Saccheri Quadrilateral - History

History

Saccheri quadrilaterals were first considered by Omar Khayyam (1048-1131) 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 was not 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.

Read more about this topic:  Saccheri Quadrilateral

Famous quotes containing the word history:

    What you don’t understand is that it is possible to be an atheist, it is possible not to know if God exists or why He should, and yet to believe that man does not live in a state of nature but in history, and that history as we know it now began with Christ, it was founded by Him on the Gospels.
    Boris Pasternak (1890–1960)

    False history gets made all day, any day,
    the truth of the new is never on the news
    False history gets written every day
    ...
    the lesbian archaeologist watches herself
    sifting her own life out from the shards she’s piecing,
    asking the clay all questions but her own.
    Adrienne Rich (b. 1929)

    Every library should try to be complete on something, if it were only the history of pinheads.
    Oliver Wendell Holmes, Sr. (1809–1894)