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Solid Geometry with Problems and Applications (Revised Edition), by H The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), by H. E. Slaught and N. J. Lennes This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Solid Geometry with Problems and Applications (Revised edition) Author: H. E. Slaught N. J. Lennes Release Date: August 26, 2009 [EBook #29807] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK SOLID GEOMETRY *** Bonaventura Cavalieri (1598–1647) was one of the most influential mathematicians of his time. He was chiefly noted for his invention of the so-called “Principle of Indivisibles” by which he derived areas and volumes. See pages 143 and 214. SOLID GEOMETRY WITH PROBLEMS AND APPLICATIONS REVISED EDITION BY H. E. SLAUGHT, Ph.D., Sc.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO AND N. J. LENNES, Ph.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MONTANA ALLYN and BACON Bo<on New York Chicago Produced by Peter Vachuska, Andrew D. Hwang, Chuck Greif and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber’s note The original book is copyright, 1919, by H. E. Slaught and N. J. Lennes. Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for printing, but may be easily recompiled for screen viewing. Please see the preamble of the LATEX source file for instructions. PREFACE In re-writing the Solid Geometry the authors have consistently car- ried out the distinctive features described in the preface of the Plane Geometry. Mention is here made only of certain matters which are particularly emphasized in the Solid Geometry. Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms are stated and applied at the precise points where they are to be used. Theorems are no longer quoted in the proofs but are only referred to by paragraph numbers; while with increasing frequency the student is left to his own devices in supplying the reasons and even in filling in the logical steps of the argument. For convenience of reference the axioms and theorems of plane geometry which are used in the Solid Geometry are collected in the Introduction. In order to put the essential principles of solid geometry, together with a reasonable number of applications, within limited bounds (156 pages), certain topics have been placed in an Appendix. This was done in order to provide a minimum course in convenient form for class use and not because these topics, Similarity of Solids and Applications of Projection, are regarded as of minor importance. In fact, some of the examples under these topics are among the most interesting and concrete in the text. For example, see pages 180–183, 187–188, 194– 195. The exercises in the main body of the text are carefully graded as to difficulty and are not too numerous to be easily performed. The concepts of three-dimensional space are made clear and vivid by many simple illustrations and questions under the suggestive headings “Sight PREFACE Work.” This plan of giving many and varied simple exercises, so effec- tive in the Plane Geometry, is still more valuable in the Solid Geometry where the visualizing of space relations is difficult for many pupils. The treatment of incommensurables throughout the body of this text, both Plane and Solid, is believed to be sane and sensible. In each case, a frank assumption is made as to the existence of the concept in question (length of a curve, area of a surface, volume of a solid) and of its realization for all practical purposes by the approximation process. Then, for theoretical completeness, rigorous proofs of these theorems are given in Appendix III, where the theory of limits is presented in far simpler terminology than is found in current text-books and in such a way as to leave nothing to be unlearned or compromised in later mathematical work. Acknowledgment is due to Professor David Eugene Smith for the use of portraits from his collection of portraits of famous mathematicians. H. E. SLAUGHT N. J. LENNES Chicago and Missoula, May, 1919. CONTENTS INTRODUCTION1 Space Concepts..........................1 Axioms and Theorems from Plane Geometry..........5 BOOK I. Properties of the Plane 10 Perpendicular Planes and Lines................. 11 Parallel Planes and Lines..................... 21 Dihedral Angles.......................... 29 Constructions of Planes and Lines................ 37 Polyhedral Angles......................... 42 BOOK II. Regular Polyhedrons 53 Construction of Regular Polyhedrons.............. 56 BOOK III. Prisms and Cylinders 58 Properties of Prisms....................... 59 Properties of Cylinders...................... 75 BOOK IV. Pyramids and Cones 85 Properties of Pyramids...................... 86 Properties of Cones........................ 98 BOOK V. The Sphere 113 Spherical Angles and Triangles................. 125 Area of the Sphere........................ 143 Volume of the Sphere....................... 150 APPENDIX TO SOLID GEOMETRY I. Similar Solids......................... 168 II. Applications of Projection.................. 183 III. Theory of Limits........................ 196 INDEX 217 PORTRAITS AND BIOGRAPHICAL SKETCHES Cavalieri......................... Frontispiece Thales............................... 52 Archimedes............................ 112 Legendre.............................. 167 SOLID GEOMETRY SOLID GEOMETRY INTRODUCTION 1. Two-Dimensional Figures. In plane geometry each figure is restricted so that all of its parts lie in the same plane. Such figures are called two-dimensional figures. A figure, all parts of which lie in one straight line, is a one-dimensional figure, while a point is of zero dimensions. 2. Three-Dimensional Figures. A figure, not all parts of which lie in the same plane, is a three-dimensional figure. Thus, a figure consisting of a plane and a line not in the plane is a three-dimensional figure because the whole figure does not lie in one plane. 3. Solid Geometry treats of the properties of three-dimensional figures. 4. Representation of a Plane. While a plane is endless in extent in all its directions, it is represented by a parallelogram, or some other limited plane figure. A plane is designated by a single letter in it, by two letters at opposite corners of the parallelogram representing it, or by any three letters in it but not in the same straight line. Thus, we say the plane M, the plane PQ, or the plane ABC. 2 SOLID GEOMETRY 5. Figures in Plane and Solid Geometry. In describing a figure in plane geometry, it is assumed, usually without special mention, that all parts of the figure lie in the same plane, while in solid geometry it is assumed that the whole figure need not lie in any one plane. Thus, in plane geometry we have the theorem: “Through a fixed point on a line one and only one perpendicular can be drawn to the line.” If all parts of the figure are not required to lie in one plane, the theorem just quoted is far from true. As can be seen from the figure, an unlimited number of lines can be drawn perpendicular to a line at a point in it. Thus, all the spokes of a wheel may be perpendicular to the axle. 6. Loci in Plane and Solid Geometry. In plane geometry, “the locus of all points at a given distance from a given point” is a circle, while in solid geometry this locus is a sphere. In plane geometry, “the locus of all points at a given distance from a given line” consists of two lines, each parallel to the given line and at the given distance from it, while in solid geometry this locus is a cylindrical surface whose radius is the given distance. INTRODUCTION 3 7. Parallel Lines. Skew Lines. In plane geometry, two lines which do not meet are parallel, while in solid geometry, two lines which do not meet need not be parallel. That is, they may not be in the same plane. Lines which are not parallel and do not meet are called skew lines. In solid geometry, as in plane geometry, the definition of parallel lines implies that the lines lie in the same plane. That is, if two lines are parallel, there is always some plane in which both lie. Thus, in the figure, l1 and l2 are parallel, as are also l1 and l3, while l3 and l4 are skew. sight work Note. In exercises 1–4 give the required loci for both plane and solid geometry. No proofs are required. 1. The locus of all points six inches distant from a given point. 2. The locus of all points ten inches distant from a given point. 3. The locus of all points at a perpendicular distance of four inches from a given straight line. 4. The locus of all points at a perpendicular distance of nine inches from a given straight line. 5. Find the locus of all points one foot from a given plane. Is this a problem in plane or in solid geometry? 6. Find the locus of all points equidistant from two parallel lines and in the same plane with them. Is this a problem in plane or in solid geometry? 7. Find the locus of all points equidistant from two given parallel planes. Is this a problem in plane or in solid geometry? 8. The side walls of your schoolroom meet each other in four vertical lines. Are any two of these parallel? Are any three of them parallel? Do any three of them lie in the same plane? 9.
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