of swing expressed in degrees; hence h/(8l)
= [theta]^2/52520, and the formula becomes

/ l / [theta]^2 \
t = [pi] / --- ( 1 + -------- ).
\/ g \ 52520 /

Hence the ratio of the time of swing of an ordinary pendulum of any
length, with a semiarc of swing = [theta] degrees is to the time of
swing of a corresponding cycloidal pendulum as 1 + [theta]^2/52520 : 1.
Also the difference of time of swing caused by a small increase
[theta]' in the semiarc of swing = 2[theta][theta]' / 52520 second per
second, or 3.3[theta][theta]' seconds per day. Hence in the case of a
seconds pendulum whose semiarc of swing is 2 deg. an increase of .1 deg. in
this semiarc of 2 deg. would cause the clock to lose 3.3 X 2 X 0.1 = .66
second a day.

Huygens proposed to apply his discovery to clocks, and since the
evolute of a cycloid is an equal cycloid, he suggested the use of a
flexible pendulum swinging between cycloidal cheeks. But this was only
an example of theory pushed too far, because the friction on the
cycloidal cheeks involves more error than they correct, and other
disturbances of a higher degree of importance are left uncorrected. In
fact the application of pendulums to clocks, though governed in the
abstract by theory, has to be modified by experiment.

Neglecting the circular error, if L be the length of a pendulum and g
the acceleration of gravity at the place where the pendulum is, then
T, the time of a single vibration = [pi] sqrt(L/g). From this formula
it follows that the times of vibration of pendulums are directly
proportional to the square root of their lengths, and inversely
proportional to the square root of the acceleration of gravity at the
place where the pendulum is swinging. The value of g for London is
32.2 ft. per second per second, whence it results that the length of a
pendulum for London to beat seconds of mean solar time = 39.14 in.
nearly, the length of an astronomical pendulum to beat seconds of
sidereal time being 38.87 in.

This length is calculated on the supposition that the arc of swing is
cycloidal and that the whole mass of the pendulum is concentrated at a
point whose distance, called the radius of oscillation, from the point
of suspension of the pendulum is 39.14 in. From this it might be
imagined that if a sphere, say of iron, were suspended from a light
rod, so that its cen