Omar Khayyam · Mathematics
In his <em>Commentary on the Difficulties in the Postulates of Euclid</em> (1077), Khayyam tried to prove Euclid’s troublesome fifth postulate. Studying a special quadrilateral - now called the Khayyam-Saccheri quadrilateral - he proved the first theorems that, centuries later, would open the door to non-Euclidean geometry.
Euclid built his Elements on five postulates. Four are short and obviously acceptable: you can draw a line between two points, extend a line, draw a circle, and all right angles are equal. The fifth is different. It says, in effect, that if a line crossing two other lines makes the interior angles on one side add to less than two right angles, then those two lines, extended far enough, must meet on that side. It is long, intricate, and reads more like a theorem than a self-evident starting point. For two thousand years mathematicians felt it ought to be provable from the other four, and tried to prove it. Khayyam, in his Commentary on the Difficulties in the Postulates of Euclid of 1077, mounted one of the most searching of these attempts - and in doing so became, without knowing it, an explorer of geometries Euclid never imagined.
Picture Khayyam’s quadrilateral: a horizontal base, two equal vertical sides standing on it at right angles, and a top edge (the summit) connecting their upper ends. By symmetry, the two angles at the summit must be equal to each other - but are they right angles, obtuse angles, or acute angles? Khayyam saw that exactly three cases are possible, and that they are not idle: each corresponds to a different assumption about parallels and about the sum of a triangle’s angles. If the summit angles are right, ordinary Euclidean geometry follows and the fifth postulate holds. If they are obtuse, or if they are acute, one gets strange geometries in which the postulate fails. Khayyam proved theorems about all three, aiming to show that only the right-angle case can be true - and that the other two lead to contradiction.
Khayyam concluded that the acute and obtuse hypotheses must be rejected and the right-angle case retained, so that Euclid’s postulate is saved. Judged by his own goal, he did not fully succeed: like everyone before and after him until the nineteenth century, he could not actually prove the fifth postulate without secretly assuming something equivalent to it. But judged by history, he did something far more interesting than he intended. In working out what would follow if the summit angles were acute or obtuse, he proved some of the earliest theorems of non-Euclidean geometry. He was exploring the new lands while trying to prove they could not exist. This is a recurring pattern in mathematics: an attempt to close a question off instead cracks it open, and the ‘failure’ becomes a discovery no one was looking for.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Khayyam tried to prove Euclid’s fifth (parallel) postulate from the others, using what is now called the Khayyam-Saccheri quadrilateral, and considered three cases for its summit angles - right, obtuse, and acute. Explain his approach and why his rejection of the acute and obtuse cases unknowingly exp…
Leads to Girolamo Saccheri.
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