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ELEMENTS OF GEOMETRY.

INTRODUCTION.

1. EVERY person possesses a conception of space indefinitely extended in all directions. Material bodies occupy finite, or limited, portions of space. The portion of space which a body occupies can be conceived as abstracted from the matter of which the body is composed, and is called a geometrical solid. The material body filling the space is called a physical solid. A geometrical solid is, therefore, merely the form, or figure, of a physical solid. In this work, since only geometrical solids will be considered, we shall, for brevity, call them simply solids.

2. Definitions. In geometry, then, a solid is a limited, or bounded, portion of space.

The limits, or boundaries, of a solid are surfaces.

The limits, or boundaries, of a surface are lines.

The limits of a line are points.

3. A solid has extension in all directions; but for the purpose of measuring its magnitude, it is considered as having three specific dimensions, called length, breadth and thickness.

A surface has only two dimensions, length and breadth.

A line has only one dimension, namely, length. The intersection of two surfaces is a line.

A point has no extension, and therefore neither length, breadth nor thickness. The intersection of two lines is a point.

4. Although our first notion of a surface, as expressed in the definition above given, is that of the boundary of a solid, we can suppose

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such boundary to be abstracted and considered separately from the solid. Moreover, we may suppose a surface of indefinite extent as to length and breadth; such a surface has no limits.

Similarly, a line may be considered, not only as the limit of a surface, but as abstracted from the surface and existing separately in space. Moreover, we may suppose a line of indefinite length, or

without limits.

Finally, a point may be considered, not merely as a limit of a line, but abstractly as having only position in space.

5. Definitions. A straight line is the shortest line between two points; as AB.

A

B

Since our first conception of a straight line may be regarded as derived from a comparison of all the lines that can be imagined to exist between two points, i. e., of lines of limited length, this definition (which is the most common one) may be admitted as expressing such a first conception; but since we can suppose straight lines of indefinite extent, a more general definition is the following:

A straight line is a line of which every portion is the shortest line between the points limiting that portion.

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

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

D

B

F

E

B

If a point moves along a line, it is said to describe the line. 6. Definitions. A plane surface, or simply a

plane, is a surface in which, if any two points.

are taken, the straight line joining these

points lies wholly in the surface.

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

7. Solids are classified according to the nature of the surfaces which limit them. The most simple are bounded by planes.

8. Definitions. A geometrical figure is any combination of points, lines, surfaces, or solids, formed under given conditions. Figures formed by points and lines in a plane are called plane figures. Those formed by straight lines alone are called rectilinear, or right-lined, figures; a straight line being often called a right line.

9. Definitions. Geometry may be defined as the science of extension and position. More specifically, it is the science which treats of the construction of figures under given conditions, of their measurement, and of their properties.

Plane geometry treats of plane figures.

The consideration of all other figures belongs to the geometry of space, also called the geometry of three dimensions.

10. Some terms of frequent use in geometry are here defined.

A theorem is a truth requiring demonstration. A lemma is an auxiliary theorem employed in the demonstration of another theorem. A problem is a question proposed for solution. An axiom is a truth assumed as self-evident. A postulate (in geometry) assumes the possibility of the solution of some problem.

Theorems, problems, axioms and postulates are all called propo

sitions.

A corollary is an immediate consequence deduced from one or more propositions. A scholium is a remark upon one or more propositions, pointing out their use, their connection, their limitation, or their extension. An hypothesis is a supposition, made either in the enunciation of a proposition, or in the course of a demonstration.

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