PENDULUM. 
These are, however, which concur in ren- 
dering the application of this curve to the 
vibration of pendulums designed for the 
measures of time, the source of errors even 
greater than those which by its peculiar 
property it is intended to obviate, and it is 
now not used. 
Although the times of vibration of a 
pendulum in different arches be nearly 
eq\ial, yet if the arches differ very consi- 
derably, the vibrations will be performed in 
different times, and the difference, though 
very small, will become sensible in the 
course of one day or more. In clocks for 
astronomical purposes, the arc of vibration 
must be accurately ascertained, and ifit be 
different from that described by the pendu- 
lum when the clock keeps time, a correc- 
tion must be applied to the time shown by 
the clock. Tliis correction, expressed in 
seconds of time, will be equal to the half of 
three times the difference of the square of 
the given arc, and of that of the arc de- 
scribed by the pendulum when the clock 
keeps time, these arcs being expressed in 
degrees ; and so much will the clock gain or 
lose according as the first of these arches is 
less or greater than the second. Thus, if a 
clock keeps true time when the pendulum 
vibrates in an arch of 3“, it will lose lOi se- 
conds daily in an arch of 4°, and 24 seconds 
in an arc of 5" for 4^— 3^ X | = r x | = lOi 
and generally x | gives the time 
lost or gained. See Simpson’s Fluxions, 
vol. ii. prob. xxviii. 
In all that has been hitherto said, the 
power of gravity has been supposed con- 
stantly the same. But, if the said power 
varies, the lengths of pendulums must vary 
in the same proportion, in order that they 
may vibrate in equal times ; for we have 
shewn, that the ratio of the times of vibra- 
tion and descent through half the lengths is 
given, and consequently the times of vibra- 
tion and descent through the whole length 
is given •, but the times of vibration are 
supposed equal, therefore the times of 
descent through the lengths of the pendu- 
lum are equal. But bodies descending 
through unequal spaces, in equal times, are 
impelled by powers that are as the spaces 
described, that is, the powers of gravity are 
as the lengths of the pendulums. 
Pendulums’ length in latitude of London, 
to swing 
Inches. 
Seconds 39.2 
^ Seconds 9.8 
I Seconds $.43 
Length of Pendulums to vibrate Seconds at 
every Fifth Degree of Latitude. 
Degrees of 
Latitude. 
Length of 
Pendulum. 
Degrees ot 
Latitude. 
Length of 
l-’eudulum. 
S3 
bb'^ 
o s 
3 
^”0 
5 a 
3 4) 
Inches. 
Inches. 
Inches. 
0 
39.027 
35 
39.084 
65 
39.168 
5 
39.029 
40 
39.097 
70 
39.177 
10 
39.032 
45 
39.111 
73 
39.185 
1.5 
39.036 
50 
39.126 
80 
39.191 
20 
39.044 
55 
39.142 
85 
39.193 
25 
30 
39.057 
39.070 
60 
39.158 
90 
39.197 
Rule. To find the length of a pendulum 
to make any number of vibrations, and 
vice versa. Call the pendulum making 61) 
vibrations the standard length ; then say, 
as the square of the given number of vibra- 
tions is to the square of 60 ; so is the length 
of the standard to the length sought. If the 
lengtlj of the pendulum be given, and the 
number of vibrations it makes in a minute 
be required ; say, as the given length is to 
the standard length, so is the square of 60, 
its vibrations in a minute, to the square of 
the number required. The square root of 
which will be the number of vibrations 
made in a minute. 
The greatest inconvenience attending 
this most useful instrument is, that it is con- 
stantly liable to an alteration of its length, 
from the effects of heat and cold, which very 
sensibly expand and contract all metalline 
bodies. See Heat, Pyrometer, &c. 
To remedy this inconvenience, the com- 
mon method is by applying the bob of the 
pendulum with a screw ; so that it may beat 
any time made longer or shorter, according 
as the bob is screwed downwards or up- 
wards, and thereby the time of its vibra- 
tions kept always the same. Again, if a 
glass or metalline tube, uniform thitmghout, 
tilled with quicksilver, and 58.8 inches 
long, were applied to a clock, it would vi- 
brate seconds for 39.2 = f of 58.8, and such 
a pendulum admits of a twofold expansion 
and contraction, viz, one of the metal and 
the other of tlie mercury, and these will be 
at the same time contrary, and therefore will 
correct each other. For by what we have 
shewn, the metal will extend in length with 
heat, and so the pendulum will vibrate 
slower on that account. The mercury also 
will expand with heat, and since by this ex- 
pansion it must extend the length of the co- 
lumn upward, and consequently raise the 
