338 
HOROLOGY. 
tions of the vertical and horizontal columns, under the 
pinions made choice of, and oppofite the factors feleCt- 
ed, will be proper for the fecond portion of the move¬ 
ment. For inftance, when pinions of 8 are ufed with the 
factors g 2 | , the wheels, found-in the fpace formed by 
the interfe&ion of the vertical column under 8 and hori- 
zontul column in which ' 2 
| ftand, are 64 and 60 ; but 
if the pinions were to be affumed, one 10 and the other 8, 
then the wheels would be either 80 and 60, or otherwife 75 
and 64, accordingly as the pinion 10 is made a part of 
the ratio to reprefent the faCtor 8, or the faftor 7-t, which 
it may be taken to do indifferently ; hence the notation 
10 8 8 10 jo 8 8 
may be r- X —, or — X —, or — x —> or — x 
80 60 80 60 75 64 75 
—; the refult in point of accuracy is the fame, but in the 
conftruftion, the wheels of the largeft diameter and weight 
are required to be taken firft in the movement, becaufe 
the diameters diminilh as the train afcends, by reafon of 
the diminution of the tranfmitted force, which otherwife 
would not overcome the inertia of the wheels, which it is 
required to do at each vibration of the pendulum. Any 
two pair of wheels and pinions properly taken out of the 
correfbonding columns, when reduced to a fimple ratio by 
the umal method of multiplying the numerators together 
for one numerator, and the denominators for one deno¬ 
minator, will be found equal to —; thus— X — = 
60 80 60 
So 1 . 10 8 80 1 , . 
-= —, and— X — = —— — —, as before; there- 
4800 60 7s 64 4800 60 
fore, if the firft wheel revolve in an hour, the laft pinion 
will, in this and in every other inftance, revolve in a mi¬ 
nute. The diminution of the diameter of the third wheel 
of the movement, which is the fecond wheel of the train, 
may be effected in two ways, either by taking tjie fe- 
conds pinion fmaller than the firft from a column at the 
bottom of the Table, where the factors are nearly of equal 
value, or by taking the pinions alike, and the wheels from 
one of the higher horizontal columns where the faftors 
• • i a 7 6 
differ confiderably in value', tnus — X— from column 
60 42 
L(, and — X — from column I0 c > are each equal to — 
7 i 70 42 3 6o’ 
but in the firft portion of the movement the ratio of the 
fize of the two wheels is as 60 to 42, and in the other as 
70 to 42, from which mode of companion of the columns 
it will be feen that a decreafe in the diameters of almoft 
any given ratio may be adopted from this Table, fo com- 
prehenfive in its extent, as it relates both to the variety 
of 11 pinions, and to the choice ofiao pair of faftors, which 
begin with the ratio 4 to 15, or 1 to 4, and end with 7-t 
to 8. - The principal care to be taken is, that every wheel 
be taken under its own pinion and oppofite its own fac¬ 
tor; for another inftance, in the horizontal column of 
faftor ^ ^ , the pinions of 9 will do for wheels 81 and 
60 ; or otherwife, wheel 90 may be taken with pinion 10; 
and wheel 60 with pinion 9 ; that is, any wheel may be 
taken oppofite its own faftor, provided the pinion in the 
fame column over it be ufed as its fellow; fo' that exam¬ 
ples may be given to a very great extent, which will af¬ 
ford abundant means of afcertainirig by experiment the 
beft poftible numbers for this part of the movement, with¬ 
out the trouble of calculation. 
Table III. contains the third portion f a dock-movement , 
or fecond portion of the train, which is that on which 
the number of vibrations per fecond depends. When a 
vibration is performed in an exaCt fecond, one wheel 
only is neceffary for this purpofe, with 30 teeth, becaufe 
one tooth completely efcapes the pallets at every fecond 
vibration, which wheel mult be placed in this cafe on the 
arbor of the feconds-hand, and is ufually denominated 
the fwing-wheel; alfo a wheel of 60, fimilarly placed, will 
be proper for half-feconds, 75 for 2| vibrations per fe¬ 
cond, and 90 for 3 per fecond; but thefe laft numbers are 
found too high for portable clocks, time-pieces, See. 
therefore a wheel and pinion are introduced in addition 
to the fcapement-wheel, in order to diminilh its diameter 
into a convenient fize for a portable conftruftion, as well 
as to render it light, fo as to have but little inertia to be 
overcome by the diminilhed force, which acts at this part 
of the movement. In ordinary pieces the two wheels are, 
one a contrate-wheel, and the other called the crown¬ 
wheel, with the pinion on its arbor; but the wheels may 
all have the ordinary form of the wheels of a clock that 
fwings feconds, it being not the lhape, but the numbers 
of the teeth, of the wheels and pinions that determine the 
frequency of the vibrations. The uppermoft horizontal 
column in this Table, like that in the two preceding Ta¬ 
bles, contains all the variety of pinions from 6 to 15 in- 
clufively ; the pinion of 16 being omitted, however, to 
make room for two additional vertical columns at the right 
hand of the Table, the firft for the number of vibrations 
per fecond made by the pendulum, and the other for the 
length of the correfpondiqg pendulum in inches and de¬ 
cimal parts, meafured from the centre of fufpenfion to 
the centre of ofcillation, which two data once fixed upon 
determine the horizontal column out of which the wheels 
muft be taken with a given pinion; the left-hand vertical 
column is that in which one of the two wheels is found, 
and the number Handing on the fame horizontal line, 
under the.given pinion, is the other; for as the produft 
of the two wheels, divided by the pinion under which 
one of them ftands, is always equal to 60 in the higheft 
large column, or column of 2 vibrations per fecond, it is 
of no importance to the accuracy which of the two wheels 
is made the pallet-wheel; the determination of this point 
being a matter of practical convenience; in the fame 
manner, in the fecond large horizontal column, the quo¬ 
tient arifing from the product of any two wheels Handing 
in the fame line taken horizontally, one in the firft vertical 
column, and the other under the chofen pinion, divided 
by the pinion, is always 75; in the third parallel column 
the quotient fo obtained is 90 ; in the fourth 105, and fo 
on ; hence any of the combinations adopted will be equal 
in value refpeftively to thofe large fimple numbers ufed 
as pallet-wheels without fuch combination, which, it has 
been faid, are objectionable in praftice. 
Let our firft example be to afeertain the requifites con¬ 
tained in the Table for a half-feconds pendULum, where a 
pinion of 2 is ufed. 
In the firft place, 12 with 40 may be taken as the re¬ 
quired numbers for the two wheels; in the fecond, 15 with 
32; in the third,’ 20 with 24; and, laftly, 30 with t 6; 
any one of which wheels may be the pallet-wheel, as the 
nature of the fcapement may require; if the crown-wheel 
fcapement is ufed, which requires an odd number of teeth 
for its action to take place at oppofite fides of the wheel, 
15 muft neceffarily be the. pallet-wheel, and 32 the other, 
with the given pinion of 8 ; but, whichfoeVer of thefe cou¬ 
plets of wheels is adopted, the effective length of the pen¬ 
dulum muft, according to the laft vertical column, be 9-80 
inches. Again, for a pendulum that is required to vi¬ 
brate 3 times per fecond with a pinion of 12 ;• we have, 
in the firft place, 15 with 72; lecondly, 18 with 60; 
thirdly, 30 with 35 ; and, laftly, 45 with 24; fo that either' 
j 5 or 45 may be in this inftance the crown-wheel, and for 
a different fcapement any of the eight numbers mention¬ 
ed ; and the choice is equally extenfive with any other 
pinion that may-be chofen from 6 to 15 ; the pendulum 
in this cafe is only 4^35 inches. 
In this way, by means of the Table before us, the third 
portion of a clock-movement may be varied almoft at 
pleafure without affecting the other two portions, and a 
pendulum of any of the calculated lengths may be ap¬ 
plied'to works previoufly conftrufted; which circumftance 
will 
