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

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, ɔr 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

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

F

B

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 theo

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 propositions.

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.

PLANE GEOMETRY.

BOOK I.

RECTILINEAR FIGURES.

THE STRAIGHT LINE.

1. AXIOM. There can be but one straight line between the same two points.

2. Postulate. A straight line can be drawn between any two points; and any straight line can be produced (i. e., prolonged) indefinitely. 3. Axiom. If two indefinite straight lines coincide in two points, they coincide throughout their whole extent, and form but one line. Hence two points determine a straight line; and a straight line may be designated by any two of its points.

C

4. Different straight lines drawn from the same point are said to have different directions; as OA, OD, etc. The point from which they are drawn, or at which they commence, is often called the origin.

If any one of the lines, as OA, be produced through O, the portions OA, OB, on opposite

B

sides of O, may be regarded as two different lines having opposite directions reckoned from the common origin 0.

Hence, also, every straight line AB has two opposite directions, namely, from A toward B (A being regarded as

its origin) expressed by AB, and from B toward

A

B

A (B being regarded as its origin) expressed by BA. If a line AB is to be produced through B, that is, toward

C, we should express this by saying that
AB is to be produced; but if it is to be

D Α

C

produced through A, that is toward D, we should express this by saying that BA is to be produced.

ANGLES.

5. Definition. An angle is a figure formed by two straight lines drawn from the same point; thus OA, OB form an angle at O. The lines OA, OB are called the sides of the angle; the common point O, its vertex.

B

An isolated angle may be designated by the letter at its vertex, as "the angle 0;" but when several angles are formed at the same point by different lines, as OA, OB, OC, we designate the angle intended by three letters; namely, by one letter on each of its sides, together with the one at its vertex, which must be written between the other two. Thus, with these lines there are formed three different angles, which are distinguished as AOB, BOC and AOC. Two angles, such as AOB, BOC, which have the same vertex O and a common side OB between them, are called adjacent.

6. Definition. Two angles are equal when one can be placed upon the other so that they shall coincide. Thus, the angles AOB and A' O'B' are equal, if A' O'B' can be superposed upon AOB so that while O'A' coincides with OA, O'B' shall also coincide with OB. The equality of the two angles is not affected by producing the sides; for the coincident sides con

ла

tinue to coincide when produced indefinitely (3).* Thus the magnitude of an angle is independent of the length of its sides.

B

7. A clear notion of the magnitude of an angle will be obtained by supposing that one of its sides, as OB, was at first coincident with the other side OA, and that it has revolved about the point O (turning upon O as the leg of a pair of dividers turns upon its hinge) until it has arrived at the position OB. During this revolution the movable side makes with the fixed side a varying angle, which increases by insensible degrees, that is, continuously; and the revolving line is

* An Arabic numeral alone refers to an article in the same Book; but in referring to articles in another Book, the number of the Book is also given.