After Euclid 9 and failed to come up with proofs that the fifth postulate follows logically from the other four, or at least to replace it with a more truly self-evident postulate. This quest, in fact, has motivated much of the development of geometry since Euclid. The earliest attempt to prove the fifth postulate that has survived to modern times was by Proclus (412–485 CE), a Greek philosopher and mathematician who lived in Asia Mi- nor during the time of the early Byzantine empire and wrote an important commentary on Euclid’s Elements [Pro70]. (This commentary, by the way, contains most of the scant biographical information we have about Euclid, and even this must be considered essen- tially as legendary because it was written at least 700 years after the time of Euclid.) In this commentary, Proclus opined that Postulate 5 did not have the self-evident nature of a postulate and thus should be proved, and then he proceeded to offer a proof of it. Unfortu- nately, like so many later attempts, Proclus’s proof was based on an unstated and unproved assumption. Although Euclid defined parallel lines to be lines in the same plane that do not meet, no matter how far they are extended, Proclus tacitly assumed also that parallel lines are everywhere equidistant, meaning the same distance apart (see Fig. 1.3). We will see in Chapter 17 that this assumption is actually equivalent to assuming Euclid’s fifth postulate. Fig. 1.3. Proclus’s assumption. After the fall of the Roman Empire, the study of geometry moved for the most part to the Islamic world. Although the original Greek text of Euclid’s Elements was lost until the Renaissance, translations into Arabic were widely studied throughout the Islamic em- pire and eventually made their way back to Europe to be translated into Latin and other languages. During the years 1000–1300, several important Islamic mathematicians took up the study of the fifth postulate. Most notable among them was the Persian scholar and poet Omar Khayyam (1048–1123), who criticized previous attempts to prove the fifth postulate and then offered a proof of his own. His proof was incorrect because, like that of Proclus, it relied on the unproved assumption that parallel lines are everywhere equidistant. With the advent of the Renaissance, Western Europeans again began to tackle the problem of the fifth postulate. One of the most important attempts was made by the Italian mathematician Giovanni Saccheri (1667–1733). Saccheri set out to prove the fifth postulate by assuming that it was false and showing that this assumption led to a contradiction. His arguments were carefully constructed and quite rigorous for their day. In the process, he proved a great many strange and counterintuitive theorems that follow from the assumption that the fifth postulate is false, such as that rectangles cannot exist and that the interior angle measures of triangles always add up to less than 180ı. In the end, though, he could not find a contradiction that measured up to the standards of rigor he had set for himself. Instead, he punted: having shown that his assumption implied that there must exist parallel lines that approach closer and closer to each other but never meet, he claimed that this result is “repugnant to the nature of the straight line” and therefore his original assumption must have been false.

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