Let O A, _fig. 77._, be a horizontal line, and let O B be a circle placed below this line, and in contact with it. If this circle be rolled upon the line from O towards A, a point upon its circumference, which at the beginning of the motion is placed at O, will during the motion trace the curve O C A. This curve is called a _cycloid_. If the circle be supposed to roll in the opposite direction towards A′, the same point will trace another cycloid O C′ A′. The points C and C′ being the lowest points of the curves, if the perpendiculars C D and C′ D′ be drawn, they will respectively be equal to the diameter of the circle. By a known property of this curve, the arcs O C and O C′ are equal to twice the diameter of the circle. From the point O suppose a flexible thread to be suspended, whose length is twice the diameter of the circle, and which sustains a pendulous body P at its extremity. If the curves O C and O C′, from the plane of the paper, be raised so as to form surfaces to which the thread may be applied, the extremity P will extend to the points C and C′, when the entire thread has been applied to either of the curves. As the thread is deflected on either side of its vertical position, it is applied to a greater or lesser portion of either curve, according to the quantity of its deflection from the vertical. If it be deflected on each side until the point P reaches the points C and C′, the extremity would trace a cycloid C P C′ precisely equal and similar to those already mentioned. Availing himself of this property of the curve, Huygens constructed his cycloidal pendulum. The time of vibration was subject to no variation, however the arc of vibration might change, provided only that the length of the string O P continued the same. If small arcs of the cycloid be taken on either side of the point P, they will not sensibly differ from arcs of a circle described with the centre O and the radius O P; for, in slight deflections from the vertical position, the effect of the curves O C and O C′ on the thread O P is altogether inconsiderable. It is for this reason that when the arcs of vibration of a circular pendulum are small, they partake of the property of isochronism peculiar to those of a cycloid. But when the deflection of P from the vertical is great, the effect of the curves O C and O C′ on the thread produces a considerable deviation of the point P from the arc of the circle whose centre is O and whose radius is O P, and consequently the property of isochronism will no longer be observed in the circular pendulum. CHAP. XII. OF SIMPLE MACHINES.