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IN writing this book the authors have been guided by two main purposes:

(a) That pupils may gain by gradual and natural processes the power and the habit of deductive reasoning.

(b) That pupils may learn to know the essential facts of elementary geometry as properties of the space in which they live, and not merely as statements in a book.

The important features by which the Plane Geometry seeks to accomplish these purposes are:

1. The simplification of the first five chapters by the exclusion of many theorems found in current books. These five chapters correspond to the usual five books, and the most important omissions are the formal treatment of the theory of limits, the incommensurable cases, maxima and minima, and numerous other theorems, together with the deduction of complicated algebraic formulæ, such as the area of a triangle and the radii of the inscribed, escribed, and circumscribed circles, in terms of the three sides.

Chapter VI contains a graphic representation of certain important theorems and an informal presentation of incommensurable cases and limits. The treatment of limits is based upon the graph, since the visual or graphic method appeals more directly to the intuition than the usual abstract processes. Chapter VII is devoted to advanced work and to a review of the preceding chapters.

2. The subject has been enriched by including many applications of special interest to pupils. Here an effort has been made to include only such concrete problems as come fairly within the observation and comprehension of the average pupil. This led to the omission, for example, of problems relating to machinery and technical industries, which might appeal to an exceptional boy, but which are entirely inappropriate for the average student. On the other hand, free use is made of certain sources of problems which may be easily comprehended without extended explanation and which involve varied and simple combinations of geometric forms. Such problems pertain to decoration, ornamental designs, and architectural forms. They are found in tile patterns, parquet floors, linoleums, wall papers, steel ceilings, grill work, ornamental windows, etc., and they furnish a large variety of simple exercises both for geometric construction and proofs and for algebraic computation. They are not of the puzzle type, but require a thorough acquaintance with geometric facts and develop the power to use mathematics.

These problems form an entirely new type of exercises, and while they require more space in the text-book than the more difficult "originals" stated in the usual abstract terms, they excel the latter in interest for the pupil and in helping to train his mathematical common sense. Many of these exercises are simple enough to be solved at sight, and such solution should be encouraged whenever possible. All the designs are taken from photographs or from actual commercial patterns now in use. By thus showing that the abstract theorems of geometry find concrete expression in a multitude of familiar objects, it is sought to make the subject a permanent part of the pupil's mental equipment.

3. Persistent effort is made to vitalize the content of the definitions and theorems. It is well known that pupils often study and recite definitions and theorems without really comprehend

ing their meaning. It is sought to check this tendency by giving definitions only when they are to be used, and by immediately verifying both definitions and theorems in concrete cases. The figure on page 4 is the basis for a large number of questions of this type. For example, see § 25, Ex. 3; § 30, Ex. 1; § 34, Exs. 1, 2; § 36, Ex. 3; §§ 322, 324.

In this connection special attention is called to the emphasis placed upon those theorems which are of fundamental importance both in the logical chain and in their immediate use in effecting constructions and indirect measurements otherwise difficult or impossible. For example, see the theorems on congruence of triangles, §§ 31-43, the constructions of §§ 44-58, and the theorems on proportional segments, §§ 243-254. Compare especially § 34, Ex. 5, § 244, Ex. 2, and § 254, Exs. 4, 5.

The summaries at the close of the chapters, which are to be made by the pupil himself, will vitalize the theorems as no made-to-order summaries can possibly do.

4. The student is made to approach the formal logic of geometry by natural and gradual processes. He is expected to grow into this new, and to him unusual, way of thinking. The treatment is at the start informal, leading through the congruence theorems directly to concrete applications and geometric constructions. The formal development then follows gradually and is characterized by a judicious guidance of the student, by questions, outlines, and other devices, into an attitude of mental independence and an appreciation of clear reasoning.

This informality of treatment, most frequent in the earlier parts, is used throughout wherever occasion seems to justify it. See §§ 318, 319. An effort has been made to vary the methods of attack and to avoid monotony. Some theorems are proved in full; some are outlined; in some, hints and suggestions are given. Any uniform method would make it impossible to leave that to the pupil which he can do for himself, and at the same time to give full assistance where

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