This section is from the book "The Tinman's Manual And Builder's And Mechanic's Handbook", by Isaac Ridler Butt. Also available from Amazon: The Tinman's Manual And Builder's And Mechanic's Handbook.
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 arrive at the line again, making just one revolution, and thereby measuring out a (straight line ABA equal to the circumference of a circle, while the point A in the circumference traces out a curve line ACAGA: then this curve is called a cycloid; and some of its properties are contained in the following lemma.
If the generating or revolving circle he placed in the middle of the cycloid, its diameter coinciding with the axis AB, and from any point there he drawn the tangent CF, the ordinate CDE perpendicular to the axis, and the chord of the circle AD; then the chief properties are these:
The right line CD equal to the circular arc AD;
The cycloidal arc AC equal to double the chord AD;
The semi-cycloid ACA equal to double the diameter AB, and
The tangent CF is parallel to the chord AD. This curve is the line of swiftest descent, and that best suited for the path of the ball of a pendulum.