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the evolution of geometric notions

In the beginning, geometry was only considered as a rule for calculating the areas, lengths, and volumes that were used for navigations and constructions by Babylonians and Egyptians. The information was then passed on to the Greeks. In 300 BC, Euclid of Alexandra wrote a book, “The Elements,” containing all the geometrical knowledge till then. Everything was not guaranteed, but everything was derived cautiously based on self-evident assertions. However, if one compares the concepts in that book with modern-day concepts, imperfections are found since he tried to provide a definition of everything in terms of common notions. He put forth five postulates, which are mentioned below:

1. A straight line can be drawn from any given point.

2. A straight line can be stretched to form an infinite straight line

3. Having the segment (radius) and the center, a circle can be easily drawn.

4. All the angles that are right to each other are equal.

5. If there are two lines that intersect the third line in such a way that the sum of the inner angles is less than 180 degrees, then those two lines must intersect each other if extended indefinitely.

The last one caught much attention from the above postulates since it contains a hidden assumption that has yet to be proved. In 1763, a man called Kugel studied all the proofs that had been done for the fifth postulate, but out of 28 proofs, he found none to be satisfactory. The work of a man, Sachheri, caught his interest, who, in the try to negate and contradict the fifth postulate, made a lot of new results that formed the basis of what we now call hyperbolic geometry. From 1777 to 1855, Gauss came up with the view that geometry varying from the one postulated by Euclid may exist. He secretly started working on those ideas and was finally able to negate the fifth postulate, and his results were published in 1829 and 1832. In the later years, there were other mathematicians like Beltrami and Klein worked on Euclidean geometry within hyperbolic geometry, and it was then finally found out that both Euclidean and hyperbolic geometry are free of contradiction while proof of parallel postulate was impossible. Due to the fifth postulate, two branches of geometry have emerged. The non-dependence of the theorem on parallel postulates made it an absolute geometry that is valid both in Euclidean and hyperbolic geometry. Affine geometry constituted theorems that depended only on postulates I, II, and V.

It was through the central projection study in mathematics that artists like Leonardo Da Vinci overcame the perspective problems. The eye acts as the center of projection, whereas the image on the canvas serves as the projection. The possibility that the geometry of the original can be recognized on the canvas is made possible only through projection geometry, which was discovered by the French engineer Poncelet when he had no books in jail and was imprisoned in Russia. Projective geometry and affine geometry are closely related because the study of the invariance of projections under parallel postulate makes affine geometry. This similarity was first identified by Euler (1707-1783).

With the introduction of analytic geometry, a lot of mathematical complications became easier. For example, the treatment of comics got easier. Analytic geometry was introduced by Descartes. The geometry scope was then enlarged by Reiman, who introduced new geometries such as double elliptic geometry. He realized that geometry surfaces can provide new geometries. Small line segments can make up a circle, and so on. Reiman and Schlafi introduced Euclidean and spherical surfaces in higher dimensions. Later on, Einstein also used Reiman’s geometry.

Euclid postulates were based on the true facts of the physical world and to free them from it, Hilbert reformulated them. Then gradually Lie Groups were introduced by Sophie Lie which catered for the transformations of geometry. Lie Groups and differential geometry are still active research area in mathematics.

References

“Euclids Postulates.” From Wolfram MathWorld, mathworld.wolfram.com/EuclidsPostulates.html.

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