By Harold E. Wolfe

Creation to NON-EUCLIDEAN GEOMETRY by way of HAROLD E. WOLFE . PREFACE: This ebook has been written in an try to supply a passable textbook for use as a foundation for simple classes in Non-Euclid ean Geometry. the necessity for this sort of quantity, certainly meant for lecture room use and containing mammoth lists of routines, has been glaring for your time. it's was hoping that this one will meet the re quirements of these teachers who've been educating the topic tegularly, and in addition that its visual appeal will inspire others to institute such classes. x the advantages and facilities of a proper examine of Non-Euclidean Geometry are as a rule famous. not just is the subject material itself beneficial and very interesting, worth the time of any scholar of arithmetic, yet there's most likely no trouble-free path which indicates so in actual fact the character and value of geometry and, certainly, of arithmetic generally. notwithstanding, an insignificant cursory acquaintance with the topic won't do. One needs to keep on with its improvement no less than a bit option to see how issues pop out, and take a look at his hand at demonstrating propositions lower than situations such that instinct now not serves as a advisor. For lecturers and potential academics of geometry within the secondary colleges the examine of Non-Euclidean Geometry is important. without it there's powerful probability that they're going to no longer comprehend the genuine nature of the topic they're educating and the import of its purposes to the translation of actual area. one of the first books on Non-Euclidean Geometry to seem in English was once one, scarcely greater than a pamphlet, written in 1880 via G. Chrystal. Even at that early date the worth of this learn for these getting ready to coach used to be well-known. within the preface to this little brochure, Chrystal expressed his wish to deliver pangeometrical speculations less than the attention of these engaged within the educating of geometry He wrote it is going to now not be intended that I suggest the creation of pan geometry as a college topic it's for the trainer that I recommend one of these examine. it's a nice mistake to consider that it's adequate for the instructor of an undemanding topic to be simply sooner than his students. not anyone could be a stable simple instructor who can't deal with his topic with the clutch of a grasp. Geometrical perception and wealth of geometrical rules, both usual or obtained, are necessary to an outstanding instructor of geometry and that i understand of no greater approach of cultivat ing them than through learning pan geometry. inside of contemporary years the variety of American schools and uni versities which provide classes in complex Euclidean Geometry has elevated swiftly. there's proof that the standard of the educating of geometry in our secondary faculties has, for this reason, tremendously superior. yet complicated research in Euclidean Geometry isn't the in simple terms needful for the great instructing of Euclid. The learn of Non-Euclidean Geometry takes its position beside it as an quintessential a part of the educational of a well-prepared instructor of highschool geometry. This e-book has been ready basically for college kids who've accomplished a path in calculus. notwithstanding, even if a few mathe matical adulthood may be chanced on necessary, a lot of it may be learn profitably and with realizing through one that has accomplished a secondary institution path in Euclidean Geometry. He desire in basic terms disregard Chapters V and VI, which utilize trigonometry and calcu lus, and the latter a part of bankruptcy VII. In Chapters II and III, the historic historical past of the topic has been taken care of rather absolutely. it's been acknowledged that no topic, while separated from its background, loses greater than arithmetic. this can be rather actual of Non-Euclidean Geometry...

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Of these attempts and their failures we shall have much to recount later, much had been proved without For the for they have an all-important bearing upon our subject. of the substitutes for the Fifth some to examine we wish present Postulate. 1 1 . Substitutes (or the Fifth Postulate. When, in the preceding chapter, attention was directed to the importance of the Fifth Postulate in elementary geometry and in what is to follow here, the reader may have been disturbed by an inability to recall any previous encounter with the Postulate.

As a matter of fact, it was essentially Saccheri 's proof which we used above to definitions others and postulates. who show that the assumption of Wallis is equivalent to the Postulate. To prepare for the application of his method, Saccheri made use of a figure with which we are already acquainted. This is the isosceles quadrilateral with the two base angles right angles. Assuming that, in quadrilateral ABCD (Fig. n), AD and BC were equal and that the angles at A and B were right angles, Saccheri without using the Fifth Postulate or its consequences, that the angles at C and were equal and that the line joining the of AB DC and was midpoints perpendicular to both lines.

It is NON-EUCLIDEAN GEOMETRY 30 possible to construct another triangle similar to Then he argues essentially as follows: AB CD it and of any size. DHG less n), cut by the transversal EF in and with the sum of angles BGH and //, respectively, than two right angles. It is to be proved that AB and CD will meet if sufficiently Given lines G points easy to It is and (Fig. and show produced. that LEGE > ZG//D. Then, to it, if segment HG is until //coincides HD rigidly attached position of G, HD takes the moved along EF, with with the initial Hence, during its motion, position GI, lying entirely above GB.