3/2 
of a circle is to its diameter. So that,. If ♦ 3= 
l()j2 feet denotes the space a heavy body falls in 
the first second of time, p = 3.1416 tlie circum- 
ference of a circle whose diameter is 1, and 
r = AC the length of the pendulum ; then, be- 
cause, by the nature of descents by gravity, 
\/g ! Vi 2- *! l v \ V~p~> that is, the time in 
•which a body will fall through \r, or half the 
length of the pendulum ; therefore, by the above 
proportion, asl * /> ** y' — 
p\/ — , which 
% 
is the time of an entire oscillation in the cy- 
cloid. 
And this conclusion is abundantly confirmed 
by experience. For example: if we consider the 
time of a vibration as 1 second, to find the 
length of the pendulum that will so oscillate in 
1 second ; this will give the equation p \/ ^ - 
- % 386 
= 1 : which reduced, gives r = — y- — 
’ 6 p l 3.1416 2 
inches — 39.11, or 39-|- inches, for the ' length 
ef the second’s pendulum ; which the best ex- 
periments shew to be about 39-| inches. 
3. Hence also, we have a method of deter- 
mining.from the experiment the length of a pen- 
dulum, 'the space a heavy body will fall per- 
pendicularly through in a given time ; for, since 
p \/ — - = 1, therefore, by reduction, _§• = \p l r 
is the space a body will fall through in the first 
second of time, when r denotes the length of the 
second’s pendulum . and as constant experience 
shews that this length is nearly 39-§ inches, in 
the latitude of London, in this Case g, or \p"r, 
becomes \ X 3.1416 2 X 39f = 193.07 inches = 
1 6xV f eet > very nearly, for the space a body 
will fall in the first second of time, in the lati- 
tude of London : a fact which has been abund- 
antly confirmed by experiments made there. 
And in the same manner, Mr. Huygens found 
the same space fallen through at Paris, to be 15 
French feet. 
The whole doctrine of pendulums oscillating 
between two semicycloids, both in theory and 
practice, was delivered by that author, in his 
Horologium Oscillatorium, sive Demonstrationes 
de Motu Pendulorum. And every thing that 
regards the motion of pendulums has since been 
demonstrated in different ways, and particularly 
by Newton, who has given an admirable theory 
on the subject, in his Principia, where he has 
extended to epicycloids the properties demon- 
strated by Huygens of the cycloids. 
4. As the cycloid may be considered as coin- 
ciding in A, with any small arc of a circle de- 
scribed from the centre C, passing through A, 
where it is known the two curves have the same 
radius and curvature ; therefore the time in the 
small arc of such a circle, will be nearly equal 
to the time in the cycloid ; so that the times in 
very small circular arcs are equal, because these 
small arcs may be considered as portions of the 
cycloid, as well as of the circle. And this is 
one great reason why the pendulums of clocks 
are made to oscillate in as small arcs as possible, 
viz. that their oscillations may be the nearer to 
a constant equality. 
This may also be deduced from a comparison 
©f the times of vibration in the circle,, and in 
the cycloid, as laid down in the foregoing arti- 
cles. It has there been shewn, that the times of 
vibration in the circle and cycloid are thus, viz. 
r a 
time in the circle nearly p <v / — X (1 + g r )> 
PENDULUM, 
Y t 
time in the cycloidal arc p \/ — ; where it is evi- 
dent that the former always exceeds the latter 
a 
in the ratio of 1 -}- — to 1 ; but this ratio al- 
8 A 
ways approaches nearer to an equality, as the 
arc, or as its versed sine a, is smaller ; till at 
length, when it is very small, the term — may 
8r 
be omitted, and then the times of vibration be- 
come both the same quantity, viz, p 
% 
Farther, by the same comparison, it appears, 
that the time lost in each second, or in each vi- 
bration of the seconds pendulum, by vibrating 
in a circle, instead of a cycloid, is — — , or 
1 8 r 
— — ; and consequently the time lost in a 
whole day of 24 hours, is |D 2 nearly. In like 
manner, the seconds lost per day by vibrating 
in the arc of A degrees, is |-A 2 . Therefore, if 
the pendulum keeps true time in one of these 
arcs, the seconds lost or gained per day, by vi- 
brating in the other, will be y (D 2 — A 2 ). So, 
for example, if a pendulum measures true time in 
an arc of 3 degrees, on each side of the lowest 
point, it will lose lly seconds a day by vibrat- 
ing 4 degrees ; and 2 6y seconds a day by vi- 
brating 5 degrees ; and so on. 
5. The action of gravity is less in those parts 
of the earth where the oscillations of the same 
pendulum are slower, and greater where these 
are swifter ; for the time of oscillation is reci- 
procally proportional to >Jg. And it being found 
by experiment, that the oscillations of the same 
pendulum are slower near the equator, than in 
places farther from it ; it follows that the force 
of gravity is less there; and consequently the 
parts about the equator are higher or farther 
from the centre, than the other parts ; and the 
shape of the earth is not a true sphere, but 
somewhat like an oblate spheroid, flatted at the 
poles, and raised gradually towards the equator. 
And hence also the times of the vibration of the 
same pendulum, in different latitudes, afford a 
method of determining the true figure of the 
earth, and the proportion between its axis and 
the equatorial diameter. 
Thus, M. Richer found by an experiment 
made in the island of Cayenne, about 4 degrees 
from the equator, that a pendulum 3 feet 83- 
lines long, which at Paris vibrated seconds, re- 
quired to be shortened a line and a quarter to 
make it vibrate seconds And many other ob- 
servations have confirmed the same principle. 
See Newton’s Principia, lib. iii. prop. 20. By 
comparing the different observations of the 
French astronomers, Newton apprehends that 
2 lines may be considered as the length a se- 
cond’s pendulum ought to be decreased at the 
equator. 
From some observations made by Mr. Camp- 
bell, in 1731, in black-river, in Jamaica, 18° 
north latitude, it is collected, that if the length 
of a simple pendulum that swings seconds in 
London, is 39.126 English inches, the length of 
one at the equator would be 39.00, and at the 
poles 39.206. 
And hence Mr. Emerson has computed the 
following Table, shewing the length of a pen- 
dulum that swings seconds at every 5th degree 
of latitude, as also the length of the degree of 
latitude there, in English miles. 
Degrees of 
Latitude. 
Length of Pen- 
dulum. 
Length of the 
Degree. 
inches, 
miles. 
0 
39.027 
68.723 
5 
39 029 
68.730 
10 
39.032 
68.750 
15 
39.036 
68.723 
20 
39.044 
68.830 
25 
39.057 
68.882 
30 
39.070 
68.950 
35 
39.084 
69.020 
40 
39.097 
69.097 
4.5 
39.111 
69.176 
50 
39.126 
69.256 
55 
39.142 
69.330 
60 
39.158 
69 401 
65 
39.168 
69 467 
70 
39.177 
69.522 
75 
39.185 
69.568 
80 
39.191 
69.601 
85 
39.195 
69.620 
90 
39.197 
69.628 
6. If two pendulums vibrate in similar arcs, 
the times of vibration are in the sub-duplicate 
ratio of their lengths. And the lengths of pen- 
dulums vibrating in similar arcs, are in the 
duplicate ratio of the times of a vibration di- 
rectly ; or in the reciprocal duplicate ratio of 
the number of oscillations made in any one and 
the same time. For, the time of vibration t 
. r 
being as / , where p and g are constant or 
% 
given, therefore t is as \/r, and r as t 2 . Hence 
therefore the length of a half-second pendulum 
39X 
will be \r, or — — 9.781 inches ; and the 
length of the quarter-second pendulum will be 
39A 
rrrr — — - — 2.445 inches ; and so of others. 
16 16 
7. The foregoing laws, &c. of the motion 
of pendulums, cannot strictly hold good, 
unless the thread that sustains the ball is 
void of weight, and the gravity of the whole 
ball is collected into a point. In practice, 
therefore, a very line thread, and a small ball, 
but of a very heavy matter, are to be used. 
But a thick thread, and a bulky ball, disturb 
the motion very much; for in that case, the 
simple pendulum becomes a compound one; 
it being much 'the same thing, as if several 
weights w ere applied to the same inflexible 
rod in several places. 
8. Mr. Kratit, in the new Petersburgh Me- 
moirs, vols. 6 and 7, has given the result of 
many experiments upon pendulums, made 
in. different parts of Russia, with deductions 
•from them, from whence he derives this theo- 
rem: ifx is the length of a pendulum that 
swings seconds in any given latitude /, and in 
a temperature of 10 degrees of Reaumur’s 
thermometer, then will the length of that 1 
pendulum, for that latitude, be thus ex- 
pressed, viz. 
x = (439 - 178 -{- 2‘321 -}- sin. 2 /) lines of a French 
foot. And this expression agrees very nearly, 
not only with xd the experiments made on 
the pendulum in Russia, but also with those 
of Mr. Graham, and those of Mr. Ljous in 
79° 50' north latitude, where he found its. 
length to be 44l 38 lines. 
Pendulum, simple, in mechanics, an ex- 
pression commonly used among artists, to., 
distinguish such pendulums as have no. pro- 
vision for correcting the effects of heat and 
