BOOK I.

PLANE GEOMETRY.

1. Space is indefinite extension in

every direction.

2. A material substance is anything, large or small, solid, liquid, or aeriform, visible or invisible, that occupies a portion of space.

3. It therefore follows that material substances have limited extension in every direction.

4. For purposes of measurement, extension in three directions only are considered, called, respectively, length, breadth (or width), and thickness; they are also called collectively dimensions.

5. Magnitude, in general, means size, and is applied to anything of which greater or less can be predicated, as time, weight, distance, etc.; a geometrical magnitude is that which has one or more of the three dimensions.

6. A geometrical point has position merely; i.e. it has no magnitude.

The dots made by pencil and crayon are called points, but they are really small substances used to indicate to the eye the location of the geometrical point.

7. A geometrical line has only one dimension; i.e. length. The lines made by pencil and crayon are substances, and

1

may be called physical lines which serve to show the position of the geometrical lines.

8. A straight line is one that lies evenly between its extreme points.

This is the definition as given by Euclid. The majority of modern geometers, however, have substituted the following as stated by Newcomb; viz. :

"A straight line is one which has the same direction throughout its whole length."

Each is designed to express the idea of straightness, and not to convey it, for it is assumed that the idea already exists in the pupil's mind prior to the beginning of this study.

9. A curved line, or simply curve, is one no part of which is straight.

10. Material substances have one or more faces which separate them from the rest of space. These faces are called surfaces, and have, obviously, only two dimensions; i.e. length and breadth.

11. The surface considered apart from the substance is called a geometrical surface.

12. A plane is a geometrical surface such that if any two points in it be selected at random, the straight line joining them will lie wholly in that surface.

13. A curved surface is a geometrical surface no portion of which is a plane.

14. A physical solid is the material composing it, and which we perceive through the medium of the senses; while the geometrical solid is the space, simply, which the physical solid occupies.

15. A geometrical figure is the term applied to combinations of points, lines, and surfaces, when reference is had to their form or outline simply.

16. A plane figure is one whose points and lines all lie in the same plane.

17. "A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line." — EUCLID.

"An angle is a figure formed by two straight lines drawn from the same point.". -CHAUVENET.

"When two straight lines meet together, their mutual inclination, or degree of opening, is called an angle." — LOOMIS.

18. The lines which form an angle are called the sides of the angle, and the point from which they are drawn is called the vertex of the angle.

19. When two plane angles have the same vertex and a common side, neither angle being a part of the other, they are said to be adjacent angles.

20. When two angles have the same vertex and the sides of one are the extensions of the sides of the other, they are called vertical angles.

21. An angle is named by a letter or number placed at its vertex. If, however, there are two or more angles with the same vertex, other letters are placed at the extremities of their sides, and the three letters are used to name the angle, the letter at the vertex always coming between the other two.

22. Let us consider the point B, in the straight line AC, a pivot, and BD another starting from the position BC, and

A

B

Fig. III.

turning about B, keeping always in the same plane. It is evident that, as soon as it has started, it forms

two angles with the line AC, of which DBC, Fig. I., is the

smaller. If it continue to revolve, however, it will finally reach a position as DB, Fig. II., in which the angle DBC is the larger. Hence, in passing from the first position to the second, it must have reached a position, Fig. III., where the two angles DBC and DBA were equal.

23. Hence, when one straight line meets another so as to form equal adjacent angles, each of the angles is called a right angle, and the lines are said to be perpendicular to each other.

24. It is also evident that the sum of the angles formed by any one position of the line BD is equal to the sum of the angles formed by any other position; for what is taken from one angle by the revolution of the line BD is added to the other.

25. Hence, when one straight line meets another so as to form two angles, the sum of these two angles equals two right angles.

26. An angle that is less than a right angle is called an acute angle.

27. An angle that is greater than one right angle and less than two is called an obtuse angle.

28. Both acute and obtuse angles are designated as oblique angles as contrasted with right angles.

29. A straight angle is a term recently adopted by prominent English and German mathematicians to express, by a single unit, the sum of two right angles.

30. When the sum of two angles is equal to a straight angle, they are said to be supplementary; i.e. each is the supplement of the other.

31. When the sum of two angles is equal to one right angle, they are said to be complementary; i.e. each is the complement of the other.