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A Phenomenon in Geometric Analysis M 23 Indian Journal of Science and Technology Vol.2 No 4 (Apr. 2009) ISSN: 0974- 6846 A phenomenon in geometric analysis M. Sivasubramanian Department of Mathematics, Dr.Mahalingam College of Engg. & Technology, Pollachi, Tamil Nadu 642003, India [email protected] Abstract: It has been established that it is impossible to He recognized that three possibilities arose from omitting deduce Euclid V from Euclid I, II, III, and IV. The Euclid's Fifth; if two perpendiculars to one line cross investigations devoted to the parallel postulate gave rise another line, judicious choice of the last can make the to a number of equivalent propositions to this problem. internal angles where it meets the two perpendiculars Also, while attempting to prove this statement as a equal (it is then parallel to the first line). If those equal special theorem Gauss, Bolyai and Lobachevsky internal angles are right angles, we get Euclid's Fifth; independently found a consistent model of first non- otherwise, they must be either acute or obtuse. He Euclidean geometry namely hyperbolic geometry. persuaded himself that the acute and obtuse cases lead Gauss’s student Riemann developed another branch of to contradiction, but had made a tacit assumption non-Euclidean geometry which is known as Riemannian equivalent to the fifth to get there. geometry. The formulae of Lobachevskyian geometry Nasir al-Din al-Tusi (1201-1274), in his Al-risala al- widely used tot sty the properties of atomic objects in shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion quantum physics. Einstein’s general theory of relativity is Which Removes Doubt about Parallel Lines) (1250), nothing but beautiful application of Riemannian geometry. wrote detailed critiques of the parallel postulate and on Einstein derived these field equations by analyzing Khayyám's attempted proof a century earlier. Nasir al-Din geometry of space-time. In this study the author re-visited attempted to derive a proof by contradiction of the parallel the parallel postulate and by protecting himself under postulate He was also one of the first to consider the Saccheri’s umbrella found a consistent geometric result cases of elliptical geometry and hyperbolic geometry, which challenged the previous contributions in this field. though he ruled out both of them Euclidean, elliptical and Keywords: Euclid, elements, postulates, non-Euclidean hyperbolic geometry. The Parallel Postulate is satisfied geometries physical applications only for models of Euclidean geometry. MSC; 51M04; PACS: 02.40.Dr. Nasir al-Din's son, Sadr al-Din (sometimes known as Introduction "Pseudo-Tusi"), wrote a book on the subject in 1298, For two thousand years, many attempts were made based on Nasir al-Din's later thoughts, which presented to prove the parallel postulate using Euclid's first four one of the earliest arguments for a non-Euclidean postulates. The main reason that such a proof was so hypothesis equivalent to the parallel postulate. "He highly sought after was that the fifth postulate isn't self- essentially revised both the Euclidean system of axioms evident unlike the other postulates. If the order the and postulates and the proofs of many propositions from postulates were listed in the Elements is significant, it the Elements. His work was published in Rome in 1594 indicates that Euclid included this postulate only when he and was studied by European geometers. This work realised he could not prove it or proceed without it. marked the starting point for Saccheri's work on the Ibn al-Haytham (Alhazen) (965-1039), an Iraqi subject. mathematician, made the first attempt at proving the Giordano Vitale (1633-1711), in his book Euclide parallel postulate using a proof by contradiction, where he restituo (1680, 1686), used the Khayyam-Saccheri introduced the concept of motion and transformation into quadrilateral to prove that if three points are equidistant geometry. He formulated the Lambert quadrilateral, which on the base AB and the summit CD, then AB and CD are Boris Abramovich Rozenfeld names the "Ibn al-Haytham– everywhere equidistant. Girolamo Saccheri (1667-1733) Lambert quadrilateral", and his attempted proof also pursued the same line of reasoning more thoroughly, shows similarities to Playfair's axiom. correctly obtaining absurdity from the obtuse case Omar Khayyám (1050-1123) made the first attempt at (proceeding, like Euclid, from the implicit assumption that formulating a non-Euclidean postulate as an alternative to lines can be extended indefinitely and have infinite the parallel postulate, and he was the first to consider the length), but failing to debunk the acute case (although he cases of elliptical geometry and hyperbolic geometry, managed to wrongly persuade himself that he had). though he excluded the latter The Khayyam-Saccheri Where Khayyám and Saccheri had attempted to quadrilateral was also first considered by Omar Khayyam prove Euclid's fifth by disproving the only possible in the late 11th century in Book I of Explanations of the alternatives, the nineteenth century finally saw Difficulties in the Postulates of Euclid. Unlike many mathematicians exploring those alternatives and commentators on Euclid before and after him (including discovering the logically consistent geometries which Giovanni Girolamo Saccheri), Khayyam was not trying to result. In 1829, Nikolai Ivanovich Lobachevsky published prove the parallel postulate as such but to derive it from an account of acute geometry in an obscure Russian an equivalent postulate: "Two convergent straight lines journal (later re-published in 1840 in German). In 1831, intersect and it is impossible for two convergent straight János Bolyai included, in a book by his father, an lines to diverge in the direction in which they converge”. appendix describing acute geometry, which doubtlessly, Research article “Geometry” Sivasubramanian Indian Society for Education and Environment (iSee) http://www.indjst.org Indian J.Sci.Technol. 24 Indian Journal of Science and Technology Vol.2 No 4 (Apr. 2009) ISSN: 0974- 6846 he had developed independently of Lobachevsky. Carl From equations (3)&(6) we get sides EF & BC equal (7) Friedrich Gauss had actually studied the problem before Discussion that, but he did not publish any of his results. However, From (7) we get that the summit angle FBC is a right upon hearing of Boylai's results in a letter from Bolyai's angle (Effimov, 1972).Consequently this establishes the father, Farkas Bolyai, he stated: "If I commenced by parallel postulate (Effimov, 1972; Smilga, 1972) But the saying that I am unable to praise this work, you would mere existence of consistent models of hyberpolic and certainly be surprised for a moment. But I cannot say elliptic geometries demonstrate that the parallel postulate otherwise. To praise it would be to praise myself. Indeed cannot b deduced from the first four postulates. Since our the whole contents of the work, the path taken by your result is consistent, there is something hidden. Only son, the results to which he is led, coincide almost further studies will unlock this mystery. The author has entirely with my meditations, which have occupied my found three more results (Sivasubramanian & Kalimuthu, mind partly for the last thirty or thirty-five years." 2008; Sivasubramanian et al., 2008; Sivasubramanian, The resulting geometries were later developed by 2009). S.Kalimuthu has proved that there exists a Lobachevsky, Riemann and Poincaré into hyperbolic spherical quadrilateral whose interior angle sum is equal geometry (the acute case) and spherical geometry (the to 360 degrees (http://wbabin.net/physics/ obtuse case). The independence of the parallel postulate kalimuthu10.pdf). Also, he has established applying linear from Euclid's other axioms was finally demonstrated by algebraic equations to Euclidean geometry that the sum Eugenio Beltrami in 1868. of the interior angles of a triangle is equal to two right Construction angles. His construction and proof can be easily extended Construct Sachheri quadrilateral ABCD as shown in Fig. 1. Sides AD & BC are equal. The Fig. 1 (Euclidean) Fig. 2 (Euclidean) angles at C& D are right angles. Locate the mid F J B points E and F of CD and AB respectively. Join L A E and F. Sachheri showed that the angles at E & F are also right angles and the summit F angles at A & B are equal (Effimov,1972; A B Smilga, 1972). L J Results Case i Let us assume that EF is smaller than BC. On the extension of EF, take a point L such that EL = BC. On the production of EC, make a point H such that DC = CH. At H, erect a perpendicular HJ equal to EL. Join L and J. D E C H D E C H Now by SASAS correspondence, Sachheri to hyperbolic and spherical geometries quadrilaterals ABCD & HELJ are congruent. So, the (http://wbabin.net/physics/kalimuthu15.pdf) (Kalimuthu, summit angles at A, B, L & J are equal. 2009). Kalimuthu’s spherical geometry theorem and his i.e angles DAB = CBA = ELJ = HJL (1) general algebraic theorem CANNOT be questioned. So, Join B & L. Now, ECBL is an another Sachheri the author’s finding is consistent. quadrilateral. So, the summit angles at B & L are equal. References i.e. angle CBL = angle ELB (2) 1. Effimov NV (1972) Higher geometry. Mir Publishers, Comparing (1) and (2) we get a contradiction. This Moscow. implies that our assumption that EF is smaller than BC is 2. Kalimuthu S (2009) The parallel postulate- return of NOT applicable (3) the roaring lion. Indian J. Sci. Technol. 2 (4), x-x. Cse ii Domain:http://www.indjst.org. Let us assume that EF is greater than BC. Now look 3. Sivasubramanian M (2009) Application of at Fig. 2. On EF choose a point such L that EL = BC. On Sivasubramanian Kalimuthu hypothesis to triangles. the extension of EC take a point H.
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