PEN 
PEN.EA, a plant of the tetranclria mono- 
gynia class, with a monopetalous campani- 
ibnn flower; and a quadragonal capsule for 
its fruit, containing four cells, with two ob- 
long seeds in each. There are nine species. 
This plant lias been erroneously suppos- 
ed to have produced the sarcocolla of the 
shops. 
PENDANT, an ornament hanging at the 
ear, frequently consisting of diamonds, pearls, 
and other precious stones. 
Pendants, in heraldry, parts hanging 
down from the label, to the number of three, 
four, live, or six. at most, resembling the 
drops in the Doric frieze. 
Pendants of a ship, are those streamers 
or long colours which are split and divided 
into two parts ending in points, and hung at 
the head of masts, or at the yard-arm 
ends. 
PENDULUM, in mechanics, any heavy 
body, so suspended as that it may swing 
backwards and forwards, about some fixed 
point, by the force of gravity. 
These alternate ascents and descents of the 
pendulum, are called its oscillations, or vi- 
brations ; each complete oscillation being 
the descent from the highest point on one 
side, down to the lowest point of the arch, 
and so on, up to the highest point on the 
other side. The point round which the pen- 
dulum moves, or vibrates, is called its centre 
of motion, or point of suspension ; and a 
right line drawn through the centre of mo- 
tion, parallel to the horizon, and perpendicu- 
lar to the plane in which the pendulum 
moves, is called the ''axis of oscillation. 
There is also a certain point within every 
pendulum, into which, if all the matter th 't 
composes the pendulum were collected, or 
■condensed as into a point, the times in which 
the vibrations would be performed, would 
not be altered by such condensation ; and 
this point is called centre of oscillation. 
The length of the pendulum is usually esti- 
mated by the distance of this point below the 
centre of motion ; being a'hvavs near the 
bottom of the pendulum; but in a cylinder, 
or any other uniform prism or rod, it is at 
the distance of one third from the bottom, 
or two thirds from and below the centre of 
motion. 
The length of a pendulum, so measured 
to its centre of oscillation, that it will per- 
form each vibration in a second of time, 
thence called the second’s pendulum, lias, in 
the latitude of London been generally taken 
at 39J>_ or 39-t inches ; but by some very 
ingenious and accurate experiments, the late 
celebrated Mr. George Graham found the 
true length to be 39AAU, inches, or 391- 
inches very nearly. 
The length of the pendulum vibrating se- 
conds at Paiis, was found by Varin, Des 
liays, I)e Glos, and Godin, to be 440 A lines; 
by Picard 440-t lines ; and bv Mairan 440-V4 
lines. 
Galileo was the first who made use of a 
heavy body annexed to a thread, and sus- 
pended by it, for measuring time, in his ex- 
periments and observations. Put according 
to Sturmius, it was Riccioii who first observ- 
ed the isochronism of pendulums, and made 
use of them in measuring time. After him, 
Tycho, Langrene, Wendeline, Mersenne, 
PEN 
Kircher, and others, observed the same 
thing; though it is said, without any inti- 
mation of what had been done by Iviccioli. 
But it was the celebrated Huygens who first 
demonstrated the principles and properties 
of pendulums, and probably the first who 
applied them to clocks. He demonstrated, 
that if the centre of motion whs perfectly 
fixed and immoveable, and all manner of 
friction, and resistance of the air, &c. re- 
moved, then a pendulum, once set in mo- 
tion, would for ever continue to vibrate with- 
out any decrease of motion, and that ail its 
vibrations would be perfectly isochronal, or 
performed in the same time. Hence the 
pendulum has universally been considered as 
the best chronometer or measurer of time. 
And as all pendulums of the same length 
perform their vibrations in the same time, 
without regard to their different weights, it 
has been suggested, by means of them, to 
establish an universal standard for all coun- 
tries. 
Pendulums are either simple or com- 
pound; and each of these may be considered 
either in theory, or as in practical mechanics 
among artisans. 
A simple pendulum, in theory, consists 
of a single weight, as A, Plate Miscel. fig. 183. 
considered as a point, and an inflexible right 
line AC, supposed void of gravity or weight, 
and suspended from a fixed point or centre, 
C, about which it moves. 
A compound pendulum, in theory, is 
a pendulum consisting of several weights 
moveable about one common centre of mo- 
tion, but connected together so as to retain 
the same distance both from one another, 
and from the centre about which they vi- 
brate. 
The doctrine and laws of pendulums. 1 . 
A pendulum raised to B, through the arc of 
the circle AB, will fall and rise again, 
through an equal arc, to a point equally high, 
as D ; and thence will tall to ‘A, and again 
rise to B; and thus continue rising and fall- 
ing perpetually. For it is the same thing, 
whether the body falls down the inside of the 
curve BAD, by the force of gravity, or is 
retained in it by the action of the string; 
for they will both have the same effect ; 
and it is otherwise known, from the oblique 
descents of bodies, t hat the body will descend 
and ascend along the curve in the manner 
above described. 
Experience also confirms this theory, in 
any finite number of oscillations. But if 
they are supposed infinitely continued, a dif- 
ference will arise. For the resistance of the 
air, and the friction and rigidity of the 
string about the centre C, will take off part 
of the force acquired in falling ; whence it 
happens that it will not rise precisely to the 
same point from whence it fell. 
Thus, the ascent continually diminishing 
the oscillation, this will be at last stopped, 
and the pendulum will hang at rest in its 
natural direction, which is perpendicular to 
the horizon. 
Now, as to the real time of oscillation in a 
circular arc BAD ; it is demonstrated bv mathe- 
P E N 3/1 
height of the arch of vibration : then the time 
of each oscillation in the arc BAD, will be 
equal to p X 'nto the infinite series 
1 + £ + ^ + * c vvhere , 
U 2 2 d ^ 2 2 . 4 V 2 ^ 2* . 4 2 . 6 V 
— 2 >• is the diameter of the arc described, or 
twice the length of the pendulum. 
And here, when the arc is a small one, as in 
the case of die vibrating pendulum of a clock, 
all the terms of this series after the 2d may be 
omitted, on account of their smallness: and then, 
the time of a whole vibration will be nearly 
equal to pj— x (1 -f 
So that the 
times of vibration of a pendulum in different 
small arcs of the same circle, are as 8r -j- a; or 
8 times the radius, added to the versed sine of 
the semi-arc. 
And farther, if D denotes the number of de- 
grees in the semi-arc AB, whose versed sine is a, 
then the quantity last mentioned, for the time of 
a whole vibration, is changed to p y/- 
% 
52524' 
). And therefore the times of vi* 
(1 
bralion in different small arcs, are as 52524 -{- 
D 2 , or as the number 52524 added to the squat © 
of the number of degrees in the semi-arc AB. 
2. Let CB he a semicycioid, havirig its 
base EC parallel to the horizon, and its ver- 
tex B downwards ; and let CD be the other half 
of the cycloid, in a similar position to the for- 
mer. Suppose a pendulum-string, of the same 
length with the curve of each semicycioid BC, 
or CD, having its end fixed in C, and the thread 
applied all the way close to the cycloidal curve 
BC, and consequently the body or pendulum- 
weight coinciding with the point B. If now the 
body is let go from B, it will descend bv its 
own gravity, and in descending it will unwind 
the string from off the arch BC, as at the posi- 
tion CGH; and the ball G will describe a semi- 
cycioid BI1A, equal and similar to BGC, when 
it. has arrived at the lowest point A ; after which, 
it will continue its motion, and ascend, by an- 
other equal and similar semicycioid AKD, to 
the same height D, as it fell from at B, the string 
now wrapping itself upon the other arch C1D* 
Front D it will descend again, and pass aloixg 
the whole cycloid DAB, to the point B; and 
thus perform continual successive oscillations 
between B and D, in the curve of a cycloid ; as 
it before oscillated in the curve of a circle, in 
the former case. 
'Ehis contrivance to make the pendulum oscil- 
late in the curve of a cycloid, is the invention 
of the celebrated Huygens, to make the pendu- 
lum perform all its vibrations in equal times, 
whether the arch, or extent of the vibration, is 
great or small ; which is not the case in a circle, 
where the larger arcs take a longer time to run 
through them than the smaller ones do, as is 
well known both from theory and practice. 
The chief properties of the cycloidal pendu- 
lum then, as demonstrated by Huygens, are tha 
following: 1st. That the time of an oscillation 
in all arcs, whether larger or smaller, is always 
the same quantity, viz. whether the body begins 
to descend from the point B, and describes the 
semiarch BA; or that it begins at H, and de- 
scribes the arch HA ; or that it sets out front 
any other point ; as it will still descend to the 
lowest point A in exactly the same time. And 
it is farther proved, that the tune of a whole vi- 
maticians, that if p — 2.1416', denote the cir- bratiwn through any double arc BAD, or HAK 
cumfereuce of a circle whose diameter is 1 | & c . i s i n proportion to the time in which a 
1 feet, or 193 inches, the space a heavy body ; heavy body will freely fall, by the force of o T a- 
falls in the first second of time ; and r — CA, ! vity, through a space equal to 4; AC, half the 
the length of the pendulum, also a =. AE, the I length of. the pendulum, as the circumference 
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