The Parallel Problem
MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

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Euclid

Born: about 325 BC
Died: about 265 BC in Alexandria, Egypt
Biography

Of Euclid's life it is known only that he taught at and founded a school at Alexandria in the time of Ptolemy I Soter, who reigned from 323 to 285/283 BC. Medieval translators and editors often confused him with the philosopher Eucleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Writing in the 5th century AD, the Greek philosopher Proclus told the story of Euclid's reply to Ptolemy, who asked whether there was any shorter way in geometry than that of the Elements: "There is no royal road to geometry." Another anecdote relates that a student, probably in Alexandria, after learning the very first proposition in geometry, wanted to know what he would get by learning these things, whereupon Euclid called his slave and said, "Give him threepence since he must needs make gain by what he learns" (Encyclopedia Britannica).

Euclid’s The Elements was probably one of the greatest donation to Geometry, as well as other types of mathematics. The Elements consists of thirteen books in which I will look at the first book. Except for the Bible, there is no other single scientific, philosophical, or literary book that has had so many translations, editions, and commentaries as The Elements (Calinger, 109).

Postulates

Postulate 1 - To draw a straight line from any point to any point.
Postulate 2 - To produce a finite straight line continuously in a straight line.
Postulate 3 - To describe a circle with any center and radius.
Postulate 4 - That all right angles equal one another.
Postulate 5 (Also known as the Parallel Postulate) - That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

These five postulates, along with twenty-three definitions and five common notions, are all that Euclid uses to prove the forty-eight propositions in Book 1. To use the postulates in this manner, they must be able to stand-alone. Postulate five was thought, by many, to be a combination of the other four, because it is so lengthy and wordy. Euclid does not even seem to like the postulate, because he does not even use this postulate until proposition 28; therefore many think that he also did not fully understand what had written. As everything else in mathematics, the mathematicians headed out to prove that the fifth postulate was not independent, but dependent on the other four postulates.

 

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Parallel - Part 2 Parallel - Part 3 Parallel - Part 4

 
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