it with two statements, which he attributes to Aristotle, that are both simpler and more self-evident: two lines that converge must intersect, and two lines that converge can never diverge in the direction of convergence. Khayyam then replaces Euclid’s 29th proposition, the first in which the parallel postulate is used, with a new sequence of eight propositions. Khayyam’s insertion amounts to determining that the so-called Saccheri quadrilateral (one with two altitudes equal in length, both emerging at right angles from a base) is in fact a rectangle. Khayyam believed his approach to be an improvement on that of his predecessor Ibn al-Haytham because his method does not rely on the concept of motion, which should be excluded from geometry. Apparently Naṣīr al-Dīn al-Ṭūsī agreed, since he followed Khayyam’s path a century or two later.