DOCSLIB.ORG

Plane and Solid Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Plane and Solid Geometry PLANE AND SOLID GEOMETRY BY C. A. HART INSTRUCTOR OF MATHEMATICS, WADLEIGH HIGH SCHOOL, NEW YORK CITY AND DANIEL D. FELDMAN HEAD OF DEPARTMENT OF MATHEMATICS, ERASMUS HALL HIGH SCHOOL, BROOKLYN WITH THE EDITORIAL COOPERATION OF J. H. TANNER AND VIRGIL SNYDER PROFESSORS OF MATHEMATICS IN CORNELL UNIVERSITY NEW YORK ":" CINCINNATI ":" CHICAGO AMERICAN BOOK COMPANY H-3 Copyright, 1911, 1912, by AMERICAN BOOK COMPANY. Entered at Stationers' Hall, London. H.-F. PLANK AND SOLID GEOMETRY. W. P. I v" PREFACE This book is the outgrowth of an experienceof many years in the teaching of mathematics in secondary schools. The text has been used by many different teachers, in classes of all stages of development, and under varying conditions of sec-ondary school teaching. The proofs have had the benefit of the criticisms of hundreds of experienced teachers of mathe-matics throughout the country. The book in its present form is therefore the combined product of experience,classroom test, and severe criticism. The following are some of the leading features of the book : Tlie student is rapidly initiated into the subject. Definitions are given only as needed. The selection and arrangement of theorems is such as to meet the generaldemand of teachers,as expressed through the Mathe-matical Associations of the country. Most of the proofs have been given in full. In the Plane Geometry, proofs of some of the easier theorems and construc-tions are left as exercises for the student, or are given in an incomplete form. In the Solid Geometry, more proofs and parts of proofs are thus left to the student ; but in every case in which the proof is not complete, the incompleteness is specificallystated. The indirect method of proof is consistentlyapplied. The usual method of proving such propositions,for example, as Arts. 189 and 415, is confusing to the student. The method used here is convincing and clear. The exercises are carefully selected. In choosing exercises, each of the following groups has been given due importance : (a) Concrete exercises,including numerical problems and problems of construction. (6) So-called practicalproblems, such as indirect measure-ments of heights and distances by means of equal and similar triangles,drawing to scale as an applicationof similar figures, ..problemsfrom physics,from design, etc, 304026 iv PREFACE (c)Traditional exercises of a more or less abstract nature. The arrangement of the exercises is pedagogical Compara-tively easy exercises are placedimmediatelyafter the theorems of which they are applications,instead of being grouped to-gether without regard to the principlesinvolved in them. For the benefit of the brighter pupils,however, and for review classes,long lists of more or less difficultexercises are grouped at the end of each book. The definitionsof closed figures are unique. The student's natural conceptionof a plane closed figure,for example, is not the boundary line only,nor the plane only,but the whole figure composed of the boundary line and the plane bounded. All definitions of closed figuresinvolve this idea,which is entirely consistent with the higher mathematics. The numerical treatment of magyiitudes is explicit,the funda-mental principlesbeing definitelyassumed (Art. 336, proof in Appendix, Art. 595). This novel procedure furnishes a logical,as well as a teachable, method of dealing with incommensurables. The area of a rectangleis introduced by actuallymeasuring it, thereby obtaining its measure-number. This method permits the same order of theorems and corollaries as is used in the parallelogramand the triangle. The correlation with arithmetic in this connection is valuable. A similar method is employed for introducingthe volume of a parallelopiped. Proofs of the superpositiontheorems and the concurrent line theorems will be found exceptionallyaccurate and complete. The many historical notes ivill add lifeand interest to the work. The carefullyarranged summaries throughout the book, the collection of formulas of Plane Geometry, and the collection of formulas of Solid Geometry, it is hoped, will be found helpful to teacher and student alike. This Argument and reasons are arranged in parallelform. arrangement gives a definite model for proving exercises,ren-ders the careless omission of the reasons in a demonstration impossible,leads to accurate thinking,and greatlylightensthe labor of reading papers. PREFACE v Every construction figure contains all necessary construction lines. This method keeps constantlybefore the student a model for his construction work, and distinguishesbetween a figurefor a construction and a figurefor a theorem. TJie mechanical arrangement is such as to give the student every possibleaid in comprehending the subjectmatter. The following are some of the specialfeatures of the Solid Geometry : The vital relation of the Solid Geometry to the Plane Geometry is emphasized at every point. (SeeArts. 703,786, 794, 813, 853, 924, 951, 955, 961, etc.) The student is given every possible aid in forming his early space concepts. In the early work in Solid Geometry, the average student experiences difficultyin fullycomprehending space relations,that is,in seeing geometric figuresin space. The student is aided in overcoming this difficultyby the intro-duction of many easy and practicalquestionsand exercises,as well as by being encouraged to make his figures. (See " 605.) As a further aid in this direction,reproductions of models made by students themselves are shown in a group (p.302) and at various points throughout Book VI. Tlie student's knowledge of the thingsabout him is constantly draivn upon. Especiallyis this true of the work on the sphere, where the student's knowledge of mathematical geography has been appealed to in making clear the terms and the relations of figuresconnected with the sphere. The same logicalrigor that characterizes the demonstrations in the Plane Geometry is used throughout the Solid. The treatment of the polyhedral angle (p. 336), of the prism (p. 345), and of the pyramid (p.350) is similar to that of the cylinder and of the cone. This is in accordance with the recommendations of the leading Mathematical Associations throughout the country. The gratefulacknowledgment of the authors is due to many friends for helpfulsuggestions; especiallyto Miss Grace A. Bruce, of the Wadleigh High School, New York; to Mr. Edward B. Parsons, of the Boys' High School,Brooklyn ; and to Professor McMahon, of Cornell University. CONTENTS PLANE GEOMETRY PAGE SYMBOLS AND ABBREVIATIONS viii INTRODUCTION 1 BOOK I. RECTILINEAR FIGURES 19 Necessity for Proof 21 Polygons. Triangles 22 Superposition 25 Measurement of Distances by Means of Triangles 32 ... Loci 42 Parallel Lines 65 Quadrilaterals. Parallelograms 84 Concurrent Line Theorems 100 Construction of Triangles 106 Directions for the Solution of Exercises 110 Miscellaneous Exercises Ill BOOK II. THE CIRCLE 113 Two Circles 131 . Measurement 133 Miscellaneous Exercises 156 BOOK III. PROPORTION AND SIMILAR FIGURES 161 . Similar Polygons 176 Drawing to Scale 188 Miscellaneous Exercises 206 BOOK IV. AREAS OF POLYGONS 209 Transformation of Figures 223 Miscellaneous Exercises 239 . .... BOOK V. REGULAR POLYGONS. MEASUREMENT OF THE CIRCLE 245 Measurement of the Circumference and of the Circle 258 . Miscellaneous Exercises 274 vi CONTENTS vii PAGE FORMULAS OF PLANE GEOMETRY .282 . APPENDIX TO PLANE GEOMETRY 284 Maxima and Minima 284 Variables and Limits. Theorems 291 Miscellaneous Theorems ........ 294-298 SOLID GEOMETRY BOOK VI. AND ANGLES IN SPACE 299 LINES, PLANES, . Lines and Planes * 301 . Dihedral 322 Angles . * .336 Polyhedral Angles . .. BOOK VII. POLYHEDRONS .343 . ... Prisms 345 Pyramids 350 Mensuration of the Prism and Pyramid 354 Areas 354 Volumes 358 . Miscellaneous Exercises 381 BOOK VIII. CYLINDERS AND CONES 383 Cylinders 383 Cones " 388 Mensuration of the Cylinder and Cone 392 Areas 392 Volumes 406 Miscellaneous Exercises 414 BOOK IX. THE SPHERE 417 Lines and Planes Tangent to a Sphere 424 SphericalPolygons 429 Mensuration of the Sphere 444 Areas 444 . , Volumes 457 Miscellaneous Exercises on Solid 467 Geometry .... FORMULAS OF SOLID GEOMETRY .471 . APPENDIX TO SOLID GEOMETRY 474 SphericalSegments 474 The Prismatoid 475 Similar Polyhedrons 477 .INDEX 481 SYMBOLS AND ABBREVIATIONS = equals, equal to, is equal to. =fc does not equal. " greater than, is greater than. " less than, is less than. =0= equivalent, equivalent to, is equiva-lent to. is similar to. ~ similar, similar to, S^ is measured by. JL perpendicular, perpendicular to, is perpendicular to. Js perpendiculars. || parallel, parallel to, is parallel to. ||s parallels. and so on (sign of continuation). v since. .-, therefore. AB. /-n arc ; AB, arc "7, OD parallelogram, parallelograms. O, (D circle, circles. Z, A angle, angles. A, A triangle, triangles. be q.e.d. Quod erat demonstrandum, which was to proved. done. q.e.f. Quod erat faciendum, which was to be have the as in The +, " X + same meanings algebra. signs , , viii PLANE GEOMETRY INTRODUCTION 1. The Subject Matter of Geometry. In geometry, although we shall continue the use of arithmetic and algebra, our main work will be a study of what will later be defined (" 13) as geometric figures. The student is already familiar with the physical objects about him, such as a ball or a block of wood. he be led By a careful study of the following exercise, may to see the relation of such physical solids to the geometric figures with which he must become familiar. it ? Exercise. Look at a block of wood (or a chalk box). Has weight the color ? taste ? shape ? size ? These are called properties of solid. What do we call such a solid ? A physical solid. Can think of the of this jj you properties , solid apart from the block of wood ? Imagine the block removed. Can you imagine the space which it occupied ? What name would you give to this space ? A geometric solid. What properties has it that the block possessed ? Shape and size. What is it that separates this geometric solid from surrounding space ? How Where thick is this surface ? How many surfaces has the block ? do wide the they intersect ? How many intersections are there ? How are intersections ? how long ? What is their name ? They are lines.
Load more

Recommended publications
  • Geometry Around Us an Introduction to Non-Euclidean Geometry
    Geometry Around Us An Introduction to Non-Euclidean Geometry Gaurish Korpal (gaurish4math.wordpress.com) National Institute of Science Education and Research, Bhubaneswar 23 April 2016 Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 1 / 15 Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's original proof was (published in 1924) stemmed from his results on algebraic curves tangent to a given number of lines. It arose from the consideration of cardioids [(x 2 + y 2 − a2)2 = 4a2((x − a)2 + y 2)].
    [Show full text]
  • Holley GM LS Race Single-Plane Intake Manifold Kits
    Holley GM LS Race Single-Plane Intake Manifold Kits 300-255 / 300-255BK LS1/2/6 Port-EFI - w/ Fuel Rails 4150 Flange 300-256 / 300-256BK LS1/2/6 Carbureted/TB EFI 4150 Flange 300-290 / 300-290BK LS3/L92 Port-EFI - w/ Fuel Rails 4150 Flange 300-291 / 300-291BK LS3/L92 Carbureted/TB EFI 4150 Flange 300-294 / 300-294BK LS1/2/6 Port-EFI - w/ Fuel Rails 4500 Flange 300-295 / 300-295BK LS1/2/6 Carbureted/TB EFI 4500 Flange IMPORTANT: Before installation, please read these instructions completely. APPLICATIONS: The Holley LS Race single-plane intake manifolds are designed for GM LS Gen III and IV engines, utilized in numerous performance applications, and are intended for carbureted, throttle body EFI, or direct-port EFI setups. The LS Race single-plane intake manifolds are designed for hi-performance/racing engine applications, 5.3 to 6.2+ liter displacement, and maximum engine speeds of 6000-7000 rpm, depending on the engine combination. This single-plane design provides optimal performance across the RPM spectrum while providing maximum performance up to 7000 rpm. These intake manifolds are for use on non-emissions controlled applications only, and will not accept stock components and hardware. Port EFI versions may not be compatible with all throttle body linkages. When installing the throttle body, make certain there is a minimum of ¼” clearance between all linkage and the fuel rail. SPLIT DESIGN: The Holley LS Race manifold incorporates a split feature, which allows disassembly of the intake for direct access to internal plenum and port surfaces, making custom porting and matching a snap.
    [Show full text]
  • Analytic Geometry
    STATISTIC ANALYTIC GEOMETRY SESSION 3 STATISTIC SESSION 3 Session 3 Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). 2D Shapes Activity: Sorting Shapes Triangles Right Angled Triangles Interactive Triangles Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite Interactive Quadrilaterals Shapes Freeplay Perimeter Area Area of Plane Shapes Area Calculation Tool Area of Polygon by Drawing Activity: Garden Area General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more: Pentagon Pentagra m Hexagon Properties of Regular Polygons Diagonals of Polygons Interactive Polygons The Circle Circle Pi Circle Sector and Segment Circle Area by Sectors Annulus Activity: Dropping a Coin onto a Grid Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Here is a short reference for you:
    [Show full text]
  • Constructive Solid Geometry Concepts
    3-1 Chapter 3 Constructive Solid Geometry Concepts Understand Constructive Solid Geometry Concepts Create a Binary Tree Understand the Basic Boolean Operations Use the SOLIDWORKS CommandManager User Interface Setup GRID and SNAP Intervals Understand the Importance of Order of Features Use the Different Extrusion Options 3-2 Parametric Modeling with SOLIDWORKS Certified SOLIDWORKS Associate Exam Objectives Coverage Sketch Entities – Lines, Rectangles, Circles, Arcs, Ellipses, Centerlines Objectives: Creating Sketch Entities. Rectangle Command ................................................3-10 Boss and Cut Features – Extrudes, Revolves, Sweeps, Lofts Objectives: Creating Basic Swept Features. Base Feature .............................................................3-9 Reverse Direction Option ........................................3-16 Hole Wizard ..............................................................3-20 Dimensions Objectives: Applying and Editing Smart Dimensions. Reposition Smart Dimension ..................................3-11 Feature Conditions – Start and End Objectives: Controlling Feature Start and End Conditions. Reference Guide Reference Extruded Cut, Up to Next .......................................3-25 Associate Certified Constructive Solid Geometry Concepts 3-3 Introduction In the 1980s, one of the main advancements in solid modeling was the development of the Constructive Solid Geometry (CSG) method. CSG describes the solid model as combinations of basic three-dimensional shapes (primitive solids). The
    [Show full text]
  • A Historical Introduction to Elementary Geometry
    i MATH 119 – Fall 2012: A HISTORICAL INTRODUCTION TO ELEMENTARY GEOMETRY Geometry is an word derived from ancient Greek meaning “earth measure” ( ge = earth or land ) + ( metria = measure ) . Euclid wrote the Elements of geometry between 330 and 320 B.C. It was a compilation of the major theorems on plane and solid geometry presented in an axiomatic style. Near the beginning of the first of the thirteen books of the Elements, Euclid enumerated five fundamental assumptions called postulates or axioms which he used to prove many related propositions or theorems on the geometry of two and three dimensions. POSTULATE 1. Any two points can be joined by a straight line. POSTULATE 2. Any straight line segment can be extended indefinitely in a straight line. POSTULATE 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. POSTULATE 4. All right angles are congruent. POSTULATE 5. (Parallel postulate) If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
    [Show full text]
  • Geometry Course Outline
    GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid
    [Show full text]
  • Geometry (Lines) SOL 4.14?
    Geometry (Lines) lines, segments, rays, points, angles, intersecting, parallel, & perpendicular Point “A point is an exact location in space. It has no length or width.” Points have names; represented with a Capital Letter. – Example: A a Lines “A line is a collection of points going on and on infinitely in both directions. It has no endpoints.” A straight line that continues forever It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified because it has arrows on the ends. It is named by “a single lower case letter”. – Example: “line a” A B C D Line Segment “A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. “ A straight line that stops It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by points at the ends It is named by the Capital Letter End Points – Example: “line segment AB” or “line segment AD” C B A Ray “A ray is part of a line. It has one endpoint and continues on and on in one direction.” A straight line that stops on one end and keeps going on the other. It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by a point at one end and an arrow at the other. It can be named by saying the endpoint first and then say the name of one other point on the ray. – Example: “Ray AC” or “Ray AB” C Angles A B Two Rays That Have the Same Endpoint Form an Angle. This Endpoint Is Called the Vertex.
    [Show full text]
  • Analytical Solid Geometry
    chap-03 B.V.Ramana August 30, 2006 10:22 Chapter 3 Analytical Solid Geometry 3.1 INTRODUCTION Rectangular Cartesian Coordinates In 1637, Rene Descartes* represented geometrical The position (location) of a point in space can be figures (configurations) by equations and vice versa. determined in terms of its perpendicular distances Analytical Geometry involves algebraic or analytic (known as rectangular cartesian coordinates or sim- methods in geometry. Analytical geometry in three ply rectangular coordinates) from three mutually dimensions also known as Analytical solid** geom- perpendicular planes (known as coordinate planes). etry or solid analytical geometry, studies geometrical The lines of intersection of these three coordinate objects in space involving three dimensions, which planes are known as coordinate axes and their point is an extension of coordinate geometry in plane (two of intersection the origin. dimensions). The three axes called x-axis, y-axis and z-axis are marked positive on one side of the origin. The pos- itive sides of axes OX, OY, OZ form a right handed system. The coordinate planes divide entire space into eight parts called octants. Thus a point P with coordinates x,y,z is denoted as P (x,y,z). Here x,y,z are respectively the perpendicular distances of P from the YZ, ZX and XY planes. Note that a line perpendicular to a plane is perpendicular to every line in the plane. Distance between two points P1(x1,y1,z1) and − 2+ − 2+ − 2 P2(x2,y2,z2)is (x2 x1) (y2 y1) (z2 z1) . 2 + 2 + 2 Distance from origin O(0, 0, 0) is x2 y2 z2.
    [Show full text]
  • Unsteadiness in Flow Over a Flat Plate at Angle-Of-Attack at Low Reynolds Numbers∗
    Unsteadiness in Flow over a Flat Plate at Angle-of-Attack at Low Reynolds Numbers∗ Kunihiko Taira,† William B. Dickson,‡ Tim Colonius,§ Michael H. Dickinson¶ California Institute of Technology, Pasadena, California, 91125 Clarence W. Rowleyk Princeton University, Princeton, New Jersey, 08544 Flow over an impulsively started low-aspect-ratio flat plate at angle-of-attack is inves- tigated for a Reynolds number of 300. Numerical simulations, validated by a companion experiment, are performed to study the influence of aspect ratio, angle of attack, and plan- form geometry on the interaction of the leading-edge and tip vortices and resulting lift and drag coefficients. Aspect ratio is found to significantly influence the wake pattern and the force experienced by the plate. For large aspect ratio plates, leading-edge vortices evolved into hairpin vortices that eventually detached from the plate, interacting with the tip vor- tices in a complex manner. Separation of the leading-edge vortex is delayed to some extent by having convective transport of the spanwise vorticity as observed in flow over elliptic, semicircular, and delta-shaped planforms. The time at which lift achieves its maximum is observed to be fairly constant over different aspect ratios, angles of attack, and planform geometries during the initial transient. Preliminary results are also presented for flow over plates with steady actuation near the leading edge. I. Introduction In recent years, flows around flapping insect wings have been investigated and, in particular, it has been observed that the leading-edge vortex (LEV) formed during stroke reversal remains stably attached throughout the wing stroke, greatly enhancing lift.1, 2 Such LEVs are found to be stable over a wide range of angles of attack and for Reynolds number over a range of Re ≈ O(102) − O(104).
    [Show full text]
  • Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
    Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers.
    [Show full text]
  • Geometry by Its History
    Undergraduate Texts in Mathematics Geometry by Its History Bearbeitet von Alexander Ostermann, Gerhard Wanner 1. Auflage 2012. Buch. xii, 440 S. Hardcover ISBN 978 3 642 29162 3 Format (B x L): 15,5 x 23,5 cm Gewicht: 836 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 The Elements of Euclid “At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B. Russell, quoted from K. Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K. H. added: An innocent look at a page of contemporary the- orems is no doubt less likely to evoke feelings of “first love”.) “At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the In- troductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew. Math. 4 (1829) p. 402; transl. from the French) “The year 1868 must be characterised as [Sophus Lie’s] break- through year.
    [Show full text]
  • The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes
    S S symmetry Article The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes Mutahir Ali *, Farhad Ali, Abdus Saboor, M. Saad Ghafar and Amir Sultan Khan Department of Mathematics, Kohat University of Science and Technology, Kohat 26000, Khyber Pakhtunkhwa, Pakistan; [email protected] (F.A.); [email protected] (A.S.); [email protected] (M.S.G.); [email protected] (A.S.K.) * Correspondence: [email protected] Received: 5 December 2018; Accepted: 22 January 2019; Published: 13 February 2019 Abstract: This research provides second-order approximate Noether symmetries of geodetic Lagrangian of time-conformal plane symmetric spacetime. A time-conformal factor is of the form ee f (t) which perturbs the plane symmetric static spacetime, where e is small a positive parameter that produces perturbation in the spacetime. By considering the perturbation up to second-order in e in plane symmetric spacetime, we find the second order approximate Noether symmetries for the corresponding Lagrangian. Using Noether theorem, the corresponding second order approximate conservation laws are investigated for plane symmetric gravitational waves like spacetimes. This technique tells about the energy content of the gravitational waves. Keywords: Einstein field equations; time conformal spacetime; approximate conservation of energy 1. Introduction Gravitational waves are ripples in the fabric of space-time produced by some of the most violent and energetic processes like colliding black holes or closely orbiting black holes and neutron stars (binary pulsars). These waves travel with the speed of light and depend on their sources [1–5]. The study of these waves provide us useful information about their sources (black holes and neutron stars).
    [Show full text]