geometría de los sólidos-en inglés

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 L A T E X 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 INTRODUCTION 1 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 [...]... 25 Two triangles are equal if two sides and the included angle of one are equal respectively to two sides and the included angle of the other 26 Two triangles are equal if two angles and the included side of one are equal respectively to two angles and the included side of the other 27 Two triangles are equal if three sides of one are equal respectively to three sides of the other 28 Two points each... added to equals, the sums are equal 12 If equals are subtracted from equals, the remainders are equal 13 If equals are multiplied by equals, the products are equal 14 If equals are divided by equals, the quotients are equal 15 If equals are added to unequals, the sums are unequal and in the same order 16 If unequals are added to unequals, in the same order, then the sums are unequal and in that order... of its base and altitude 59 Two parallelograms have equal areas if they have equal bases and equal altitudes 60 The area of a triangle is equal to one half the product of its base and altitude 61 If a is a side of a triangle and h the altitude on it and b another side and k the altitude on it, then ah = bk 62 The area of a trapezoid is equal to one half the product of its altitude and the sum of its... angles 32 If two adjacent angles are supplementary, their exterior sides lie in the same straight line 33 If in two triangles two sides of one are equal respectively to two sides of the other, but the third side of the first is greater than the third side of the second, then the included angle of the first is greater than the included angle of the second 34 Two lines which are perpendicular to the same... proportion the antecedents are equal, then the consequents are equal and conversely 52 In a series of equal ratios the sum of any two or more antecedents is to the sum of the corresponding consequents as any antecedent is to its consequent 53 If a line cuts two sides of a triangle and is parallel to the third side, then any two pairs of corresponding segments form a proportion 54 If two sides of a triangle... (1) if both pairs of opposite sides are equal; or (2) if two opposite sides are equal and parallel 44 Opposite sides of a parallelogram are equal 45 Two parallelograms are equal if an angle and the two adjacent sides of one are equal respectively to an angle and the two adjacent sides of the other 46 The segment connecting the middle points of the two non-parallel sides of a trapezoid is parallel to... 1 Proof : Draw BD and make DE ⊥ DB Take points A and E so that BA = DE, and draw AD, AE, and BE Now prove: (1) ABD = BDE and ∴ AD = BE; (2) ADE = ABE and ∴ ∠ADE = ∠ABE = Rt ∠ ∴ DC, DA, DB, and BA all lie in the same plane § 79 AB CD § 34 Given (2) AB CD and AB ⊥ M Fig 2 To prove that CD ⊥ M Proof : If CD is not ⊥ M let C D be ⊥ M Then C D AB by case (1), and C D coincides with CD § 21 and ∴ CD ⊥... parallel to the third side, a triangle is formed which is similar to the given triangle 55 In two similar triangles corresponding altitudes are proportional to any two corresponding sides 56 Two triangles are similar if an angle of one is equal to an angle of the other and the pairs of adjacent sides are proportional 57 Two triangles are similar if their pairs of corresponding sides are proportional 58... is drawn incidentally in making a proof or constructing a figure, is marked in long dashes if it is in full view (3) Any line whatever which is behind a part of the figure is marked in short dashes or dots, or sometimes is not shown at all (4) Where a figure is shaded it is usually regarded as opaque and the lines behind it cannot be seen at all (5) In some cases a shaded surface is regarded as translucent... different planes are determined by the points A, B, C, D? 4 How many planes are determined by any four points which do not all lie in one plane? 5 How many planes are determined by the points A, B, C, D, E, if A, B, C lie in a straight line and C, D, E lie in another straight line? 6 How many planes are determined by five points, no four of which lie in the same plane? 7 How many planes are determined by three . equal if two sides and the included angle of one are equal respectively to two sides and the included angle of the other. 26. Two triangles are equal if two angles and the included side of one are. altitudes. 60. The area of a triangle is equal to one half the product of its base and altitude. 61. If a is a side of a triangle and h the altitude on it and b another side and k the altitude on. respectively to two sides of the other, but the third side of the first is greater than the third side of the second, then the included angle of the first is greater than the included angle of the second. 34.