Euclidean and non-Euclidean 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:

  1. [it is possible] To draw a straight line from any point to any other.
  2. [it is possible] To produce a finite straight line continuously in a straight line.
  3. [it is possible] To describe a circle with any centre and distance.
  4. That all right angles are equal to each other.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

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 (410-485AD) 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:

  1. that the summit angles (angles C and D) are larger than the right angle
  2. that they are smaller than the right angle
  3. that they are equal to the right angle.

He came across many theorems of what few centuries later became known as non-Euclidean 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, JŠnos Bolyai wrote a seminal work on the matter. In 1823 he wrote to his father that he

“discovered things so wonderful that I was astounded… out of nothing I have created a strange new world”.

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:

  1. the Euclidean (where parallel lines never meet)
  2. the hyperbolic geometry (in which you can draw infinitely many lines through a point which are all ‘parallel’ to the given line, but they converge or diverge)
  3. absolute geometry (Euclidean geometry but without the Fifth Postulate - in other words anything that depends on the Fifth Postulate is disregarded).

After János, another Eastern European mathematician came to the similar or same conclusions after studying the possibilities of non-Euclidean 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.

Euclid

Bolyai

Lobachevsky

 

 

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