Mechanics.

118Fig. 4.

PlateVI,Fig. 4.

Second Pendulum 9,8 Inches. (7.) At;uniform homogeneous Body BG, as a Rod,Staff', &c. which is one third Part longerthan a Pendulum AD, will vibrate in thefame Time with it (XXVIII), (8.) This

Centre

(XXVIII) x. As the Doctrine of Pendulums andTime-keeping Instruments depends in a great Measureupon the Cycloid, I think it necessary here to shew theNature and Use of that Curve, with regard thereto. Ifa Circle ABC, insisting on a right Line AL, begin torevolve in the Manner of a Wheel, from A towards L,the Point A will by its twofold Motion describe theCurve A C DIL, while the Circle makes one Revolu-tion from A to L,

2. This Curve is called the Cycloid, and from theDefinition his evident (1.) That the Base of the Cy-cloid AL is equal to the Periphery of the generatingCircle ABC. (2.) The Axis of the Cycloid FD isequal to the Diameter of the said Circle. (3.) Thatthe Part of the Base KL is equal to the Arch of theCircle IK. (4.) Therefore KF (—ME—IG) is equajto the remaining Arch IH, or GD. (5.) That theChord of the Circle KI is perpendicular to the Cycloidin the Point I; and (6.) Therefore the Chord HI(being at right Angles with IK) is a Tangent to theCurve in the Point I. (7.) The said Tangent HI isparallel to the Chord DG.

3. Parallel to El draw el infinitely near, and Inperpendicular thereto ; then will the Triangles DGE,DGF, I ni, be similar, and so we have DE : DG : : DG:DF : : I« : 1 /; that is, (putting DF — a, and DE = x,

DI=z,) x : 4/a .v : : ^/a x : a :: x : i — 1 / _ - —

Vtf x

Fluxion of the Arch DI, whose Fluent is 2 4/ax =2DG — the Arch DI; and consequently, the Semicy-cloid DIL = 2DF, the Diameter of the generatingCircle,

4. Let