Thursday, April 06, 2006

Hyperbolic Geometry and it's Rise

Omar Khayyám the mathematician(6 april 2006 Wikipedia)

He was famous during his lifetime as a mathematician, well known for inventing the method of solving cubic equations by intersecting a parabola with a circle. Although his approach at achieving this had earlier been attempted by Menaechmus and others, Khayyám provided a generalization extending it to all cubics. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry.

Giovanni Girolamo Saccheri(6 April 2006 Wikipedia)

Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin from 1694 to 1697, and philosophy, theology, and mathematics at Pavia from 1697 until his death. He was a protege of the mathematician Tommaso Ceva and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).

Of course the question as to "Victorian" was on mind. Is non-euclidean held to a time frame, or not?

Victorian Era(wikipedia 6 April 2006)

It is often defined as the years from 1837 to 1901

Time valuations are being thought about here. In regards too, non euclidean geometry and it's rise. Shows, many correlations within that time frame. So that was suprizing, if held to a context of the victorian socialogical time frame. But we know this statement is far from the truth?

Seminar on the History of Hyperbolic Geometry, by Greg Schreiber

We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.

History (wikipedia 6 April 2006)

Hyperbolic geometry was initially explored by Giovanni Gerolamo Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by János Bolyai, Karl Friedrich Gauss, and Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.

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