ELEMENTS OF GEOMETRY.

GENERAL PRINCIPLES.

DEFINITIONS.

1. Space is unlimited extension in all directions. A limited portion of space is called a geometrical solid. A physical solid is the material occupying the space. Geometry treats only of the form and magnitude of the solid. The term solid, in this work, will signify a geometrical solid. 2. A solid has extension in all directions from any point within; but it is sufficient to consider three dimensions, length, breadth, and thickness.

3. The limits of a solid are called surfaces. They have only two dimensions, length and breadth; and are, therefore, no part of the solid.

4. The limits of a surface are called lines. They have only one dimension, length; and are no part of the surface which they bound.

5. The entire limit of a surface is called its perimeter. 6. The limits of a line are called points. They are no part of the line, and have neither length, breadth, nor thick

ness.

7. We may consider a point as merely a position in space; a line, as the path of a moving point; a surface, as generated by a moving line; and a solid, as generated by a moving surface.

8. The direction in which the point is supposed to have moved in generating a line may be considered positive; the opposite direction, negative. This may be indi

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cated by the order of the letters; as, AB = — BA. 9. A straight line is one that does not change its direction at any point. It is sometimes called simply a line.

10. A curved line is one that changes its direction at every point.

11. A broken line is a line composed of straight lines lying in diferent directions.

12. A surface is called a plane when the straight line joining any two of its points lies wholly in the surface.

13. A curved surface is a surface which is neither a plane, nor composed of planes; as the surface of a ball, pipe, &c.

14. A geometrical figure is a combination of points, lines, surfaces, or solids, formed under certain conditions. A plane figure is a combination of points and lines which are confined to one and the same plane.

15. A rectilinear figure is one formed by straight lines. 16. Geometry is that part of mathematics which treats of the properties, measurement, and construction of figures. 17. Plane geometry treats of plane figures.

18. Solid geometry treats of figures which represent points, lines, surfaces, and solids that are not confined to the same plane.

TERMS.

1. An axiom is a self-evident truth.

2. A theorem is a truth requiring demonstration. 3. A problem is a question requiring a solution.

4. A postulate is a problem the possibility of whose solution is self-evident.

5. A lemma is an auxiliary theorem.

6. Axioms, theorems, problems, and postulates are called propositions.

7. A corollary is an obvious consequence of one or more propositions or other premises.

8. A scholium is a remark made upon one or more propositions, showing their connection, use, extension, or limitation.

9. An hypothesis is a supposition made either in the statement of a proposition or in the course of a demonstration.

AXIOMS.

1. Things which are equal to the same thing are equal to each other.

2. If equals be added to equals, the sums will be equal. 3. If equals be subtracted from equals, the remainders will be equal.

4. If equals be added to unequals, the sums will be unequal. 5. If equals be subtracted from unequals, the remainders will be unequal.

6. If equals be multiplied by equals, the products will be equal.

7. If equals be divided by equals, the quotients will be equal.

8. The whole is greater than any of its parts.

9. The whole is equal to the sum of all its parts.

Illustrate these axioms with the equations a=b, c=d, &c.

TABLE OF SYMBOLS USED IN THIS WORK.

is the sign of addition; thus, ab indicates that b is to be added to a, and is read a plus b.

is the sign of subtraction; thus, a b indicates that b is to be subtracted from a, and is read a minus b. X is the sign of multiplication.

a

is the sign of division; also indicates that a is to be divided by b.

b

a2, a3, . . . indicates that a is to be raised to the second,

third, ✔a,

a, . . . indicates that the square root, cube root, .. of a is to be taken.

is the sign of equality; thus, a=b is read a is equal to b.

> or < is the sign of inequality; thus, a > b is read, a is greater than b, and a ≤ b is read, a is less than b.

is used for the word angle, p. 6.

R is used for the word right-angle, p. 6. ▲ is used for the word triangle, p. 15. designates perpendicularity, p. 6.

|| designates parallelism, p. 9.

designates congruity, p. 14.

~designates similarity, p. 83.

In the references, Roman numerals refer to books, and Arabic numerals to articles; thus (I., 38) refers to Book I., Article XXXVIII. (18) refers to Article XVIII. of the same book.

PLANE GEOMETRY.

BOOK I.

LINES AND ANGLES.

I. POSTULATES.

1. A straight line may be drawn between any two points. 2. A straight line may be prolonged to any length.

3. From the longer of two straight lines a part may be cut off equal to the less.

4. A straight line may be bisected; that is, divided into two equal parts.

II. - AXIOMS.

1. A straight line is the shortest distance between two points.

2. But one straight line can be drawn between two points. 3. Two straight lines that have two points common coincide throughout their whole extent, and form one and the same straight line.

4. Two straight lines can cut each other in but one point.

III.

DEF. 1. An angle is the difference of direction of two straight lines drawn from the same

point; as, AOB. The common

point, O, is called the vertex; the lines OA and OB, the sides of the angle.

B

A