WELLS AND HART'S MATHEMATICS FIRST YEAR ALGEBRA A one year course NEW HIGH SCHOOL ALGEBRA A three semester course SECOND COURSE IN ALGEBRA Appears in a brief and in an enlarged edition. PLANE GEOMETRY Provision is made for a brief or an extended course SOLID GEOMETRY D. C. HEATH & CO., PUBLISHERS COPYRIGHT, 1915, BY D. C. HEATH & Co. 115 Gift 12-29 PREFACE GREEK GEOMETRY, the finest product of deductive thinking which high school pupils encounter, has come down to us through twenty-two centuries practically unchanged in essential content or form. It has been presented in texts, each built upon a preceding one and each good in its day, which have sought to present the great science in accord with the ideals of their time. This text is a thorough revision of Wells's Essentials of Geometry in accord with current scientific and pedagogical thought. The scientific ideal is represented the world over by Hilbert's Foundations of Geometry. (Translated by Townsend, Open Court Pub. Co., Chicago.) The pedagogical ideals are represented in this country by the Report of the National Committee of Fifteen. (See Mathematics Teacher, Dec., 1912; School Science and Mathematics, 1911; Proceedings of N. E. A., 1911.) These ideals and the personal experience of one of the authors in teaching high school geometry in recent years have been the determining factors in the making of this text. Permit us to direct attention to some of its features. In each Book, the fundamentally important theorems are given first. These theorems present a safe and sane minimum course. These are followed in each Book by one or more groups of theorems or applications which are strictly supplementary, material which either has long appeared in geometries in some form or has been introduced in recent years to add to the pupils' interest. Teachers will find no difficulty in making selection from this material, and, on the other hand, will not be embarrassed by omitting any of it. (See pp. 172, 210, and 245.) The introduction presents only the immediately necessary concepts, notation, and terminology. Emphasis is upon the acquisition of these and of skill in the use of tools, and above all upon the acquisition of the important point of view presented in §§ 48-50. The fundamental constructions are placed early in Book I so that pupils can be required to construct their figures; they are not placed earlier because they cannot be proved earlier. Authorities and details of demonstrations' which pupils can supply are increasingly omitted from the demonstrations, and often only suggestions are given. The resulting proofs are an incentive to real thinking for all the members of the class; they do not consume time that can be spent more profitably upon exercises and other valuable supplementary material. Pupils are encouraged to plan their proofs instead of plunging blindly into a demonstration. (See §§ 69 and 117.) Unnecessary corollaries have been omitted, and dignity and importance is given to those which are included in the text. (See §§ 71, 96, 101.) The stages of the proof are plainly marked, the steps are numbered, the reasons are given in full, and the proofs are arranged attractively on the page. Carefully selected exercises follow most of the propositions. Notice exercises such as Exercises 2, 23, 45, 63, of Book I, designed to teach concretely and inductively the theorems which immediately follow. Notice also the illustrative exercises which set a standard for the pupils' solution. (See pp. 31, 32, 157.) Enough exercises are provided for a minimum course. Besides these, there are miscellaneous exercises at the close of each Book, depending upon only the theorems of the minimum course. Finally there will be found from time to time a note like that on page 52, referring to supplementary exercises at the end of the text. (See pp. 52, 59, 83.) Suggestions are given with exercises where experience has shown that a majority of a class require such assistance in order to do effective. work. (See Book I, Ex. 128, 131, etc.) Simple applied problems (see Book I, Ex. 15, 37, 39, 40, 41, etc.) and artistic designs (see pp. 1, 47, 50, etc.) exhibit to the pupils some of the uses of geometry. Only simple applications are included in the minimum course. Other applications are introduced among the supplementary exercises at the end of the text and among the supplementary topics at the close of certain of the Books. (See pp. 138, 172, 174, 246, etc.) A brief history of geometry is included in the introduction, and other historical references are introduced from time to time throughout the text. (See pp. 29, 36, 46, 240, etc.) Axioms are defined in the accepted modern form (p. 22). They are introduced only as they become necessary. (See pp. 22, 29, 50, 82.) In the introduction, their meaning is made clear by suitable preliminary exercises. (See Introduction, Ex. 22, 23, 24, 37, etc.) The definitions also are modern and consistent, even though they are in some cases different from those ordinarily given. (See §§ 1, 2, 4, 5, 47, the note on p. 27, etc.) For example, after defining a circle as a line, which is correct, there is every reason for also defining a polygon as a line, instead of defining it as a portion of a plane. It may seem strange at first also not to find in the first paragraph of the text the attempted distinction between a physical and a geometrical solid, something that is psychologically impossible for beginners, but the authors believe firmly that there is much to recommend their own informal statements in § 1. The incommensurable cases are dismissed with a mere remark on pages 113, 149, and 194, and are treated fully only after the theory of limits is given on page 260. The mensuration of the circle is treated informally at first on page 238. The treatment involves nevertheless the basic ideas which are developed more fully in the formal treatment of the same topic which appears as one of the supplementary topics of Book V on page 248. This treatment is as elemen tary as the difficulty of the subject permits; to give less would render the treatment either incomprehensible or incomplete. On the other hand, the treatment is as sound as an elementary presentation renders possible; to give more would certainly render the subject distasteful to an average high school class. |