HOROLOGY. 
^ I. Of, toe PENDULUM. 
Pendulums -are of two kinds, fimple and compound. 
The fimplc pendulum is.fuppofed to have all its weight 
united in one point. The compound pendulum has more 
or lefs weight in its different parts. In the Horology 
Plate VII. fig. 31 reprelents the affion of the Ample pen¬ 
dulum. C is the point of fufpenfion ; C A the rod, or 
wire, which carries the vibrating body, or bob , A. This 
body may defcribe arcs of greater or lefs extent round 
the point C; one of which arcs is BAD. The motion 
of the pendulum is caufed by the gravity of the body A, 
for A moved to B and then fet at liberty, would fall in a 
direction B H, perpendicular to the centre of the earth; 
but, being retained by the rod C A always at an uniform 
diftance from C, the fall muft be in the line B A; and, 
being arrived at A, the body has acquired a velocity 
equal to what it would have gained.by falling perpendi¬ 
cularly from I to A; and by this velocity the body is 
enabled to rife from A to D in a time equal to that of 
the femi-vibration B A- The body, arrived at D, cannot 
remain at reft, but will fall again to A, and thence rife 
to B. Thus the vibrations will continue, til! home exte¬ 
rior force puts an end to them. The fpecific weight of 
the vibrating body does not fennbly afferit the duration 
of the vibrations; nor would it affeft them at all if the 
pendulum vibrated in a vacuum. 
Ifochonifm, or an equality in the duration of the greater 
and lefs arcs of vibration of a pendulum, is not obtained 
without difficulty when the pendulum makes large vibra¬ 
tions. Huygens found, that, by making the pendulum 
vibrate in a cycloidal curve, inftead of circular arcs, the 
l^rge and final 1 vibrations were performed ih equal times. 
To produce thefe cycloidal vibrations, he employed the 
following method. In fig. 32, CE and CF reprefent two 
cycloids, defended by a circle whofe diameter is half the 
length of the pendulum. The wire CA, being extreme¬ 
ly thin and flexible, in its, courfe bends round the cy¬ 
cloids, and thence A deferibes the cycloidal curve E A F, 
inftead of.the circular arc BAD. The two cycloids 
muft be made of materials fufficiently ftrongnot to yield 
to the vibrating force of C A. Though this principle 
be mathematically true in theory, it is not poifible to re¬ 
duce it efteftuaiiy to practice, the difficulty of giving the 
two curves' a'-perfe&ly-cycloida! fhape is one great obfta- 
cle.; and, even were that obftacle removed, it is evidently 
very inconvenient to fufpend a pendulum by a thread or 
very thin wire. But We can attain this object by a me¬ 
thod extremely fimple. By looking at the cycloid and 
the circle in the figure, it is evident that the two lines 
coincide from I to G, their curvatures not having any 
fenfible difference for a-certain fpace. Therefore, by cauf- 
ing the pendulum to vibrate very fmail acs, the vibrations 
may be coMidered as cycloidal ; and thus we obtain ifo* 
chroftifrn by the moft fimple means. 
Experience has proved that two pendulums -of equal 
length will not perform their vibrations in equal times 
if placed in different parts of the earth. Time is mea- 
fured by the' gravity of the vibrating body and by the 
length of the pendulum: and, when gravity increal'es or 
diminifhes, it is evident that the times of the vibrations 
muft. differ. The earth being flatted near the poles, gra¬ 
vity is ftronger there, as being nearer the centre, and pro¬ 
duces quicker vibrations ; while, towards the equator, 
gravity is weaker, and.a pendulum of the fame length 
will make-flower vibrations. • As gravity diminifhes, cen¬ 
trifugal force increal'es, and the contrary ; fo that two 
caufes combine'to'produce quicker or flower vibrations. 
But in the fame latitude gravity cannot alter; conle- 
quently pendulums'of equal length will vibrate in equal 
times in every place under the fame degree of latitude. 
A pendulum to vibrate once in a fecond, in the latitude 
of London, fhould be 39-?- inches long; at Paris, 36^ 
inches, or 44.0!'lines, Paris rqeafure. 
Mr. Emerfon has computed the following Table, fhow- 
307 
ing the length of a pendulum that fwings feconds at 
every 5th degree of latitude, as alio the length of the de¬ 
gree of latitude there, in Englifh miles. 
Degrees of 
L:;t. 
Length or the Pen¬ 
dulum in Inches. 
Length ot the De¬ 
gree in Miles. 
0 
.39-027 
68-723 
5 
39-029 
68.730 
10 
39-032 
68-750 
is 
39-036 
68-783 
- 20 
39’°44 
68'S3o 
25 
39-057 
68-882 
30 
39 “070 
68-950 
35 
39-084 
69-020 
4 ° 
39-097 
69-097 
45 
3 9 -iii 
69-176 
5 ° 
39-126 
69-256 
55 
39-142 
69- 330. 
60 
3 y' 1 5 8 
69'4oi 
65 
39-168 
69-467 
70 
39 ’i 77 
69-522 
75 
39 ' i ^5 
69-568 
80 
39-191 
69-601 
85 
39’ 1 95 
69-620 
90 
39-197 
69-628 
The mathematical rules as to the vibrations and lengths' 
of pendulums are thefe: 1. That the vibrations are per¬ 
formed in times proportioned to the fquare-root of their 
lengths. 2. That the lengths are in proportion to the 
i'quare of the times of the vibrations.* Whence it follows, 
3. That the number of vibrations in a given time is in 
the inverfe proportion of the fquure-root of the lengths ; 
and, 4. That the lengths are in an inverfe ratio of the 
i'quare of the vibrations ; fo that a pendulum, to make 
twice the number of vibrations in an equal time, muft be 
reduced to one fourth of its original length. 
The time of the vibrations of a pendulum, and its 
length, being given, we may thence difeover the length 
of another pendulum to make vibrations in a given time .; 
or, the length being given, we may difeover the number 
of vibrations. 
Example- It is required to know liow many vibrations 
a pendulum of 10 inches or 120 lines in length will make, 
in an hour.—We know that a pendulum 39-i ■ inches or 
469^ lines makes one vibration (at London) in a fecond, 
or 3600 vibrations in an hour.. Now>the vibrations'of the 
long pendulum, 3600, • are to the required vibrations of 
the Ihort pendulum, as the lquare-roots of the lengths of 
the two pendulums are to each other. The fquare-root 
of 120 is 10-95-; the fquare-root of 469-t is 21-67 nearly. 
Therefore, As 10-95 : 21-67 :: 3600 : 7124. A pendu¬ 
lum of 10 inches long will make about 7124, vibrations 
in an hour in tire latitude of London . 
To find the length of a pendulum that' fhall'make a 
given number of vibrations in a given time, we recur to. 
the fecond rule above recited ; or, if we obl’erve a marked- 
proportion between the given number of vibrations and 
3600, we may fhorten the labour of calculation by no¬ 
ticing the remark at the end of the 4th rule. Thus, if’ 
the length of a pendulum which ftiould vibrate 7200 times 
in an hour were required, we ftiould directly obferve,. 
that, 7200 being double 3600, the required length of the 
pendulum mult be one-fourth of our common feconds 
pendulum, or 117-4 lines, neai'ty inches. For, as the 
i'quare of 7200 is to the i'quare of 3600, fo is 469-5. lines 
to the length required of the ihort pendulum ; or, As 
518400 129600 -.: 469-5 : 117-4. Thus we fee that it is 
not difficult to proportion the lengths of pendulums to 
the number of vibrations required. And upon theie Am¬ 
ple principles Tables of the lengths of pendulums havd 
been calculated. 
The refiftance which a body in motion meets with from 
the air is in proportion to its furface and to the iquare of. 
its. velocity. A body moving at the-fame rate with.ano¬ 
ther 
