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. C D The word curve is frequently used to designate a curved line. 7. A BROKEN LINE is one which is composed of straight lines, not lying in the same direction; as the line EF. E F 8. A MIXED LINE is one which is composed of straight lines and of curved lines. 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. 12. A SOLID, or VOLUME, is that which has length, breadth, and thickness. 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 A D B 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 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. B C A D Two adjacent angles, as CAB and BAD, have a common vertex, as A; and a common side, as A B. 15. A RIGHT ANGLE is one which is formed by a straight line and a perpendicular to it; as the angle CAB. 16. An ACUTE ANGLE is one which is less than a right angle; as the angle DEF. C B F E 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. 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 C A -B C -D E G |