h o rf o 
teetli, how many teeth fliould the pinion have ? Anfwer, 
As 30 : 3 :: 120 : 12. The pinion therefore mult have 
iz teeth, or leaves as they are commonly called. 
Now, the number of the teeth of wheels being always 
in the ratio of the diameters, it is evident that vve can 
.exp refs the fize of the wheel by the number ot teeth, and 
confequently ufe thefe numbers in calculating the revo¬ 
lutions of wheels. 
The number of revolutions made by the laft wheel of a 
train of wheels and pinions may be found by multiply¬ 
ing the teeth of all the wheels (except that laft) together, 
and dividing it by the produdt of the teeth of all the pi¬ 
nions.—Example. Required the number of revolutions 
which the lalt or fcapement-wheel will make while the 
fir ft or fufee- wheel makes one revolution; the train con¬ 
fining of five wheels and four pinions ; the firlt wheel 
having 100 teeth, the fecond 80, the third 60, the fourth 
50; the firlt pinion having 20 leaves, the fecond 16, the 
third 10, the fourth S. Anfwer, 100X 80X 60X 502= 
24.000000; and 20X16X 10X8 = 25600; 24000000 di¬ 
vided by 25600 gives a quotient of 9375, being the pro¬ 
portionate velocity of the lalt wheel of the train, (of what¬ 
ever number of teeth it may confilt,) or the number of 
revolutions it will make while the firlt wheel makes one. 
Or, Divide the number of the teeth in each wheel by the 
number of teeth in the pinion it drives ; then, by multi¬ 
plying all thefe quotients together, you will have the lame 
refult. Thus: 100-2-20=25; 80-7-1622:5; 60-2-102=6; 
50-2-82=6^ : and 5 X 5 X 6X 6^2= 937^, as before. 
M. le Roy, a celebrated artilt of Paris, pointed out 
fome inconveniences attending the ufual difpofition of 
the wheels and pinions of watches, and propol'ed a method 
of remedying them. Every wheel of a watch is fixed in 
an arbor, or Item, which terminates in two pivots, and 
thefe turn in holes drilled in the plates of the watch. 
Each arbor is charged both with a wheel and a pinion ; 
and it is the pinion which receives the aftion of the im¬ 
mediately preceding wheel, and tranfmits it to the wheel 
fixed on its own arbor. M. le Roy obferves, that a watch- 
wheel placed near the middle of its arbor is in the molt 
advantageous polition, efpecially if its pinion be nearly in 
the lame pofition ; becaufe then, the efi’ort it receives is 
diltributed equally between the two pivots ; the pivot- 
holes in the plates wear equally and on the fame fide, and 
their enlargement will always let the wheel continue pa¬ 
rallel to the plates 5 and therefore, the pofition of the 
planes of the wheels fuffering no alteration by fuch wear¬ 
ing, with refpeft to one another, they drive one another 
on without any alteration as to the pitching or the friction. 
But it will be otherwife when the wheel or the pinion 
is near one of the extremities of the arbor, becaule the 
friftion arifing from the aftion of the wheel is no longer 
equal on both the pivots; that which is nearell the pinion 
receiving almoll the whole effort of the preceding wheel, 
whiHl the other is very flightly affefted by it. Hence it 
will follow, that the hole of fuch pivot mult wear much 
more, and in a fhorter time, than the other; the pofi¬ 
tion of the arbor will be altered, and confequently that 
of the parallelifm of the planes of the wheels. This is the 
great defeft of common watches ; the pinion of the fmall 
middle wheel, or third 'wheel, and that of the contrate 
wheel, are fo near one of their pivots, that it is often ne- 
ceffary to fcop or bufh up their holes, and drill them anew', 
in a year or two. M. le Roy fet himfelf about remedying 
this inconvenience ; in order to which he was under a 
neceffity of inverting the fufee, fo that the w'ide bafe 
lhould be at top, and the little end at bottom, near the 
great wheel ; and thus the middle wheel might be made 
to aft upon the pinion of the contrate wheel near the 
middle of its arbor. M. le Roy makes the lower pivots 
to turn, not in the plate of the frame, but in another 
little plate or cap* called a bar , placed on the outfide of 
the frame-plate; by which contrivance he renders the ef¬ 
fort of the pivots nearly equal, and keeps the oil from 
quitting the pivots, as it is too apt to do in the common 
Vol.X. No. 663. 
LOGY. sti 
conltruftion. By the inVerfion of the fufee, its bafe be¬ 
ing on the fame fide as the fquare of the great pivot, the 
diameters of the pivots are proportioned to the friftion 
they fuller; whereas in Englifh watches, the great pivot 
is on the fide where the chain draw's nearell the centre, or 
on the wrong fide. 
It is hardly neceflary to infill upon the necefiity of 
having all the teeth of a wheel perfeftly equal, both for 
dillance and for fize, and that their extremities, or points, 
be equally diftant from the centre. All this is equally 
efiential, perhaps more fo, in regard to the pinions. Pi¬ 
nions Ihould be very hard and well polifned, to lelfen 
friftion and facilitate motion ; and it is advantageous to 
have the wheels as light as is confident with llrength, in 
order that the moving pow'er may more eafily overcome 
the refiilance caufed by their inertia. 
The fize of pinions may be determined, in general, by 
the following Rule, though there are particular cafes to 
which it wdll not exaftly apply: As the number of teeth 
in the wheel which drives the pinion is to the number of 
leaves in the pinion itlelf, fo is the femi-diameter or ra¬ 
dius of the wheel to that of the pinion. Suppofe a wheel 
with 50 teeth, and its radius 222 20, to drive a pinion of 
10: the radius of the pinion mull be = 4; for, As 50 : 
10 : : 20 : 4. 
The teeth of wheels and leaves of pinions, require great 
care and judgment in their formation, that they may 
neither clog the machinery, with unneceflary friftion, nor 
aft fo irregularly as to produce any inequalities in the 
motion, and a conlequent wearing-away of one part be¬ 
fore another is much affefted. 
Several eminent mathematicians upon the continent, 
and a few in England and Scotland, have direfted their 
invelligations towards a fubjeft lb efiential to the perfec¬ 
tion of machinery : yet, although Roemer, Varignon, De 
la Hire, Camus, Euler, Kaellner, and Robilon, have 
turned their thoughts to this objeft, and have llruck out 
fome rules of ready application in praftice, it is to be re¬ 
gretted that thele rules have been little followed by prac¬ 
tical mechanics, moll of whom have in this caie been 
more inclined to follow in their conltruftions the rules of 
Imifon and others, though completely deltitute of me¬ 
chanic principle. As the conllruction of teeth of a pro¬ 
per form is exceedingly ealy, we beg to recommend it 
earnestly to practical men ; and it may not be amils to en¬ 
ter a little into detail, availing ourfelves) for the molt part, 
of the judicious remarks lately publilhed by Mr. Brewdler, 
It has been long known to mathematicians, and need 
not here be demonltrated, that one wheel will not drive 
another wdth uniform velocity, unlefs the teeth of one or 
of both wheels have their faces formed into a curve ge¬ 
nerated after the manner of an epicycloid, comprehending-, 
under curves of this kind, thole which are formed by 
evolving the circumferences of circles. But, in order to- 
enfui'ean uniformity of prefl'ure and velocity in the aftion 
of one wheel upon another, it is not abfolutely neceflarv 
that the teeth either of one or both wheels be exaftly 
epicycloids, in the fienfe to which geometricians commonly 
reftrift that term. If the teeth of one of them be either 
circular or triangular, with plain fides, or like a triangle 
with its fides converging to the centre of the wheel, or, 
in Ihort, of any other form, this uniformity of force and 
motion will be attained, provided that the teeth of the 
other wheel have a figure which is compounded of that 
of an epicycloid and the figure of the teeth of the firlt 
wheel. But, as it is often difficult to deferibe this com¬ 
pound curve, and fometimes impofiible to difeover its 
nature, we fhall endeavour to feleft fuch a form for the 
teeth as may be eafily deferibed by the practical mechanic, 
while it enlures an uniformity of prefl'ure and velocity. 
In order to avoid circumlocution and obfcurity, we lhall 
call, as is cullomary with praftical men, the fmall wheel 
(which is l'uppoled always to be driven by a greater one) 
the pinion, and -its teeth, the leaves of the pinion. The 
line which joins the centres of the wheel and pinion m->y 
+ N ' be 
