Two people are called Omar Khayyam and they are the same person, which is the whole reason this chapter is in Book IX rather than Book VII. One wrote the Treatise on Demonstration of Problems of Algebra around 1070 and solved the cubic by intersecting conics. The other — or the same one — is the voice of the quatrains, the wine and the potter's shop and the bowl they call the sky.
The corpus's own standard obliges a warning here, and it is a real one. The attribution of the Rubáiyát is a mess. Quatrains accumulated under his name for centuries; the manuscripts are late; scholars have argued the authentic core down to a few dozen and up again. FitzGerald's English of 1859 is a Victorian poem that made him famous in a language he never read. Whether Khayyam was a Sufi, a sceptic, or a court astronomer who wrote verse to amuse himself, is contested and this page does not settle it. CITED OPEN
What is not contested is that he wrote philosophical treatises on existence and on being, that he worked as an astronomer for Malik-Shah, and that he built a calendar. The practice and the metaphysics were not separate departments of his life. That is the test for this gallery.
The root he could not compute, so he built it
Take $x^3 + ax = b$ with $a, b > 0$. Khayyam had no formula — nobody would for another four hundred and fifty years — so he built the root instead of computing it.
A parabola and a circle. Their second intersection has abscissa the real root.
It is exact. For $a = 2$, $b = 5$ the intersection sits at $x = 1.328268855669$ and the real root of $x^3 + 2x - 5$ is $1.328268855669$ — agreeing to every digit double precision carries, across every case tested. COMPUTED
And here the chapter joins Hypatia's, five hundred years earlier and two thousand miles west. She preserved Apollonius; Khayyam used him. The conic sections are the only curves available to him, and he classifies the cubics by which pair of conics is needed — fourteen types, because he has no negative coefficients and must treat each arrangement separately. The classification is forced by the constraint, not chosen.
The sentence a modern paper would have to be brave to write
Then he says something a modern paper would have to be brave to say.
We have tried to express these roots by algebra but have failed. It may be that those who come after us will succeed.
Khayyam, on the general algebraic solution of the cubic · paraphrase of the standard translation · CITED
He was right, and he was right about the shape of the future too: del Ferro, Tartaglia and Cardano got there in the 1530s, and Cardano's chapter in Book VII picks the story up. What matters here is the sentence itself. He marked his own result OPEN. He had a method that worked, he knew its boundary, he stated the boundary, and he named it as work for other people. That is the epistemic standard this corpus tries to hold, written in the eleventh century by someone who had no reason to and did it anyway.
Thirty-three years
In 1079 he led the reform that produced the Jalali calendar. Its rule puts 8 leap years in every 33, which makes the mean year $365 + 8/33$ days.
| Rule | Mean year | Error against the tropical year | One day adrift in |
|---|---|---|---|
| Jalali, 8/33 | 365.242424242 | +0.000234 d/yr | 4 269 yr |
| Gregorian, 97/400 | 365.242500000 | +0.000310 d/yr | 3 226 yr |
| Julian, 1/4 | 365.250000000 | +0.007810 d/yr | 128 yr |
Five centuries before Gregory, and better. COMPUTED But the interesting part is not that it wins; it is why it wins, and the reason is not astronomy.
The continued fraction of the tropical-year fraction $0.242190$ is $[0;4,7,1,3,24,\ldots]$, and its convergents are $\tfrac14,\ \tfrac{7}{29},\ \tfrac{8}{33},\ \tfrac{31}{128},\ \tfrac{752}{3105},\ldots$ — $8/33$ is one of them and $97/400$ is not.
A convergent is a best rational approximation: no fraction with a smaller denominator comes closer. The Jalali rule is not merely accurate, it is optimal for a cycle that short. And $31/128$ is better still, with a denominator smaller than $400$ — so the Gregorian rule is not even the best available at its own size. COMPUTED
Whether Khayyam's committee arrived at $8/33$ through the continued-fraction algorithm or through observation and arithmetic is not known to this page, and it would be an invention to say. OPEN What is known is which number they chose.
The wheel
So the two halves close. The quatrains circle one figure over and over: the wheel, the bowl inverted overhead, the turning that returns to where it began and finds the drinker gone. Whether that is Sufi doctrine or a sceptic's consolation is the contested question this page leaves open.
What is not contested is the shape. A man who spent his working life on curves that close and cycles that repeat — the circle cutting the parabola, the thirty-three-year wheel of intercalation — wrote verse about a wheel that turns and does not give anything back. The mathematics and the metaphysics are the same figure seen twice, and he did not have two vocabularies for it because in the eleventh century nobody did.
Place in the Series
| Element | In this chapter | In dm³ |
|---|---|---|
| C | no negative coefficients — fourteen types, forced | compression: the constraint that issues the classification |
| K | the circle swept until it meets the parabola again | approach to the crossing |
| F | the second intersection — the root, constructed not computed | the fold SHOWN |
| U | $8/33$ — the cycle that closes, and closes best | a convergent SHOWN |
Verification
Every number on this page is produced by omega/ch-khayyam-verify.py, which records in its own closing block what it establishes and what it does not.
References
‘Umar al-Khayyāmī, Risāla fī l-barāhīn ‘alā masā’il al-jabr wa-l-muqābala (Treatise on Demonstration of Problems of Algebra), c. 1070.
R. Rashed and B. Vahabzadeh, Omar Khayyam, the Mathematician, Bibliotheca Persica, 2000 — the critical edition and translation.
D. S. Kasir, The Algebra of Omar Khayyam, Columbia, 1931.
E. S. Kennedy, “The Persian Calendar”, Vistas in Astronomy 31, 1988.
E. FitzGerald, Rubáiyát of Omar Khayyám, 1859 — a Victorian poem, and the reason for the fame.
A. Dashti, In Search of Omar Khayyam, trans. L. P. Elwell-Sutton, Allen & Unwin, 1971 — on how few quatrains survive attribution.