This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
My math history class is currently studying non-Euclidean geometry, which means we’ve studied quite a few “proofs” of Euclid’s fifth postulate, also known as the parallel postulate. I’ve written about this postulate before. There are many statements that are equivalent to the parallel postulate, including the fact that parallel lines in a plane are equidistant. This postulate is independent from the rest of Euclid’s assumptions, but for centuries, mathematicians tried to prove that the parallel postulate followed from the rest of Euclidean geometry.
Omar Khayyam, the eleventh and twelfth century Persian polymath who is probably best known in the West as a poet (“a jug of wine, a loaf of bread—and thou…”), is one of the many mathematicians who tackled the parallel postulate. In 1077, he wrote a Commentary on the Difficulties of Certain Postulates of Euclid’s Work, which deals with the parallel postulate and some of Euclid’s work on ratios and proportions.
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Through the magic of interlibrary loan, I was able to get my hands on a copy of the hard-to-find book Omar Khayyam the Mathematician by R. Rashed and B. Vahabzadeh. Among other things, it contains an English translation of Khayyam’s commentary, so my class and I were able to see exactly how Khayyam approached the theory of parallels. (Or at least as nearly as possible without reading Arabic.)
Unlike many of his predecessors, Khayyam did not try to show that Euclid’s fifth postulate followed from the rest of the postulates and axioms; instead, he says that Euclid should have started with different postulates that Khayyam believed were more self-evident. He proposes a total of five new postulates that should be added to Euclid's Elements. From these, he proves eight new propositions to replace Euclid’s propositions 29 and 30, the first two propositions of the Elements that require the parallel postulate. Khayyam didn’t know it, but his first three propositions are some of the very first theorems in non-Euclidean geometry. But that’s not what this post is about. This post is about some passages from the commentary that I found delightfully grumpy.
Khayyam mostly seems grumpy about what he views as Euclid’s inconsistency in his assumptions. Euclid demonstrates some things that Khayyam thinks are obvious and doesn’t demonstrate some things that Khayyam thinks need demonstration. The first grumpy passage comes early in the commentary. Khayyam has just given a heuristic proof that Euclid's parallel postulate follows from the nature of straight lines and right angles. (Emphasis from the translation by Rashed and Vahabzadeh.)
So it is because of this that Euclid was of the opinion that the cause of the meeting of the lines...is the two angles being less than two right angles.And this opinion is true, but one cannot build upon it except after other demonstrations, for it is them which induced Euclid to admit this premise and build upon it without demonstration. But upon my life! these are very hypothetical propositions,...
And how can one allow Euclid to postulate this proposition [the parallel postulate] because of this opinion while he has demonstrated many things much easier than these? (As when he proves in the third Book, that equal angles at the centres of equal circles cut off from the circumference equal arcs. But this idea is well known from the principles, for equal circles apply to one another, and equal angles likewise. Consequently the arcs will inevitably apply to one another; therefore they will be equal. So one who proves something like this, how much will he stand in need of proving something like that!—And as when he proves in the fifth Book, that the ratio of the same magnitude to two equal magnitudes is the same. But as ratio falls within magnitude qua magnitude, why should this need a demonstration? Since the two equal magnitudes are equal qua measure, there is no difference whatsoever between them; therefore they are from this viewpoint truly the same: there is no alterity whatsoever between them, except the alterity of number and no more.)
Khayyam must have been particularly bothered by Euclid's proofs of theorems involving circles and equal angles because he came back to them later in the commentary.
Do you not see that whosoever conceives the reality of the circle, the reality of the angle, and the reality of the ratio between magnitudes, will know with a modicum of reflection, that the ratio of the angles which stand at the center is equal to the ratio of the arcs on which they stand? Yet this notion has been demonstrated by Euclid in proposition 36 of Book VI.
After outlining the five postulates Khayyam believes Euclid neglected to include but should have, he wrote:
And there are more obvious primary premises than this; but Euclid has not produced most of them in the beginning of the work, although he has produced primary things which one could very well do without. And he had to either not produce them at all, or produce all of them without excluding any of them, even if they are obvious.
I admit that I am probably reading too much into Omar Khayyam’s mood; after all, we are separated by more than 900 years, a couple of continents, and several rounds of translation. It's a bit presumptuous for me to claim any knowledge of his emotional state. But imagining him writing his commentary grumpily helps me see him as a person the same way listening to Peter Schickele read Bach's grumpy letters about spilled wine and low pay makes me feel like maybe Bach and I aren't so different. Omar Khayyam—he's just like us! Geometry sometimes makes him grumpy.