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Unit 8

Geometries Beyond Euclid

8.3 Non-Euclidean Geometry

OF QUESTIONED NECESSITY

Euclid's parallel postulate bothered mathematicians for many years. Everyone agreed that the first four postulates were completely obvious, but it seemed to be asking too much to say that the fifth was equally as obvious. It was more complicated than the other postulates, and, in fact, many thought that it could actually be proved from them. This idea led to numerous attempts to show that the fifth postulate was not independent of the other four and that, therefore, all of geometry could be built upon only four fundamental ideas. One of the most famous of these attempts was undertaken by an Italian Jesuit named Girolamo Saccheri in the early eighteenth century.

Saccheri wanted to show that the parallel postulate was not necessary (i.e., that it was a derivative of the other four), and if the title of his book, "Euclid Cleared of Every Flaw," is any indication, he truly believed that he had accomplished this. He, like many others before, believed that the parallel postulate could be proved from the other four. After numerous unsuccessful attempts to find a direct proof of his claim, he tried a different tack. His renewed efforts were somewhat influenced by the work of one of Genghis Khan's numerous grandchildren, Nasir Eddin. Eddin had attempted to prove the fifth postulate almost 500 years earlier by looking at quadrilaterals and making assumptions that he hoped would lead to contradictions. We've seen examples of similar "proofs by contradiction" before, such as Euclid's proof of the infinitude of primes back in unit one.

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SACCHERI'S QUADRILATERALS

Saccheri's approach was similar to that of Eddin. He began by considering a quadrilateral whose base angles are both 90 degrees:

Quadrilateral

He then showed that the summit angles must be equal to each other without using the fifth postulate. In renouncing the parallel postulate in the construction of this quadrilateral, Saccheri had to consider two possibilities—either:

  1. There are no lines parallel to a given line, or
  2. There are at least two lines parallel to a given line

These cases presented three optional scenarios: 1) that the summit angles are acute, (which would allow for more than one parallel line); 2) that the summit angles are right angles (indicating that there is only one parallel line); and 3) that the summit angles are obtuse (and, therefore, there are no parallel lines). He hoped to show that the only possible arrangement would be the second scenario, because it can play the role of the parallel postulate. Consequently, if it turned out to be the only case that works, then he would have shown that the parallel postulate is not necessary (recall that he arrived at this arrangement without its help).

Summit Angles

Saccheri was able to eliminate the obtuse angle scenario by assuming (implicitly) that straight lines can be extended forever. If this were not true, it would violate Euclid's second postulate—that lines may be extended indefinitely. This in turn makes the obtuse case useless as far as an indicator of the necessity of the parallel postulate. Ruling out obtuse summit angles leaves just two options—they must be either acute or right.

To show that the acute case was incorrect, Saccheri set about trying to derive the propositions found in The Elements, assuming that the summit angles were less than 90 degrees. He was hoping that this path would consistently lead to contradictions. To his chagrin, he found that he was able to derive many of Euclid's propositions using this assumption and yet still avoid contradictions. He was onto something, but at the time he was unaware that he was indeed building a logically consistent universe in which the summit angles of such a quadrilateral could each be acute.

Acute Angles

He was so sure that Euclid had to be correct, however, and that the acute angle case could not stand, that he twisted his own logic to accommodate what he had hoped to find, namely that the summit angles must be 90 degrees. Saccheri basically invented a contradiction where none existed in order to fit his preconception. Not surprisingly, his arguments were not convincing, even to the mathematicians of his day. He published another work after "Euclid Cleared of Every Flaw," attempting to clarify his so-called proof, but to no avail. Mathematicians of the time believed that Saccheri had neither proven nor disproven the necessity of Euclid's fifth postulate.

Saccheri's work was not in vain; he simply did not recognize what he had found. In his dogged effort to prove his preconception, he missed out on his claim to one of the great discoveries in geometry: that Euclid's system of geometry is not the only possible self-consistent geometry.

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Next: 8.4 Spherical and Hyperbolic Geometry


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