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PLANE GEOMETRY.

CHAPTER I.

RECTILINEAR FIGURES.

Definitions.

1. The block represented in the accompanying figure occupies a limited portion of space. If we imagine the block to be removed, its form or shape

can still be retained in the mind.

is true of any object or body.

[graphic]

This

FIG. I.

The space conceived to be occupied by an object or body as distinguished from the substance of which it is made, is a geometrical solid. The matter or substance of which a body or object is composed is a physical solid. Hence a geometrical solid is the shape or form of a physical solid, or some form or figure conceived by the mind. A geometrical solid is a limited portion of space, and has length, breadth, and thickness.

The term solid will be used hereafter to signify a geometrical solid.

2. When space is divided into distinct portions or geometrical solids, the boundaries of these portions or solids are surfaces. Distinct portions of the bounding surface are faces.

Surface has length and breadth, but no thickness.

3. When a surface is divided into distinct portions, the boundaries of these portions are lines. In the solid, represented in Fig. 1, the edges, or boundaries of the faces, are lines. These lines, being the intersection of faces which have no thickness, can themselves have neither breadth nor thickness.

A line has length, but neither breadth nor thickness.

4. When a line is divided into distinct portions, the limits of these portions are points. In the solid, represented in Fig. 1, the corners, or limits of the edges, are points. These points, being the intersections of lines. which have neither breadth nor thickness, can themselves have neither length, breadth, nor thickness.

A point has position, but neither length, breadth, nor thickness.

5. A surface can be conceived of apart from a solid, a line apart from a surface, and a point apart from a line. If a point is conceived to move, the path in which it moves is a line. Hence a line is the path, or locus, of a moving point.

A line can be thought of as generated by a point in motion; surface can be thought of as generated by a line in motion; a solid, as generated by a surface in motion. 6. A geometrical figure is a combination of points, lines, surfaces, or solids.

Geometrical figures are ideal, that is, they are mental conceptions, but they can be represented to the eye only by material substances. For instance, a line can be represented by a mark made by a pencil or crayon; a solid can be represented by a drawing, by a block of wood, or by some other material of any given shape.

To avoid multiplying words, the material representation of geometrical figures will be generally referred to as standing for the mental conceptions themselves. The pupil will be able to tell by the context whether the word 'figure" refers to a geometrical figure or to the material representation of a figure.

7. A straight line is a line such that any part of it, however placed, lies wholly in any other part if its extremities lie in that part. Let O B, which is any part of A B, be placed upon some other part in any way, except

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that O and B shall lie upon that part, for instance with O at M and B at N. If O B exactly coincides with M N, A B is a straight line. Illustrate by lines represented by wood, paper, or other material.

B

A line is read by naming letters placed at its extremities, as line A B in Fig. 3; or by naming a single letter placed upon it, as line O in Fig. 9.

8. A broken line is a line made up of a succession of different straight lines, as A B C D E, in Fig. 4.

9. A curved line, or a curve, is a line no portion of which is straight,

as C D, in Fig. 5.

FIG. 3.

B

FIG. 4.

FIG. 5.

D

E

10. A plane surface, or a plane, is a surface such that if any two of its points be joined by a straight line the line lies wholly in the plane surface. If a carpenter wishes to determine whether or not the surface of a board

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