The History of Mathematics: A Brief Course

4. GEOMETRY IN THE MUSLIM WORLD 337

converging lines must intersect. Although none of the writings now attributed

to Aristotle contain such an argument, Gray (1989, p. 47) points out that Omar

Khayyam may have had access to Aristotelian treatises that no longer exist. In

any case, he concluded on the basis of Aristotle's argument that two lines that

converge on one side of a transversal must diverge on the other side. With that,

having proved correctly that the perpendicular bisector of the base of a Thabit

quadrilateral is also the perpendicular bisector of the summit, Omar Khayyam

concluded that the base and summit could not diverge on either side, and hence

must be equidistant. Like Thabit ibn-Qurra's proof, his proof depended on building

one Thabit quadrilateral on top of another by doubling the common bisector of the

base and summit, then crossing its endpoint with a perpendicular which (he said)

would intersect the extensions of the lateral sides. Unfortunately, if that procedure

is repeated often enough in hyperbolic geometry, those intersections will not occur.

All of these mathematicians were well versed in the Euclidean tradition of

geometry. In the preface to his book on algebra, Omar Khayyam says that no one

should attempt to read it who has not already read Euclid's Elements and Data and

the first two books of Apollonius' Conies. His reason for requiring this background

was that he intended to use conic sections to solve cubic and quartic equations

geometrically. This book contains Euclidean rigor attached to algebra in a way

that fits equally well into the history of both algebra and geometry. However, in

other places it seems clear that Omar Khayyam was posing geometric problems for

the sake of getting interesting equations to solve. For example (Amir-Moez, 1963),

he posed the problem of finding a point on a circle such that the perpendicular

from the point to a radius has the same ratio to the radius that the two segments

into which it divides the radius have to each other. If the radius is r and the

length of the longer segment cut off on the radius is the unknown x, the equation

to be satisfied is x^3 + rx^2 4- r^2 x = r^3. Without actually writing out this equation,

Omar Khayyam showed that the geometric problem amounted to using the stated

condition to find the second asymptote of a rectangular hyperbola, knowing one of

its asymptotes and one point on the hyperbola. However, he regarded that analysis

as merely an introduction to his real purpose, which was a discussion of the kinds

of cubic equations that require conic sections for their solution. After a digression

to classify these equations, he returned to the original problem, and finally, showed

how to solve it using a rectangular hyperbola. He found the arc to be about 57°,

so that χ ~ rcos(57°) = 0.544r. Omar Khayyam described χ as being about 30|

pieces, that is, sixtieths of the radius. We reserve the discussion of this combination

of algebra and geometry for Chapter 14.

As his work on the parallel postulate shows, Omar Khayyam was very interested

in logical niceties. In the preface to his Algebra and elsewhere (for example, Amir-

Moez, 1963, p. 328) he shows his adherence to Euclidean standards, denying the

reality of a fourth dimension:

If the algebraist were to use the square of the square in measur-

ing areas, his result would be figurative [theoretical] and not real,

because it is impossible to consider the square of the square as a

magnitude of a measurable nature... This is even more true in the

case of higher powers. [Kasir, 1931, p. 48]

4.6. Nasir al-Din al-Tusi. The thirteenth century was as disruptive to the Is-

lamic world as the fifth century had been to the Roman world. This was the time