One Too Many? The Role of Euclid’s Fifth Postulate in the Development of non-Euclidean Geometries

This essay will show that non-Euclidean geometries arose as a direct consequence of a weakness in one of Euclid's own axiomatic postulates, specifically Postulate Five. Further, that should Euclid have omitted this contentious Fifth Postulate, the very possibility of non-Euclidean geometries would be challenged. This paper will trace the history of non-Euclidean geometries by exploring how the generations of post-Euclidean philosophers and scientists struggled with the Fifth Postulate, and how the work that was performed on this task led directly to the formulation of these alternate geometries. In this way it will be proven that without the debate over the Fifth Postulate, non-Euclidean geometries would be unnecessary because, as we will see, the only modifications these geometries make is to the Fifth Postulate, with the rest remaining almost wholly intact. The discussion will encompass the work of many great thinkers, including Euclid, Kant, Gauss, Bolyai, Lobachevsky, Sacchieri, and Poincare.

 

Euclid and The Greeks

 

Non-Euclidean geometries were viewed by many as casting doubt on the correctness of Euclidean theory for the first time. Much of the power of the non-Euclidean geometries came from the shaky foundations of Euclid's Fifth Postulate as published in Book One of The Elements. This postulate states: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." (Huggett, p.16). This so-called "parallel postulate" was thought by many to be unnecessary and deducible from the other nine postulates. Euclid himself was not completely satisfied with this last postulate, for it seemed to lack the "intuitive" quality of the first four. In fact, he proved his first twenty-eight propositions without invoking the Fifth Postulate once. His peers also left records of their struggles with this postulate: Proclus (410-485) wrote a commentary on The Elements where he notes attempted proofs to deduce the Fifth Postulate from the other four, in particular he finds that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did leave us the following hypothesis which is equivalent to Euclid's Fifth Postulate: "Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.(van den Berg).

 

Saccheri


Gerolamo Saccheri (1667-1733) brilliantly attempted to prove that Postulate Five was deducible from the other propositions through a reductio ad absurdum (reduction to absurdity) argument. According to Larry Sklar in Space, Time, and Spacetime, "To demonstrate the deducibility of the Fifth Postulate, it would be sufficient to show that both the hypothesis that through a point outside a given line no such nonintersecting line could be drawn and the hypothesis that more than one nonintersecting line could be drawn would, in combination with the remaining nine basic propositions, lead to self-contradictions or logical absurdities."(Sklar, p.17). Saccheri was able to show that the hypothesis of a line having no nonintersecting straight lines led to self-contradiction. In order to find this, he had to assume that Postulate Two "...To produce a finite straight line continuously in a straight line"(Huggett, p.16) meant one could extend the line indefinitely in either direction, and this being so, the idea of there being a line that could be drawn parallel to any given line was shown to be deducible from other postulates. The second part of the proof, the uniqueness of such a line, was more difficult. In fact, he could not produce logical contradictions from the assertion of more than one parallel line. This is due to the fact that there seems to be multiple nonintersecting lines that can be drawn for any given extended line, and this seemed to be in accordance with the former Euclidean propositions.


Gauss


The first person to really come to understand the problem of the parallels was Karl Friedrich Gauss. He began work on the Fifth Postulate in 1792 while only 15 years old, at first attempting to prove the Fifth Postulate from the others. By 1817 Gauss had become convinced that the Fifth Postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Surprisingly, Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry was synthetic a priori knowledge, and Gauss did not wish to challenge Kant's considerable authority. In 1827 Gauss published his crucial discovery that the curvature of a surface can be defined intrinsically, meaning solely in terms of properties defined within the surface and without reference to the surrounding Euclidean space. This result was to be decisive in the acceptance of non-Euclidean geometry, because it highlighted the possibility of alternate geometries. If there was a logically consistent geometry differing from Euclid's only because it made a different assumption about the behavior of parallel lines, it too could apply to physical space, and so the truth of Euclidean geometry could no longer be assured a priori, as Kant had thought. Once again, the Fifth Postulate was under attack, yet Gauss would not publish his work out of fear of repercussions. The job of promulgating the new geometric possibilities fell on the shoulders of two men some say Gauss secretively coached through written correspondence: Janos Bolyai and Nicolai Lobachevsky.


Bolyai & Lobachevsky


Now that the Fifth Postulate has been shown to be unclear and open to interpretation, a review of the proposed alternatives to this postulate is in order. One alteration of the original was formulated almost simultaneously by Gauss, Schweikart, Bolyai, and Lobachevsky, and is commonly associated with hyperbolic geometry because it applies to spherical objects or those with a constant negative curvature. In this revision of the postulate, the "one parallel" clause was replaced by a "many parallel postulate" in which more than one parallel line can be drawn from a given point for a given line. Both Bolyai and Lobachevsky made an assumption about parallel lines that differed from Euclid's and proceeded to draw out its consequences. This way of working under assumptions cannot guarantee the consistency of one's findings, so strictly speaking they could not prove the existence of a new geometry in this way. Both men described a three-dimensional space different from Euclidean space, couching their findings in the language of trigonometry, but were poorly recieved by their peers, mostly because Gauss was still reluctant to publicly voice his approval. The turn toward acceptance only came in the 1860s after Bolyai and Lobachevsky had died. The Italian mathematician Eugenio Beltrami decided to investigate Lobachevsky's work and to place it, if possible, within the context of differential geometry as redefined by Gauss, thereby moving independently in the direction already taken by Bernhard Riemann. Beltrami investigated the surface of constant negative curvature and found that on such a surface triangles obeyed the formulas of hyperbolic trigonometry that Lobachevsky had discovered were appropriate to his form of non-Euclidean geometry. Thus he strengthened the Bolyai-Lobachevsky hypothesis, and gave the first rigorous description of a geometry other than Euclid's.

 

Reimann


In 1854 mathematician Bernhard Reimann took up a subject that much interested Gauss, Euclid's Fifth Postulate. Riemann took his inspiration from Gauss's discovery that the curvature of a surface is intrinsic, and he argued that one should therefore ignore Euclidean space and treat each surface by itself. A geometric property, he argued, was one that was intrinsic to the surface. To do geometry, it was enough to be given a set of points and a way of measuring lengths along curves in the surface. For this, traditional ways of applying calculus to the study of curves could be made to suffice. But Riemann did not stop with surfaces. He proposed that geometers study spaces of any dimension in this spirit, even, he said, spaces of infinite dimension. Several profound consequences followed from this view. It essentially dethroned Euclidean geometry, which now became just one of many possible geometries. It allowed the geometry of Bolyai and Lobachevsky to be recognized as the geometry of a surface of constant negative curvature, thus resolving doubts about the logical consistency of their work. Lastly, Riemann's work ensured that any investigation of the geometric nature of physical space would thereafter have to be partly empirical. One could no longer say that physical space is Euclidean because there is no geometry but Euclid's. This finally destroyed any hope that questions about the world could be answered by a priori reasoning.

 

Poincaré


Poincaré's ideas on the foundation of geometry most probably stem from his research on functions defined by differential equations, whereby he actually used non-Euclidean geometry. He found that Lobachevskian geometry could easily prove several geometric properties, while in terms of Euclidean geometry their proof is difficult to achieve. Also, Poincaré knew Beltrami's research on Lobachevskian geometry, whereby Beltrami proved the consistency of Lobachevskian geometry with respect to Euclidean geometry by means of a translation of every term of Lobachevskian geometry into an equivalent term of Euclidean geometry. The translation is carefully chosen so that every axiom of non-Euclidean geometry is translated into a theorem of Euclidean geometry. Beltrami's translation and Poincaré's study of functions led Poincaré to assert that:

  • Non-Euclidean geometries have the same logical and mathematical legitimacy as Euclidean geometry.

  • All geometric systems are equivalent and thus no system of axioms may claim that it is the true geometry.

  • Axioms of geometry are neither synthetic a priori judgments nor analytic ones; they are conventions or 'disguised' definitions.


For Poincaré, all geometric systems deal with the same properties of space, although each of them employs its own language, whose syntax is defined by the set of axioms. In other words, geometries differ in their language, but they are concerned with the same reality, for one geometry can be translated into another geometry. There is only one criterion according to which we can select a geometry, namely a criterion of economy and simplicity. This is the very reason why we commonly use Euclidean geometry: it is the simplest. However, due to the reformulations of the Fifth Postulate, we may at times, and with respect to certain problems, choose non-Euclidean geometry in order to obtain the desired result with less effort.


In conclusion, various alternatives to standard Euclidean geometry were made possible simply by the existence of Euclid's Fifth Postulate. Sklar notes this correlation in speaking of German mathematician David Hilbert's axiomatic study of both Euclidean and non-Euclidean geometries; "Hilbert's formalization also makes much clearer the extent to which non-Euclidean geometries depend in their specification on the changes in the parallel postulate... ...[the Fifth Postulate] uniquely characterizes Euclidean geometry, the No-Parallel Postulate Reimannian geometry, and the Many-Parallel Postulate Lobachevskian geometry." (Sklar, p.27). We can clearly see that the Fifth Postulate is the primary source of differentiation between these three forms of geometry. This being true, it necessarily follows that if that source of differentiation were to be removed (ie: if Euclid had omitted it) then there would simply be no need for these alternate geometries. Reimann, Bolyai, Lobachevsky, Gauss and Poincaré would all have been describing viable possibilities, but these formulations would all have been grounded solidly within Euclidean geometric parameters.


 

Texts Consulted

 

 

Sklar, L., Space, Time, and Spacetime

University of California Press, Berkeley 1977

 

 

 

Huggett, N., Space From Zeno to Einstein Classic Readings with a Contemporary Commentary

The MIT Press, Cambridge 1999


van den Berg, R. M. (June 19, 1998) [Online]

Available WWW :http://www.leidenuniv.nl/philosophy/proclus/ [March 11,2000]1

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