Born in dusseldorf to prussian parents, he received his education at the university of bonn. In two papers titled on the socalled noneuclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant. His 1872 erlangen program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of. Thanks also to euclid, henri poincare, felix klein, janos bolyai, and all other pioneers in the field of geometry. Euclidean geometry is not a necessary precondition for a consistent description of the spatial aspects of the physical world. Albert einsteins theory of special relativity illustrates the power of klein s approach to geometry. In each case the common features that, in kleins opinion, made them geometries were that there were a. Noneuclidean geometry is not not euclidean geometry. Poincare discovered a model made from points in a disk and arcs of circles orthogonal to the boundary of the disk. Non euclidean geometry is not not euclidean geometry. Tait particular plane point of view present principal numbers problem professor of mathematics projective geometry pure quantity quartic. Lectures on the icosahedron and the solution of the fifth. Special relativity, says einstein, is derived from the notion that the laws of nature are invariant with respect to lorentz transformations.
The aim of the thematic afternoon, the legacy of felix klein was to inform, think about, and discuss the meaning of the work and ideas of felix klein both currently and in the future. Very short prehistory of noneuclidean geometry the euclidean postulates were from beginning the object of research due to a long and complicated formulation of the 5th postulate on parallels. The isometries of beltramis half plane model are readily described in terms of linear fractional transformations in fact, the harmony is even more marked in three. Beltrami, klein, and the acceptance of noneuclidean geometry. So pythagoras theorem is a theorem of euclidean geometry. Look at the lines of longitude, say 30 degrees w and 40 degrees w, note that at the equator these lines are parallel, no look at either of the poles and you will see that they intersect. This booklet and its accompanying resources on euclidean geometry represent the first famc course to be written up. In two papers titled on the socalled noneuclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1 that is, hyperbolic, euclidean and spherical geometry within the realm of projective geometry.
Felix christian klein april 25, 1849 june 22, 1925 was a german mathematician, known for his work in group theory, function theory, noneuclidean geometry, and on the connections between geometry and group theory. Felix klein always emphasised the great importance of teaching at the university, and he strongly promoted the modernisation of mathematics in the classrooms. It is one type of noneuclidean geometry, that is, a geometry that discards one of euclids axioms. Euclidean geometry is not a necessary precondition for a consistent description of. On the side ab of 4abc, construct a square of side c. Elementary mathematics from a higher standpoint volume ii. Theory of functions and geometry felix klein, the evanston colloquium.
The project gutenberg ebook noneuclidean geometry, by. In two papers titled on the socalled non euclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1 that is, hyperbolic, euclidean and spherical geometry within the realm of projective geometry. Klein went further than this and gave models of other non euclidean geometries such as riemanns spherical geometry. What is and what might be the legacy of felix klein. The project gutenberg ebook noneuclidean geometry, by henry. This is a text on 3d euclidean computational geometry intended to be used in engineering applications. His published works include elementary mathematics from an advanced standpoint. Noneuclidean geometry, literally any geometry that is not the same as euclidean geometry. Cayleyklein geometry, noneuclidean geometry, projectivemetric spaces, kleins model. His 1872 erlangen program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Cayley discovered that euclidean geometry can be considered as a special case of projective geometry1 which led him to the famous statement that descriptive geometry his term for projective geometry is all geometry. Felix klein 18491925 was one of the most influential mathematicians and mathematics educators of his time. Felix klein moved on from his work on the klein quadric and moved on physically from bonn. Many attempts failed until the ideas of non euclidean geometry appeared in 1829 by n.
Beltramis work on a model of bolyai lobachevskys non euclidean geometry was completed by klein in 1871. Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries hyperbolic and spherical that differ from but are very close to euclidean geometry see table. Felix klein, seminar on the psychological foundations of mathematics translated by eugene chislenko draft contents n. Felix kleins life shows us that a sensitised person is always necessary in order to recognise problems, think in a visionary manner, and act effectively. Elementary mathematics from a higher standpoint volume. Even so, klein saw a fundamental unity in the subject of non euclidean geometry. Very short prehistory of non euclidean geometry the euclidean postulates were from beginning the object of research due to a long and complicated formulation of the 5th postulate on parallels.
Einstein and minkowski found in noneuclidean geometry a. The present paper is dedicated to felix klein 18491925, one of the leading german mathematicians in the second half of the 19th century. In 1871, klein completed the ideas of noneuclidean geometry and gave the solid underpinnings to the subject. Felix klein 18491925 proposed systematizing all these geometries in grouptheoretic terms. Pdf on kleins socalled noneuclidean geometry researchgate. Kleins work was based on a notion of distance defined by cayley in 1859 when he proposed a generalised definition for. Elementary mathematics from an advanced standpoint. The project gutenberg ebook non euclidean geometry, by henry manning this ebook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. German mathematician felix klein 18491925 specialized in function theory, group theory, and non euclidean geometry. As euclidean geometry lies at the intersection of metric geometry and affine geometry, noneuclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. Felix klein, seminar on the psychological foundations of. In his erlanger programm klein proposed that euclidean and non euclidean geometry be regarded as special cases of projective geometry. Klein s work was inspired by ideas of cayley who derived the distance between two points and the angle between two planes in.
Klein model of noneuclidean geometry, establish ing that the consistency of noneuclidean geometry is equivalent to the consistency of euclidean ge ometry. Xl the most recent researches in noneuclidean geometry. As euclidean geometry lies at the intersection of metric geometry and affine geometry, non euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. Now here is a much less tangible model of a noneuclidean geometry. Noneuclidean geometry mactutor history of mathematics. Noneuclidean geometry proves to constitute a part of this unifying process. Papadopoulos, on kleins socalled noneuclidean geometry, in sophus lie and felix klein. Old and new results in the foundations of elementary plane. In that article, he gave the classification of them that now stands, saying that the points at infinity on a. He shows that there are essentially three types of geometry. Christian felix klein 25 april 1849 22 june 1925 was a german mathematician and mathematics educator, known for his work in group theory, complex analysis, noneuclidean geometry, and on the connections between geometry and group theory. Felix christian klein april 25, 1849 june 22, 1925 was a german mathematician, known for his work in group theory, function theory, non euclidean geometry, and on the connections between geometry and group theory.
In mathematics, noneuclidean geometry consists of two geometries based on axioms closely related to those specifying euclidean geometry. In two papers titled on the socalled noneuclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1. To see this think of the globe and two lines of longitude. Sources of hyperbolic geometry history of mathematics, v. German mathematician felix klein 18491925 specialized in function theory, group theory, and noneuclidean geometry.
Apr 21, 2020 made by the german mathematician felix klein and published at erlangen in 1872. Early in the novel two of the brothers, ivan and alyosha, get reacquainted at a tavern. Moreover, the discovery of noneuclidean geometry, about 40 years earlier, had just been demonstrated to be a true geometry on the same footing as euclidean geometry. Fyodor dostoevsky thought noneuclidean geometry was interesting enough to include in the brothers karamazov, first published in 1880. A brief history of geometry geometry mike hitchman. On the other hand, the methods of whiteheads algebra enable us to readily deal with euclidean and non euclidean spaces of any dimension. Lastly, there are three little texts by poincare, from a third viewpoint. Like so much of mathematics, the development of non euclidean geometry anticipated applications. The project gutenberg ebook noneuclidean geometry, by henry manning this ebook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. Euclidean geometry is the study of plane and solid gures which is based on a set of axioms formulated by the greek mathematician, euclid, in his books, the elements. In a noneuclidean geometry such as spherical geometry, two lines can be parallel and still intersect. In mathematics, non euclidean geometry consists of two geometries based on axioms closely related to those specifying euclidean geometry. Old and new results in the foundations of elementary plane euclidean and non euclidean geometries marvin jay greenberg by elementary plane geometry i mean the geometry of lines and circles straightedge and compass constructions in both euclidean and non euclidean planes. Many attempts failed until the ideas of noneuclidean geometry appeared in 1829 by n.
In a non euclidean geometry such as spherical geometry, two lines can be parallel and still intersect. Hyperbolic geometry is an imaginative challenge that lacks important features of euclidean geometry such as a natural coordinate system. In two major papers of 1871 and 1873, and also in his erlangen program 1872, felix klein unified most of the existing geometries, including noneuclidean. Euclid was born around 300 bce and not much is known about. In his erlanger programm klein proposed that euclidean and noneuclidean geometry be regarded as special cases of projective geometry. Pdf a survey of the development of geometry up to 1870. Indeed, beltramis projective disc metric begs to be interpreted in terms of projective. The gottingen mathematician felix klein 18491925, for example, in his 18891890 lectures on noneuclidean geometry, elaborated projective methods in great detail, wasting little time on other approaches klein 189092. Pdf in two papers titled on the socalled noneuclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature. The gottingen mathematician felix klein 18491925, for example, in his 18891890 lectures on non euclidean geometry, elaborated projective methods in great detail, wasting little time on other approaches klein 189092. Einstein and minkowski found in noneuclidean geometry a this work was supported in part by the geometry center, university of minnesota, an stc. Hyperbolic geometry was created in the rst half of the nineteenth century in the midst of attempts to understand euclids axiomatic basis for geometry. Felix klein demonstrates that elliptical spherical geometry is logically consistent.
In each case the common features that, in klein s opinion, made them geometries were that there were a. In two papers titled on the socalled noneuclidean geometry, i and ii 32 and 34, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1 that is, hyperbolic, euclidean and spherical geometry within the realm of projective geometry. Similarly, projective geometry subsumes spherical and euclidean geometry as well. Non euclidean geometry, literally any geometry that is not the same as euclidean geometry. Klein s work was inspired by ideas of cayley who derived the distance between. The transcription and translation are to run on facing pages. Jun 27, 2014 in two papers titled on the socalled non euclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1 that is, hyperbolic, euclidean. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison.
The most recent researches in noneuclidean geometry. In two papers titled on the socalled noneuclidean geometry, i and. However, theodosius study was entirely based on the sphere as an object embedded in euclidean space, and never considered it in the noneuclidean sense. Ii 32 and 34, felix klein proposed a construction of the spaces of. Instead of differential geometry, klein approches the subject from the point of view of projective geometry. Klein s work was based on a notion of distance defined by cayley in 1859 when he proposed a generalised definition for. Noneuclidean geometry special topics a brief course. Old and new results in the foundations of elementary plane euclidean and noneuclidean geometries marvin jay greenberg by elementary plane geometry i mean the geometry of lines and circles straightedge and compass constructions in both euclidean and noneuclidean planes. Kleins work was inspired by ideas of cayley who derived the distance between two points and the angle. This booklet and its accompanying resources on euclidean geometry. The arrival of noneuclidean geometry soon caused a stir in circles outside the mathematics community.
The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of euclidean geometry in a complete system such as hilberts. Klein went further than this and gave models of other noneuclidean geometries such as riemanns spherical geometry. The felix klein protocols american mathematical society. His 1872 erlangen program, classifying geometries by their basic symmetry groups, was an. Even so, klein saw a fundamental unity in the subject of noneuclidean geometry. On the other hand, the methods of whiteheads algebra enable us to readily deal with euclidean and noneuclidean spaces of any dimension. In two papers titled on the socalled noneuclidean geometry, i and ii, felix klein proposed a construction of the spaces of constant curvature 1, 0 and and 1 that is, hyperbolic, euclidean. His 1872 erlangen program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of. Darboux arranged for the first translation of a paper by klein. Beltramis work on a model of bolyai lobachevskys noneuclidean geometry was completed by klein in 1871. Felix christian klein biography facts, childhood, family.