3j<5 
HOROLOGY. 
an epicycloidal tooth is ufed, the diftance of the pitch- 
line from the end of the tooth is equal to -J of its breadth ; 
and, if we fuppofe the tooth and lpace cut to be recipro¬ 
cally equal, we (hall have the true afting diameter of any 
wheel or pinion greater than the geometrical diameter, 
which Camus calls all'o the primitive diameter, by ^ of a 
tooth or fpace on each tide of the centre, or in the 
whole diameter. Let now a fpace or a tooth be called a 
meafure, and there will be double the number of meafures 
as teeth in any wheel; alfo let thefe meafures of the cir¬ 
cumference be reduced into meafures of the diameter by 
the ufual ratio of y 1416 to 1, and then it added to fuch 
geometrical meafures of the diameter will give the pro¬ 
per adding diameter, which may be exprefled in inches 
and parts when the meafures per inch are known. For 
8 
inftance, let our great wheel and its pinion — be taken 
at 12 teeth per inch at the pitch-line, which may in prac¬ 
tice be more or lefs, according to the thicknefs of the me¬ 
tal compared with the maintaining power as modified at 
the wheel’s circumference; the number of meafures of 
the great wheel is 192, viz. 96 teeth and 96 fpaces, each 
meafuriilg i-24th of,an inch; then, as 3‘i4i6 : 1 :: 192 : 
6i"i; therefore, if to the geometrical diameter exprefled by 
6ri meafures there be added 1*5, the fum 62-6 or 62^ 
will be the aiding diameter in the fame denomination, 
which are fo many 24th parts of an inch ; but ^ 2 --- gives 
^ 4 - 
2-5 inches for the full a£Hng diameter of the wheel in 
queftion. Again, the pinion 8 has 16 fimilar meafures in 
its circumference, and by the fame proportion the diame¬ 
ter will be 5-09 meafures; to which if 1-5 be added, the 
aiding diameter will be 5-091-5 — 6-59, or with fuffi- 
cient accuracy (if,y, which divided by 24 as before, will 
give the fame of an inch, or fomewhat more than a 
quarter, for the aiding diameter of the pinion. Upon 
thefe principles Hatton (Introduidion to the Mechanical 
Part of Clock and Watch-work, page 334) has conftru«d- 
ed a Table of the fizes of the pinions meafured diametri¬ 
cally, and compared by a paid' of callipers with a given 
number of teeth and fpaces in their correfponding wheels, 
which many workmen copy in praidice; but, as his cal¬ 
culations are founded on a l'uppofition that the ends of 
the teeth are circular, requiring unity as the fupplemen- 
tal portion, we find them differ eflentially from Ber- 
thoud’s determination in his Eflai fur l’Horlogerie, tom. i. 
and fhall therefore inlert here a Table which we find in 
Dr. Rees’s Cyclopaedia, calculated on a fuppofition that 
the curve is epicycloidal, and that the circumference is 
to the diameter as 3 to 1, inftead of 3’1416 to 1; the re- 
fult of which mode agrees very nearly with Berthoud’s 
experiments on the proper lizes of wheels and pinions, 
and therefore we recommend it to the notice of the ac¬ 
curate workman. 
Table of the true pra&ical Sizes of Pinions. 
Teeth in the 
Pinions. 
Meafures of the 
Wheel for a Dia- 
meterof thePinion. 
Teeth in the 
Pinions. 
Meafures of the 
Wheel for a Dia¬ 
meter ofthePinion. 
3 
3’5 
IO 
8 -i 
4 
4 ’l 
I I 
8-8 
5 
4'8 
12 
9’5 
6 
5 ‘ 5 
13 
10*1 
7 
6 -x 
14 
io - 8 
8 
6"8 
15 
”’5 
9 
7'5 
l6 
12*1 
The procefs by which this Table is calculated is fimply 
this: Multiply the pinion by 2 for the meafures in the cir¬ 
cumference; divide by 3 for the diameter, and add thereto 
11 for the acting fize ; thus for the diameter of apinion of 6, 
it is;c.X 2 -4- 3 +"iT) = 5 l or S'5 5 namely, 6 X 2 =z 12 , 
I 2 
and —- = 4, and 4 4 * i‘S = S'5 for the meafures; which 
3 
laft quantity, taken by the callipers acrofs the extreme edge 
of the wheel, will be 3 teeth and fpaces, or 3 fpaces 
and teeth, which are here fuppofed to be cut, but not 
rounded. 
The application of this Table, it is prefumed, cannot 
be eafily miftaken by any workman who underftands that 
the figures in the fecond column, to the left of the deci¬ 
mal point, mean fo many meafures, either teeth or fpaces, 
and the figure to the right of the faid point, fo many 
tenth parts more of a meafure to be added to the integral 
meafures. It may be proper to add here, that a propor¬ 
tioned pinion mult be made fomewhat fmaller for a fmall 
wheel than for a large one, and alfo fmaller when driven 
than when it is the driver. 
We now know the numbers of our movement, and alfo 
that, whatever the diameters of the wheels may be, our 
pinions of 8 mult be turned in the lathe till their diame¬ 
ters are precilely each 6-8 meafures, or three teeth and 
very nearly four fpaces (taken by a pair of pinion calli¬ 
pers from their refpedtive wheels, in a Itraight line acrofs 
the ends of the teeth,) either before or after they are flit, 
as the operation of dividing and cutting is called by the 
workmen. The diameters of the wheels are ulually made 
to diminilh as the train alcends, probably becaufe the 
force to overcome their inertia diminilhes, and the fric¬ 
tion alfo is lefs in fine teeth with flender pivots, than in 
coarle ones with thick pivots; indeed there feems to want 
a ftandard rule for the guide of workmen in this particu¬ 
lar. Having taken the great wheel at 12 teeth per inch, 
meafured at the pitch-line, we will take the centre wheel 
of 64 at 14, and the fecond wheel of 60 at 16, which will 
make fomething like a regular diminution in the fizes in 
the afcent of the train, and allow us room enough in our 
plates for the reprefentation. From thefe data, by the 
help of the foregoing directions, we readily al’certain the 
requifites for drawing the calliper as exprefled in the fub- 
joined 
Table of Wheels and Pinions. 
Wheels. 
Teeth 
per 
Inch. 
Afting Diame¬ 
ters in Inches. 
Geometrical Diame- 
cers meafured Iron 
the Pitch Lines. 
Great wheel 
96 
I 2 
2’6o 
2 ’ 5 S 
Its pinion 
8 
12 
0-273 
0*21 
Centre wheel 
64 
14 
1-514 
1-46 
Its pinion 
8 
14 
0-234 
o"i8 
Second wheel 
60 
l6 . 
1*24. 
1-19 
Its pinion 
8 
l6 
0*207 
o‘i6 
Swing wheel 
60 
l6 
1*24 
Pennington, of Camberwell, the ingenious mechanic 
who conftruCted Mr. Mudge’s time-piece, and gave the 
drawings in Mr. Mudge’s pamphlet, has paid particu¬ 
lar attention to the lubject of fizing wheels and pinions, 
and has publifhed a fmall pamphlet, recommending the 
ufe of his method of calculation by a feftor of a peculiar 
conftruition ; but we do not find that its ufe has become 
general. His practice is, to add 2-*- meafures of the geo¬ 
metrical diameter to the wheel, and 1^ to the pinion, in 
watch-work, when the wheel is the driver; and to 
each, when the pinion is the driver. The rule, it will be 
feen, differs very little from the theory above laid down, 
where we obferved that the driver ought to be fomewhat 
larger than the proportion affigned by the addition of im¬ 
parts or meafures to each, and more particularly where 
the wheels have fmall diameters, as is the cafe in watch- 
work. But, as the good aftion of wheels and pinions 
depends upon their being duly proportioned and calli¬ 
pered, as well as on a proper thape being given to the 
teeth, we will add moreover the method of fizing pinions 
praitifed and recommended by F. Berthoud, in bis Eflai 
fur l’Horlogerie, which is as follows: viz. 
i ,No. 
