Omar Khayyam - 2

Khayyam-Saccheri quadrilateral

The Khayyam–Saccheri quadrilateral was first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):

Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.

Khayyam then considered the three cases (right, obtuse, and acute) that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.

It wasn't until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

As a Astronomer

Like most Persian mathematicians of the period, Omar Khayyam was also famous as an astronomer. In 1073, the Seljuk Sultan, Sultan Jalal al-Din Malekshah Saljuqi (Malik-Shah I, 1072-92), invited Khayyam to build an observatory, along with various other distinguished scientists. One is Shamse Tabrizi, his mentor and the father of Kimia Khatoon, who he fell in love with. Eventually, Khayyam and his colleagues measured the length of the solar year as 365.2425 days. Omar's calendar was more accurate than 500 years later the Gregorian calendar. The modern Iranian calendar is based on his calculations

Calendar reform

Omar Khayyam was part of a panel that introduced several reforms to the Persian calendar, largely based on ideas from the Hindu calendar. On March 15, 1079, Sultan Malik Shah I accepted this corrected calendar as the official Persian calendar.

This calendar was known as Jalali calendar after the Sultan, and was in force across Greater Iran from the 11th to the 20th centuries. It is the basis of the Iranian calendar which is followed today in Iran and Afghanistan. While the Jalali calendar is more accurate than the Gregorian, it is based on actual solar transit, (similar to Hindu calendars), and requires an Ephemeris for calculating dates. The lengths of the months can vary between 29 and 32 days depending on the moment when the sun crossed into a new zodiacal area (an attribute common to most Hindu calendars). This meant that seasonal errors were lower than in the Gregorian calendar.

The modern-day Iranian calendar standardizes the month lengths based on a reform from 1925, thus minimizing the effect of solar transits. Seasonal errors are somewhat higher than in the Jalali version, but leap years are calculated as before.

Omar Khayyam also built a star map (now lost), which was famous in the Persian and Islamic world.

Heliocentric theory

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It is said that Omar Khayyam also estimated and proved to an audience that included the then-prestigious and most respected scholar Imam Ghazali, that the universe is not moving around earth as was believed by all at that time. By constructing a revolving platform and simple arrangement of the star charts lit by candles around the circular walls of the room, he demonstrated that earth revolves on its axis, bringing into view different constellations throughout the night and day (completing a one-day cycle). He also elaborated that stars are stationary objects in space which, if moving around earth, would have been burnt to cinders due to their large mass.

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Poet

He is believed to have written about a thousand four-line verses or quatrains (rubaai's). In the English-speaking world, he was introduced through the Rubaiyat of Omar Khayyam which is rather free-wheeling English translations by Edward FitzGerald (1809-1883).

Other translations of parts of the Rubaiyat (Rubaiyat meaning "quatrains") exist, but FitzGerald's are the most well known. Translations also exist in languages other than English.

Ironically, FitzGerald's translations reintroduced Khayyam to Iranians "who had long ignored the Nishapuri poet." A 1934 book by one of Iran's most prominent writers, Sadeq Hedayat, Songs of Khayyam, (Taranehha-ye Khayyam) is said have "shaped the way a generation of Iranians viewed" the poet.

Omar Khayyam's personal beliefs are not known with certainty, but much is discernible from his poetic oeuvre.

P o e t r y

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(These poems were translated by Edward FitzGerald and are potentially more

revealing of the thoughts of Edward than Omar.)

And, as the Cock crew,
Those who stood before
The Tavern shouted - "Open then the Door!
You know how little time we have to stay,
And once departed, may return no more."

Alike for those who for TO-DAY prepare,
And that after a TO-MORROW stare,
A Muezzin from the Tower of Darkness cries
"Fools! Your reward is neither Here nor There!"

Why, all the Saints and Sages who discuss'd
Of the Two Worlds so learnedly, are thrust
Like foolish Prophets forth; their Words to Scorn
Are scatter'd, and their mouths are stopt with Dust.

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.

With them the Seed of Wisdom did I sow,
And with my own hand labour'd it to grow:
And this was all the Harvest that I reap'd –
"I came like Water, and like Wind I go."

Into this Universe, and why not knowing,
Nor whence, like Water willy-nilly flowing:
And out of it, as Wind along the Waste,
I know not whither, willy-nilly blowing.

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.

And that inverted Bowl we call The Sky,
Where under crawling coop't we live and die,
Lift not thy hands to It for help - for
It Rolls impotently on as Thou or I.

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