S 0 $ H ORO 
thc-r whole Airfare is double, will experience but half the 
refiftance of the larger body. In fixing upon a proper 
ihape for a pendulum, that is, for the part commonly 
called the bob , two things are particularly to be attended 
to: the fliape which is bell calculated for meeting with 
the lead refiftance from the air; and the weight which is 
beft adapted for preferring the ifochroriifln of the vibra¬ 
tions with a given maintaining power. Each of thefe 
confiderations has been the fubjeft of much invelligation. 
A fphere is a folid, the furface of which bears the lead 
proportion to its folidity of any other, and a cube is one 
with great extent of furface compared to its folid content; 
conlequently, the .former (hape has obtained in cannon¬ 
balls, intended to pal’s through the air with as little ob¬ 
it r lift ion as pofiible, and lias fometimes, for the fame reafon, 
been applied as a bob for a pendulum. Such a ihape is 
better adapted for a large bob than a fmail one ; for the 
increafe of the furface is in proportion to the 1’quare of the 
diameter, whereas the increafe of weight or quantity of 
matter is us the cube. Mr. B. Martin (Mathematical In- 
llitutions, vol. ii. p. 417.) propofed two equal frultra of 
limilar cones, to be joined together at the bales, as a fi¬ 
gure approximating nearly to a folid of lead refiftance, 
and reaoinmended it as that out of which the middle one 
of three legments, cut longitudinally, will conilitute the 
1)611 Ihape'for. an appended weight, to anl'wer the purpofe 
of avoiding refiftance. Such a left ion, he conceived, 
would dilplace but a fmail quantity of air in one vibration, 
and the impulfes of that quantity, being made obliquely 
on the doping furfaces of the ends, would produce but a 
fmail effedl. The lenticular (hape, however, compofed of 
two legments of a large giobe joined to the line of feftion, 
has been generally adopted by clock-makers, from the 
confideration, that, as the pendulum moves but in two 
directions, the aftion of the air can be but on two fides 
of it; though others think the lenticular bob has been 
adopted chiefly becaufe two plates of brafs of that Ihape 
can be eafily foldered together, and left hollow, fa as to 
be filled with lead, of other heavy metal, in order to ob¬ 
tain a due degree of weight without adding to the thick- 
nefs, which is an advantage that brafs or copper cannot 
have of itfelf in any drape. M. Ferdinand Berthoud 
(ElTai fur l’Horlorgerie, tom. ii. chap, xiii.) made fonre 
experiments with bobs of both a fpherical and lenticular 
drape, of equal weight, attached fuccellively to the fame 
free pendulum, from which it appeared, that, in the fame 
temperature, the latter continued to vibrate feconds much 
longer than the former before they arrived at the quiefcent 
ftate, which experiment was comidered as a proof that the 
lenticular Ihape has the advantage in efcaping the effeft of 
refiftance ; it was difcovered, however, on a repetition of 
the experiment, with feconds and half-l’econds pendulums, 
that the friction at the point of fufpenlion occalioned by 
heavy weights, particularly when vibrating in long arcs, 
made confiderable alterations in the refults, and proved 
itfelf a fecond fource of refiftance to the free motion of 
the pendulum. 
The fecond confideration to be attended to in making 
the bob, as has been faid, is the weight which a given 
pendulum requires with a given maintaining power. No 
.theory is adequate to determine this defideratum of itfelf, 
becaufe the diminution of the maintaining power by the 
friftion of the pivots and the ftate of the oil, the duration 
of the impulfe on the pallets, the nature of the fcape- 
ments in other refpefts, the refiftance of the air, and at 
the point of fufpenfion, and particularly the nature and 
quantity of momentum of the pendulum, mult all enter 
into the calculation ; and thefe are data, many of which 
arq conftantly varying. The momentum, or whole quan¬ 
tity of motion of any pendulum, is the weight multiplied 
into the l’quare of its velocity, fo that a large arc with a 
fmail weight, and a fmail arc with a correfponding large 
weight, caiteris paribus, ought to have an equal effeft 
upon the ifochronilm of the fame pendulum. But the 
theory is not perffeft 5 for firft, large circular arcs deviate 
LOGY. 
confiderably from cycloidal ones, which, it has been ile- 
monftrated by Huygens and others lince, are thofe which 
have the ifochronal property in an uniformly denle me¬ 
dium ; and fecondly, they require a greater maintaining 
power than calculation gives: for inftance, where the arc 
is io° from the point of reft, the addition to be made to 
the maintaining power beyond calculation is 57 parts in 
100, according to Berthoud’s experiments, which confi¬ 
deration induces him to conclude, that irregularities in 
the maintaining power will affeft the momentum princi¬ 
pally compofed of velocity more than the momentum 
principally compofed of weight. Each kind of momen¬ 
tum, however, has its peculiar difadvantage ; for great 
velocity is fubjeft to great refiftance from the air, and 
great weight to much friction at the point of fufpenfion. 
Mr. Alexander Cummings was an advocate for a large arc 
with a fmail bob; but modern practice is in favour of a 
large ball or bob with a ftiort arc of vibration, as more 
nearly coinciding with the cycloidal curve. 
The length of the pendulum-rod, of whatever metal it 
may be, increafes with heat, and leffens with cold. But 
different metals dilate and condenle at different rates, 
and, according to M. Berthoud’s pyrometer, in the fol¬ 
lowing proportions : 
Lead - - - 193 
Tin - 160 
Zink - - 148 
Brafs - - 121 
Silver - - - 119 
Copper - - 107 
Gold-wire - - 94 
Some experimentalifts give different proportions from 
the above; but it is to be obferved, that metals will net 
always dilate and contraft in the fame proportion ; much 
depends upon the quality of the metal, efpecially of cop¬ 
per : fo that a workman muff not rely in all cafes upon 
the above Table, nor upon any other ; but rnuft verify 
all by aftu.al experiment upon the materials he is about 
to ul’e. 
Another confideration muft be taken into the calcula¬ 
tion ; which is, that the metal pendulum-rod will dilate 
and condenle in proportion to its length only. An iron 
rod 3 feet long, and one line fquare, will dilate in the fame 
degree as another of the fame length, but of twice or 
thrice the thicknefs ; but the thickelt rod will longeft refill 
the influence of a change of temperature. By knowing, 
therefore, the degree of dilatation? of a rod of a given 
length, it is eafy to find the degree of dilatation of a rod 
of any other length. Suppofe two rods, one of 10, the 
other of 15, inches long; and that the dilatation of the 
former is half a line in a certain temperature ; we fhall then 
have the following proportion : As ia : 15 :: A : J. A 
rod of 15 inches long, without regard to its thicknefs, 
will dilate three quarters of a line in the fame tempera¬ 
ture. 
We have now to {peak of the aftual effeft of tempera¬ 
ture upon the action of the pendulum. We have feen. 
that the pendulum lengthens by heat, and Ihortens by 
cold ; and we have alfo fliown the proportions in which 
pendulums of different metals dilate or condenle in a fimi- 
iar temperature; but the temperature is ever varying, 
efpecially in our country ; and thus the length of a pen¬ 
dulum muft be alfo continually varying. The duration 
of the olcillations depends on the length of the pendulum ; 
a clock will generally therefore go fafter in the winter, 
becaufe, as the pendulum Ihortens by cold, the vibrations 
are quicker ; and in lummer the contrary. But we may 
obferve, that in clocks for common ufe; efpecially if in a 
room where a fire is kept in winter, the variation is not 
worth notice. Where greater exaftnefs is required, a 
thermometer Ihould be placed within the clock-cafe at a 
height nearly equal to that of the middle of the pendulum ; 
and its height, for this purpofe, fnould be examined at 
leaft once a-day. Now by a table conllrufted to exhibit 
the daily quantity of accelcr ation or retardation of the 
’ 1 clock 
Gold annealed - 82 
Iron, hammered - - 78 
Steel, tempered - - 77 
Iron, annealed - 75 
Steel, annealed - - 74 
Platina and Glafs - 62 
