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.