4. The proof of the most difficult as well as of the early theorems is given in full, but, as the work progresses, a gradual elimination of parts of proofs is made as the needs and capacity of the learner increase. Thus a middle course is adopted in the use of the suggestive method. In this way, the student thoroughly comprehends the subject-matter and at the same time is strengthened by doing work within his grasp.

5. The work has been planned so that practical applications may be made as early in the course as possible. To this end, the chapter on areas is placed before that on similarity because it furnishes a great variety of applications to real life.

The traditional division into five parts is made, but these are named "chapters" because the word "book" has only historical value in this connection.

In addition to the five cardinal points outlined above, attention is called to the following features of the text.

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To a great extent, terms and expressions are used which carry their own meaning. For example, "equal" is used instead of congruent," at first because the word requires no explanation and adds no obstacle to the beginner. Some words like "corollary," "homologous," and "scholium," which are of doubtful value, are either used sparingly or omitted.

All basic material for meeting college requirements or for use in advanced mathematics has been given adequate treatment. A few topics, such as the steel square, the plane-table, screwthreads, and trigonometric ratios, are included. These are to be studied as occasion may demand.

The experiments may be performed on the laboratory or supervised study plan, with the pupils working under the guidance of the teacher. This makes possible a thorough understanding of what is expected and the attainment of most satisfactory results. For convenience, a combination ruler and protractor accompanies the book. It is suggested that blackboard protractors, also, may be used to advantage.

Incommensurable quantities and the theory of limits are

treated in a rational manner. The consideration of incommensurable numbers is introduced informally in the treatment of areas. The discussion of the theory of limits is postponed until Chapter V, where it is treated in connection with the measurement of the circle.

The more important theorems are printed in heavy type. This is in accordance with the recommendation of the Committee of Fifteen of the National Education Association, whose report has been given careful consideration in the preparation of the entire work.

Some of the ideas in this text and the material for many of the exercises have been obtained through a wide reading of authorities found in various university and technical libraries. The numerous practical applications have been gleaned from a variety of sources covering a period of many years and pertain to nearly all phases of life.

Acknowledgment is due to the McGraw-Hill Book Co., Inc., for permission to use exercises from Palmer's Practical Mathematics; to Prof. James H. Breasted, and to the publishers, for the illustration on page 8, from A History of Egypt, copyright, 1905, 1909, published by Charles Scribner's Sons; to Mr. Lawrence C. Irwin of the Joliet Township High School, and to Mr. A. A. Ellsworth, of the public schools of Grayling, Michigan, for many helpful criticisms and suggestions; to Mr. F. G. Taylor of the public schools of Oregon, Illinois, who taught the material in manuscript form, and who has given advice of great value; to Miss Margaret Willcox of Oak Park, Illinois, who critically read all the proof, and to many students of the Oak Park High School whose response to the study of the manuscript gave encouragement in the preparation of the work.

Chicago, August, 1915.

C. I. PALMER.
D. P. TAYLOR.

PRELIMINARY STATEMENT

Any physical body, such as a book, a tree, or a tract of land, has certain qualities that refer to shape, size, and relative position with respect to other bodies. These qualities are known as the geometric properties of a body, and the science which deals with them is called geometry. Arithmetic centers about computation with numbers, and algebra about the equation. Geometry makes use of both of these subjects and applies them in determining the form of objects and in comparing their relative position and magnitude.

It is apparent on every hand that geometric principles play an important part in everyday life. Their use is seen in mosaic tiling and parquet flooring, in the designs of stained-glass windows, in the architecture of buildings, in engineering constructions, and in the measurement of land by the surveyor.