ELEMENTS OF GEOMETRY.

BOOK I.

ELEMENTARY PRINCIPLES.

DEFINITIONS.

1. GEOMETRY is the science of Position and Extension. The elements of position are direction and distance. The dimensions of extension are length, breadth, and height or thickness.

2. MAGNITUDE, in general, is that which has one or more of the three dimensions of extension.

3. A POINT is that which has position, without magnitude.

4. A LINE is that which has length, without either breadth or thickness.

5. A STRAIGHT LINE, or RIGHT LINE, is one which has the same direction in its whole extent; as the line A B.

A

-B

The word line is frequently used alone, to designate a straight line.

6. A CURVED LINE is one which

continually changes its direction; as the line CD.

The word curve is frequently used to designate a curved line.

D

7. A BROKEN LINE is one which is composed of straight lines, not lying in the same direction; as the line EF.

8. A MIXED LINE is one which is composed of straight lines and of curved lines.

E

9. A SURFACE is that which has length and breadth, without height or thickness.

10. A PLANE SURFACE, or simply a PLANE, is one in which any two points being taken, the straight line that joins them will lie wholly in the surface.

11. A CURVED SURFACE is one that is not a plane surface, nor made up of plane surfaces.

ANGLES AND LINES.

F

12. A SOLID, or VOLUME, is that which has length, breadth, and thickness.

13. A PLANE ANGLE, or simply an ANGLE, is the difference in the direction of two lines, which meet at a point; as the angle A.

A

C

A

B

The point of meeting, A, is the vertex of the angle, and the lines A B, AC are the sides of the angle. An angle may be designated, not only by the letter at its vertex, as C, but by three letters, particularly when two or more angles have the same vertex; as the angle ACD or DCB, the letter at the vertex always occupying the middle place.

C

D

B

The quantity of an angle does not depend upon the length, but entirely upon the position, of the sides; for the angle remains the same, however the lines containing it be increased or diminished.

14. Two straight lines are said to be perpendicular to each other, when their meeting forms equal adjacent angles; thus the lines A B and CD are perpendicular to each other.

15. A RIGHT ANGLE is one which is formed by a straight line and a perpendicular to it; as the angle CAB.

C

-D

A

Two adjacent angles, as CAB and BAD, have a common vertex, as A; and a common side, as A B.

с

16. An ACUTE ANGLE is one which is less than a right angle; as the angle DEF.

An OBTUSE ANGLE is one which is greater than a right angle; as the angle EFG.

18. When a straight line, as EF, intersects two parallel lines, as AB, CD, the angles formed A by the intersecting or secant line take particular names, thus:

INTERIOR ANGLES ON THE SAME SIDE are those which lie within the parallels, and on the same

E

C

E

A

C

A

E

B

G

F

Acute and obtuse angles have their sides oblique to each other, and are sometimes called oblique angles. 17. PARALLEL LINES are such as, being in the same plane, cannot meet, however far either way both of them may be produced; as the lines A B, CD.

G

H

D

F

B

F

B

-D

B

D

20. When the boundary lines are straight, the space they enclose is called a RECTILINEAL FIGURE, or POLYGON; as the figure ABCDE.

ALTERNATE INTERIOR ANGLES lie within the parallels, and on different sides of the secant line, but are not adjacent to each other; as the angles BGH, GH C, and also AGH, GHD.

ALTERNATE EXTERIOR ANGLES lie without the parallels, and on different sides of the secant line, but not adjacent to each other; as the angles EG B, CHF, and also the angles AGE, DHF.

E

B

OPPOSITE EXTERIOR and INTERIOR ANGLES lie on the same side of the secant line, the one without and the other within the parallels, but not adjacent to each other; as the angles EGB, GHD, and also EGA, GHC, are, respectively, the opposite exterior and interior angles.

The boundary of any figure is called its perimeter.

D

D

PLANE FIGURES.

19. A PLANE FIGURE is a plane terminated on all sides by straight lines or curves.

B

C

A

21. A polygon of three sides is called a TRIANGLE; one of four sides, a QUADRILATERAL; one of five, a PENTAGON; one of six, a HEXAGON; one of seven, a HEPTAGON; one

of eight, an OCTAGON; one of nine, a NONAGON; one of ten, a DECAGON; one of eleven, an UNDECAGON; one of twelve, a DODECAGON; and so on.

22. An EQUILATERAL TRIANGLE is one which has its three sides equal; as the triangle ABC.

An ISOSCELES TRIANGLE is one which has two of its sides equal; as the triangle DEF.

A SCALENE TRIANGLE is one which has no two of its sides equal; as the triangle G H I.

23. A RIGHT-ANGLED TRIANGLE is one which has a right angle; as the triangle J K L.

B

E

H

26. A RECTANGLE is any parallelogram whose angles are right angles; as the parallelogram A B C D.

A

D

D

G

A

C

F

J

K

The side opposite to the right angle is called the hypothenuse; as the side JL.

I

24. An ACUTE-ANGLED TRIANGLE is one which has three acute angles; as the triangles A B C and DEF, Art. 22. An OBTUSE-ANGLED TRIANGLE is one which has an obtuse angle; as the triangle G H I, Art. 22.

Acute-angled and obtuse-angled triangles are also called oblique-angled triangles.

L

25. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel.

C

B