Omar Khayyam and Euclid's 5th
You are at a point P. From P, walk a mile South, then a mile West, and then a mile North. How far are you now from P?
Most school children will say “1 mile!”, visualizing the motion as being on three sides of a square. The correct answer, of course, is that 'it depends.' Consider the North Pole. Going a mile South from the Pole brings you on the 89.99o N latitude, each point of which is located a mile South from the North Pole. Walking straight West, you stay on the same parallel, and therefore at the same distance from the Pole. If you turn and move back a mile north, you will be back at the Pole, i.e. your distance from P would be 0 in this case. In other words, the answer depends on the curvature "omega" of the space you want to find the solution in; the omega value for flat space is 1, that for spherical space greater than 1, and that for hyperbolic space less than 1.
What can wandering around the Pole possibly have to do with Uzbekistan? The connection is in the person of a mathematician who probably has more bars, wines, dance clubs, gentlemen's clubs or oases of dissolution named after him than any other man (or woman) of science – Omar Khayyam. Born in Nishapur in Khorasan (now Iran) in year 1048, Omar Khayyam lived in his childhood in Balkh, and was educated in Samarkand. His discomfort with Euclidean Geometry grew during a 10-year stay in Bukhara. Eventually, he published a famous treatment of the problem -- Sharh ma ashkala min musadarat kitab Uqlidis (Explanations of the Difficulties in the Postulates of Euclid.)
Euclid derived much of his geometry from five postulates:
1. A straight line may be drawn between any two points.
2. A straight line may be extended indefinitely.
3. A circle may be drawn with any given radius at an arbitrary center.
4. All right angles are equal.
5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
The Fifth postulate refers to the diagram above. If the sum of two angles 1 and 2, formed by a line C crossing two lines A and B, add up to less than two right angles then lines A and B will meet somewhere on the side of angles 1 and 2 if continued indefinitely.
Compared to the first four, the Fifth postulate immediately looks elaborate and contrived. Euclid himself possibly had mixed feelings about it, as he did not make use of it until Theorem 29. The postulate looks more like another Theorem than a basic truth. It was attacked almost from the beginning. Over the centuries, many mathematicians attempted to take away the Fifth, but all ended up making some assumption or the other that inherently implied the Fifth postulate. Proclus Diadochos, who lived around 450 (i.e. 700 years after Euclid) mentions Ptolemy's attempts in the second century to prove the postulate, and demonstrates that Ptolemy had unwittingly assumed (what in later years became known as Playfair's Axiom) that through a point only one straight line can be drawn parallel to a given straight line, which is just another way of stating the Fifth postulate. Proclus left a proof of his own, but the latter rested on the assumption that parallel lines are always a bounded distance apart, and this assumption can also be shown to be equivalent to the Fifth postulate.
The great Ibn al-Haytham (Alhazen) (965-1039) of Basra, during the middle-ages called the Second Ptolemy in Europe, made an attempt at proving the parallel postulate using a proof-by-contradiction. He, too, alas, needed Playfair's Axiom. What was missing from these attempts was the recognition that Euclid's postulates fix only one kind of geometry; if you start relaxing them, you can get many more, often far lovelier, and at least as consistent.
Euclid's postulates had been based on our intuition about geometric objects on flat planes. For example, mathematicians had assumed that the second postulate, viz. “A straight line may be extended indefinitely”, also meant that straight lines were infinitely long. How about a “straight” line of latitude or longitude on the surface of a sphere, say a great circle? Circles can be extended indefinitely since they have no ends; going in circles means exactly this: doing something with no end in sight. However, circles (and great circles) are of finite extent – they can be extended indefinitely, round and round, and Euclid's postulate is not violated if they are not, as a result, of infinite length. The spherical-polar solution to the 1-mile-walk problem shows that there is a perfectly reasonable triangle with two right angles at the base and a nonzero angle at the top, i.e. a triangle whose angles sum up to more than 180o.
Omar Khayyam made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate and he was the first to consider the cases of elliptical geometry and hyperbolic geometry. The Khayyam-Saccheri quadrilateral was also first considered by him. Khayyam, and Saccheri 700 years later, recognized that three possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another line, a choice of the orientation of the last line can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse.
These cases anticipate the non-Euclidean geometries of Gauss, Bolyai, Lobchevsky and Riemann. Omar Khayyam did not get that far; he eventually persuaded himself that the acute and obtuse cases lead to contradiction, but not without a tacit assumption equivalent to the Fifth to get there. It took till the 19th century for mathematicians to start diving into those alternatives Omar Khayyam had listed but recoiled from, and discovering the logically consistent geometries which result. In 1823, then 21-year-old budding Hungaro-Romanian mathematician Janos Bolyai wrote to his father: "Out of nothing I have created a new universe"; by which he meant that starting from the first four postulates, by relaxing the Fifth, he had developed a geometry that, although quite unusual, did not lead to any logical contradiction. Bolyai Sr. consulted with Gauss. In a letter of 1824 Gauss wrote:
The assumption that (in a triangle) the sum of the three angles is less than 180o leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction.
In 1829, Nikolai Lobachevsky published an account of acute geometry in an obscure Russian journal in Kazan. (His work remained largely unrecognized for many years, though before his death Lobachevsky did get a citation from the Tsar's government, for a new way he had developed for processing wool!) Lobachevsky and Bolyai built their geometries on the assumption that through a point not on the line there exist more than one parallel to the line. Riemann's geometry, on the other hand, (among other things) models a space where there are no parallel lines -- the great circles on the surface of a sphere always meet at the poles. The General Theory of Relativity famously uses Riemannian geometry to model curvatures in space-time due to the effects of gravity; inertial particles follow the geodesics of Riemannian space.
Omar Khayyam developed parts of the general binomial theorem, many centuries before Pascal. He writes:
From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 12, 22, 32 (etc.) and their respective products, i.e. 2 × 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic.
Khayyam was also able to solve certain types of cubic equations using conic sections. See here for an interesting illustration.
Given this oeuvre in mathematics (to say nothing about his contributions to philosophy and the politics of pacifism), it is almost a pity that Omar Khayyam is remembered mostly for his verse:
Oh, come with old Khayyam, and leave the Wise
To talk; one thing is certain, that Life flies;
One thing is certain, and the Rest is Lies;
The Flower that once has blown for ever dies.
Myself when young did eagerly frequent
Doctor and Saint, and heard great Argument
About it and about: but evermore
Came out of the same Door as in I went.
And if the Wine you drink, the Lip you press,
End in the Nothing all Things end in — Yes —
Then fancy while Thou art, Thou art but what
Thou shalt be — Nothing — Thou shalt not be less.
And that inverted Bowl we call The Sky,
Whereunder crawling coop't we live and die,
Lift not thy hands to It for help - for It
Rolls impotently on as Thou or I.
The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.