nous. The qualification "If the arc
of swing is small" is introduced because, as was discovered by
Christiaan Huygens, the arc of vibration of a truly isochronous pendulum
should not be a circle with centre O, but a cycloid DM, generated by
the rolling of a circle with diameter DQ = 1/2OD, upon a straight line QM.
However, for a short distance near the bottom, the circle so nearly
coincides with the cycloid that a pendulum swinging in the usual
circular path is, for small arcs, isochronous for practical purposes.

[Illustration: FIG. 6.]

The formula representing the time of oscillation of a pendulum, in a
circular arc, is thus found:--Let OB (fig. 6) be the pendulum, B be
the position from which the bob is let go, and P be its position at
some period during its swing. Put FC = h, and MC = x, and OB = l. Now
when a body is allowed to move under the force of gravity in any path
from a height h, the velocity it attains is the same as a body would
attain falling freely vertically through the distance h. Whence if v
be the velocity of the bob at P, v = sqrt(2gFM) = sqrt(2g(h - x)). Let
Pp = ds, and the vertical distance of p below P = dx, then Pp =
velocity at P X dt; that is, dt = ds/v.

ds l l
Also -- = -- = ---------------,
dx MP sqrt(x(2l - x))

ds ldx 1
whence dt = -- = --------------- . ---------------
v sqrt(x(2l - x)) sqrt(2g(h - x))

1 / l dx 1
= --- / --- . -------------- . ----------------
2 \/ g sqrt(x(p - x)) sqrt(1 - (x/2l))

Expanding the second part we have

1 / l dx / x \
dt = --- / --- . -------------- . ( 1 + --- + ... ).
2 \/ g sqrt(x(h - x)) \ 4l /

If this is integrated between the limits of 0 and h, we have

/ l / h \
t = [pi] / --- . ( 1 + --- + ... ),
\/ g \ 8l /

where t is the time of swing from B to A. The terms after the second
may be neglected. The first term, [pi] sqrt(l/g), is the time of swing
in a cycloid. The second part represents the addition necessary if the
swing is circular and not cycloidal, and therefore expresses the
"circular error." Now h = BC^2/l = 2[pi]^2[theta]^2l / 360^2, where
[theta] is half the angle