Parallel lines

- Indra N Arjyal

Nov 9, 2015-

How does one write the 14th letter of the Roman alphabet? It comes in three stages: first,  first draw a short vertical line followed by another, a little to the right, and finally join the tip of first with the base of other creating ‘N’. There is no confusion for us as the first two happen to be parallel lines. But for Euclid (300 BC), the creator of Euclidean geometry it was equivalent to a never-ending marathon. He proved as many theorems as possible but he could not proceed further without the concept of parallel lines. He, instead, put up a long statement in the form of a ‘supposition’ called  ‘parallel postulates’. Euclid made hundreds of attempts to prove the parallel postulate as a theorem of the converse with the help of other postulates but did not succeed.

The parallel postulates stood out as an outcrop in the smooth field of Euclidean geometry. Of the many substitutes that tried to address the shortcoming the oneby Scottish mathematician John Playfair in 1795 was the best. It goes like: “Through a given point, not on a given line can we draw only one line parallel to the given line”.

Further, if you look at ‘N’ the angle ’zig’ is alternately equal to the angle ’zag’. With the help of this concept we are able to prove three angles in a triangle together add up to 180 degrees. Euclid’s geometry represents the unrivalled and universally acceptable system of thoughts.

But by abandoning Euclid’s postulate about parallelism Janos Bolyai created, a kind of, non-Euclidian geometry we now know as  ‘Hyperbolic geometry’ dealing with a ’trumpet’ shaped surface. Of the other two who created non-Euclidian geometry one was Carl Friedrich Gauss, a German and theother Nicolas Lobachevski, a Russian.

Both Lobachevski and Bolyai, though lived in different place thought likewise to create geometry on the basis of parallelism contradicting Euclid. The axiom of their choice runs as: “Through a given point, not on a given line, there pass two lines (left and right) parallel to the given line”. One of the theorems thus proved is: “The three angles of triangle are together always less than (<) 180 degrees”.

Euclid was not at ease with his axiom which says a line segment can be extended indefinitely in  either direction. As an alternate to that, as proposed by Bernhard Riemann, assumes all lines are finite but endless as is the Equator. As like two longitudinal lines, which stand perpendicular to equator meet in the pole making a triangle. But the sum of the angles, within, adds up to greater than 180 degrees. This is true for non-Euclidian or Elliptic geometry.

Thus we have three geometrical doctrines namely, hyperbolic, parabolic and elliptic. While the sum of the angles within the triangle in the first being less than (<)180, the middle, Euclidian, being exactly (=) 180 and the last being greater than (>)180 degrees. This makes Euclid’s version, finally, lie comfortably between the Hyperbolic and Elliptic geometries.

Published: 09-11-2015 08:31

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