DOCSLIB.ORG

János Bolyai, Non-Euclidean Geometry, and the Nature of Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
János Bolyai, Non-Euclidean Geometry, and the Nature of Space Book Review János Bolyai, Non-Euclidean Geometry, and the Nature of Space Reviewed by Robert Osserman János Bolyai, Non-Euclidean Geometry, and the that is still very Nature of Space much alive today Jeremy J. Gray and whose contem- MIT Press, 2004 porary shape might Paperback, 256 pages, $20.00 profit from a deeper ISBN 0-262-57174-9 historical perspec- tive. This attractive little volume consists of two Gray starts by dis- major components, together with a number of cussing the way shorter sections. The major components are philosophers and labeled “Introduction” and “Appendix” respec- scientists, as well as tively, both designations grossly understating humankind in gen- their content. The “Appendix” consists of Bolyai’s eral, pictured the revolutionary tract, with the subtitle “THE SCIENCE physical space we in- ABSOLUTE OF SPACE Independent of the Truth or habit—the shape of Falsity of Euclid’s Axiom XI (which can never be the universe as a decided a priori)”. It appears here both as a whole—and the way facsimile in the original Latin and also in Halstead’s that such pictures 1896 English translation. It was indeed originally influenced and were influenced by the mathemat- published as an appendix to his father’s two- ical constructs that became known as “geometry”. volume treatise on mathematics, whence its name. In particular, the use made by Newton of the for- Jeremy Gray’s “Introduction”—or to give it its full malism of Euclidean geometry, together with the name, “Bolyai’s Appendix: An Introduction”—is in overwhelming success of his approach to physics fact a 122-page historical survey of Euclidean and astronomy based on the model of Euclidean and non-Euclidean geometry that focuses on space, enshrined Euclidean geometry and cloaked Bolyai and his appendix but goes far beyond that. it in a certainty and inevitability that made ques- Among other things, it includes a number of tioning it appear to be a sign of mental instability. fascinating historical remarks on the debates And indeed, that was the reaction by the Russian over the course of many centuries about how to mathematical establishment, the only ones to read teach Euclidean geometry in school—a debate Lobachevskii’s first articles on what was to be- come non-Euclidean geometry, published starting in 1829 in the obscure Russian journal the Kazan Robert Osserman is the special projects director of the Messenger. The news of this work apparently did Mathematical Sciences Research Institute in Berkeley. His email address is [email protected]. not reach the Bolyais until 1844, which is 1030 NOTICES OF THE AMS VOLUME 52, NUMBER 9 astonishing on many grounds, not the least of given his endorsement of the work publicly and which involves the shameful role played by Gauss brought it to wider attention, it would have changed in the sad story of János Bolyai’s brilliant beginnings the course of János’s life and career. But he kept and bitter end. his silence,1 and János not only got no public The story of Gauss and the Bolyais starts back support but was convinced that his father had before Gray picks it up, with the close friendship betrayed him and revealed what he had been doing between Gauss and János’s father, Farkas (also to Gauss, who was now claiming it as his own. known by his German name of Wolfgang), dating When Gauss later became aware of Lobachevskii’s back to their student days together in Göttingen. work, he made no attempt to inform his old friend, The family obsession with Euclid’s parallel postu- Farkas. late apparently dates to that time, and indeed, Recognition of Bolyai’s achievement did not father Bolyai published his first work on the sub- come until too late. Gauss died in 1855, Farkas and ject, The Theory of Parallels, in 1804. He was only Lobachevskii the following year, and János in 1860. one of many who tried to prove that the parallel As Gauss’s correspondence became public, his postulate follows from Euclid’s other axioms. views on non-Euclidean geometry finally became Gray describes in some detail the history of such known. Bolyai’s appendix was translated into Ital- attempts throughout the eighteenth and nineteenth ian in 1868, the year and the place that were to be centuries, even well after their futility had been fully decisive for the future of the new geometry that he demonstrated. Among the prominent mathemati- had invented. cians who fell into the trap were Legendre, who per- Between 1832, when it was originally published, sisted in publishing false proofs over many years, and 1868, when it became more widely known, the and none other than Lagrange, who did not go so critical events for the fate of Bolyai’s appendix far as publishing any but did have the embarrass- were, as already mentioned, the publication ment of presenting one at the prestigious Institute between 1860 and 1865 of the correspondence of France. As for Farkas Bolyai, it was his friend between Gauss and Schumaker, including a letter Gauss who pointed out the error in his argument, Gauss wrote in 1846 praising Lobachevskii’s work but he persisted for at least ten more years before on non-Euclidean geometry and saying that he had giving up in despair. No wonder then, when his son, shared the same views for fifty-four years (since János, who had turned out to be something of a 1792, when Gauss was fifteen), and the presenta- mathematical prodigy, appeared as a teenager to tion of Riemann’s Habilitationsvortrag, entitled have already been bitten by the parallel-postulate “On the hypotheses that lie at the foundation bug, Farkas wrote him the often-quoted feverish ad- of geometry” in 1854. Not that Riemann refers monishment to profit from his own example and directly to Bolyai or even to non-Euclidean geom- guard against this will-o-the-wisp: “I have traversed etry as it is usually understood. Rather, Riemann this bottomless night, which extinguished all light proposes a radical rethinking of the entire subject and joy in my life. I entreat you, leave the science of geometry, not based on axioms in the fashion of parallels alone.” of Euclid, Bolyai, and Lobachevskii, but on the As it turned out, Bolyai Sr. was right, but not for perhaps shakier but far more flexible foundations the reasons he thought. János soon concluded that of measurement, the calculus, and the whole field proving the parallel postulate was hopeless, and he of differential geometry as developed by Euler, gradually became convinced that one could con- Gauss, and the French school. struct a perfectly consistent geometry in which it Gray’s “Introduction” devotes about twenty-five was not true. By 1825, at the age of twenty-three, pages to Bolyai’s “Appendix” itself and provides an János was able to show his achievement to his excellent overview of Bolyai’s approach, with its father but not to convince him that there were no emphasis on an “absolute” geometry of space, hidden flaws. It was not until 1832 that Farkas with parallel (if one may use the word) results in agreed to publish his son’s work in the form of an appendix to his own book. When Gauss received a 1This was not the first time that Gauss’s silence on the sub- copy of the appendix, he wrote back a letter that ject of non-Euclidean geometry had a devastating effect. effectively ended János’s brilliant career, as well Another of Gauss’s correspondents, named Taurinus, pub- as, Gray tells us, the relationship between father lished a brochure in 1826 in which he derived a number and son, who did not speak to each other for years of the same trigonometric formulae as Bolyai. In the pref- afterward. In the letter Gauss gives an infuriat- ace to the brochure he asked Gauss to state his views on the subject, after which Gauss terminated the correspon- ingly mixed message, saying that he had himself dence. In the book by B. A. Rosenfeld, A History of Non- carried out much the same program but had Euclidean Geometry: Evolution of the Concept of a written very little of it down, and how pleased he Geometric Space, English translation Springer-Verlag, was that it should turn out to be the son of his 1988, p. 219), the author tells us, “Gauss’s reaction reduced old friend who had written it down, thereby spar- Taurinus to despair, and he burned all copies of the ing him the trouble of doing it himself. Had Gauss brochure in his possession.” OCTOBER 2005 NOTICES OF THE AMS 1031 both Euclidean and non-Euclidean geometry, but point that start out along a given two-dimensional also special results, such as the possibility of section. That surface will have a curvature whose “squaring the circle” in the non-Euclidean case, a value is given explicitly by a formula due to Gauss. construction taking up the last ten sections of The Gauss curvature of the surface at the given Bolyai’s appendix and described in detail by Gray. point is by definition the Riemannian (sectional) cur- It is in the description of the contributions by vature of the manifold at the point in the given Riemann and his successors Beltrami and Poin- two-dimensional direction. As an illustration, he caré that I find Gray’s version of the history of the gives explicit formulae for Riemannian metrics of subject to be somewhat lacking. After noting that constant positive, negative, and zero curvature in Riemann chose not to limit his differential geom- any number of dimensions.
Load more

Recommended publications
  • Five-Dimensional Design
    PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 50, NO. 1, PP. 35–41 (2006) FIVE-DIMENSIONAL DESIGN Elek TÓTH Department of Building Constructions Budapest University of Technology and Economics H–1521 Budapest, POB. 91. Hungary Received: April 3 2006 Abstract The method architectural and engineering design and construction apply for two-dimensional rep- resentation is geometry, the axioms of which were first outlined by Euclid. The proposition of his 5th postulate started on its course a problem of geometry that provoked perhaps the most mistaken demonstrations and that remained unresolved for two thousand years, the question if the axiom of parallels can be proved. The quest for the solution of this problem led Bolyai János to revolutionary conclusions in the wake of which he began to lay the foundations of a new (absolute) geometry in the system of which planes bend and become hyperbolic. It was Einstein’s relativity theory that eventually created the possibility of the recognition of new dimensions by discussing space as well as a curved, non-linear entity. Architectural creations exist in what Kant termed the dual form of intuition, that is in time and space, thus in four dimensions. In this context the fourth dimension is to be interpreted as the current time that may be actively experienced. On the other hand there has been a dichotomy in the concept of time since ancient times. The introduction of a time-concept into the activity of the architectural designer that comprises duration, the passage of time and cyclic time opens up a so-far unknown new direction, a fifth dimension, the perfection of which may be achieved through the comprehension and the synthesis of the coded messages of building diagnostics and building reconstruction.
    [Show full text]
  • Lecture 3: Geometry
    E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-2015 Lecture 3: Geometry Geometry is the science of shape, size and symmetry. While arithmetic dealt with numerical structures, geometry deals with metric structures. Geometry is one of the oldest mathemati- cal disciplines and early geometry has relations with arithmetics: we have seen that that the implementation of a commutative multiplication on the natural numbers is rooted from an inter- pretation of n × m as an area of a shape that is invariant under rotational symmetry. Number systems built upon the natural numbers inherit this. Identities like the Pythagorean triples 32 +42 = 52 were interpreted geometrically. The right angle is the most "symmetric" angle apart from 0. Symmetry manifests itself in quantities which are invariant. Invariants are one the most central aspects of geometry. Felix Klein's Erlanger program uses symmetry to classify geome- tries depending on how large the symmetries of the shapes are. In this lecture, we look at a few results which can all be stated in terms of invariants. In the presentation as well as the worksheet part of this lecture, we will work us through smaller miracles like special points in triangles as well as a couple of gems: Pythagoras, Thales,Hippocrates, Feuerbach, Pappus, Morley, Butterfly which illustrate the importance of symmetry. Much of geometry is based on our ability to measure length, the distance between two points. A modern way to measure distance is to determine how long light needs to get from one point to the other. This geodesic distance generalizes to curved spaces like the sphere and is also a practical way to measure distances, for example with lasers.
    [Show full text]
  • Feb 23 Notes: Definition: Two Lines L and M Are Parallel If They Lie in The
    Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal. We define alternate interior angles, corresponding angles, alternate exterior angles, and interior angles on the same side of the transversal using various betweeness and half-plane notions. Suppose line l intersects lines m and n at points B and E, respectively, with points A and C on line m and points D and F on line n such that A-B-C and D-E-F, with A and D on the same side of l. Suppose also that G and H are points such that H-E-B- G. Then pABE and pBEF are alternate interior angles, as are pCBE and pDEB. pABG and pFEH are alternate exterior angles, as are pCBG and pDEH. pGBC and pBEF are a pair of corresponding angles, as are pGBA & pBED, pCBE & pFEH, and pABE & pDEH. pCBE and pFEB are interior angles on the same side of the transversal, as are pABE and pDEB. Our Last Theorem in Absolute Geometry: If two lines in the same plane are cut by a transversal so that a pair of alternate interior angles are congruent, the lines are parallel. Proof: Let l intersect lines m and n at points A and B respectively. Let p1 p2. Suppose m and n meet at point C. Then either p1 is exterior to ªABC, or p2 is exterior to ªABC. In the first case, the exterior angle inequality gives p1 > p2; in the second, it gives p2 > p1.
    [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]
  • Absolute Geometry
    Absolute geometry 1.1 Axiom system We will follow the axiom system of Hilbert, given in 1899. The base of it are primitive terms and primitive relations. We will divide the axioms into five parts: 1. Incidence 2. Order 3. Congruence 4. Continuity 5. Parallels 1.1.1 Incidence Primitive terms: point, line, plane Primitive relation: "lies on"(containment) Axiom I1 For every two points there exists exactly one line that contains them both. Axiom I2 For every three points, not lying on the same line, there exists exactly one plane that contains all of them. Axiom I3 If two points of a line lie in a plane, then every point of the line lies in the plane. Axiom I4 If two planes have a point in common, then they have another point in com- mon. Axiom I5 There exists four points not lying in a plane. Remark 1.1 The incidence structure does not imply that we have infinitely many points. We may consider 4 points {A, B, C, D}. Then, the lines can be the two-element subsets and the planes the three-element subsets. 1 1.1.2 Order Primitive relation: "betweeness" Notion: If A, B, C are points of a line then (ABC) := B means the "B is between A and C". Axiom O1 If (ABC) then (CBA). Axiom O2 If A and B are two points of a line, there exists at least one point C on the line AB such that (ABC). Axiom O3 Of any three points situated on a line, there is no more than one which lies between the other two.
    [Show full text]
  • Information to Users
    INFORMATION TO USERS This manuscrit has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The qnali^ of this reproduction is dependent upon the qnali^ of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins^ and inqnoper alignment can adversefy afreet reproductioiL In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beghming at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photogrsphs included in the original manuscript have been reproduced xerographically in this copy. Higher quali^ 6" x 9" black and white photographic prints are available for aiy photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313.'761-4700 800/521-0600 ARISTOHE ON MATHEMATICAL INFINITY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Theokritos Kouremenos, B.A., M.A ***** The Ohio State University 1995 Dissertation Committee; Approved by D.E.
    [Show full text]
  • Chapter 4 Euclidean Geometry
    Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles De¯nition 4.1 Two distinct lines ` and m are said to be parallel ( and we write `km) i® they lie in the same plane and do not meet. Terminologies: 1. Transversal: a line intersecting two other lines. 2. Alternate interior angles 3. Corresponding angles 4. Interior angles on the same side of transversal 56 Yi Wang Chapter 4. Euclidean Geometry 57 Theorem 4.2 (Parallelism in absolute geometry) If two lines in the same plane are cut by a transversal to that a pair of alternate interior angles are congruent, the lines are parallel. Remark: Although this theorem involves parallel lines, it does not use the parallel postulate and is valid in absolute geometry. Proof: Assume to the contrary that the two lines meet, then use Exterior Angle Inequality to draw a contradiction. 2 The converse of above theorem is the Euclidean Parallel Postulate. Euclid's Fifth Postulate of Parallels If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, the lines will meet on that side of the transversal. In e®ect, this says If m\1 + m\2 6= 180; then ` is not parallel to m Yi Wang Chapter 4. Euclidean Geometry 58 It's contrapositive is If `km; then m\1 + m\2 = 180( or m\2 = m\3): Three possible notions of parallelism Consider in a single ¯xed plane a line ` and a point P not on it.
    [Show full text]
  • Chapter 3 Foundations of Geometry 2
    Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis De¯nition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices) are taken, in pairs, from a set of three noncollinear points. Thus, if the vertices of a triangle are A; B; and C, then its sides are AB; BC; AC, and the triangle is then the set de¯ned by AB [ BC [ AC, denoted by ¢ABC. The angles of ¢ABC are \A ´ \BAC, \B ´ \ABC, and \C ´ \ACB. Equality and Congruence Equal(=): identically the same as. For examples, 1) Two points are equal, as in A = B, we mean they coincide. 2) AB = CD only if the set of points AB is the exact same set of points denoted by CD. Congruence(»=): is an equivalence relation. (to be de¯ned later). Congruence for Segments and angles 25 Yi Wang Chapter 3. Foundations of Geometry 2 26 AB »= XY i® AB = XY \ABC »= \XYZ i® m\ABC = m\XYZ Congruence for triangles Notation: correspondence between two triangles Given ¢ABC and ¢XYZ, write ABC $ XYZ to mean the correspondence between vertices, sides and angles in the order written. Note: There are possible six ways for one triangle to correspond to another. ABC $ XY Z ABC $ XZY ABC $ YXZ ABC $ Y ZX ABC $ ZXY ABC $ ZYX De¯nition 3.2 (Congruence for triangles) If, under some correspondence between the vertices of two triangles, corresponding sides and corresponding angles are congruent, the triangles are said to be congruent. Thus we write ¢ABC »= ¢XYZ whenever AB »= XY; BC »= YZ; AC »= XZ; \A »= \X; \B »= \Y; \C »= \Z Notation: CPCF: Corresponding parts of congruent ¯gures (are congruent).
    [Show full text]
  • Neutral Geometry
    NEUTRAL GEOMETRY The goal of this book is to provide a pleasant but thorough introduction to Euclidean and non-Euclidean (hyperbolic) geometry. Before I go any further, let me clear up something that could lead to confusion on down 1the road. Some mathematicians use the term non-Euclidean geometry to mean any of a whole host of geometries which fail to be Euclidean for any number of reasons. The kind of non-Euclidean geometry that we will study in these lessons, and the kind that I mean when I use the term non- Euclidean geometry, is something much more specific– it is a geometry that satisfies all of hilbert’s axioms for euclidean geometry except the parallel axiom. it turns out that that parallel axiom is absolutely central to the nature of the geometry. the euclidean geometry with the parallel axiom and the non-Euclidean geometry without it are radically different. Even so, Eu- clidean and non-Euclidean geometry are not polar opposites. As different as they are in many ways, they still share many basic characteristics. Neu- tral geometry (also known as absolute geometry in older texts) is the study of those commonalities. 9 1. OUR DUCKS IN A ROW the axioms of incidence and order 12 lesson 2 From Euclid to Hilbert You pretty much have to begin a study of Euclidean geometry with at least some mention of euclid’s Elements, the book that got the ball rolling over two thousand years ago. The Elements opens with a short list of definitions. as discussed in the previous chapter, the first few of these definitions are a little problematic.
    [Show full text]
  • The Centroid in Absolute Geometry
    The centroid in absolute geometry Ricardo Bianconi∗ Instituto de Matem´atica e Estat´ıstica Univeridade de S˜aoPaulo Caixa Postal 66281, CEP 05311-970, S˜ao Paulo, SP, Brazil e-mail: [email protected] Abstract We give a synthetic proof in absolute geometry (Birkhoff axioms with- out the parallel postulate) that given any triangle, its medians are con- current. This means that the same proof is valid in both euclidean and hyperbolic geometry. We also indicate how to generalize this result. 1 Introduction It is true both in (real) euclidean and hyperbolic geometry that the medians of a triangle are concurrent (and this point of concurrence is called the centroid of the triangle). The proof in the euclidean case uses similarity of triangles and in the hyperbolic case one uses the Klein model inside the projective plane (see [1, pp. 30-31 and 229-230], and [5, Exercise 105]). These two geometries differ only by the parallel axiom, so we can conclude that this result is a theorem of pure absolute geometry (Birkhoff axioms without the parallel postulate; the interested reader can provide a justification of this claim). Here we present a proof of this fact using only the axioms of pure absolute geometry. The idea is simple. We project a triangle onto an isosceles one, for which it is easy to prove the result and then go back to the original triangle. We prove the necessary results on the needed projection in the same system by proving a particular configuration of Desargues’ theorem. This is done in section 2.
    [Show full text]
  • Geometry Illuminated an Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry
    AMS / MAA TEXTBOOKS VOL 30 Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry Matthew Harvey Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2015936098 Print ISBN: 978-1-93951-211-6 Electronic ISBN: 978-1-61444-618-7 Printed in the United States of America Current Printing (last digit): 10987654321 10.1090/text/030 Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry Matthew Harvey The University of Virginia’s College at Wise Published and distributed by The Mathematical Association of America Council on Publications and Communications Jennifer J. Quinn, Chair Committee on Books Fernando Gouvea,ˆ Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor Matthias Beck Richard E. Bedient Otto Bretscher Heather Ann Dye Charles R. Hampton Suzanne Lynne Larson John Lorch Susan F. Pustejovsky MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence Calculus Deconstructed: A Second Course in First-Year Calculus, Zbigniew H. Nitecki Calculus for the Life Sciences: A Modeling Approach, James L. Cornette and Ralph A. Ackerman Combinatorics: A Guided Tour, David R. Mazur Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Differential Geometry and its Applications, John Oprea Distilling Ideas: An Introduction to Mathematical Thinking, Brian P.Katz and Michael Starbird Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven G.
    [Show full text]
  • Absolute Geometry
    Absolute geometry Circles (properties) Properties are taken from Kay, p 105. Fill in the missing items – note that if a fill in number repeats within a problem (e.g. you see [Fill in (2)] in two places in the same property), it means that that exact statement is being repeated. Property 1: The center of a circle is the midpoint of any diameter. Proof: Statement: Consider a circle centered at O with diameter PQ . Reason: Given. Statement: Because P and Q are points on the circle, segments OP and OQ are both radii of the circle. Therefore: [Fill in (1)] (What can you say about their lengths?) Reason: Definition of radius. Statement: Since PQ is a diameter, OPQ∈ (definition), and [Fill in (2)] (What betweenness relationship holds?) Statement: Since [Fill in (1) from above] and [Fill in (2) from above], O is the [Fill in (3)] of segment PQ . Reason: Definition of a [Fill in (3) from above]. Property 2: The perpendicular bisector of any chord of a circle passes through the center. Proof: Statement: Let a circle centered at O with radius r, and with chord PQ given. Let RS be the perpendicular bisector of PQ . (Be careful with appearances here – you don't know that RS passes through O, although it certainly appears to. That's the thing you're trying to prove.) Reason: Given. Now, there's a theorem on p 144 you want to look at. By this theorem, the perpendicular bisector of PQ is the set of all points which are an equal distance from the endpoints P and Q.
    [Show full text]