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BOOK VIII.

THE SPHERE.

Circles of the Sphere and Tangent Planes...

Spherical Triangles and Polygons....

Relation of a Spherical Polygon to a Polyedral Angle

Symmetrical Spherical Triangles.....

Polar Triangles.....

Relative Areas of Spherical Figures

Exercises. Theorems..

Numerical Exercises..

Problems..

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BOOK IX.

THE THREE ROUND BODIES.

The Cylinder.

The Cone...

The Sphere...

Exercises. Theorems..

Numerical Exercises .

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· 388

ELEMENTARY GEOMETRY.

INTRODUCTION.

DEFINITIONS. 1. SPACE is indefinite extension in every direction. All material bodies occupy limited portions of space, and have length, breadth, thickness, form, and position. The material body occupying any portion of space is called a physical solid. The part of space which is or may be occupied by a material body is called a geometric solid. A physical solid is therefore a real body, while a geometric solid is only the form of a physical solid, and is the one treated of in Geometry. The term solid will be used for brevity to denote a geometric solid.

2. A solid is a limited portion of space, and has length, breadth, and thickness. Length, breadth, and thickness are called the three dimensions of the solid.

3. A surface is the limit or boundary of a solid, and has only two dimensions, length and breadth.

A surface has no thickness, for if it had any, however small, it would form part of the solid, and would be space of three dimensions.

4. A line is the limit or boundary of a surface, and has only one dimension, namely, length.

A line has no breadth, for if it had any, however small, it would form part of the surface, and would be space of two dimensions; and if in addition it had any thickness, it would be space of three dimensions; hence a line has neither breadth nor thickness.

5. A point is the limit or extremity of a line, and has position, but neither length, breadth, nor thickness.

A point has no length, for if it had any, however small, it would form part of the line of which it is the extremity; and it can have neither breadth nor thickness because the line bas none.

6. If we suppose a solid to be divided into two parts which touch each other, the division between the two parts is a surface. This surface can have no thickness, for if it had a thickness, however small, it would be a part either of the one solid or the other, and would therefore be a solid and not a surface.

Again, if we suppose a surface cut into two parts which touch each other, the division between the two parts is a line. This line can have no thickness, because the surface has none, and it can have no breadth, for it forms no part of either surface.

If we suppose a line cut into two parts which touch each other, the division between the two parts is a point. This point can have neither breadth nor thickness, because the line has none, and it can have no length, for it forms no part of either line.

Euclid regarded a point merely as a mark of position, and he attached to it no idea of size and shape.

Similarly, he considered that the properties of a line arise only from its length and position, without reference to that minute breadth which every line must really have if actually drawn, even though the most perfect instruments are used.

We cannot make the points, lines, and surfaces of Geometry. A dot, made on paper or on the blackboard, will have length, breadth, and thickness, and hence will not be a real point. Yet the dot may be taken as an imperfect representation of the real point. So also a line, drawn on paper or on the blackboard, will have breadth and thickness, and hence will not be a real line. Yet the line which we draw may be taken to represent the real line.

7. We have considered a surface as the boundary of a solid, a line as the boundary of a surface, and a point as the limit of a line. On the other hand, inversely, we may re

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