Mechanics.

I 2 I

Vibration in the Cycloid is to the Time of Descent thro'half its Length, as the Circumference of a Circle to itsDiameter, or as 3,14159 to 1.

10. If A C be the Length of a Pendulum vibratingSeconds, it may be easily found, since it is known bvmost accurate Experiments, that a Body descends freelyby Gravity thro’ 193s Inches in the first Second ofTime ; therefore by making this Analogy, as 3, 14159*: T : : 193! : 19,6 = A K, and so A 0 = 39,2 Inches,the Length of a second Pendulum, as required. And sincethe Lengths are as the Squares of the Times of Vibra-tions, we have 4.1: : 39,2 : 9,8 Inches, the Lengthof an half-fecond Pendulum.

11. Since the Length of a Pendulum A C, for vi-brating Seconds, is to considerable, the Bob, withoutthe Cycloidal Cheeks A L and A D, will indeed de-scribe the Arch of a Circle, as f C e, when it vibrates,but then this circular Arch will, for some Distance oneach Side the lowest Point C, coincide with the Cy-cloid very nearly; as for Example, to g and h-, andhence it follows that a common Pendulum, vibratingthrough very small Arches g h will perform all its Vi-brations in equal Times. And hence the above-men-tioned Cycloidal Checks came into Disuse,

12- Hence the Time of Descent through the ChordC g is to the Time of Descent through the Arch of the

Circle or Cycloid belonging to it as 1/4 a (supposingi p

A C =: 2 a) to 2 ——, or as 2 « to f P, that is, as a to* a

3A4£59

0,7854, or as 1 to 0,7854, which is

the Proportion of a Square to its inscribed Circle.

13. In all that has been hitherto said, the Power ofGravity has been supposed constantly the fame. But ifthe said Power varies, the Lengths of Pendulums mustvary in the fame Proportion, in order that they rnayvibrate in equal Times; for we, have shewn that theRatio of the Times of Vibration and Descent throughhalf the Lengths is given, and consequently the Timesof Vibration and Descent through the whole Length isgiven ; But the Times of Vibration are supposed equal,L there-