GEOMETRY.

DEFINITIONS.

1. If a block of wood or stone be cut in the shape represented in Fig. 1, it will have six flat faces.

Each face of the block is called a surface; and if these faces are made D smooth by polishing, so that, when a straight-edge is applied to any one of them, the straight edge in every part will touch the surface, the faces are called plane surfaces, or planes.

A

FIG. 1. B

2. The edge in which any two of these surfaces meet is called a line.

3. The corner at which any three of these lines meet is called a point.

4. For computing its volume, the block is measured in three principal directions:

From left to right, A to B.

From front to back, A to C.

From bottom to top, A to D.

These three measurements are called the dimensions of the block, and are named length, breadth (or width), thickness (height or depth).

A solid, therefore, has three dimensions, length, breadth, and thickness.

5. The surface of a solid is no part of the solid. It is simply the boundary or limit of the solid. A surface, therefore, has only two dimensions, length and breadth. So that, if any number of flat surfaces be put together, they will coincide and form one surface.

6. A line is no part of a surface. It is simply a boundary or limit of the surface. A line, therefore, has only one dimension, length. So that, if any number of straight lines be put together, they will coincide and form one line.

7. A point is no part of a line. It is simply the limit of the line. A point, therefore, has no dimension, but denotes position simply. So that, if any number of points be put together, they will coincide and form a single point.

8. A solid, in common language, is a limited portion of space filled with matter; but in Geometry we have nothing to do with the matter of which a body is composed; we study simply its shape and size; that is, we regard a solid as a limited portion of space which may be occupied by a physical body, or marked out in some other way. Hence,

A geometrical solid is a limited portion of space.

9. It must be distinctly understood at the outset that the points, lines, surfaces, and solids of Geometry are purely ideal, though they can be represented to the eye in only a material way. Lines, for example, drawn on paper or on the blackboard, will have some width and some thickness, and will so far fail of being true lines; yet, when they are used to help the mind in reasoning, it is assumed that they represent perfect lines, without breadth and without thickness.

C

D

10. A point is represented to the eye by a fine dot, and named by a letter, as A (Fig. 2); a line is named by two letters, placed one at each end, as BF; a surface is represented and named by the lines which bound it, as BCDF; a solid is represented by the faces which bound it.

A

B

F

FIG. 2.

11. By supposing a solid to diminish gradually until it vanishes we may consider the vanishing point, a point in space, independent of a line, having position but no extent.

12. If a point moves continuously in space, its path is a line. This line may be supposed to be of unlimited extent, and may be considered independent of the idea of a surface.

13. A surface may be conceived as generated by a line moving in space, and as of unlimited extent. A surface can then be considered independent of the idea of a solid.

14. A solid may be conceived as generated by a surface in motion.

D

H

E

C

G

B

F

FIG. 3.

Thus, in the diagram, let the upright surface ABCD move to the right to the position EFGH. The points A, B, C, and D will generate the lines AE, BF, CG, and DH, respectively. The lines AB, BC, CD, and AD will generate the surfaces AF, BG, CH, and AH, respectively. The surface ABCD will generate the solid AG.

15. Geometry is the science which treats of position, form, and magnitude.

16. Points, lines, surfaces, and solids, with their relations, constitute the subject-matter of Geometry.

17. A straight line, or right line, is a line which has the same direction throughout its A

whole extent, as the line AB.

B

20. A mixed line is a line composed of straight and curved lines, as the line GH.

A straight line is often called simply a line, and a curved line, a curve.

21. A plane surface, or a plane, is a surface in which, if any two points be taken, the straight line joining these points will lie wholly in the surface.

22. A curved surface is a surface no part of which is plane.

23. Figure or form depends upon the relative position of points. Thus, the figure or form of a line (straight or curved) depends upon the relative position of the points in that line; the figure or form of a surface depends upon the relative position of the points in that surface.

24. With reference to form or shape, lines, surfaces, and solids are called figures.

With reference to extent, lines, surfaces, and solids are called magnitudes.

25. A plane figure is a figure all points of which are in the same plane.

26. Plane figures formed by straight lines are called rectilinear figures; those formed by curved lines are called curvilinear figures; and those formed by straight and curved lines are called mixtilinear figures,

27. Figures which have the same shape are called similar figures. Figures which have the same size are called equivalent figures. Figures which have the same shape and size are called equal figures.

28. Geometry is divided in two parts, Plane Geometry and Solid Geometry. Plane Geometry treats of figures all points of which are in the same plane. Solid Geometry treats of figures all points of which are not in the same plane.

STRAIGHT LINES.

29. Through a point an indefinite number of straight lines be drawn. These lines will have different directions.

may

30. If the direction of a straight line and a point in the line are known, the position of the line is known; in other words, a straight line is determined if its direction and one of its points are known. Hence,

All straight lines which pass through the same point in the same direction coincide, and form but one line.

31. Between two points one, and only one, straight line can be drawn; in other words, a straight line is determined if two of its points are known. Hence,

Two straight lines which have two points common coincide throughout their whole extent, and form but one line.

32. Two straight lines can intersect (cut each other) in only one point; for if they had two points common, they would coincide and not intersect.

33. Of all lines joining two points the shortest is the straight line, and the length of the straight line is called the distance between the two points.