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Plane and Solid Geometry DUKE UNIVERSITY LIBRARY vv i\\S FOUR DISCOVERERS AND FOUNDERS OF GEOMETRY (See pages 4-90-498) PLANE AND SOLID GEOMETRY BY FLETCHER DURELL, Ph.D. HEAD OP THE MATHEMATICAL DEPARTMENT, THE LAWRENCEVILLE SCHOOL NEW YORK CHARLES E. MERRILL CO, wm r DURELL’S MATHEMATICAL SERIES ARITHMETIC Two Book Series Elementary Arithmetic Teachers’ Edition Advanced Arithmetic Three Book Series Book One Book Two Book Three ALGEBRA Two Book Course Book One Book Two Book Two with Advanced Work Introductory Algebra School Algebra GEOMETRY Plane Geometry Solid Geometry Plane and Solid Geometry, TRIGONOMETRY Plane Trigonometry and Tables Plane and Spherical Trigonometry and Tables. • Plane and Spherical Trigonometry with Surveying and Tables Logarithmic and Trigonometric Tables Copyright. 1904, by Charles E. Merrill Cft It o o UV 5 / 3 .^ PREFACE One of the main purposes in writing this book has been to try to present the subject of Geometry so that the pupil shall understand it not merely as a series of correct deductions, but shall realize the value and meaning of its principles as well. This aspect of the subject has been directly presented in some places, and it is hoped that it per- vades and shapes the presentation in all places. Again, teachers of Geometry generally agree that the most difficult part of their work lies in developing in pupils the power to work original exercises. The second main purpose of the book is to aid in the solution of this difficulty by arranging original exercises in groups, each of the earlier groups to be worked by a distinct method. The pupil is to be kept working at each of these groups till he masters the method involved in it. Later, groups of mixed exercises to be worked by various methods are given. In the current exercises at the bottom of the page, only such exercises are used as can readily be solved in connection with the daily work. All difficult originals are included in the groups of exercises as indicated above. Similarly, in the writer’s opinion, many of the numeri- (iii) V iv PREFACE cal applications of geometry call for special methods of solution, and the thorough treatment of such exercises should be taken up separately and systematically. [See pp. 304-318, etc.] In the daily extempore work only such numerical problems are included as are needed to make clear and definite the meanr'ng and value of the geometric principles considered. Every attempt has been made to create and cultivate the heuristic attitude on the part of the pupil. This has been done by the method of initiating the pupil into original work described above, by queries in the course of proofs, and also at the bottom of different pages, and also by occasional queries in the course of the text where definitions and discussions are presented. In the writer opinion, the time has not yet come for the purely heuristic study of Geometry in most schools, but it is all-important to use every means to arouse in the pupil the attitude and energy of original investigation in the study of the subject. In other respects, the aim has been to depart as little as possible from the methods most generally used at present in teaching geometry. The Practical Applications (Groups 88-91) have been drawn from many sources, but the author wishes to ex- press his especial indebtedness to the Committee which has collected the Real Applied Problems published from time time in School Science and Mathematics and of which to , Professor J. F. Millis of the Francis W. Parker School of Chicago is the chairman. Page 360 is due almost en- tirely to Professor William Bets; of the East High School of Rochester, N. Y. FLETCHER DURELL. 1904. Lawekncevulie. N. J. , Sept. 1, TO THE TEACHER 1. In working original exercises, one of the chief dif- , ficulties of pupils lies in their inability to construct the figure required and to make the particular enunciation from it. Many pupils, who are quite uuable to do this preliminary work, after it is done can readily discover a proof or a solution. In many exercises in this book the figure is drawn and the particular enunciation made. It is left to the discretion of the teacher to determine for what other exercises it is best to do this for pupils. 2. It is frequently important to give partial aid to the pupil by eliciting the outline of a proof by questions such as the following: "On this figure (or, in these two tri- angles) what angles are equal, and why?” "What lines are equal, and why?” etc. 3. In many cases it is also helpful to mark in colored crayon pairs of equal lines, or of equal angles. Thus, in the figure on p. 37 lines AB and BE may be drawn with red crayon, AG and BE with blue, and the angles A and D marked by sinaU arcs drawn with green crayon. If colored crayons are not at hand, the homologous equa. I parts may be denoted by like symbols placed on them . thus: c F In solving theorems concerning proportional lines, it is Occasionally helpful to denote the lines in ~ proportion SYMBOLS AND ABBREVIATIONS plus, increased by. + or Adj. adjacent. — minus, or diminished by. Alt. alternate. X multiplied by. Art. article. -4- divided by. Ax. axiom. — equals; is (or are) equal to. Constr. construction , = approaches (as a limit). Cor. corollary. =o= is (or are) equivalent to. Def. definition. is (or are) greater than. Ex. exercise. <( is (or are) less than. Ext. exterior. .’. therefcre. Fig. figure. perpendicular perpendicular to, Hyp. hypothesis. J. , or, is perpendicular to. Ident. identity. A perpendiculars. Int. interior. or, is parallel to. Post. postulate. || parallel, . Prop. proposition. || s parallels. angle, angles Et. right. Z , A A, A triangle, triangles. Sug. suggestion. supplemental CO , CO parallelogram, parallelograms. Sup. G, © circle, circles. St. straight. Q. E. d. quod erat demonstrandum; that is, vrhieh was to be proved. q. E. f. quod erat faciendum; that is, which was to be made. A few other abbreviations and symbols will be introduced and their meaning indicated later on. , — DEFINITIONS AND FIRST PRINCIPLES INTRODUCTORY ILLUSTRATIONS. DEFINITIONS 1. Computation of an area. Practical experience has taught men that certain ways of dealing with objects in the world about us are more advantageous than others; thus, if it be desired to find the num- 1 1 —1 — 1 ber of square yards in the area of a 1 ! Li 1 floor, we do not mark off the floor 1 — — 1 ""I-” 1 — 1 1 1 j— 1 L 1 into actual square yards and count 1 “ T the number of square yards thus 1 j ! J 1 made, but pursue the much easier 1 » s i i ! course of measuring two lines, the length and breadth of the floor, obtaining 7 yards, say, as the length, and 5 yards as the width, and multiplying the length by the breadth. The area is thus found to be 35 square yards. Let the pupil determine the area of some convenient floor in each of these two ways, and compare the labor of the two processes. 2. Computation of a volume. Similarly, for example, in order to determine the number of cubic feet which a box contains, instead of filling the box with blocks of wood, each of the size of a cubic foot, and counting the number of blocks, we pur- sue the much easier course of measuring the number of linear feet in each inside edge of the box and multiplying together the three dimensions obtained; thus, if ( 9 ) 10 PLANE GEOMETRY the inside dimensions are 5, 4 and 3 feet, the volume is 5 X 4 X 3, or 60 cubic feet. Similarly, the direct method of measuring the number of bushels of wheat in a bin is to fill a bushel measure with wheat from the bin, time after time, till the bin is exhausted, and count the number of times the bushel measure is used. But a much less laborious method is to measure the three dimensions of the bin in inches and divide their product by the number of cubic inches in a bushel. Let the pupil in like manner compare the labor of finding the number of feet of lumber in a given block of wood by actually saw- ing the block up into lumber feet, with the labor of measuring the dimensions of the block and computing the number of lumber feet by taking the product of the linear dimensions obtained. 3. Unknown line determined from known lines. The student is also probably familiar with the fact that, by computations based on the relations of cer- tain lines whose lengths are known, the lengths of other unknown lines may be determined without the labor of measuring these unknown lines. Thus, for instance, if a ladder 5 yards long lean against a wall and have its foot 3 yards from the wall, the height of the top of the ladder from the ground may be determined thus: 2 (Height) = 52 — 32 = 25 — 9 = 16. .'. height = 4 yards. 4. Economies in representing surfaces, lines, etc. Other principles of advantage of an even more general character occur in dealing with geometric objects. Thus, since only one straight line can be passed through two given points, the two points may be taken as a highly economized symbol or representative of the line, by the use of which much labor is saved in dealing with lines, and new results are made attainable. Geometric magnitudes 11 Similarly, since only one flat, or plane surface can be passed through three given points (not in the same straight line), these three points may be taken as a symbol or representative of the flat surface.
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