Work Of Gauss Bolyai And Lobachevsky Philosophy
In around 300BC the mathematician Euclid developed the first geometry based on postulates, proofs and definitions which were detailed in his Elements, a series of thirteen books. Two thousand years on in the early 19th Century several mathematicians had attempted and failed to prove Euclid's Parallel Postulate and from this non-Euclidean geometry was discovered. Rosenfeld [R1] suspects its discovery was first published by the mathematician NikolaÄ Lobachevsky in the paper "On the principles of geometry". If this is the case, what were Lobachevsky's findings and how many other mathematicians contributed to the conception of non-Euclidean geometry?
Coxeter [C1] suggests it was Gauss who conceived the name 'non-Euclidean' to describe a geometry which stemmed from the assumption that we could prove the independence of the Parallel Postulate. Artemiadis [A1] adds that this geometry would still preserve all the principles of Euclidean geometry. In place of the axiom of parallelism mathematicians introduced this principle [A1] "In a plane that contains a given straight line and a given point, it is possible from the point to draw more than one straight line that does not intersect the given straight line". Gray [G1,p.43] describes Gauss as the 'acutest critic of mathematics of his day' however his notion of a geometry independent of the Fifth Postulate was only revealed after the works of Lobachevsky and Boyai were published [Bonola B2]. Miller [M1] suggests we can only infer from Gauss' letters the results of his study because he never published his findings.
Boyer [B1] regards Lobachevsky as the 'Copernicus of geometry' for his revolution of a new geometry which showed that Euclidean geometry was not the absolute truth that we believed it to be for two million years. Lobachevsky was the first mathematician to take what Boyer [B1] describes as a revolutionary step to publish a geometry built on an assumption in direct conflict with the Parallel Postulate. Let us explore the findings that were so contrary to common sense that even Lobachevsky himself described it as an 'imaginary geometry' [B1].
Lobachevsky chose to build his geometry on the basis of a Parallel Postulate contradicting to Euclid's. Euclid had said in his Fifth Postulate that through a given point there is only one parallel to a given line. However, as we have already said many mathematicians were suggesting alternatives to this postulate instead of trying to prove it. Vucinich [V1] looks at how Lobachevsky constructed his geometry on this proposition: through a given point there are an infinite number of parallels to a given line. Boyer [B1] suggests that with this new postulate Lobachevsky was able to create a geometric structure that had no inherent contradictions.
Rosenfeld [R1] explores how from establishing a foundation for his 'imaginary geometry' Lobachevsky defined the angle of parallelism. Lobachevsky dropped a perpendicular length 'a' from a point B to a straight line and draws through B a parallel to that line. From this Lobachevsky showed that the angle of parallelism was the angle between the perpendicular line and the parallel. When we compare this to Euclidean geometry we note that this angle is always equal to Ï€/2, however, in Lobachevskian geometry it is acute and is a function of 'a' which as we will see Lobachevsky later denotes as âˆ(x).
Chandrasekhar [C1] considers that Lobachevskian geometry widened the structure of the Euclidean system. Lobachevsky showed that in his geometry the sum of angles in a triangle is less than two right angles where the sides of the triangles are arcs of circles. If we produced these they would cut orthogonally the fundamental plane. Lobachevsky also went on to show that the sum of the angles of a triangle is not constant; in fact he suggested that it reduces as the area of the triangle increases. STUPID MAN??!!
Lobachevsky's most interesting results involving trigonometry in the non-Euclidean plane were unknown to other mathematicians such as Saccheri and Lambert. Katz [K1] looks at how Lobachevsky developed a complex argument involving spherical triangles and triangles on the non-Euclidean plane to evaluate the function ÐŸ (x) in the form tan½âˆ(x)=e-x . This result can be considered as fundamentally the same as the result mathematician Taurinius found. Lobachevsky continued to show that it followed that ÐŸ (0) = n/2 and that the limit of ÐŸ (x) as x tended to zero was equal to zero. From his findings he could then derive a relationship between the sides a,b,c and the opposite angles A, B, C on any arbitrary Euclidean triangle.
Lobachevsky derived his trigonometric formulae from studying the horocycle, a circle of infinite radius. During his study he re-discovered a theorem belonging to Wachter which said that the geometry of horocycles on a horosphere is identical with the geometry of straight lines in the Euclidean plane [C1]. From this discovery Lobachevsky continued to argue that geometry should be based on ideas about bodies and their motion. Lobachevsky showed that when deriving ideas about straight lines his argument demonstrated that they need not be as Euclid had defined them to be. Gray [G1] proposes that it was from this that Lobachevsky believed the foundations of Euclidean geometry were flawed because terms such as 'line', 'surface' and 'position' were more obscure than fundamental.
From examining the work of Lobachevsky can we consider his non-Euclidean to be with out fault? Gray [G1] notes that Lobachevsky's concluding argument was what convinced him that his geometry was not self-contradictory. Lobachevsky expressed each of his theorems in the terms of trigonometry and calculus. He deeply believed that geometry was about measurement and we can connect measurements and numbers through formulas. It was with this understanding that he deliberately sought out formulae to communicate his geometry. Lobachevsky's work demonstrated as Vucinich [V1] states that there are geometries which differ from Euclid's but they are just as valid.
Around the same time as Lobachevsky another mathematician, Bolyai, was also working on developing his own non-Euclidean geometry independent of the Fifth Postulate. Boyer [B2] suggests Bolyai came to the same conclusions as Lobachevsky in 1829 a few years later. Bolyai developed what he called 'Absolute Science of space' starting from the assumption that through a point infinitely many lines can be drawn in the plane, each parallel to a given line. Bolyai's findings were sent to his father who published them in the form of an appendix to a treatise he had completed. His findings are described by Gray [G1] as a brilliant for changing our fundamental ideas about space.
Bolyai set out to discover as many theorems that were only true if the parallel postulate was false. Gray [G1] offers some of the theorems that hold if the parallel postulate is false:
1. All the lines b parallel to the axis meet the surface F at right angles.
2. Any plane containing the axis a meets the surface F in a curve L.
We can consider these theorems as being true even when the Parallel Postulate is true, however, they are deemed trivial by Gray [G1] because in that case the surface F is a plane. Gray [G1] also proposes that in the first theorem the surface F could be a sphere of infinite radius, something that Gauss and Wachter may also have had in mind.
One of Bolyai's most important results is true whether the parallel postulate is true or false, however, it is more interesting to mathematicians when the postulate is false.
3. On any surface F, if two curves L cross a third and the sum of the interior angles is less than two right angles, then the two L curves intersect.
This final result can be equated to a Parallel Postulate for curves L on a surface F. It then follows that if the L curves are straight lines, then Euclidean geometry holds on the surface F, whether or not we assume the Parallel Postulate. Gray [G1] describes how Bolyai called his result an 'absolute' theorem, one which is true even when the Parallel Postulate is false. Further to this Gray [G1] proposes that when Bolyai studied his geometry he could also relate it in a natural way to Euclidean geometry as we have just seen.
One of Bolyai's most exciting discoveries is discussed by Gray [G1] when he showed that if the Parallel Postulate is false then one can construct a square which is equal in area to a given circle. In this sense Bolyai had squared the circle something regarded as being synonymous with achieving the impossible since at least 450 B.C.E. We can infer than it would be with some enthusiasm that Boyai realised in his new geometry one could indeed square the circle.
Bolyai did not publish anymore because of lack of recognition and because of being disheartened by the publication of Lobachevsky's work in 1840. However, we can commend Bolyai's progress for creating possibilities within his geometry to achieve what could not have been done before in Euclidean geometry.
Gauss, Lobachevsky and Bolyai were not the only mathematicians to develop their own non-Euclidean geometry. Wolfe [W1] recognises the achievements of Saccheri an Italian Jesuit priest one-hundred years earlier. In 1773 Saccheri published a book in Milan called 'Euclid Free of Every Flaw'. Saccheri created a non-Euclidean geometry but had to disguise it from Ignatius Viceomes, Provincial of the Jesuits. Halsted [H1] explains Saccheri's motivations for disguising his work as being a way of preserving his life. Another writer DeMorgan describes Saccheri's actions as being similar to writing another New Testament, an idea which is completely unorthodox and would never have been condoned at the time.
So what did Saccheri achieve which could mean he is described as one of the discoverers of non-Euclidean geometry opposed to Lobachevsky, Gauss or Bolyai? Bonola [B2] explains how the majority of Saccheri's work was devoted to proving Euclid's fifth postulate. Saccheri used a particular method of reasoning similar to Euclid's which by assuming as hypothesis that the proposition which is to be proved is false, one is brought to the conclusion that it is true, today we describe this as proof by contradiction.
However, because of a deep belief in Euclid's hypothesis as the absolute truth and being bound by tradition Saccheri's imagination was restricted. We must also consider that the context within which he lived was not conducive to developing a new geometry. To this end I am in agreement with Daus [D1] that without these factors Saccheri would have anticipated the discovery of non-Euclidean geometry by a century, and for this reason we should recognise Saccheri for his work in this field.
Let us also consider that the ideas brought forward by Boyai and Lobachevsky were not recognised for many years. In fact it was Riemanns publication of his famous dissertation in 1866 which was a catalyst for non-Euclidean geometry to be seriously studied. Daus [D1] finds is remarkable that is was not until after the publication of Bolyai and Lobachevsky's work no one working in the field chose to examine the hypothesis of an obtuse angle. In addition we also have the assumption that a straight line is unbounded and yet of finite length. We owe this concept to Riemann who in his dissertation of 1854 laid the analytical foundation for non-Euclidean geometry to be studied.
We can consider Bolyai and Lobachevsky as great contributors to the development of non-Euclidean geometry. However, we must also recognise the important work that Saccheri completed despite being restricted in what he could write and publish. Further to this we should consider that it was not until Riemann's dissertation that other mathematicians chose to seriously study non-Euclidean geometry, in this sense Lobachevsky and Bolyai and indebted to Riemann. However, I conclude in agreement with Boyer [B1. p.522] 'the lion's share of the credit for the development of non-Euclidean geometry consequently belongs to Lobachevsky".
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