DOCSLIB.ORG

Projective Geometry from 1822-1918

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Projective Geometry from 1822-1918 University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1930 Projective geometry from 1822-1918 John B. Lennes The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Lennes, John B., "Projective geometry from 1822-1918" (1930). Graduate Student Theses, Dissertations, & Professional Papers. 8077. https://scholarworks.umt.edu/etd/8077 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. PR0JECTI7E GEOMETRY from 1822 to 1918 by J. Burr Lennea Presented, in partial fulfillment of the requirement for the degree of Master of Arts, State Unirer^y of Montana 1930 Approved EXAMHÎIUG COIMITTEE Chairman Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38878 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OisssrtaMon RjMishing UMI EP38878 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COEfTSM?S Page Introaiiot ion 1 Chapter I. Greek Geometry with Regard to the Origin of Projective Geometry 4 Chapter II. Geometry in Europe from 1500 to 1800, with Regard to the Origin of Projective Geometry. 12 Chapter III. Projective Geometry from 1822 to 1906. Poncelet 46 Ton Stattdt 88 S tein e r 122 Reye 124 Chapter IT. The Reorganization of Pro­ jective Geometry in the First Part of the Present Century. 131 Chapter T. Resulting conclusions. I. The nature of Intellectual Progress 165 II. The nature, Uses, etc., of the Deductive logical Structure. 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "Bn observant oe q^ne les r^sulta*ts partic-uliers avaient de oornmin entre enz, on est suocesslvement parvenu k des résultats fort étendus, et les sciences mathématiques sont a la fols devenues plus générales et plus sim ples." Laplace tOncted by Steiner on the title page of his "Systematische Bntwiclc/iielung", ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction Although many of the theorems which now form the sub­ ject matter of projective geometry have been known for hun­ dreds of years, and some of them for more than two thousand years, it was not until the nineteenth century that they became unified into a separate, well-defined branch of math­ ematics. It is our object to trace the origin of the methods and propositions which now constitute projective geometry, thus outlining the gradual development of the subject, and to give a picture of modern projective geometry, with special attention to the fundamental significance of the logical pro­ cedure. Thus, it is hoped, we may throw new light upon the early discoveries, revealing more clearly their importance in the history of thought and their position in modern geometry. The subject divides itself naturally into four parts. The first is a brief résumé of what there was in Greek geom­ etry that has become fundamental in projective geometry. The second part covers the period including the seventeenth and sixteenth centuries, when the French and English math­ ematicians were most active in this as in nearly every other branch of mathematics. The third is the account of the jrogress in the nineteenth century, when the subject assumed definite shape and developed more rapidly than in any earlier period. In this period most of the work was done in France and Germany. The fo u rth part is an account of p ro jectiv e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. geometry in the last thirty years. Since 1900 the whole subject has been given new, rigorous treatment and has been unified in a clear and fundamental way, with full conscious­ ness of the underlying logical foundation. There have been certain major changes in the develop­ ment of geometry which it will be well to enumerate at the o u tse t. One of the most important of these was the swing from the rigorous proofs of the Greeks to the Icgical laxness of the mathematicians of the seventeenth, eighteenth, and early nineteenth centuries, and then back to the very highly de­ veloped rigor of modern p ro jectiv e geometry. Another was the steady change of emphasis from the solution of individual prob­ lems to the development of connected unified theories - that is, the change from the particular to the general. The last Important change was the change from nstrie to non-metric geometry. Greek geometry was essentially metric, whereas mod­ ern projective geometry is essentially non-metric. Completely non-metric geometry is a very recent development in the history of mathematics. Such general tendencies as these give interest and sig­ nificance to the particular historical facts. The facts pre­ sented here have been selected with a view to including all those which were important, especially those which seem to have had an influence on the histoiy of the subject, and to rather rigidly excluding all which from this view point were Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. less relevant. heedless to say, not all the original documents, especial­ ly in regard to the Greeks, are available. And even in the parts dealing with the suhjeot from 1800 to the present time, where the original sources have been used, not all the facts have been considered, since only the major works were studied. It is hoped, however, that nothing of really great significance has been overlooked. The last chapter of this thesis (chapter 7) contains ma­ terial less strictly mathematical and more general in its sig­ nificance. We have seen how painfhlly slow and halting was the development of a subject whiah in the finished state seems relatively simple, and that in spite of the fact that many of the world’s greatest geniuses gave it very insistent and pro­ longed effort. Why did tiiey fail to hit upon some of the later and much more powerful methods? Again the very existence of huge logical structure purely deductive in character is cer­ tainly of general philosophical significance. 7/hat is the nature of such a deductive logical structure, and is it as barren as some have claimed it to be? What is the function in thought of purely deductive logic? If final answers to some of these questions cannot be given, it is believed that they are formulated in a manner to bring out their significance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter I Plato (430—347 B.C.) is the first of the Greek mathe­ maticians In whose work we find anything related at all di­ rectly to oar subject. It was he who studied conic sections, which foim a considerable part of the subject matter of pro­ jective geometry. There is, however, no evidence that Plato developed many of the properties of these curves. He was the first to femulate the idea of geometrical loci, which is ingortant in all branches of mathematics. Euclid, about 300 B. C., is famous for his "Elements" in which he formulated rather completely the classical Greek geometry, with considerable extension and improvement of his own. This is the basis of ordinary modern elementary geometry. The method of "reductio ad absurdum" which is valuable in projective geometry as everywhere else in logic, was for­ mulated by Euclid. Among his works which have not come down to us there are four books on conic sections, the theory of which he had considerably augmented. Apollonius of Perga, about 247 B. C., perhaps the great­ est of a n the Greek mathematicians, wrote, besides other minor works, a treatise of eight books on conic sections. It was Apollonius who first considered conics as. sections of any obliq[ue cone with a circular base; until then they had been conceived only as sections of right circular cones, and the cutting plane had always been supposed perpendicular to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one of the lines of the cone passing throngh the vertex. This required, three cones of different vertex angles to form the three conic sections. These curves were designated by the words "section of the acute-angled cone", "section of the right-angled cone", "section of the obtuse-angles cone"; they first took definitely the names ellipse, parabola, and hyperbola in the work of Apollonius, although the words ellipse and parabola were known to Archimedes. The whole of Apollonius’ treatise on conics was based on a unique property of conic sections: Suppose an oblique cone with a circular base to be out by a plane through its vertex and through that diameter of the circle at its base which makes this plane perpendicular to the base.
Load more

Recommended publications
  • The Dual Theorem Relative to the Simson's Line
    THE DUAL THEOREM RELATIVE TO THE SIMSON’S LINE Prof. Ion Pătrașcu Frații Buzești College, Craiova, Romania Translated by Prof. Florentin Smarandache University of New Mexico, Gallup, NM 87301, USA Abstract In this article we elementarily prove some theorems on the poles and polars theory, we present the transformation using duality and we apply this transformation to obtain the dual theorem relative to the Samson’s line. I. POLE AND POLAR IN RAPPORT TO A CIRCLE Definition 1. Considering the circle COR(,), the point P in its plane PO≠ and uuur uuuur the point P ' such that OP⋅= OP' R2 . It is said about the perpendicular p constructed in the point P ' on the line OP that it is the point’s P polar, and about the point P that it is the line’s p pole. Observations M 1. If the point P belongs to the circle, its polar is the tangent in P to the circle U COR(,). uuur uuuur P' Indeed, the relation OP⋅= OP' R2 gives that PP' = . OP 2. If P is interior to the circle, its polar p is a V line exterior to the circle. 3. If P and Q are two points such that p m(POQ )= 90o , and p,q are their polars, from the definition results that p ⊥ q . Fig. 1 Proposition 1. If the point P is external to the circle COR( , ) , its polar is determined by the contact points with the circle of the tangents constructed from P to the circle Proof Let U and V be the contact points of the tangents constructed from P to the circle COR( , ) (see fig.1).
    [Show full text]
  • 9.-PARABOLA-THEORY.Pdf
    10. PARABOLA 1. INTRODUCTION TO CONIC SECTIONS 1.1 Geometrical Interpretation Conic section, or conic is the locus of a point which moves in a plane such that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. (a) The fixed point is called the Focus. (b) The fixed straight line is called the Directrix. (c) The constant ratio is called the Eccentricity denoted by e. (d) The line passing through the focus and perpendicular to the directrix is called the Axis. (e) The point of intersection of a conic with its axis is called the Vertex. Sections on right circular cone by different planes (a) Section of a right circular cone by a plane passing through its vertex is a pair of straight lines passing through the vertex as shown in the figure. (b) Section of a right circular cone by a plane parallel to its base is a circle. (c) Section of a right circular cone by a plane parallel to a generator of the cone is a parabola. (d) Section of a right circular cone by a plane neither parallel to a side of the cone nor perpendicular to the axis of the cone is an ellipse. (e) Section of a right circular cone by a plane parallel to the axis of the cone is an ellipse or a hyperbola. 3D View: Circle Ellipse Parabola Hyperbola Figure 10.1 1.2 Conic Section as a Locus of a Point If a point moves in a plane such that its distances from a fixed point and from a fixed line always bear a constant ratio 'e' then the locus of the point is a conic section of the eccentricity e (focus-directrix property).
    [Show full text]
  • A Survey of the Development of Geometry up to 1870
    A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.
    [Show full text]
  • The Rise of Projective Geometry II
    The Rise of Projective Geometry II The Renaissance Artists Although isolated results from earlier periods are now considered as belonging to the subject of projective geometry, the fundamental ideas that form the core of this area stem from the work of artists during the Renaissance. Earlier art appears to us as being very stylized and flat. The Renaissance Artists Towards the end of the 13th century, early Renaissance artists began to attempt to portray situations in a more realistic way. One early technique is known as terraced perspective, where people in a group scene that are further in the back are drawn higher up than those in the front. Simone Martini: Majesty The Renaissance Artists As artists attempted to find better techniques to improve the realism of their work, the idea of vertical perspective was developed by the Italian school of artists (for example Duccio (1255-1318) and Giotto (1266-1337)). To create the sense of depth, parallel lines in the scene are represented by lines that meet in the centerline of the picture. Duccio's Last Supper The Renaissance Artists The modern system of focused perspective was discovered around 1425 by the sculptor and architect Brunelleschi (1377-1446), and formulated in a treatise a few years later by the painter and architect Leone Battista Alberti (1404-1472). The method was perfected by Leonardo da Vinci (1452 – 1519). The German artist Albrecht Dürer (1471 – 1528) introduced the term perspective (from the Latin verb meaning “to see through”) to describe this technique and illustrated it by a series of well- known woodcuts in his book Underweysung der Messung mit dem Zyrkel und Rychtsscheyed [Instruction on measuring with compass and straight edge] in 1525.
    [Show full text]
  • Maximum Area Ellipses Inscribing Specific Quadrilaterals
    Proceedings of The Canadian Society for Mechanical Engineering International Congress 2016 2016 CCToMM M3 Symposium June 26-29, 2016, Kelowna, British Columbia, Canada MAXIMUM AREA ELLIPSES INSCRIBING SPECIFIC QUADRILATERALS M. John D. Hayes Department of Mechanical and Aerospace Engineering Carleton University Ottawa, ON., Canada [email protected] Abstract— In this paper synthetic and analytic projective and isotropy of the mechanism, while the polygon is defined by the Euclidean geometry are used to map the unit circle inscribing reachable workspace of the mechanism [2]. There, the approach the unit square to an ellipse possessing the largest area inscribing is a numerical maximization problem, essentially fitting the special prescribed convex quadrilaterals; namely parallelograms largest area inscribing ellipse starting with a unit circle. and trapezoids. The transformation of the unit circle to the There are only a handful of papers that report investigations inscribing ellipse is the inverse transformation that maps the ver- into determining maximum area ellipses inscribing arbitrary tices of the convex quadrilateral to the vertices of the unit square. polygons, to the best of the author’s knowledge. The dual We describe the computational geometric details of determining problem, that is the problem of determining the polygons of the transformation matrix that is used in the mapping. Addi- greatest area inscribed in an ellipse is reported in [3]. While tionally, employing synthetic geometric arguments, we prove interesting, this dual problem is not germane to determining that the resulting inscribing ellipses are the ones possessing the the maximum area ellipse inscribing a polygon. Three papers maximum area in the corresponding one parameter pencil of by the same author [8-10] appear to lead to a solution to the ellipses inscribing the prescribed convex quadrilaterals.
    [Show full text]
  • The Synthetic Treatment of Conics at the Present Time
    248 SYNTHETIC TEEATMENT OF CONICS. [Feb., THE SYNTHETIC TREATMENT OF CONICS AT THE PRESENT TIME. BY DR. E. B. WILSON. ( Read before the American Mathematical Society, December 29, 1902. ) IN the August-September number of the Jahresbericht der Deutschen Mathernatiker-Vereinigung for the year 1902 Pro­ fessor Reye has printed an address, delivered on entering upon the rectorship of the university at Strassburg, on u Syn­ thetic geometry in ancient and modern times," in which he sketches in a general manner, as the nature of his audience necessitated, the differences between the methods and points of view in vogue in geometry some two thousand years ago and at the present day. The range of the present essay is by no means so extended. It covers merely the different methods which are available at the present time for the development of the elements of synthetic geometry — that is the theory of the conic sections and its relation to poles and polars. There is one method of treatment which was much in use, the world over, half a century ago. At the present time this method is perhaps best represented by Cremona's Projective Geometry, translated into English by Leudesdorf and published by the Oxford University Press, or by RusselPs Pure Geom­ etry from the same press. This treatment depends essentially upon the conception and numerical measure of the double or cross ratio. It starts from the knowledge the student already possesses of Euclid's Elements. This was the method of Cre­ mona, of Steiner, and of Chasles, as set forth in their books.
    [Show full text]
  • The Common Self-Polar Triangle of Concentric Circles and Its Application to Camera Calibration
    The Common Self-polar Triangle of Concentric Circles and Its Application to Camera Calibration Haifei Huang1,2, Hui Zhang2,3, and Yiu-ming Cheung1,2 1Department of Computer Science, Hong Kong Baptist University, Hong Kong SAR, China 2United International College, BNU-HKBU, Zhuhai, China 3Shenzhen Key Lab of Intelligent Media and Speech, Shenzhen, China {mikehuang,amyzhang}@uic.edu.hk [email protected] Abstract struction [13, 12], robot navigation [3] and image-based rendering [11]. The issue of camera calibration includes In projective geometry, the common self-polar triangle the estimation of internal parameters, external parameters has often been used to discuss the position relationship of and lens distortion [22, 28]. Many calibration methods two planar conics. However, there are few researches on have been presented in past decades [19, 28, 29, 5, 8, 14]. the properties of the common self-polar triangle, especially Traditional object-based calibration involves highly accu- when the two planar conics are special conics. In this pa- rate tailor-made 3D objects, which usually consist of two per, we explore the properties of the common self-polar tri- or three orthogonal planes [19]. These calibration meth- angle, when the two conics happen to be concentric cir- ods can achieve a high calibration accuracy, but they re- cles. We show there exist infinite many common self-polar quire an expensive calibration apparatus, and an elaborate triangles of two concentric circles, and provide a method setup [28]. Recently, sphere, a common simple 3D object, to locate the vertices of these triangles. By investigating has been used in calibration [18, 1, 25, 27].
    [Show full text]
  • Advanced Algebra and Geometry
    Advanced Algebra and Geometry Paul Yiu Department of Mathematics Florida Atlantic University Fall 2016 November 14, 2016 Contents 12 Conics 301 12.1 Parabolas ...................................301 12.1.1 Chords and tangents . ........................302 12.1.2 Triangle bounded by three tangents . ................304 12.2 Ellipses . ...................................306 12.2.1 The directrices and eccentricity of an ellipse . ............306 12.2.2 The auxiliary circle and eccentric angle . ............307 12.2.3 Tangents of an ellipse . ........................307 12.2.4 Ellipse inscribed in a triangle . ....................308 12.3 Hyperbolas . ...............................311 12.3.1 The directrices and eccentricity of a hyperbola . ........311 12.3.2 Tangents of a hyperbola . ....................312 12.3.3 Rectangular hyperbolas ........................313 12.3.4 A theorem on the tangents from a point to a conic . ........314 13 General conics 317 13.1 Classification of conics . ........................317 13.2 Pole and polar . ...............................318 13.3 Condition of tangency . ........................319 13.4 Parametrized conics . ........................320 13.5 Rectangular hyperbolas and the nine-point circle . ............321 14 Conic solution of the (O, H, I) problem 325 14.1 Ruler and compass construction of circumcircle and incircle ........325 14.2 The general case: intersections with a conic ................325 14.3 The case OI = IH .............................328 Chapter 12 Conics 12.1 Parabolas Given a point F (focus) and a line L (directrix), the locus of a point P which is equidistant from F and L is a parabola P. Let the distance between F and L be 2a. Set up a Cartesian coordinate system such that F =(a, 0) and L has equation x = −a.
    [Show full text]
  • Desargues' Configuration in a Special Layout
    Journal for Geometry and Graphics Volume 7 (2003), No. 2, 191{199. Desargues' Con¯guration in a Special Layout Barbara Wojtowicz DG&EG Division A-9, Dept. of Architecture, Cracow University of Technology Warszawska 24, PL 31-155 Krak¶ow, Poland email: [email protected] Abstract. In the paper a special case of Desargues' con¯guration will be dis- cussed, where one triangle is inscribed into a conic and the other is circumscribed about the same conic. These two triangles are in a correspondence called a De- sargues collineation KD. Three theorems have been formulated and proved. One of them characterizes the central collineation KD. In a Desargues collineation the base conic will be transformed into another conic. Di®erent cases are dis- cussed. In the case of a base circle the center of the collineation KD coincides with the Gergonne point. Key words: Desargues' con¯guration, collineation MSC 2000: 51M35, 51M05 1. Introduction If two triangles are in Desargues' con¯guration then the straight lines connecting pairs of corresponding vertices intersect at a single point while the three points of intersection between corresponding sides are collinear (Fig. 1). This planar ¯gure consists of ten points and ten straight lines. This ¯gure creates a con¯guration [103] as each point of this con¯guration coincides with three di®erent straight lines and simultaneously on each line there are three speci¯c points creating a Desargues' con¯guration [1]. In this work a special case of Desargues' con¯guration will be discussed where one of the two triangles is inscribed into the other.
    [Show full text]
  • Circles Yufei Zhao
    IMO Training 2008 Circles Yufei Zhao Circles Yufei Zhao [email protected] 1 Warm up problems 1. Let AB and CD be two segments, and let lines AC and BD meet at X. Let the circumcircles of ABX and CDX meet again at O. Prove that triangles OAB and OCD are similar. 2. (Miquel's theorem) Let ABC be a triangle. Points X; Y , and Z lie on sides BC; CA, and AB, respectively. Prove that the circumcircles of triangles AY Z; BXZ; CXY meet at a common point. 3. (Also Miquel's theorem) Let a; b; c; d be four lines in space, no two parallel, no three concurrent. Let !a denote the circumcircle of the triangle formed by lines b; c; d. Similarly define !b;!c;!d. (a) Prove that !a;!b;!c;!d pass through a common point. This is called the Miquel point. (b) Continuing the above notation, prove that the centers of !a;!b;!c;!d, and the Miquel point all lie on a common circle. 4. (Simson line) Let ABC be a triangle, and let P be another point on its circumcircle. Let X; Y; Z be the feet of perpendiculars from P to lines BC; CA; AB respectively. Prove that X; Y; Z are collinear. 5. Let \AOB be a right angle, M and N points on rays OA and OB, respectively. Let MNPQ be a square such that MN separates the points O and P . Find the locus of the center of the square when M and N vary. 6. Let ABCD be a convex quadrilateral such that the diagonals AC and BD are perpendicular, and let P be their intersection.
    [Show full text]
  • On Polar Duality, Lagrange and Legendre Singularities and Stereographic Projection to Quadrics Ricardo Uribe–Vargas
    On Polar Duality, Lagrange and Legendre Singularities and Stereographic Projection to Quadrics Ricardo Uribe–Vargas Universit´e Paris 7, Equipe´ G´eom´etrie et Dynamique. UFR de Math. Case 7012. 2, Pl. Jussieu, 75005 Paris. [email protected] http://www.math.jussieu.fr/∼uribe/ Abstract. In [16], V.D. Sedykh has shown that there is a relation between La- grangian and Legendrian singularities by stereographic projection into sphere in the Eu- clidean space. We show that the theorems of [16] hold for any stereographic projection ρ of a hyperplane Rn ×{b}⊂Rn × R into a quasi-revolution quadric Q (defined below) in Rn × R,whereRn denotes a Euclidean space. We show that this relation between Lagrangian and Legendrian singularities is the most natural one and that the calcula- tions and formulas are simpler when the quadric is a paraboloid of revolution. Using the classical theory of poles, polars and polar duality, we give an explicit construction of the (Lagrangian) Normal map of a submanifold Γ in Rn ×{b} as the composition of 3 maps, namely: 1.– the (Legendrian) Tangential map of the manifold ρ(Γ), 2.– the polar duality map and 3.– the ‘vertical’ projection of Rn+1 = Rn × R to Rn ×{b} from the pole of the stereographic projection. We apply our result to give a formula to calculate the vertices of a curve in the Euclidean space Rn. This can be applied to calculate and to study umbilic points of surfaces in the Euclidean space R3. Introduction Here, the symbol Rn denotes always an n−dimensional Euclidean space.
    [Show full text]
  • Non-Euclidean Geometry H.S.M
    AMS / MAA SPECTRUM VOL AMS / MAA SPECTRUM VOL 23 23 Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and conti- Non-Euclidean Geometry nuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, begin- ning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations.' Following a NON-EUCLIDEAN recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing in elliptic or hyperbolic polarity which transforms points into lines GEOMETRY (in two dimensions), planes (in three dimensions), and vice versa. Sixth Edition An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss. This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. The present (sixth) edition clari es some obscurities in the fth, and includes a new section 15.9 on the author's useful concept of inversive distance. H.S.M. Coxeter This book presents a very readable account of the fundamental principles of hyperbolic and elliptic geometries.
    [Show full text]