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In the preparation of this text the author acknowledges joint authorship with Robert L. Short.

Geometry is approached from the constructive side, all methods of construction needed for drawing any figure in Books I or II being given in the introduction. In cases where a geometric principle is used in any construction, a note at the end tells where the principle is proved. In all figures in the early portions of Book I, the construction lines and arcs are given; afterwards these are dispensed with.

In Props. II and III, Book I, and in other places, colored diagrams are given in addition to the regular figures, in which the equal parts in the given triangles are represented by lines of the same color; this scheme will be found of great assistance to the pupil in the earlier portions of the work.

Below each figure is a paragraph in smaller type, giving full directions for the construction of the diagram in accordance with the statement of the theorem. This gives the pupil a familiarity with the figure, and what in it is given, and what to be proved, that is of great value, and lessens the tendency to memorize. Many figures are omitted, but complete directions for their construction are given in each case.

In all figures given in connection with theorems and problems, given and required lines are made heavy.

Attention is invited to the order of theorems in Book I; in this case, the pupil begins with the easier proofs. From the start the student has practice in representing angles and lines by small letters.

Only the outline of the proof is given after Book II, except in the more difficult demonstrations. In this outline work the pupil has explicit directions, but develops the demonstration himself. In all portions of the work proofs are omitted in


cases where they should present no difficulty; usually, in such cases, hints are given as to the method of demonstration.

The numbering of the steps of the proof makes them more easy of reference.

In Book I, and in the first part of Book VI, the authority for each statement of the proof will be found directly below, in smaller type, enclosed in brackets, with the number of the section where it is to be found. In other portions of the work only the section number is given, and in some cases only an interrogation point. In all such cases the pupil should be required to give the authority as fully as if it were actually printed on the page.

No principles are given as immediate consequences of theorems except such as actually belong in this category. Separate propositions are made for all truths which are not immediate consequences of preceding theorems.

The originals are new, and very largely of a practical nature. They are not too difficult, will make the pupil think, and are more than five hundred in number. The smaller number is compensated for by the fact that the pupil has to do some original work in almost every proof after Book II. Exercises coming under Book I will be found scattered through all the following Books of the Plane Geometry; and a similar remark applies to exercises under Book II, Book III, etc.

At the end of Book I will be found a list of principles proved, which will be of great assistance in solving originals. A similar list in regard to similar triangles is given in § 265.

The authors thought that it would be an improvement to put all work under Polyedrons, including their measurement, in Book VII, devoting Book VIII to the Cylinder and Cone and Book IX to the Sphere and its measurement.

The authors wish to thank the many teachers whose advice and criticism have been useful in preparing a treatise that should stand the test of class-room work. They are also indebted to Mr. C. W. Sutton for a part of the exercises.


BOSTON, 1908.


IN studying the opening propositions, in Geometry, beginners have difficulty in fixing clearly in mind just what parts, in the figure, are given. To aid this, in Props. II and III, Book I, and in other places, diagrams are given in addition to the regular cuts, in which the equal given parts are printed in the same color.

This color-scheme may be advantageously followed in the class-room, in connection with all figures in the earlier portions of Book I. If colored crayons are not available, the beginner may designate equal lines by the marks, ",", etc., and equal angles by single, double, or triple arcs.

In solving the non-numerical exercises, while it is not practicable to give very much assistance to the pupil, the following suggestions may be found of service:

1. Draw an accurate figure, showing all given and required lines.

This sometimes suggests the method of proof.

2. Be sure that the figure is the most general one allowable.

Thus, in exercises relating to triangles, do not draw them right, isosceles, or equilateral, unless the exercise calls for a right, isosceles, or equilateral triangle; in exercises relating to quadrilaterals, do not draw them with two sides equal or parallel, unless the exercise calls for such a construction.

3. Write down carefully what is given, and what is to be proved.

4. Look up all previous theorems which may possibly have a bearing on what is to be proved.

Thus, if two lines are to be proved equal, refer to all previous theorems regarding equal lines. (Compare § 141.)

In solving exercises in construction, it is advantageous to regard the problem as solved, and draw the given and required lines. Studying the relations between these will frequently suggest the method of construction to be employed.

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