GG4 
MECHANIC S. 
AB be a portion of the wheel on which the tooth is to 
be fixed, and let A pa be a thread lapped round its cir¬ 
cumference, having a loop-hole at the extremity a. In 
this loop-hole fix the pin a, and with it defcribe the curve 
or involute abed eh, by unlapping the thread gradually 
from the circumference A pm. This curve will be the 
proper fliape for the teeth of the wheel whole diameter is 
A E. Dr. Robifon observes, that, as the form admits of 
feveral teeth to be afting at the fame time (twice the num¬ 
ber that can be admitted in M..de la Hire’s method), the 
prefl'ure is divided among .feveral teeth, and the quantity 
upon any one of them is fo diminished, that thofe dents 
and impreflions which they unavoidably make upon each 
other are partly prevented. He candidly allows, how¬ 
ever, that the teeth thus.formed are not completely free 
from hiding and friction, though this Aide is only -J g th 
of an inch, when a tooth three inches long fixed on a 
wheel ten feet in diameter drives another wheel whofe 
diameter is two feet. 
Nothing can be of greater importance to the practical 
mechanic than to have a method of drawing epicycloids 
with facility and accuracy. The following, we truft, is 
the moft fimple mechanical method that can be employed. 
-—Take a piece of plain wood Q H, fig. 98. and fix upon 
it another piece of wood E, having its circumference mb 
of the fame curvature as the circular bafe upon which the 
generating circle AB is to roll. When the generating 
circle is large, the fegment B will be ftifficient. In any 
part of the circumference of this fegment, fix a fliarp- 
pointed nail a, Hoping in fuch a manner that the diltance 
of its point from the centre of the circle may be exaftly 
equal to its radius 5 and fatten to the board G H a piece 
of thin brafs, or copper, or tin, a b, difiinguifhed by the 
dotted lines. Place the fegment B in fuch a pofition that 
the point of the nail a may be upon the point b, and roll 
the fegment towards G, fo that the nail a may rife gradu¬ 
ally, and the point of contact between the two circular 
fegments may advance towards m ; the curve a b described 
upon the brafs plate will be an accurate exterior epicycloid. 
In order to prevent the fegments from Hiding, their peri¬ 
pheries fhould be rubbed with rofin or chalk, or a num¬ 
ber of fmall iron points may be fixed on the circumference 
of the generating fegment. Remove, with a file, the part 
of the brafs on the left hand of the epicycloid ; and the re¬ 
maining concave arch or gage a b will be a pattern-tooth, 
by means of which all the reft may be eafily formed. 
When an interior epicycloid is wanted, the concave fide of 
its circular bafe mutt be ufed. The method of deferibing 
it is reprefented in fig. 99. where C D is the generating 
circle, F the concave circular bafe, MN the piece of 
wood on which the bafe is fixed, and cd the interior epi¬ 
cycloid formed upon the plate of brafs, by rolling the ge¬ 
nerating circle C, or the generating fegment D, towards 
the right hand. The cycloid, which is ufeful in forming 
the teeth of rack-work, is generated precifely in the fame 
manner, with this difference only, that the bafe on which 
the generating circle rolls mutt be a ttraight line. 
In order that the teeth may not embarrafs one another 
before their aftion commences, and that one tooth may 
begin to aft upon its correfponding leaf of the pinion 
before the preceding tooth has ceafed to aft upon the pre¬ 
ceding leaf, the height, breadth, and diftance, of the 
teeth, mutt be properly proportioned. For this purpofe 
the pitch-line or circumference of the wheel, which is re¬ 
prefented in figs. 94. and 95, by the dotted arches, mutt 
be divided into as many equal fpaces as the number of 
teeth which the wheel is to carry. Divide each of thefe 
fpaces into 16 equal parts ; allow 7 of thefe for the greateft 
breadth of the teeth, and nine for the diltance between 
each ; or the diftance of the teeth may be made equal to 
their breadth. If the wheel drive a trundle, each fpace 
fhould he divided into 7 equal parts, and 3 of thefe al¬ 
lotted for the thicknefs of the tooth, and 3-J for the di¬ 
ameter of the cylindrical ltave of the trundle. If each of 
the fpaces already mentioned, or if the diltance between 
the centres of each tooth, be divided into three equal 
parts, the height of the teeth mutt be equal to two of 
thefe. Thefe diftances and heights, however, vary ac T , 
cording to the mode of adtion which is employed. The 
teeth fhould be rounded off at the extremities, and the. 
radius of the wheel made a little larger than that which 
is deduced from the rules jufFgiven ; but, when the pi¬ 
nion drives the wheel, a fmall addition fhould be made to 
the radius of the pinion. 
Of the Nature of Bevelled Wheels. Plate VII. 
The principle of bevelled wheels was pointed out by 
De la Hire, fo long ago as the end of the feventeenth cen¬ 
tury. It confilts in one fluted or toothed cone acting 
upon another, as is reprefented in fig 100. where the cone 
OD drives the cone O C, conveying its motion in the 
direction OC. - If thefe cones be cut parallel to their 
bafes as at A and B, and if the two fmall cones between 
A B and O be removed, the remaining parts A C and 
BD may be confidered as two bevelled wheels, and BD 
will aft upon A C in the very fame manner, and with the 
fame effeft, that the whole cone O D afted upon the whole 
cone O C. If the feftion be made nearer the bafes of the 
cones, the fame effeft will be produced : this is the C 3 fe 
in fig. iot. where CD and DE are but very fmall por¬ 
tions of the imaginary cones A C D and A D E. 
In order to convey motion in any given direction, and * 
determine the relative fize and fituation of the wheels for 
this purpofe, let A B, fig. 102. be the axis of a wheel, and 
CD the given direftion in which it is required to con¬ 
vey the motion by means of a wheel fixed upon the axis 
A B, and acting upon another wheei'fixed on the axis 
C D ; and let us fuppofe that the axis C D mutt have four 
times the velocity of A B, or mutt perform four revolu¬ 
tions while A B performs one. Then the number of 
teeth in the wheel fixed upon AB mutt be four times 
greater than the number of teeth in the wheel fixed upon 
C D, and their radii mutt have the fame proportion. Draw 
cd parallel to C D at any convenient diftance, and draw’ 
a b parallel to A B at four times that diftance ; then the 
lines im and in, drawn perpendicular to AB and C D re- 
fpeftively, will mark the fituation and fize of the wheels 
required. In this cafe the cones are O n i and O mi, and 
srni, rpmi, are the portions of them that are employed. 
The formation of the teeth of bevelled wheels is more 
difficult than one would at firtt imagine. The teeth of 
fuch wheels, indeed, mutt be formed by the fame rules 
which have been given for other wheels ; but, fince dif¬ 
ferent parts of the fame tooth are at different diftances 
from the axis, thefe parts mutt have the curvature of their 
afting furfaces proportioned to that diftance. Thus, in 
fig. 102. the part of the tooth at r niuft be more incur- 
vated than the part at i, as is evident from the infpeftion 
of fig. 101. and the epicycloid for the part 7 mult he 
formed by means of circles whofe diameters are im and 
Ff, while the epicycloid for the part r mult be generated 
by circles whofe diameters are C n and D d. For, let us 
fuppofe a plane to pafs through the points O, A, D; the 
lines A B, AO, will evidently be in this plane, which 
may be called the plane of centres. Now, when the-teeth 
of the wheel D E, which is fuppofed to drive CD the 
fmalleft of the two, commence their aftion on the teeth 
of C D, when they arrive at the plane of centres, and 
continue their aftion after they have patted this plane, the 
curve given to the teeth of C D at C, fhould be a portion 
of an interior epicycloid formed by any generating circle 
rolling on the concave fuperficies of a circle w-hofe dia¬ 
meter is twice C n perpendicular to C A ; and the curva¬ 
ture of the teeth at i fhould be part of a fimilar epicy¬ 
cloid, formed upon a circle whole diameter is twice im. 
The curvature of the teeth of the’ wheel D E at D, fhould 
be part of an exterior epicycloid formed by the fame ge¬ 
nerating circle rolling upon the concave circumference 
of a circle whofe diameter is twice D d perpendicular to 
D A j and the epicycloid for the teeth at F is formed in 
