![]() ![]() Of all the reformulations, one proves to be most useful. There is no upper bound to the area of a triangle.If in a quadrilateral a pair of opposite sides are equal and if the angles adjacent to the third side are right angles, then the other two angles are also right angles.There exists a pair of similar, non-congruent triangles.There exists a pair of coplanar straight lines, everywhere equidistant from one another.One of most important by-products of the efforts to derive Euclid's fifth postulate were simpler, alternative formulations of the postulate that could be used in place of Euclid's original. Alternative Formulations of Euclid's Fifth Postulate InĬhapter, we will give an illustration of what it is like to do geometry in a space governed by an alternative to Euclid's fifth postulate. If one has a prior background in Euclidean geometry, it takes a little while to be comfortable with the idea that space does not have to be Euclidean and that other geometries are quite possible. But would it still work if we surveyed volumes of space on the cosmic scale? And what are to we to make of Kant's assurance that space has to be Euclidean, a synthetic a priori fact? It certainly worked where ever we looked. The success of Euclidean geometry was something to be discovered. It gradually became clear that geometry did not have to be Euclidean. The import of this realization was profound. These facts seemed odd simply because they belonged in a geometry different from that of Euclid. Rather they were defining a new geometry.The conclusions they drew were merely facts in the new geometry. When the earlier geometers had posited an alternative to Euclid's fifth postulate, they were not creating a contradiction. In the nineteenth century, the reason for this frustrating failure was finally recognized by Gauss, Riemann, Bolyai, Lobachevsky and others. None flatly asserted "A and not-A." It was as if the geometers had struggled past many dangers but were perpetually trapped one step short of the end of their journey. It was maddening in that none of the results, no matter how odd, was actually a flat-out contradiction. It was encouraging in that all sorts of very odd results followed. The work was both encouraging and maddening. The postulates needed for Euclid's geometry would thereby be reduced to the first four. And that just means that we have inferred the truth of the fifth postulate from the other four. Or, more carefully, as long as the first four postulates are true, then the fifth is true. Stripping out the double negative (".negation.false") we just have that the fifth postulate is true. That is, the finding of a contradiction showed that the negation of the fifth postulate was false. So the false presumption had to be the negation of the fifth postulate. Four of these were just the first four of Euclid's postulates, which were taken to be secure. The candidates for the false presumption were the five assumptions of the starting point. Euclid'sĪrriving at a contradiction would show that a false presumption had been made somewhere. The goal was to demonstrate that a contradiction followed. Then the geometer would proceed to explore the consequences of these five assumptions. The procedure was to take the first four postulates and add the negation of the fifth to them. ![]() Then, geometry would need only to posit the first four postulates the fifth would be deduced from them.Īn indirect strategy was used in the efforts to derive the fifth postulate from the other four. The goal was to derive the fifth postulate from the other four. In the eighteenth century, as in the centuries before, the project had been to rid Euclid's geometry of this flaw. Unlike the other four postulates, the fifth postulate just did not look like a self-evident truth. The one blemish was the artificiality of the fifth postulate. Alternative Formulations of Euclid's Fifth PostulateĮuclid's Postulates and Some Non-Euclidean Alternativesįrom the Eighteenth to the Nineteenth CenturyĬhapter that the earlier centuries brought the nearly perfect geometry of Euclid to nineteenth century geometers.From the Eighteenth to the Nineteenth Century. ![]()
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