Fig. 18.

Cycloid, a curve much used in mechanics. It is thus formed: If the circumference of a circle be rolled on a right line, beginning at any point A, and continued till the same point A arrives at the line again, making just one revolution, and thereby measuring out a straight line A B A equal to the circumference of a circle, while the point A in the circumference traces out a curve line A C A G A: then this curve is called a cycloid; and some of its properties are contained in the following lemma.

If the generating or revolving circle be placed in the middle of the cycloid, its diameter coinciding with the axis A B, and from any point there be drawn the tangent C F, the ordinate C D E perpendicular to the axis, and the chord of the circle A D; then the chief properties are these:

The right line C D equal to the circular arc A D;

The cycloidal arc A C equal to double the chord A D;

The semi-cycloid AC A equal to double the diameter A B, and The tangent C F is parallel to the chord A D. This curve is the line of swiftest descent, and that best suited for the path of the ball of a pendulum.