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there is also an important scientific gain from it. It is commonly regarded as bad scientific practice to allow unlimited use of hypothetical constructions. With beginners in geometry it is always a question how far it is desirable to permit them to give proofs that involve constructions which they have not yet learned. For example, before the student has learned how to bisect an angle he may be allowed, openly or tacitly, to assume that a bisector can be drawn, while he is proving the theorem about the base angles of an isosceles triangle. The real question is how far shall this sort of practice be allowed. The prevailing view is that it should be practiced no longer than necessary. Too long continued, it blunts the learner's perception of the nature of real geometrical proof. By including in the preliminary work such constructions as are needed early in proofs, the need for hypothetical constructions is reduced to a minimum. The present organization thus scores a real gain in soundness of early geometrical thinking in the very act of reaching an important pedagogical end.

To the beginner no motives are so wholesomely appealing as the all-around usefulness of the study. "What benefit can be derived from the subject?" is the question most on his mind and lips. A good pedagogical organization must so marshal its ideas for presentation that the most attractive ones are in the foreground. This important principle has been kept prominent in the authors' minds in the organization of the subject as a whole, and also of its important parts. The exercise lists are exceptionally rich and varied in really practical problems, without at any point failing to include an ample supply of the standard geometric type as well. Many of the abstract exercises are followed at once by practical problems that give a life-setting to the geometric principles involved. An earnest attempt has been made to give a great variety to the types of practical exercise. This feature will assist in reaching a wide range of taste and aptitude among pupils, and also in keeping up the spirit of study through a many-sided motivation. This

cannot fail to appeal to those who hold the view that it is of maximum importance to make the learner's work seem worth while to him while he is yet in school.

The text lays no claim to logical precision beyond that which good students can appreciate. It recognizes that boys and girls can prove things long before they can demonstrate them. That is to say, they can bring to bear enough relevant evidence to heighten materially the feeling of certainty of the truth of a theorem before they can intelligently master the technique of formal deductive proof. This treatment recognizes the learner's right to acquire the ability to demonstrate through the exercise of his ability to prove. To do otherwise is to endanger needlessly the very spirit of geometric reasoning in a vain show of mechanical steps and logical technique. In gauging this treatment, pedagogical rather than logical standards should be applied. Logic has been intentionally sacrificed to insight whenever it was believed that the general geometric interests of the student would be thereby materially subserved.

Finally, this book has been prepared by actual teachers of long and successful high school experience. They know high school boys and girls. Visionary theories about what ought to be, but cannot be, have been ruthlessly set aside. They have kept an eye single to what is feasible, practicable, and remunerative in the class-room. Much of the manuscript was used for years in mimeograph form and the roughnesses have been smoothed off through class-room findings. The plan has also worked successfully in the hands of other teachers than the authors and in much less practicable form than that in which it is here given to the public. The editor feels that the text has an important rôle to play in the attempt now being made to bring school geometry into closer conformity with the needs, standards, and possibilities of those who are to study it.

Chicago, August, 1915.



The purpose of this book is to present the essentials of geometry together with some of their applications. There is no subject in the high school curriculum which can be made of more practical value to the average student. The work has been so arranged as to give suitable mathematical training as well as to appeal to the vital interests of the growing intellect.

The main aim of the authors in the preparation of this text has been to approach abstract reasoning by a method that is natural and comprehensible to the youthful mind, and to vitalize the subject-matter,-making it both interesting and useful through a wide range of practical applications.

The plans which have been adopted for the attainment of this end are outlined in the following paragraphs.

1. The method of presentation is psychological rather than that of pure deductive logic. While most of the subject is presented in the traditional form, the experimental or inductive method is employed with some of the fundamental propositions, especially in the early pages. In this way, the student is enabled to comprehend the exact meaning of these propositions and to apply them at once to the solution of exercises. Later, after he has acquired some skill in using the method of formal logic, all of these theorems are proved deductively.

2. Actual work in geometry is begun at once without a formidable array of definitions, axioms, and principles. In the Introduction, only an informal treatment of terms is given and only such topics are discussed as are necessary for the study of the early propositions and exercises. It should be noted that no attempt is made at this point to give a technical definition of any of the terms considered. Fundamental definitions and basic discussions are presented in the latter part of the first chapter. This plan avoids the confusion arising from the

presentation of abstract material which the pupil does not comprehend at this stage, but which causes him little difficulty after he has had a partial survey of the subject.

Axioms and basic principles are introduced experimentally before they are expressed in a formal statement. In general, definitions and axioms are placed in the text at the point where they are first needed.

3. Especial attention has been given to the selection and arrangement of original exercises.

(a) They are introduced almost at the beginning and are very carefully graded. At first a few are proved in detail, in order that the student may understand what is desired in the solution of exercises and to give a model for writing out the proof. (b) In the first chapter, a large proportion of the exercises are accompanied by figures. This enables the student to concentrate his attention upon the proof and to economize time in solving exercises. It also makes possible the solution of many exercises at sight during the recitation, a training which develops rapid thinking through concerted effort among the members of the class. As the work advances, the student is required with increasing frequency to construct the figures from the text.

(c) The large number and great variety of exercises are carefully distributed and range from some quite elementary to others of considerable difficulty. This enables the teacher to adapt the book to any group of students, whether in a classical or a technical high school. The abstract exercises give mental training and application of the basic theorems, while the practical exercises are used to correlate geometric facts with real life. For the most part, the problems pertaining to shop work are for use in technical schools. The exercises marked with a star (*) are usually more difficult or of special application. (d) Many of the exercises involve an application of arithmetic and algebra to geometry. Formulas are given in connection with the theorems and form the basis of many literal exercises and of much computation.

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