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own demonstrations frequently proves unprofitable as well as arduous, and engenders in the learner a distaste for a study in which he might otherwise take delight. This text does not aim to produce accomplished geometricians at the completion of the first book, but to aid the learner in his progress throughout the volume, wherever experience has shown that he is likely to require assistance. It is designed, under good instruction, to develop a clear conception of the geometric idea, and to produce at the end of the course a rational individual and a friend of this particular science.

3. The theorems and their demonstrations the real subjectmatter of Geometry - are introduced as early in the study as possible.

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4. The simple fundamental truths are explained instead of being formally demonstrated.

5. The original exercises are distinguished by their abundance, their practical bearings upon the affairs of life, their careful gradation and classification, and their independence. Every exercise can be solved or demonstrated without the use of any other exercise. Only the truths in the numbered paragraphs are necessary in working originals.

6. The exercises are introduced as near as practicable to the theorems to which they apply.

7. Emphasis is given to the discussion of original constructions. 8. The summaries will be found a valuable aid in reviews.

9. The historical notes give the pupil a knowledge of the development of the science of geometry and add interest to the study. 10. The attractive open page will appeal alike to pupils and to teachers.

The author sincerely desires to extend his thanks to those friends and fellow teachers who, by suggestion and encouragement, have inspired him in the preparation of these pages.

EDWARD R. ROBBINS.

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PLANE GEOMETRY

INTRODUCTION

1. Geometry is a science which treats of the measurement of magnitudes.

2. A point is that which has position but not magnitude. 3. A line is that which has length but no other magnitude.

4. A straight line is a line which is determined (fixed in position) by any two of its points. That is, two lines that coincide entirely, if they coincide at any two points, are straight lines.

5. A rectilinear figure is a figure containing straight lines and no others.

6. A surface is that which has length and breadth but no other magnitude.

7. A plane is a surface in which if any two points are taken, the straight line connecting them lies wholly in that surface.

8. Plane Geometry is a science which treats of the properties of magnitudes in a plane.

9. A solid is that which has length, breadth, and thickA solid is that which occupies space.

ness.

10. Boundaries. The boundaries (or boundary) of a solid are surfaces. The boundaries (or boundary) of a surface

are lines. The boundaries of a line are points. These boundaries can be no part of the things they limit. A surface is no part of a solid; a line is no part of a surface; a point is no part of a line.

11. Motion. If a point moves, its path is a line. Hence, if a point moves, it generates (describes or traces) a line; if a line moves (except upon itself), it generates a surface; if a surface moves (except upon itself), it generates a solid. NOTE. Unless otherwise specified the word "line" means straight line.

ANGLES

< LX I

ANGLE

RIGHT ANGLES

ADJACENT
ANGLES

VERTICAL ANGLES

PERPENDICULAR

12. A plane angle is the amount of divergence of two straight lines that meet. The lines are called the sides of the angle. The vertex of an angle is the point at which the lines meet.

13. Adjacent angles are two angles that have the same vertex and a common side between them.

14. Vertical angles are two angles that have the same vertex, the sides of one being prolongations of the sides of the other.

15. If one straight line meets another and makes the adjacent angles equal, the angles are right angles.

16. One line is perpendicular to another if they meet at right angles. Either line is perpendicular to the other. The point at which the lines meet is the foot of the perpendicular. Oblique lines are lines that meet but are not perpendicular.

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