HOROLOGY. 
X 
OkJJ 
the fame, and would moreover be capable of afting more 
behind than before the line joining the centres of the 
wheel and pinion; in like manner, pinions of 6 would re¬ 
quire wheels of 48 and 45, and pinions of 12 wheels of 
96 and 90, as may be feen in Table II. farther on. 
3. The lalt portion of the movement, or 1 'econd portion 
of the train, fora half-feconds pendulum, will require only 
one wheel of 60 teeth on the feconds-arbor, properly lhaped 
for the fcaperrrent; for, as one tooth in the dead-beat and 
common anchor fcapements efcapes completely at two vi¬ 
brations of the pendulum, 60 teeth will efcape, that is, a 
■whole revolution of the feconds-hand will be made, in 
120 vibrations; if, however, the pendulum had been re¬ 
quired to vibrate feconds, the wheel in queition, called 
tilually the fwing-wheel, in oppoiition to the crown¬ 
wheel, which requires another fcapement, would have de¬ 
manded only 30 teeth for that purpofe; and, if three vi¬ 
brations had been fixed upon, the number to correlpond 
mull have been 90, otherwife there mull have been a 
. . 8 10 
wheel and pinion of the value ot 3, h.-ce— 5 .or—’ in 
24 . 3° 
addition to the ufual fwing-wheel of 30; or, which is the 
fame thing, a wheel-and pinion of the value of 6, like 
8 10 
—> or —mull have been introduced between the fe- 
4S 60 
conds arbor and a pallet or fpring-wheel of 15. See Ta¬ 
ble III. Thus all the variety in the calculation of trains, 
where feconds are indicated, is confined, as we have inti¬ 
mated, to the lad portion of the movement; and the cal¬ 
culation itfelf is fo fimple, that the mere altering of the 
numbers of the pallet-wheel will convert a clock with a 
feconds pendulum into one with half-feconds, and vice 
virfa. 
The calculation of numbers fuitable for an eight-day 
clock with a half-feconds pendulum being thus readily 
obtained by three fimple operations, which may be had 
by mere infpeftion from the three Tables which we (hall 
prefently give, the whole may be reprefented, and its 
value ellimated again, by a compound fraftion, thus : 
8 "g g”! 1 
viz. —off xH of xTT:— of 12 hours, or, which is 
96 64 6o| 00 X a 
8 8 8 1 
the fame thing in effeft, thus, -7X rXr- X 7-- = 
0 56 64 60 60 x 2 
- = ——of n h , or 86400 vibrations in 12 hours. 
44236800 86400 
which is the time of a revolution of the fulee, and great 
8640° 
wheel, 96, on its arbor, and therefore -’ or 7200 vi- 
y n. 
brations, each of half a fecond in duration, in one hour, 
conftitute the value of this train. This mode of nota¬ 
tion gives the value better than- any other perhaps that 
has been adopted ; 'but the pofition of the wheels and pi¬ 
nions will be. better underltood from the ordinary mecha¬ 
nical method of writing them down ; thus : 
Great wheel 96 
Pin. 8—64 hour-wheel 
Pin. 8—60 fecond-wlieel 
Pin. 8—60 fwing-wheel 
2 pallets. 
Indeed it is difficult to write down the movement by 
any one notation that (hall exprefs, at the fame time, both 
the value and pofition of the wheel-work; on which ac¬ 
count we recommend to the workman to write down his 
numbers by both forms, taking care, in the method by 
compound ratios, to put all the drivers under the line of 
divifion, and all the driven ones above; fo that, when an 
afcending movement is reprefented, the wheels may be 
the denominators, and, when a defcending one, the pi¬ 
nions. In our mode of calculation from the bottom of 
the train, the notation mull be, as we have made it, af- 
cepding. 
The calculation of proper numbers being made and 
noted down, the next flage of the work is proportioning 
the diameters of the wheels and of their refpeftive pi¬ 
nions, fo as to tranfmit the maintaining power from the 
fulee, or barrel in an ordinary 30-hour clock, to the pal¬ 
lets, and thence to the pendulum, to compeniate the lofs 
of motion which, when unaided, it would fuftain from 
friftion and the refiltance of the air. If a wheel and pi¬ 
nion were to be made like two rollers,, p re (Ting their edges 
againll one another, to produce a communication of ro¬ 
tatory motion, their diameters might and ought to be in 
geometrical proportion directly as their calculated num¬ 
bers of teeth ; but the force of the maintaining power 
would be too great to be fultained by mere frift ion at the 
points of contaft of fuch rolling wheels and pinions ; 
they have therefore been neceflarily indented, and their 
teeth mutually inferted fo far into their correfponding 
fpaces, as to prevent the revolution of one wheel or pi¬ 
nion without a correfponding motion produced in the 
next adjoining, which would not be the cafe with rollers, 
if a confiderable force impelled them at one end of the 
train, and at the fame time a retarding force oppofed them 
at the other; for their lurfaces, at the point of greatell 
oppofition to free motion, would mutually rub without ef- 
fefting a communication of rotatory motion beyond fuch 
point. If now we call the points of contaft of the two 
rollers, made in geometrical proportion to each other, or, 
in other words, the points where they pitch againft one 
another, the pitch-line, and conceive a number of pro- 
jefting little levers, or teeth, fixed at proper intervals 
from each other at thele points of contaft, in the circular 
pitch-line of each roller, we lhall have a true idea of two 
wheels properly proportioned to aft together; which, 
when of unequal diameters, will not now be in geome¬ 
trical proportion to each other, by reafon of an equal le¬ 
ver, or length of tooth, being added to each feparately, 
after they were in exaft geometrical proportion ; more¬ 
over, it will be eafily apprehended that the deviation from 
their original proportion, in the date of rollers, will be 
the greater, the greater their difparity of numbers. Hence 
it will be readily conceived, that the due proportioning 
of wheels and pinions is an important objeft in clock¬ 
making ; for, fuppofing the teeth refpeftively of the true 
epycycloidal forms, uniefs the rel’peftive fizes be properly 
adjusted, the tranfmilfion of the maintaining power, and 
communication of motion, will both be unequable, and 
the mechanifm lubjeft to rapid deflruftion. 
The ufual mode of proportioning, or fizing, wheels and 
pinions, as it is often called, is, firlt to make both a little 
too large for the propofed calliper, and then, having- 
rounded all the teeth of the pinion and a few of the cor¬ 
refponding wheel, to diminilh the latter in the lathe, or 
turning-frame, gradually, until, by fucceffive trials in the 
clock-frame, they are found to aft at a proper depth, 
when placed in the pivot-holes previoufiy made; this vul¬ 
gar mode we reprobate, as calculated to deltroy the due 
practical proportions, and hope to fee it banifiied from 
the workfnops by the general adoption of a better me¬ 
thod, which we have now to propofe. 
In proportioning wheels and pinions, after the num¬ 
bers of their teeth are determined upon, two particulars 
are to be attended to ; the coarfenefs or folidity, and the 
(hape, of the tooth; the former may be exprelfed by the 
number of teeth per inch in the circumference of the 
wheel; and the latter by the denomination epicycloidal; 
if a tooth were rounded in a circular (hape, which we by 
no means recommend, the pitch-line would he confidered 
as at one half the breadth of the tooth from the extreme 
edge; but when it is rounded in an epicycloidal (hape, 
or, as the workmen call it, the bay^lecif form, Hatton has 
found, from numerous experiments, that the depth, or 
diitance of the pitch-line from the circumference, w.ilLo- e . 
nerally be § of the breadth of the tooth in any .wheel or 
pinion ; and, as the epicycloidal is the belt (hape for the 
regular tranfmiffion of force and velocity, we will adopt 
it as the bell for praftice. Wejiave jull laid that, when 
an 
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