1. A material body, such as a block of wood, occupies a limited or bounded portion of space.

The boundary which separates such a body from surrounding space is called the surface of the body.

2. If the material composing such a body could be conceived as taken away from it, without altering the form or shape of the bounding surface, there would remain a portion of space having the same bounding surface as the former material body; this portion of space is called a geometrical solid, or simply a solid.

The surface which bounds it is called a geometrical surface, or simply a surface; it is also called the surface of the

solid.

3. If two geometrical surfaces intersect each other, that which is common to both is called a geometrical line, or simply a line.

Thus, if surfaces AB and CD cut each other, their common intersection, EF, is a line.

1

4. If two geometrical lines intersect A each other, that which is common to both is called a geometrical point, or simply a point.

B

Thus, if lines AB and CD cut each other, their common intersection, O, is a point.

5. A solid has extension in every direction; but this is not true of surfaces and lines.

A point has extension in no direction, but simply position in space.

6. A surface may be conceived as existing independently in space, without reference to the solid whose boundary it forms.

In like manner, we may conceive of lines and points as having an independent existence in space.

7. A straight line, or right line, is a line which has the same direction throughout its length; as AB.

A curved line, or curve, is a line no portion of which is straight; as CD.

A broken line is a line which is composed of different successive straight lines; as EFGH.

8. The word "line" will be used hereafter as signifying a straight line.

9. A plane surface, or plane, is a surface such that the straight line joining any two of its points

lies entirely in the surface.

Thus, if P and Q are any two points in

surface MN, and the straight line joining

M

P

N

P and Q lies entirely in the surface, then MN is a plane. 10. A curved surface is a surface no portion of which is plane.

11. We may conceive of a straight line as being of unlimited extent in regard to length; and in like manner we may conceive of a plane as being of unlimited extent in regard to length and breadth.

12. A geometrical figure is any combination of points, lines, surfaces, or solids.

A plane figure is a figure formed by points and lines all lying in the same plane.

A geometrical figure is called rectilinear, or right-lined, when it is composed of straight lines only.

13. Geometry treats of the properties, construction, and measurement of geometrical figures.

14. Plane Geometry treats of plane figures only.

Solid Geometry, also called Geometry of Space, or Geometry of Three Dimensions, treats of figures which are not plane. 15. An Axiom is a truth which is assumed without proof as being self-evident.

A Theorem is a truth which requires demonstration.
A Problem is a question proposed for solution.

A Proposition is a general term for a theorem or problem. A Postulate assumes the possibility of solving a certain problem.

A Corollary is a secondary theorem, which is an immediate consequence of the proposition which it follows.

A Scholium is a remark or note.

An Hypothesis is a supposition made either in the statement or the demonstration of a proposition.

16. Postulates.

1. We assume that a straight line can be drawn between any two points.

2. We assume that a straight line can be produced (i.e., prolonged) indefinitely in either direction.

17. Axioms.

We assume the truth of the following:

1. Things which are equal to the same thing, or to equals, are equal to each other.

2. If the same operation be performed upon equals, the results will be equal.

3. But one straight line can be drawn between two points. 4. A straight line is the shortest line between two points.

5. The whole is equal to the sum of all its parts.

6. The whole is greater than any of its parts.

18. Since but one straight line can be drawn between two points, a straight line is said to be determined by any two of its points.

19. Symbols and Abbreviations.

The following symbols will be used in the work :

PLANE GEOMETRY.

BOOK I.

RECTILINEAR FIGURES.

DEFINITIONS AND GENERAL PRINCIPLES.

20. An angle (4) is the amount of divergence of two straight lines which are drawn from the same point in different directions. The point is called the vertex of the angle, and the straight lines are called its sides.

A

-B

21. If there is but one angle at a given vertex, it may be designated by the letter at that vertex; but if two or more angles have the same vertex, we avoid ambiguity by naming also a letter on each side, placing the letter at the vertex between the others.

Thus, we should call the angle of § 20 "angle 0"; but if there were other angles having the same vertex, we should read it either AOB or BOA.

Another way of designating an angle is by means of a letter placed between its sides; examples of this will be found in § 71.

22. Two geometrical figures are said to be equal when one can be applied to the other so that they shall coincide throughout.

To prove two angles equal, we do not consider the lengths of their sides.

5