Euclidean and nonEuclidean geometries 

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When Euclid wrote his Elements, he set out the full system of geometry that would remain more or less unchanged for 2000 years. Although various branches of geometry were developed within that period, no one came with a concept that would shake the foundations of Euclidean geometry. The first book of Euclid's Elements (there are thirteen altogether) states the five postulates on which all the theorems are based:
If you look at the fifth postulate, you can see that it is different from others. It is, first of all, the longest one, and from studying further the Elements, it becomes more or less clear that Euclid himself didn’t like it as it is, because the first 28 propositions are proved without a reference to this postulate. This postulate  probably because it is not as short and clear as the others are  seems to have been a thorn in the eye for many mathematicians throughout the centuries to come. Proculs (410485AD) was the first to attempt to do something about it. He tried to deduce this postulate from the first four, but failed. Girolamo Saccheri in 1697 tried to vindicate Euclid, and wrote a work entitled Euclides ab Omni Naevo Vindicatus in which he attacked a problem in looking at the sums of angles in a quadrilateral drawn between two parallel lines like this: He looked at three possible cases:
He came across many theorems of what few centuries later became known as nonEuclidean geometry, but wasn’t aware of it. In any case, he thought that he vindicated Euclid, as the title of his work says. Some french mathematicians tried their luck on the same problem; Lambert, Legendre, and D’Alembert (the last two knew Monge well). Gauss also discussed the problem with his friend, the Hungarian mathematician, Farkas Bolyai, who made several attempts in trying to solve it. In the end, his son, János Bolyai wrote a seminal work on the matter. In 1823 he wrote to his father that he
János Bolyai took further two years to write a work on this strange new world and it got published in an appendix to his father’s book. This was Farkas Bolyai Tentamen, and so one of the most important mathematical discoveries is given in the 24 pages of the appendix to another work not so very important at all. What János discovered was that if the Fifth Postulate of Euclid held in one region of space, it held in every region of that space. He however showed that apart from that system, it is possible to consider geometry in which this system is not valid, but that, for example instead of two parallel straight never meeting, they can also diverge (get increasingly more away from each other), or converge (meet). There is also another type of geometry in which there is no Fifth Postulate whatsoever. So there are three types of geometries that János in effect defined:
After János, another Eastern European mathematician came to the similar or same conclusions after studying the possibilities of nonEuclidean geometries. It was Nikolai Ivanovich Lobachevsky, who was a lecturer at the University of Kazan in the Tatarstan, a state in Russia.

Click on the picture above to see Book I of Euclid's Elements, or on the picture below for a larger copy of Girolamo Saccheri's fronticepiece to his Euclides Ab Omni Naevo Vindicatus. To see the fronticepiece of Bolyai's work click on the picture below. Click on the portraits of these famous mathematicians to see their biographies. Bolyai Lobachevsky


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