Fig.40                   NEW ODONTOGRAPa
Full 8ize
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~~~~~~~








A
PRACTICAL TREATISE
ON THE
TEET'H OF WHEELS
With the Theory and the use of
ROBINSON'S ODONTOGRAPH.
BY
S. W. ROBINSON,
Professor of Mechlanical Engineering,
ILLINOIS INDUSTRIAL UNIVERSITY.
ILLUSTRATED.
NEW YORK:
D. VAN NOSTRAND, PUBLISHER,
23 MURRAY AND 2T WARREN STREET.
1 8 7 6.








PREFACE.
The present little volume, reprinted from the
pages of VAN NOSTRAND'S MAGAZINE, has resulted from an effort to bring before the mechanical public improved facilities for obtaining
correct gearing.
Correct gear teeth are essential for the economic transmission of power, as well as extremely desirable for avoiding needless rattling.
Many of the cast gears introduced into mills
and factories may properly be termed rattling
machines, and simply because the designer had
too vague a conception of the theory of gearing,
or because he sought to avoid the trouble of
laying them out.
If this manual, and the odontograph, shall
contribute toward relieving these inconveniences, the object sought, when the instrument
was devised, and an effort made to present it
in convenient form for use, will be realized.
S. W. R.
ILLINOIS INDUSTRIAL UNIVERSITY,
July, 1876.








PART I.
ON THE
THEORY OF GEAR TEETH,
AND ADAPTATION OF PARTICULAR FORMS
FOR CERTAIN PURPOSES.
THOUGH the curves which may be employed for the outlines of Inathematically correct gear teeth are infinite in
number, varying greatly in external appearance, yet all of them must conform
with one general principle, viz.: tooth
curves must transmit, from one shcaft to
the other, the same motion as wouldc the
pitch lines when acting by simp2le rolling
contact.
This truth is, in reality, axiomatic;
and for the present purpose needs no
discussion or comment.  It forms the
basis of the present investigation, the
object of which is to show how the




6
curves may differ, and the adaptation of
certain ones for special purposes.
In this article the terminology indicated below is adopted.
The pitch lines of a pair of wheels are
lines which pass near the mid-height of
all the teeth, and separate the faces
from the flanks. These lines must always exist, whether the wheels are designed with regard to them or not, and
on which the tooth spaces of one wheel
equal those of the other.
The face of a tooth is that part of the
outline of one side of a tooth which extends from the point or top of tooth to
the pitch line.
The flenk of a tooth is that part of the
outline of one side of a tooth which extends from the pitch line to the bottom
of the space between two adjacent teeth.
The length of this, reckoned in the direction of a radius, is usually made a
little greater than the length of the
face, similarly measured, to allow fol
clearance.
The pitch is the distance, reckoned




7
on the pitch line, from a point on one
tooth to the corresponding point on the
next.
The velocity-ratio is the ratio of the
angular velocities of the wheels, and may
be constant or variable. If constant, it
is equal to the number of revolutions per
minute of one wheel, divided by the
same for the other.
In the study of gearing two kinds of
contact are considered, viz.: rolling contact and sliding contact. The former
takes place between the pitch lines, and
the latter between the curves forming
the outlines of the teeth.  When the
pitch lines are not circular, the problem
of determining them so as to give true
rolling contact, with fixed axes, is quite
extensive. The present paper is confined
to the outlines of teeth, the pitch lines
being supposed given.
In the figure, let A and B represent
the centers of a pair of gears, and M N
and O P the curves of a pair of teeth in
action. The point of contact C is chosen
at one side of the line of centers, so as
to represent the general case.




Now, suppose the wheel A to swing
through the small angle C A E. The
FIG. 1.
point C will describe the elementary arc
C E. At the same time the wheel I3, to
maintain contact with A, must move
through the elementary angle C B F, and
C through the arc C F.
To determine C F, it is sufficient to
note that as the movements of the wheels
are elementary, the common tangent to
the curves at C will scarcely change its
inclination appreciably. Therefore, the
end of the radius B C will be found at
F, in a line, E F, parallel to the common
tangent.




9
Draw a common normal CD to the
curves. It will be perpendicular to E F.
Then the velocity-ratio may be obtained
as follows:
At a unit's distance fr6m A and B
draw  II I and J K respectively. These
arcs will serve to measure the angular
velocities of the wheels, and hence:
HI
Velocity-ratio =   K.
The triangle C E F is similar to C D G,
and hence:
AHl: AC::  I: CE
BJ: BC:   JK: CF
CE: CF': CG: DG
DG: B C:: AD: AB1
CG: AC:: BD: AB
Whence, observing that A H=TB J =
unity,
HI EC BC CG C
Velocity-ratio —         C J C D FD
BC AB BD BD V*
AC-ADA B-A D-    v 
* This geometrical solution is essentially the




10
if V and v represent the angular velocities of A and B.
Hence, when rotary motion is transmitted from one shaft to another by sliding contact as in gear teeth, the angular
velocities are to each other inversely as
the segments, or parts of the line of
centers, formed by its being cut by the
common normal to the curves.
It is to be observed that at the end of
same as that given in Belanger's most excellent
work on Cinematique, page 90, where also the
velocity of sliding is found to be proportional
to C D.
Those familiar with Trigonometry may prefer the following trigonometrical solution. We
have
CE=AC. V, CF=B C. v,
CE F=angle y, CFE=x, ADC=-z;
and hence:
CE sin. y —CF sin. x,
A C sin. y-A D sin, z,
BC sin. x=BD sin. z,
or        V. A D sin z=v. B D sin. z,
and               V  BD
v —A D':as before.




11
the movements C E and CF, the points
which were in mutual contact at C are
now removed a distance E F; and as
this displace!nent was accompanied with
slipping, the amount of the latter is E F.
In the similar triangles C E F and CDG,
we have the same relation between E F
and C D, as we have between C E anlld
(, G, or CF and GD.  Hence, EF can
only be zero when C D vanishes; or when
the point of contact is on the line of
centers.  For this, the  contact  has
changed from  sliding to rolling; and,
therefore, for rolling contact, tile angular
velocities are inversely as the radii of
contact.  In circular wheels the velocityratio must remain constant, because these
segments are constant; whereas, in noncircular wheels it is variable.  In the
former, the point of contact remains
fixed, while, for the latter, its position
varies along the line A B.
The contact of a pair of teeth must
continue for a time, at least, long enongh
for the next pair of teeth to come
fairly to contact; or, long enough for




12
the pair of teeth to move through a
space as great as the length of the pitch.
This contact cannot therefore be confined
to the line of centers, and hence sliding
of the curves upon each other is unavoidable*.
From the preceding discussions it follows that in all gear wheels, circular or
non-circular, the common normal to the
sliding curves, and the pitch lines, must
both maintain a common point of intersection with the line of centers; and
this condition is the only one necessary
for fulfilling the general principle stated
at the outset. The common normal, because it is the line of action of the press.
ure between the teeth, friction neglected, and, for its importance otherwise, is
called the line of action.
Suppose, for example, any curve
whatever, as b a c Fig. 2, be taken as the
* Hooke's spiral gearing admits of contact which may
be confined to the plane of the axes, and hence free from
slip. But this is due, not to the form of tooth, but
width of wheel face; or, length of wheel in direction of
axis.




13
form of the side of a tooth of a gear
wheel revolving about B. Then let the
curve d caf be found for the side of a
tooth of A, which, acting by sliding contact on b a c, will give to A the same angular motion as that transmitted by the
rolling curves DE and F G, the latter
being supposed circular or not. Let D




14
and F be two points of the rolling curves
which come to mutual contact on the
line of centers, and similarly for E and
G. Then arc C D = arc C F, and are
CE = arc C G, &c.
Now for the present point of contact
C of the pitch lines there must be a
point of contact of the tooth curves, and
because the common normal to those
curves must meet AB in C, we simply
draw C ca, normal to b a c. This is the
present point of contact of the tooth
curves, and hence a point of the curve
sought. To find another point, draw
F b normal to b ac.  Then draw D d
equal in length to F b, and making the
same angle with A D, that F b does with
B F prolonged. Then, d is a second
point of the required curve; because
when D and F are in contact on the line
A B, D d and F b will coincide in length
and direction, with d and b superposed,
forming the point of contact of the
tooth-curves. In like mannerf, and indeed any number of points may be




15
found. The curve d af; traced through
these points, is the desired curve.*
In the above example, A and B are
supposed to be the centers of rotation,
in which case D E, and F G, must be a
pair of correct rolling curves.  But the
solution is still more general. Any two
curves whatever,  without fixed  axes
even, may be used for the rolling curves.
and the sliding curves determined.
When radii do not exist, tangents to the
rolling curves may be used to  reckon
the angles by.  Such a case, however,
would but rarely occur in practice.
This solution indicates that one tooth
curve may be assumed, and the other
formed'as a dependence.  It fails, however, as being  both  theoretically  and
mechanically impossible, and so with
any theory, when the normals C a, F b,
&c., will not strike the pitch line.
Though one tooth-curve may be assumed and the other determined, the
* This solution given in Belanger's Cinematique, p. 9T,
appears to be due to De la Hire; and given by him as
early as 1694. See Willis' Princ. Mechanism, 2d ed., p. 89.




16
theory admits of other solutions; the
most common being by aid of clescribing
curves, called also generating or tracing
curves.  In this case the describing
curve is assumed and both tooth-curves
made dependent. To show that this solution depends upon one and the same
general theory, let us find a describing
curve which will generate the toothcurve b ca c, or cd acf.
In Fig. 3 draw g 1 equal to h 1, Fig. 2.
Then take I C, Fig. 3, equal 1 C, Fig. 2.
Also g C, Fig. 3, equal a C, Fig. 2. And
again C F and g F, Fig. 3, equal C F
and b F, Fig. 2, respectively, &-c., for as
many points as desired. Then trace the
curve g l C F, Fig. 3. This is the describing curve sought.  Now  suppose
this curve be placed in contactwith F G,
Fig. 2, so that g, Fig. 3, falls at g, Fig.
FIG. 3.
\,
CO    /




17
2.  Then  by  rolling  the  describing
curve along toward F, the points 1, C, F,
Fig. 3 will exactly fall at 1, C, F, Fig. 2,
because the arcs g 1, I C, C F, are equal
in the two figures. Also the construction
makes g 1, ig C, g F, &c., intersect the arc
of the describing curve at the same angles that the normals h 1, a C, b F, &c., intersect the pitch line, which is easily
seen to be true by supposing the points
g, C, F, &c., very numerous.  Hence the
radii vectors of Fig. 3 will successively
come to exact coincidence with the normals of Fig. 2, and at the same time
g7 will trace out the very curve g ]h a b.*
And similarly, by rolling the same curve
feom  i to D, Fig. 2, the curve i k ad
will be traced. In like manner a describing curve can be produced which
will generate, by rolling from g or i toward G or E, the remaining portions g 
and if of the tooth-curves.
Thus it appears that a generating
curve can always be found that will
trace  out a given  tooth-curve.  But
*De la Hire, in 1733.




18
whenever the latter are neither of them
fixed, we are at liberty to assume a describing curve, and this may be of any
form whatever which may admit of rolling on the pitch lines. The two solutions
are therefore one and the same, the difference being simply in the part assumed.
And within this are comprehended all
known forms of gearing, such as involute, epicycloidal, epitrochoidal, pingearing, Prof. Edward Sang's hour
glass curve gearing, &c., &c.
The solution of the problem is therefore reduced to simply this; any describing curve rolled upon the outside of
one pitch line, and upon the inside of the
other pitch line, beginning at mutual
contact points; will describe a pair of
faces and flanks which will work correctly.
This may be demonstrated by a direct
graphical method as follows.
Take pitch lines AD and B E, circular or non-circular, and a describing
curve CA H; or, which is the same,
K B I. Also take D and E as the pair




19
FiG. 4.
of mutual contact points, where the pair
of tooth-curves sought begin. Let C A H
roll in outside contact from D to A, describing D C by a tracing point C; and
similarly rolling the same curve from E
to B, describes the curve E K. This
rolling may be continued to any extent,
giving the full line curves D C, and E K
produced.
If the same curve be rolled on tangents at A and B, two equal curves will




20
be described which are shown in dotted
lines.  Let arc D A = are E B = arc
CA = arc KB-=AF — BG.  Then
chord A C=  chord B K, and angle
C A F = angle K B G. Hence if A and
B be placed in tangential contact, the
tangents and dotted curves will coincide
precisely; and the full line curves, or
tooth-curves, will have a point of tangency at C K. Also the curves will not
pass, one to the opposite side of the
other, because the full line curve for A
lies wholly below, and for B Mwholly
above the mutually coinciding dotted
curves. This being true for one point,
taken at random must be true for all,
giving the precise conditions required
for tooth-curves.
We might suppose for illustration that
the three curves roll upon each other
with one point A, Fig. 5, in common tan -
gency. The tracing point C, would then
trace, simultaneously, the two curves
shown, one upon the plane of each wheel,
which will serve to work correctly as
a face and flank.




21
FIG. 5.
\\         C
We are now prepared to draw the following conclusions:
Ist. The faces and flanks all have their
origins at the pitch line.
2d. Each face and flank couplet must
be generated by rolling the tracing curve
in the same direction, on the outside of
one, and inside of the other pitch line.
3d. The contact of a couplet is confined to one side of the line of centers.
4th. Couplets which do not interchange flanks and faces may be generat



22
ed by different describing curves; even
though on the same pair of wheels.
Different describing curves give different forms of gear teeth, some of  the
most important of which wTill now be
considered.*
In practice the pitch lines are usually
circular; though on shaping and slotting
machines, and others requiring a " quick
return," non-circular wheels have been
employed with advantage. In Fig. 6 is
given such a pair of non-circular wheels,
drawn  carefully  to  a scale, with  each
tooth-curve worked out for its particular
position  by  the  general theory.t  The
describing    curve used  was a circle, the
diameter of which was about one-third
that of the pitch line of A. These wheels
were designed with  special reference to
* It may be observed by comparing Figs. 2, 3 and 5 that
the generating eurve must lie on the opposite side of the
pitch line from where the inte sections of the normals
are found. From this it follows that if these intersections cross the pitch line as the rolling proceeds, so must
the describing curve. The latter will then change its direction with a cusp.
t Drawn by a former pupil, Mr. Emory Patch.




La-a''
*9 *JIJ








23
adaptation for shaping and slotting machines, in which the tool is reciprocated
by a crank and pitman. The forward
stroke is uniform, except for one-fourteenth at each end, given for slowing
down for passing the center. The return is accomplished in one-half the time
the entire forward stroke requires. The
teeth are made large, proportionately,
for the purpose of illustrating the fact
that such wheels require teeth which differ from each other in form.
Circular wheels form only a special
case of the general theory; but as these
are of almost exclusive use in practice,
with the teeth varied for different purposes, the following important classes
will be considered separately, viz.:
I. Wheels with epicycloidal teeth;
II. Wheels with involute teeth;
III. Wheels with random teeth.
Epitrochoidal curves have been suggested for tooth-curves, but as they do
not admit of tooth-contacts near the line
of centers, where most desirable, they
are not useful for general purposes.




24
I. EPICYCLOIDAL GEARING.
Tlhis name is applied because the toothcurves are epicycloids and hypocycloids.
An epicycloid is traced by a point upon
the plane of a circle, when that point is
guided by being attached to the circumference of a second circle which rolls
upon the first. If external it is an epicycloid, if internal an hypocycloid.
A curious fact relating to these curves
deserves notice, viz.: that every epicycloid and hypocycloid admits of being
generated by two different rolling circles.
To illustrate, suppose A b c to be a primitive or pitch circle, and B a rolling or
describing circle, tracing with p the epicycloid D, by rolling on A. Draw the
line pbe through the point of tangency
b. Then bp will be a normal to the epicycloid at p, because an elementary part
of the latter is described while b serves
as an instantaneous axis for the wheel B
to turn about. Now, this elementary
portion of D will be the same if c be the
center instead of b. This being true for




25
each elementary part of D, it follows
that a circle C, tangent at c, and having
a tracing point p, will trace the same
epicycloid that B will. It can be easily
FIG. 7.
B
C\                  C 
c              A
shown that the difference of the diameters of C and B is equal to the diameter
of A.*
D* ue to Duhamel, see Bour Cours de Cinematique, p.
120.




26
Fig. 8 indicates how this double generation applies to the hypocycloid, in
which A is the pitch circle and 1B the
FIG. 8.
hypocycloid.  The sum of the two diameters of the generating circles evidently equal the diameter of the pitch
circle.
That the teeth be mechanically possible, it is necessary that the generating
circles be as small, or smaller than the
pitch circles within which they roll; otherwise the teeth would be much undercut.
This range in the relation of sizes gives
variety of form of teeth, the following
of which are the most essential:




27
1st. Flanks radial.
2d. Flanks parallel, for strength.
3d. For minimum of friction.
4th. For change-gears and sets.
5th. Pin gearing.
1ST. EPICYCLOIDAL TEETH WITH RADIAL
FLANKS.
Suppose a generating circle to roll inside a pitch circle, with the diameter of
the former equal to the radius of the
latter. By reference to Fig. 8 we see
that the two generating circles would be
equal, and the hypocycloid a straight
line, a diameter of the pitch  circle.
This principle, is the foundation of the
well-known White's parallel motion.
Hence when the diameters of the generating circles equal the radii of the
pitch circles in which they roll, the
flanks will be radial, and very convenient
to construct.  In practice, therefore, we
have simply to delineate the epicycloids,
with circles half as large as the opposite
pitch circles, and strike in the tangent
radii for flanks. This form of gearing




28
is the most convenient for practical execution of any good form known, and for
this reason is the form usually adopted,
when correct delineation is attempted.
Indeed many designers have no idea of
the existence of any other correct form,
except, possibly, the involute.
Fig. 9 represents the simultaneous generation of the faces and flanks for both
wheels, the four circles being supposed
FIG. 9.




29
to roll upon each other with only one
point in common, each tracing point describing two lines.
In wheels of few teeth, the latter appear somewhat undercut, and weakened,
when the flanks are drawn wholly radial.
But it is easily demonstrated that a
face works upon only a small portion of
the flank, which also is adjacent to the
pitch line. Below this the teeth may be
thickened by clearance curves, almost
entirely removing this objection. An advantage is gained by these teeth which
probably more than compensates for the
above objection. This consists in the
great freedom from crowding of the
wheels from each other, due to the near
approach of the line of action to the perpendicular to the line of centers; and
this is evidently necessary for most
favorably  transmitting  the  rotative
forces from one wheel to the other. For
instance if the line of action be made
parallel to the line of centers no rotative
effect is secured.




30
2D. FLANKS PARALLEL.
By employing smaller generating circles than in the above the flanks may be
rendered more nearly parallel as shown
in Fig. 10; and the bases of the teeth
FIG. 10.
/ /
thickened  as  required  for strength.
Clearing curves may still be introduced
for further strengthening of the teeth.




31
No rule of proportions is prescribed: in
each case the diameters of the rolling
circles must be determined by the judgment of the designer.
The flanks are only approximately
parallel, because, in reality, they are the
slightly curved hypocycloids.  But for
that small part against which the faces
act, a straight line may be substituted
without sensible error.
The rolling circles being smaller than
for the case of radial flanks, the epicycloids will be smaller for the same pitch
lines, and hence the points of the teeth
narrower.  This results in  a greater
general inclination of the side of a tooth
to a radius, causing greater inclination of
the line of action from the perpendicular
to the line of centers, inducing increased
lateral crowding, and consequent friction.  In  a long train  of gearing it
is therefore doubtful whether this form
of tooth is any safer than the preceding,
because with greater strength there is
greater resistance.




32
3D. FOR MINIMUM OF FRICTION.
From the preceding examples, it appears that the friction is reduced by approaching the line of action to the perpendicular to the line of centers. The
line of action in epicycloidal teeth being
the chord to the generating circle which
connects the describing point with the
point of l:olling, it appears that the most
favorable condition is secured by making
the generating circles large as possible.
Again the generating circles, when
situated as in Figs. 9, 10 and 11, form
the path of the points of contact in their
journey from the point of beginning, to
the point of quitting contact, one circle
for one side and the other for the opposite side of the line of centers; so that
large circles, in both senses, appealr most
favorable.
Fig. 11 presents these conditions, and
indicates that the teeth are very much
undercut; indeed, in extreme cases, entirely cut off by reason of the convex
flanks, due to the excessively large rolling
circles.




33
FIG. 11.
/  /
It is evident that such teeth cannot be
used where great strength is require(l, as
in mill-work; but where the resistance
is slight, and smoothness of motion and
freedom of action are requisites, as in
horology, this form may be used with
great advantage.  Fig. 12 is a careful




34
FIG. 12.
____K
drawing of such a pair of wheels, with
teeth much undercut.
4TII. CHANGE GEARS, AND SETS.
The most common use of change gears
occurs with the machinist's engine lathe
for screw-cutting. In this, and most
other cases requiring such wheels, the
teeth are small; and when only approxi



35
mate, occasion no especial inconvenience.
But sets sometimes occur in mill-gearing
where correct forms of teeth are necessary.
For such cases it is only needful that
onie generating circle be employed for
the whole set.*. This circle should not
be more than half as large as the smallest wheel of the set. The teeth of the
latter will then have radial flanks, while
those of all the others have concave
flanks and thick bases.
5TH. PIN;WEAnRING.
By increasing the generating circle,
Fig. 11, the hypocycloid decreases until,
when the former coincides with the pitch
circle, the latter becomes a point, the
tracing point itself. Then the theoretical teeth are points for one wheel, and
for the other epicycloids generated by
the opposite pitch circle as shown in
Fig. 13. In practice the points are en* This solution was first given by Prof.Willis, see Princ.
of Mech., 2d ed., pp. 87, 118, and 136, the latter as in connection with the Willis Odontograph.




36
FIG. 13.
larged to pins of cylindrical form, and
the teeth reduced to lines parallel to the
epicycloids and distant from them a
space equal to the radius of the pin.
Such wheels are sometimes called pin
gearing.
These wheels should work smoothly, because the line of action is suitably situated for a minimum of friction, but they
are not free from objection'for having
the tooth contacts all on one side of the
line of centers. These one-sided contacts
serve best when the pin wheel is driven,
and in this way this gearing is much




37
used in modern clock-work.  The pinion,
called.callowoer, tb',dle or l/tnter~n wheel,
is usually made for clocks by inserting
wires between two collars; while the
wheels are cut from  plates, many of
which are stacked and cut at once.* In
this wav should the wheel be quite thin,
and wabble, the trundle may have sufficient length  of pins to accommodate.
Such gearing is cheaper than thick cut
gearings, and may explain its introduction into the inexpensive, though very
good, American clocks.
But it is probable that the very best
conditions for clock and watch gearing
would be secured by a sort of cros*s between Figs. 11 and 1., represented in
Fig. 14.  The generating  circle should
be so large that the hypocycloid, for
both pitch lines, spans a breadth just
sufficient for the tooth thickness, and the
whole hypocycloid takeni as that part of
a tooth, or pin, which lies within the
pitch line.  The balance of the tooth
* In the full sense of the term pin gearing the wheel
teeth are also pins set into the edge of a disk.




38
s]iould then extend outward, to a p,oint,
similarly as in Fig. 11.  Thus COntacts
FIG. 14.
w\-ill be secured on both sides of the liiie
of centers, as required, with the nearest
approach of the line of action to the perp1endicular to the line of ceinters consistent therewith.
These wheels could be constructed by
inserting suitably  formed pins by on1e
end into disks, and projecting toward
each other from tile tNwo whleels; or, by




39
supporting the teeth upon slender necks,
as shown, which is possible, because only
small parts of the hypocycloids  are
brought into contact with the faces.
It is plain that if the line of. action
could be straight, and perpendicular to
the line of centers, the wheels could be
approached or withdrawn without causing great changes of conditions of action
of the teeth. The form, Fig. 14, witlh
teeth supported upon clearance curvenecks, would, therefore, be one of the
best for this purpose, and nearly the
same as found on the rollers of clotheswringing machines, &c., as arrived at,
probably, by the demands of practice.
II. INVOLUTE TEETH.
The involute curve is a special case of
the epicycloid in which the generating
circle is infinite.
Let A and B represent two pitch cilrcles in contact at C. Draw any inclined
right line through C and two circles
tangent to it with centers A  and 1B.
These are taken as the base circles for




40
FIG. 15.
involutes.  The ratio of the radii of
these circles will be equal to that of the
pitch circles. Suppose the inclined line,
C D, be produced, and wound upon the
base circle A; and also separated near
C, with a tracing point attached.  As




41
the string is unwound, the tracing point
will describe an involute which may be
taken for the side of a tooth of A as
shown. A similar unwinding from B
will describe the involute tooth-curve for
it. It is plain that the contacts will all
lie on the inclined line CD, because it
will cut normally all the involutes described on the bases A and B. This is
at once the line of action, and locus of
the points of contact.
In the present case the line of action
is inclined more or less, at pleasure, but
must always pass C.  This change of
inclination may follow from varying the
diameters of either the pitch or base
circles.  The former corresponds  to
changing the distance between the axes
of the wheels, which these gears are well
known to admit of, and work correctly.
But when the distance is considerable,
the teeth become monstrous thick at
base, and objectionable; and unsuited
for such purposes as clothes-wringers.
The contact for these teeth is only possible between the points of tangency




42
of the line of action with the basecircles. Hence, by enlarging the basecircles, these limits are constricted, and,
finally, vanishing when the line of action
becomes perpendicular to the line of centers, giving no contact.'I'The best condition is a matter of judgment, and the designer must choose between objectionably  few  contacts, and  objectionable
inclination of line of action and thick
teeth.
The contact being confined to the line
of action, limited as above, it is )lainl
that no bearing can occur upon a tooth
below or inside the base circle, and usually clearance curves are arbitrarily struck
in to give the necessary depth for receiving the points of the teeth of the other
wheel.
One fact is worthy of note, serving to
put these wheels in comparison with t]he
epicycloidal. If a circle be struck with
A C as diameter, it will always pass tl e
point of tangency D. That point, in involute gearing, is the limit of contact of
teeth; but not in epicycloidal gearing




43
with radial flanks, though that will be
one position. Intermediate between D
and C the contact for involutes is on the
line C D; and for epicycloids, on the arc
C D.  The line of action in the latter
case will vary from a perpendicular to
A B, to the inclination C D, while the
point of contact moves from C to D.
Hence, the average inclination of the line
of action between C and D is much more
favorable in epicycloidal gearing than
involute, from which it appears that the
former is the preferable form of gearing.
Fig. 15 may appear to furnish an example which is inconsistent with our
general theory, because the generation is
effected by rolling the describing curve
or line on the base circle instead of the
pitch line. But this is reconciled thus:
In Fig. 16 take a pitch circle A C, base
circle B D, describing line D F, with describing point F, and a log. spiral F A.
Now  as the describing  line unwinds,
tracing the curve C F, the line will cut
the pitch circle at various points A at a
constant angle, which follows from the




44
concentricity of the pitch and base circles. As the radius-vector makes a constant angle with the are of a log. spiral
if that angle at A, be the same as DF
with the circle at A, then the radius vector'F A will coincide with the line D F
continually, and the tracing point will
describe one and the same curve whether
carried by the spiral F A or line F D.
This curve begins at C. To supply the
remaining part C B roll the same spiral
FIG. 16.
D
inside as shown. This description will
stop at 13 for the reason that the log.




45
spiral then bears on the inside of the circle where its curvature equals that of the
circle. Hence this is explained by the
general theory, and the wheel has true
flanks and faces, separated by the pitch
line, as usual.
III. RAND)OM TE:ETH.
These might be styled v'cacdom  teeth
because a tooth-curve for one wheel is
drawn at random, or at least arbitralry,
and the form  of tooth for the other
wheel found from it. The solution on
the drawing board is the same as Fig. 2,
and hence, it is in keeping with the
general theory of the teeth of wheels. It
would need no further consideration except for the exceedingly advantageous
p)ractical process for obtaining the second
curve, first made known by Prof. Willis
in 1837.
Let A and Bl represent the pitch lines,
forlmed by cutting strips to the right
curves.  A  tooth curve C is cut at
pleasure, and mounted on A so that the
card board D may play between A and




46
FJIG. 17.
0        o oiB
i     i
C. The card board D is secured to B.
The pitch lines are then placed in contact
as shown by the dotted lines, and a line
traced on D, along the edge of C. Then
FIG. 18
1C
A is rolled oil B, without slipping, to a
new position, and a second line traced
by the edge of C. Thus proceed for
various positions. An enveloping curve
traced on D will be the required tootlhcurve for B.




47
This process would apply admirably
for matching a gear already existing,
and no matter if it be worn provided it
is regular.
rATI'C'TICAIL OPERA'TIONS.
In  actually  executing  wheel-work,
there are certain' operations which contribute to the accuracy and dispatch.
The draughtsman can draw his tracing
curve in several positions, and by stel,ping with spacing  dividers arrive at
several points in the desired tooth-curves.
But in thus measuring off distances on
arcs, errors occur in that the dividers
measure chords instead of arcs. Also
each placing of the divider points involves a small error depending upon eyesight, precision of draughtsman, &c.,
Again several operations, following and
depending upon each other, between the
theory and the result, entail cumulative
errors, sometimes so great, that the result obtained  scarcely  resembles the
truth.  The fewer the intervening operations, the less will the truth be adulterated with error.




48
Templates are valuable because tliey
secure a more direct relation between the
theory and final tooth-curves.  In some
instances a still closer relation has been
obtained. To illustrate, the templates
give the curve, and then the teeth or
cutters are to be formed from it by an
eye-and-hand  process.  But in  some
watch-works, epicycloidal machines are
used in which the cutters are formed direct from the rolling curves. These cutters however are nm(ster cutters from
which the working cutters are formed.
But this last operation admits of great
precision.
In Fig. 17, the pieces A and B are
called pitch templates and a piece cut
to the form of the describing curve as in
Fig. 3, is called a describing template.
By a set of such templates the operation represented in Fig. 4 may be undertaken.  Tooth-curves may thus be drawn
on metal plates, and subsequently cut
carefully to the line.  Such a pattern
may be used for turning up a cutter by,
o-r for marking out the teeth on at gear




49
pattern of wood. In the latter case, it
may be called a scribe template.  And
it is convenient to mount this upon a rod
attached to swing around a center pin to
the gear.  Either of these operations involves two eye-and-hand processes, as, for
instance, dressing up the scribe template;
and then turning the cutter by it. lBut
it is not convenient to reduce the lttnlber further, though, by aid of an odoetograph, described  in Part II of this
number, the  inconvenience of resorting  to  pitch  and  describing  templates, is avoided from the fact that the
odontograph  serves as a ready-made
scribe template, as exact as a special
scribe template laid out and dressed up,
as above described, would be, and which
is also adapted to being mounted t(c
swing around the center of the gear.








PART II.
ON A
NEW ODONTOGRAPH:
An Instrunzentfor laying out
THE TEETH OF GEAR WHEELS.
THE WVILLIS ODONTOGRAPtI.
THOSE who have used, or even studied
the Odontograph of Professor Willis, of
Cambridge, England, cannot be otherwise than forcibly impressed with the
admirable simplicity and great utility of
the instrument. Indeed, it has attained
to such general favor that it is not only
in the hands of a large proportion of the
constructors of mill gearing in Europe
and America, but in every work of importance on Cinematics, Principles of
Mechanism, Machinery and Millwork,




52
&c., published  since its discovery in
1838. In fact, the instrument is worthy
of a place in the tool chest of every
millwright and machinist; particularly
those who have no better understanding
of the mode of delineating gear teeth,
than to use, for generating the faces of
teeth with radial flanks, describing circies whicih are the opposite pitch circles
themselves, instead of circles one-half as
harge.
Although the teeth of wheels, laid out
by the aid of the odontograph, are only
alpproximately correct, such teeth  are
not very far from correct working forms.
This fact was practically demonstrated
to the writer at the justly-celebrated
machine factory of Wim. Sellers & Co.,
of Philadelphia, where a pair of heavy
cut gears were seen in action, running
with extraordinary quietness. On inquir-y
Lis to the remarkable working quality of
these gears, it awas stated that the cutters were made expressly for them  ill
strict conformity with the Willis odonto-,r1aph; which, in the present instance,




53
it is fair to suppose, was an experiment
to test the odontograph teeth, as compared with those laid down correctly by
theory.  Teeth formed thus carefully,
even though they be only approximate,
are much to be preferred to the rattling,
guess shapes; and amply  repay  the
trouble required to secure them. It is
therefore much better, in cases where
the theory of gearing is not understood,
to resort to an odontograph, than to
jump blindly at a rattle trap.
For example: a common form among
millwrights is obtained by taking the
middle point of a tooth at the pitch line
for a center, and describing, by arcs of
circles,_the nearest flanks of the next adjacent teeth, and the further faces of the
same two teeth. Describing, thus four
arcs from the center of each tooth, all
the faces and flanks are laid out. This
gives flanks which are concave, with
radii which are less than those of the
convex faces, while the opposite relation
is known to be required in correctly laid
out teeth.  This rule probably gives




54
shapes which are worse than mere guess
shapes.
A GOOD ODONTOGRAPII DESIRA1ILE.
The delineation of correct working
gear teeth according to theory for individual cases is a very easy matter when
the method is understood, but not so
easy as to use a convenient odontograph;
and hence, a good odontograph is advalntageous to all designers of gearing.
ESSENTIAL QUALITIES'I    OF A. GOOD ODONS-TOGRLAPH.
An odontograph, to give the highest
satisfaction, should be one which leaves
nothing to be desired; neither in the degree of approximation, nor the general
form of tooth it gives.
Those who have employed the odontograph of Prof. Willis, cannot fail to
have noticed that in wheels of many
teeth the form is very different front
that obtained by laying out the teeth of




55
the epicycloidal form  with radial flanks,*
which  form  is the  most common and
usually the most acceptable. Also tlhie
odontograph must be simple, and  easy
of application.
P'ECUIARITIES OF TIHE  WILLIS OI)O1TCOGRl APII TOOTL.
A tooth of a wheel of eighty teeth, as
laid down by aid of the Willis odontograph, is shown in Fig. 1; and also, a
little  to  one  side in dotted  lilies, the
same  tooth  of epicycloidal form  with
radial flanks, as calculated to work with
a second wheel of twelve teeth. The
clearance curves are not put in.  The
odontograph tooth exhibits very marked
peculiarities, a noticeable one being the
greater general inclination of the side
of the odontograph tooth to the radius
of the wheel; giving rise to unnecessary
When these teeth would appear too weak at the root
or base, clearance curves are usually struck in, or curves
between the flanks which the points of the teeth of the
opposite wheel will just clear, up to which the roots of
the teeth may )be thickened for the purpose of greater
strength.




56
obliquity of action, and consequent mutual crowding of the wheels against
their shafts.
Fig. 1.
/             \
/!
Prof. Willis limits his odontograph to
twelve teeth for the least number, and
for this the teeth have radial flanks as
shown in Fig. 2, which works with a
wheel of eighty teeth. This limitation
causes the teeth of small wheels to be
narrow at the base, and of large wheels
to be unnecessarily wide.  In wheels
having a not unusually large number of
teeth, the flanks start out from the pitch




line towards the center with a divergence
fronm  each other approaching 300.*  In
Fig. 2.
//          \
ouher words, the odontograph teeth approach the narrowness of base of epicycloidal teeth with radial flanks when the
latter are objectionably narrow, as in
wheels of few  teeth; and much wider,
indeed  excessively wide, when, for the
same wheels, epicycloidal teeth are of
good form.  Also a marked peculiarity,
~ See Fairbairn, Mills and Millwork, Part II, Plate IX,
Fig. 1. A. Lambert, Cinematique, P1. 14. M. CH. Viry
Cours de Mkchanique, p. 194, Fig. 30.




58
as exhibited in Fig. 2, is the decided
angular junctions of the flanks and
faces.
RESTRICTION OF THE WILLIS T'IEORY, ANI)
MODIFIED FORM OF ODONTOGRAPH.
These undesirable features are, however, in the main, no fault of the general theory of Prof. Willis for the formation of teeth with circular arcs, but appear to be due to restrictions imposed,
for the purpose of simplifying the tables
for, and application of, the odontograph
itself. The restrictions which contribute
mostly to this are the fixed angle of 75~,
and the constant length, relatively to the
pitch, of the perpendicular to the line of
action. By restricting the theory in a
different manner, a certain modification
of the instrument and its application
may be effected, which gives straight
and more nearly radial flanks for all
wheels, and circle arc faces. This will
be treated further on.




59
NEW FORM OF ODONTOGRAPH.
For those who will adopt nothing
which is so incomplete as a circle are for
a tooth face, but will lay down the correct curves; and for even the less theoretical who would prefer to accept a
good practical short-cut to a correct result than to go by guess; I would offer
the new odontograph, which is now to
be described: an instrument which in no
circle arc, mode or form, resembles any
thing above referred to.
This new form of odontograph is not
only adapted to give iradial flanks, but
faces which agree with almost undiscoverable nearness with the epicycloidal face.
The face curve, as laid down by the instrument, is exactly normal to a tangent
to the pitch line drawn from the middle
point of a tooth, and it intersects the addendum circle precisely where the true
epicycloidal curve proper to the tooth in
question does.
Now, as the pitch line and its taingent,
pass either way from t]ie middle of the




60
tooth, they  depart from  each  other
slightly in distance, and more so in direction.  A circle arc, approximating to a
tooth face, will generally have a radius
considerably greater than the half thickhess of a tooth: )but for a circle arc to
be normal to the above-named tangent
and to the pitch circle at the side of a
tooth, )both at the same time, its radius
must be much less than the half thickness of a tooth, as is evident on inspection; an(l hence, the circle arc cannot
osculate closely with the epicycloid, and
at the sanue time be normal to the pitch
circle, as Figs. 1 and 2 show.  The most
-closely osculating curvNe must, therefore,
be one whicll rapidly changes curvature;
and such a curve will be very learly
normal to the above melntioned tangent
to the pitch circle.
The odontograph  should, therefore,
give a curve which changes curvature
rapidly. Now, as the epicycloidal curve
is normal to the pitch line, and very
nearly so to the tangent to the pitch
circle drlawn from the middle of a tooth,




61
it is clear that if a curve of rapidly
changing curvature be so placed as to
be normal to the tangent, as above described, and at the same time intersecting the addendum circle at the same
point that the epicycloidal curve required
for the tooth considered does, it will
lrepresent the epicycloidal tooth face
with great precision. This truth is rendered apparent to the eye in Fig. 3, in
which the full line is the odontograph
curve, and the dotted line the required
epicycloid. The flank is also drawn
in to give an idea of the conformity of
the whole tooth, as delineated by aid of
the instrument, with the true epicycloidal
tooth. These teeth belong to a wheel,
the pair of which are of equal size. The
object of making the curve perpendicular or normal to the tangent, as above
described, instead of the pitch circle, is
to render the odontograph more convenient of application; and it is found that
the curve selected, in the manner hereafter explained, meets the radius for extending the side of the tooth into the




I/
/
l /
I I
/
I  /
\  \\
\\
I0
FA-1Fe~ ~ ~ 1
C'




63
flank with an error of tangency, which
is too small to be readily detected.
With these explanations, it becomes
evident that the leading propositions of
the new odontograph are:
1st. That it give a curve of rapidly
changing curvature, having the closest
possible osculation with the epicycloid,
and at the same time be of general application.
2d. That the curve for a tooth face,
given by the instrument, be normal to a
tangent line to the pitch circle at the
middle of a tooth.
3d. That this curve intersect the addendum circle precisely where the epicycloidal curve proper to the tooth in
question does.
FORM OF CURVE ADOPTED.
The curve adopted as conforming most
closely, in general, with limited initial
portions of the epicycloid, is the logarithmic spiral.
This curve appears to possess the
highest degree of adaptation, because of




64
its uniform rate of change of curvature,
and also because this rate can be assumed at pleasure.*  Furthermore, the analytical relation of this spiral to the problem is simple.
FO-ItM AND APPLICATION OF INSTIUtMENT.
In planning this odontograph, the
leading endeavor was to make its practical application the most convenient possible, without a sacrifice from  accuracy
of form of tooth obtained by it.
That the reader may learn the leading
features of the instrument before proceeding to the details of its demonstration, its form and mode of application
will first be explained.
The odontograph is shown full size ill
FiO. 4, as of suitable capacity for laying
out all teeth below six inches pitch. Its
capacity, however, is extended to any
degree by simply prolonging the curved
edges. It should be made of metal, be* These points follow from the fact that limited portions falling within a given angle at the pole are similar
figures.




65
cause it is intended that the instrument,
when desired, may be used directly for a
scribe template, in which use it will be
subject to wear from the passes of the
scribe. It has several holes countersunk
on both sides as shown, so that it
may be attached by wood screws, or by
bolts expressly prepared, to any convenient wooden rod, in such a manner that
when the rod swings around a center-pin
to the wheel, all the faces of the teeth
may be described directly from the instrument itself. The desired result is
thus obtained directly without intervening center points and dividers.
In this manner the odontograph becomes a general or ready made tenmplate, and equally valuable for guiding
the draughtsman's right-line pen, or
the pattern maker's steel scribe.
To place the instrument in position
for drawing a tooth face, a table is used
which should accompany the instrument.
From this table a value is taken which
depends upon the diameters of the pair
of wheels, and the number of teeth in




66
the wheel for which the teeth are sought.
This tabular value, when multiplied by
the pitch, is to be found on the graduated edge AD DB, Fig. 4, of the odonto.
graph. This done, draw the tangent
D EF to the pitch line at the middle
point E of a tooth; and lay off the half
thickness ED of tooth on either the
tangent line or pitch line. Then place
the graduated edge of the odontograph
at D, and in such position that the number, and division of scale found as above,
shall come precisely on the tangent line
at D. Also get the curved edge, H F C,
so that the curve will just lie tangent to
the tangent line, as at F. Then all is
ready for tracing the curve for the tooth
face from the pitch line through D toward B as far as needed. liy turning the
instrument, which is graduated on both
sides, over, and doing likewise,.we get
the opposite face of the same tooth.
Then we have to simply draw the radial
flanks when the tooth becomes fully delineated, with the exception of limiting
the point and root. Clearance curves




67
may also be struck in, if desired, as in
any other case.  Of course, when the
setting of the instrument is once made,
it may be mounted upon a rod, as, and
for the purpose described above, for
drawing all the teeth.
FOR INVOLUTE TEETH, ETC.
The curve which this instrument gives
will closely represent initial portion of
the hypocycloid, cycloid and involute, as
well as the epicycloid, as far as required
for a tooth face; and hence the instrument is adapted for tracing teeth of these
various forms including the rack and
pinion, internal gearing and involute
teeth. For the latter form of teeth the
tables become very simple.
CONDITIONS OF TOOTH CURVES.
Before proceeding to the demonstrations, it will be necessary to understand
how the various curves, used for tooth
outlines, are conditioned.




68
To generate the epicycloidal faces for
teeth with radial3flanks, the generating
circles, which roll upon the pitch circles,
should have diameters equal to the radii
of the opposite pitch lines, as set forth
in Part I of this manual.   Hence,
as the odontograph is proposed to give
nearly the epicycloid face for radial
flanks, the above epicycloid is the one it
should approximate to.
When one of the pitch circles is infinitely enlarged, we have the case of the
rack and pinion, the epicycloid of the
former being the cycloid generated by a
circle half as large as the pinion; and its
flanks are parallel to each other. Also
the generating circle, whose diameter
equals the radius of the rack, is infinite,
and hence the generating circle for the
pinion teeth is a straight line. This
makes the faces of pinion teeth involutes,
with the pitch circle the base of the involutes.
Therefore, for the rack and pinion with
radial flanks, the faces of -the rack are
cycloids, and of the pinion involutes.




69
For the case of in.volulte teeth for involute gearing in general, the base circles
of the involutes are usually taken a little
within the pitch circles; the ratio of the
base circles being equal to that of the
pitch circles. The relation between a
lbase circle and the pitch circle containing it is arbitrary, being generally assuned, in each case, according to the
judgment of the designer, and any rule
given as regulating this simply expresses
the judgment of its author.  The table
for involute teeth is therefore adapted to
suit any assumption in this regard.  In
this case, however, it should be borne in
mind that the tangent line, used in setting the odontograph, should be drawn
to these base circles instead of the pitch
circles, because the involutes are normal
to the base circles. But the half pitch
should still be laid off on the pitch lines.
The case of internael gearing is pecllliar, in that radial flanks for the larger
wheel are impossible.* That flank which
* See Belanger, Cinematique, p. 121.




7i)
approaches nearest to radial, is therefore
adopted, and this can be readily shown
to be the involute, with the pitch circle
its base.  Observing  that the larger
wheel has its flanks outside, and faces inside the pitch circle, we see that the
flanks must be generated by rolling a
curve on its exterior. Rolling a circle
thus, we see that the epicycloid generated rises more nearly in a radial direction,
as the circle is increased in size; because
the latter curve reaches the greater
height. Carrying this to the limit, we
have the infinite generating circle or
straight line, and hence the involute
curve. But suppose we adopt a rolling circle of negative curvature, as in
Fig. 7, Part I. Now for this mode
of generation, it is easily shown that
the curve generated will be an epicycloid exactly similar and equal to that
obtained by rolling a circle of positive curvature, outside the pitch circle
as before, the diameter of which is
equal to the difference of diameters of
the generating circle of negative curva



71
ture and the pitch circle.*  Itenee, in
either case, we get similar results, and
the steepest epicycloid is the above
named limiting one, viz., the involute.
The flanks of the larger wheel are
therefore concave involutes.  Using the
same generatrix for the faces of the
pinion, the  latter will have  involute
faeces.
The faces of the larger wheel will be
hypocycloids, because internal, and generated by a circle whose diameter equals
the radius of the pinion.
In bevel wheels, the pitch circles should
be regarded as small circles of a spheret
which has its center situated at the point
of intersection of the axes, or shafts of
the wheels.  These pitch circles are not,
in the case of bevel wheels, the proper
circles to lay out the teeth  upon, but
should be regarded as the bases of cones
whose vertices are ill the axes, outside
* Bour, Cours de Cinematique, p. 120: Note. Duhamel Elements de Calcule Infinitesimal.
t Willis, p. 146; Rankine, p. 144; Fairbairn, Mill and
]Millwork, part 2, p. 38. A. Lambert, Cinematique, P1. 15.




72
the sphere, and which, developed upon a
plane surface, are the desired circular
arcs on which as pitch circles to delineate
the teeth.  The describing of teeth on
these circles, being the same as oil the
pitch circles of spur wheels, the odonto(graph applies to them in the same manner, their radii number of teeth, &c.,
being employed in connection with tlle
tables, instead of those of the real pitch
circles.
In all these cases the same rule will
apply for use of odontograph, except in
involute gearing, with which we lhave ai
exception as above explained.
PA:R'l Ct:ULA.R LORITHlAMiC SPHIZAL
ADOPTED.
In adopting the particular logarlithmic
spiral for the odontograph curve, inasmuch as this curve may have an infinite
variety of obliquities, it is evident that
the selection is not a matter of indifference.  When the obliquity, or angle be_
tween the normal and radius-vector, is
vely small, the arc of this spiral changes




73
curvature less rapidly than when the ohliquity is great: when the obliquity is
zero, the spiral becomes a circle; andl
when it is 90~, the spiral is simply a radius; neither of which approximates to,
the epicycloid.  To find that obliquity
which makes the spiral best fit the epicycloid, it will probably be most saitisfactory to assume an epicycloid  lichll
represents an average of those likely to
be used for both curves, and adapt tile
spiral to it; though any ordinary logarithmic spiral will evidently conform more
closely with it than the circle.  T''he
spiral which most closely osculates the
epicycloid for a pail of equal pitch
circles, is therefore adopted as the average spiral, because the opposite wheel
may be made larger or smaller, tll us
making a higher or lower epicycloid.
Also, it is preferable that the spiral fit
for the fewest teeth, because for this the
longest portion of the epicycloid would
be required.  The fewest number assunmed is ten.
Let A F, Fig. 5, represent the pitch




74
line, A B the epicycloid above named,
dotted beyond B, and the full line CAB,
0   0
the osculating logarithmic spiral with its
the osculating logarithmic spiral with its
pole at C. Draw normals G E and B D.
These normals will be common to both
the epicycloid and spiral, if they closely
osculate each other at A and B. Also
ED will be the evolute of both. The




75
point B is supposed to be 0.3 pitch*
above the pitch line, as the ordinary
height of tooth above pitch line.  Or
what is nearly the same, AB = 0.3 P.
Also A G is assumed = 0.25 A B. Draw
the normals or radii of curvature G E and
B D. For the particular case underconsideration we find for the epicycloid,DE=
2 GB. This fact is most conveniently proved graphically on account of the complication of the analysis. This solution was
made with great care on a drawing, and
it was also checked by an approximate
analytical method.  The results very
nearly agreed, though as that by analysis
was necessarily approximate, the graphical result was adopted.  It happens to
be also a very convenient value. The
graphical operation was, in fact, partly
by analysis thus: The normal to the
epicycloid always coincides with the
chord of the generating circle connect* In all cases in this investigation, requiring dimensions to be given to the teeth, the common and wellknown rule is adopted, viz.: Thickness of tooth, P-l- P.
Thickness of space, A-j" P. Height above pitch line, -ia
P. Depth below, -,a P. See Willis, p. 113, &c.




76
ing the point of contingence of this and
the pitch circle with the tracing point.
This determines the direction  of  G E
or B D, where B F, for instance, is a
chord.  The true radius of curvature B D,
for an epicycloid, is' B F* in the present
case for radial flanks where the pitch
circles are  equal, or generating  circle
half as large as the directing pitch circle.
This value of B D or G E, in terms of the
chord, is from the well-known formula for
tie radius of curvature for epicycloids
in general,j and hence the length is computed.
Now  the  logarithmic  spiral sought
must have
E D
D —  2-tangent of obliquity.
The evolute of the logarithmic spiral,
being an equal spiral,t makes CEI)D  a
spiral like C A B. Also it is well known
that the radii C B and C D have B C D a
right anglec  when these radii vectores
* Davies and Pecks, Math. Dic., p. 222.
t See Rankine, Mach. and Milliork, p. 60, note. Redtelnbacher Resultats, p. 3, rule 4. Willis, p. 142. Belanger,
Cinematique, p. 103, &c.
See Courtenay's Calculus, p. 181.
See Courtenay's Calculus, p. 173.




are drawn to the extremities of a radius
of curvature BD, and similarly for ECG.
Tlhis, with the second property of constant obliquity of the logarithmic spiral,
makes the two triangles B C D and G C E
similar. From  this it follows that the
triangles C B G and C D E are similar,
and hence
C 1: B G   CD: DE,
DE    CDtanget of obliquity
B G - B C =tangent of obliquity-2.
The equation of the logarithmic spiral is
log. I- aC 
in which Ca-tangent of obliquity, or a= 2;
and therefore the logarithmic spiral
sought has the equation
log. I= 2 H
or                   2
or
~101'1           2(-2
l   2'
Values of the radius vector, 1, have
been carefully computed for the purpose
of laying out the odontograph curve




78
with accuracy. These were computed
for the intervening angles of 22~ degrees,
which divides the circumference into
sixteen equal parts; and for some of
the larger values the radii subdividing
these angles as follows:
For 0, —                  2.=922~:3
H-=   - 1       i, =   0.500!)
" =H1+ 22       1,=  1.09(6i
"=,+ 45"   l1 =  2".4052
"= —+  67.1~ 0 l -         2d.
1J                 1 4
"=0+  90        11 -. a370
1= 8  + 11      I,` =2. 3(t
"-0 + 123a~   16i =-.37.5820
0 + 13- 5     1 =-55.6560
Withl these values, the odontograph,
as showni in Fig. 4, was laid out, not only
the curve A D B, but C F IT; these being
similar and equal as above explained.
From the fact that tangent obliquity — 2,
we may, when a radius I is laid off for a
point on A DB, draw a line at right
angles, and lay off on it 2 1, which will
give a point on C F H.




79
EQUATION OF THE ODO:NTOGIIAPLI CURVE.
In the analytical demonstration of the
applicability of this curve, an equation
will be needed for it, which can be combined with those for the curves which
are proposed for the outlines of teeth.
For reasons which will appear hereafter,
an equation is required which gives the
relation between the abscissas a e and
the ordinates e b, Fig. 6. Let a c be the
radius of curvature of the spiral at a.
Let fall upon this a perpendicular b e.
Also prolong a c to d. Some point d
can always be found with which, as a
center, a circle may be struck through a
and b. Now let us propose that c l =
qX e b as a relation sufficiently exact for
any possible height of tooth,  4 b, above
pitch line.
Then we will have from the property
of the circle,
a e (2 a  — e) =,  2,
or calling a e=y and e b —x, a c= — r,
we have
y I2 (' +'(I ) - 1 =,x2




80
whence
Y/         /
b?'\
If, in this equation, q proves to be
constant, we have the equation desired.
To this end the following results have
been very carefully determined graphically, this method being the only one
admissible for complexity of formulas:




81
For r=2.0 inch x —1.1 inch q —.596
r=2.0 inch x=2.0 inch q-.588
r=4.4 inch x=1.0 inch q-.561
r —4.4 inch x=2.5 inch  —.557
T=4.4 inch x —4.5 inch q=.562
— 9.0 inch x=0.5 inch q=-.595
r=-9.0 inch x=-9.0 inch q=.599
r= 9.5 inch x —2.2 inch q=.555
r=9.5 inch x —5.5 inch q —.600
Alean.. q=.579
These figures show that q is very
nearly constant; in fact, the differences
caln only be accounted for as errors of
the graphical operation; errors which
might be expected for the reason that
the point el, Fig. 6, may vary its position
considerably, and yet cause but slight
variations at b. We may therefore adopt
this, not only as constant, but as the correct value of q for the particular spiral
chosen, and this value was employed in
the formulas when computing the tables
for the odontograph.  Equation (1) is
entirely empirical, and will not apply to
the spiral for a very great range; though




82
to an extent greater than ever needed
for the present purpose, it is practically
exact. It is necessary to resort to the
empirical method, because the equation
of this spiral cannot be expressed in rectangular co- ordinates.
DEMONSTRATION FOR EPICYCLOIDAL SPUR
GEARING.
To show that the odontograph, as
above proposed, is applicable to the case
of ordinary epicycloidal gearing-teeth
with radial flanks, it will be necessary to
prove that the value of r to be found on
the scale by which the instrument is set,
can be computed when certain functional
parts of the gears in question are known;
and also, to show that it can be expressed
in so few terms, that values of it may be
computed beforehand, and arranged in
tabular form, it will be necessary to
resort to mathematical expressions for
r, by which it can be discussed, and by
which we may determine the number of
quantities required to express it.  For
instance, if it can be expressed in terms




83
of two different quantities only, one
table will suffice for all values of r. But
if it requires three quantities for its expression, a series of tables will be necessary.
By aid of the equations of the epicycloid the above co-ordinates x and y may
be expressed in quantities belonging to
the wheels.
Let R=the radius of the pitch circle
of the wheel for which the teeth
are sought, whether it be the
larger or smaller.
Let N=the number of its teeth.
Let r=the radius of the wheel which
is to work with the wheel sought,
and equal to the diameter of the
rolling circle for generating the
faces sought.
Let n=the number of its teeth.
Let P=the pitch of the teeth=- -N
Let -=the radius of curvature of the
curve of odontograph, as above,




84
the numerical value of which is.also the number to be used for
setting the instrument, also the
tabular numbers.
Let x and y=co-ordinates as in eqtuation (1) of odontograph  curve
above, x=0.3 P.
Let  = —an angle, which may be considered as an auxiliary quantity,
simply; or as the angle which locates the center of the rolling
circle.
Then the well-known * polar equation
of the epicycloid is -
(rad.vector)%=  (R +~x)-R + r  R +' 2
2R
1 —cos. 2 (  -
whence
cos.    i —     
+R~ 1+-R t
See Price's Calculus, vol. i, p. 349.




85
or
01.3 
I+
eos. —— i —- -                     (l)
-1+ 2N
This value of 0 is in terms of the two
P 
quantities R and N only, as desired.
The polar equation is here employed
instead of that referred to rectangular
co-ordinates, because the former is resolvable with respect to 0, and the latter
not. The latter, however, would have
been preferable to the extent of about
the difference between the height of b,
Fig. 6, above the tangent and above the
pitch line, because b e, which is aibout
midway between a, and the middle of
the tooth, is almost exactly equal to.-, as
given by the rectangular equation.  B'ut
this difference is very small in wheels of
the fewest teeth, say 8 or 10(, and illal,preciable in larger wheels.
An expression for y is mnost readily
obtained by employing the well-known *
* See Price's Calculus, vol. i, p. 283.




86
equation of y in terms of 0 for rectangular co-ordinates to the epicycloid.
In Fig. 7, let A B represent the face
of the tooth for the wheel whose center
is 0. Take C D the height of the tooth
above the pitch line reckoned on the cen



87
ter line of tooth, and = B F = 0.3 P,
nearly = x. Let C G represent the tangent to the pitch line at the center point
C of tooth, as above described.
Now, since the co-ordinates x and y in
equation (1) of the odontograph curve
represent the height of B above the tangent C G, and the distance from the face
along the tangent C G to the perpendicular let fall from B upon C G respectively,
or equal to B F and A a respectively,
very nearly, we have to find, by help of
the equation of the epicycloid, between
y and 0, the value of A a. But this
equation gives E B, or A b. Drawing
B b parallel to A E, and B a parallel to
D C, we have
b: FB   AC: CO
or
ab: x:  P  R
since the thickness of tooth=A- P.
Hence
5 xP  5 0.3 P2 p2
ab=         2 —    _-0.06819
22 R22  R  -R




88
fand hence, by aid of the equation of the
epicycloid,
or
N +         -- — 0.06819  (3)
This equation, together with (2), enables
Y      P     it,
us to compute  when   and -only,are
P O          N
known, 0 being eliminated between (2)
and (3).
Observing that x=o0.3 P, equation (1)
may be written
p=So  0. 1      Y(   + -P-2q)   (4)
So thatp becomes fully known when
the two quantities  - and -belonging to
iR    N
the pair of wheels under consideration
P
become known. Or, observing that t27r
=,- the most convenient forms of the
equations for computation are




89
(1     1.885)
Cos.  N   --                    (5)
_ 1 + _-\
-0.0)7958 N      2 (-  + 1 )sin.0-sin.
2 N+           0.42841    (6)
+-       )3p..
-= 0.15       +0.3       1.158 ).(7)
These formulas enable us to compute
T-     N
- when -and N are known. Table I contains most of the values P needed in practice, and obtained from these formulas.
The tabular value of p, when multiplied
by P gives r the index number for setting
the odontograph. Examples will be
given hereafter, illustrating the use of
the instrument and table in delineating
gear teeth.




90
DEMONSTRATION FOR TIlE RACK AND
PINION.
For this case formula (6) becomes indeterminate, and hence fails.  It will,
therefore, be necessary to deduce others.
The cycloid for the tooth faces on the
rack, flanks radial, will have the following values of x and y *:
x=   1 - C(- os. 
Y=(O-sin. 0)
and hence
cos. d= l-    2 r        (8):Y  J(.0 —sin.).  (9)
By expanding cos. 0 into a series and
dropping all after 02, by which we involve an error of only Xsg part for the
case of 8 teeth, and much less for a
greater number; and similarly expanding sin. 0 into a series, omitting the
Price's Calculus, vol. i, p. 280.




91
terms following O' with an opposite error of A0 for 8 teeth, giving a
residual error of about -1 we may write,
Y  I.4 n
p-o.4.          (o0)
This, combined with (7), was used for
computing the tabular values for racks
given at the bottom of table T.
For the involute faces of the pinion
we readily obtain special formulas.
Let A H, Fig. 8, be regarded as the
Tiga8.
E
A      F
pitch line, AB as a tooth face of the involute form, and B F height of tooth above
the pitch line and equal to 0.3 P -=.
We will have a b same as before.




92
y=E B-a b
as in Fig. 8.
E B —(R + B F) arc B O E nearly
=(ROt+0. 3 P) (0 —BO IH)
H B=A F H= —RL 0 —  tang. B O 11
Hence
— 1
y=EB-a b=( R+0o..3 P) (&-tang. 0)
01/    N  + 0.3 ) (0-tang.  0)
0.42841
and (
2 H =1f&-0=(R + 0.3 P)2- R2
or
2,_(,_ 0.o;~)            (,7)
02=(1 - ~)'-i.(12)
which equations, combined with (7),
were used in bringing out the tabular
quantities marked "Pinion" near the
bottom of table I.




93
It is observed that we have one line of
values only for each, the rack and pinion,
the reason for this is that the value of
p for the rack and pinion become known
in terms of one arbitrary quantity instead
of two.
DE-MONSTRATION FOR WHEELS WITH INVOLUTE TEETH.
Ha-ving a set of values of p for involute faces, we are at once prepared to lay
out any involute gear-teeth with the
odontograph, from the fact that in this
form of teeth each wheel is independent
of all others. In this case, however, we
must observe that the tangent line E D,
Fig. 4, should be drawn to the base circle instead of from the pitch circle, as
stated above under conditions for involute teeth. The half thickness of tooth
should still be laid off on the pitch line,
because this is the proper line to reckon
tooth thicknesses on. The tabular val



94
ues of Table II are therefore the same
as those for Pinion, Table I.
DEMONSTRATION FOR WHEELS IN INTERNAL GEAR.
First consider the larger wheel, in
which the faces are internal and hence
hypocycloids. Equations (2) and (3)
apply to this case by changing the signs
of r and x, and hence
2R                R
cos. - 0 =1 
r         r(     r
or
0.6 n72
COS.1 —            0
~N      Y2~l7.-    N
and            i9    (iN\  2Nj
Py2R        2  N     sin. 0-sin.
(2 N_1  )0 }+ o    0.2273 (14)




95
which, combined with (7), were used in
producing the first part of table III. The
flanks of the pinion are supposed to be
radial, and those of the wheel involutes as set forth under conditions for
these wheels.  The external involute
flanks of the wheel are laid out by aid of
the values given in table I for "pirtion,"
the tangent line E D, Fig. 4, being drawn
to the pitch line from the middle of a
space, to suit the present reversed conditions from outside to inside.
The faces of the pinion, being also
involutes, the same set of tabular values
last-named apply, which for convenience's sake are arranged in the latter
part of Table III.
CASE OF BEVEL WHEELS.
Observing the explanations, page
76, regarding bevel wheels, either
epicycloidal or involute teeth may be
laid out for these wheels.




TAB  I,  T.
FOR EPICYCLOID TEETH WITH RADIAL FLANKES.
Values of    in iaLches: or, number by which to multiply the pitch, measured in inches, to obtain
the index number for setting the odontograph.
R'
r     B
Number of teeth in wheel for which the teelhll are sought, N and it.
8     16    24    32    40    48    56i         4     72    80   88    96 112
f=.083.321.458.555.=.167.317.450.551.6834.704
~=.250.311.438.540.624.692.780.853
=.333.300.426.526.611.679.770.831. 869,.910.95   1.01711.080 1.225
-=.417.293.417.515.598.666.741.8U08.853.894.948 1.0001.(o2 1.147
-=.500.283.406.504.584.654. 22.786.834.878.930.9801.032 1 132
i=. 583.272.394.492.573.640.705.766.817.862.911.960.9951.107




i.667.266.382 1.479.558.627.690.A48.799l.845.895.946.987 1.l(89
-- 750.252.369.467.545.613.677.733.784.834.878.93231.973 1.066
=.833.245.379.456.534.602.662.721.768.820.862.905.9571.047
i+=.917.238.348.446.522.590.648.709.754.806  847.892.9381. 037
1..230.338.435.510.577.634.693   740.791.  832.8761.91911.007
ij-=1.09.221.329.423.500.563.621.679.722.774.813 i856.898.990
-=1.20.211.318..48.548.607.661.05.755.794.836.877.968
-=1.33.202.307.396.472.532.590.640.685.731.771 1.812.855.935,i=1.50.193.294.:81.452.511.571.618.663.707.727 1.789.827.896
4=1.71.183.278.362.428.487.542.591.636.678.716.76.794.848
2.174.260.336.400.458.511.560.601.642.679.715.748.795
-it=-2.40.219.304.362.421.474.523.562.97.633.663.693.739
3.265.321.379.428  -.473.08.540.572.600.624.673
4.271.328.374.409.435.467  491.521.548,.600
6.254.289.321.348.372.397.428.460.519
12.'257.292.286.3181.3611.417
For both Rack and Pinionl, iuse number of teeth in pinioun, for N, above.
Rack....338.513.657.777.884.982 1.071  1.155 1.233 1.304 1.378 1.44411.573
Piniol..32:,  443.551.(2'718.785   847.897.941.990  03.02 67 t.1 22
_   __                                                                   I.....................  




TABLE II.
FOR INVOLUTE TEETH.
tl~?es of T in inches or, number by which to multiply the pitch, measured in inches, to obtain
the index number for setting the odontograph.
Number of teeth in wheel: or N snd m2.
8    16    24    32    40    48    56    64    72    80          88     96      112.323.443.551.642.718.785.847.897.941 1.990   1.032  1.067  1.122
N. B. —The tangent should lie drawn to the. base circles.




99
EXAMPLES.
To show the application of the tables, the following examples are given:
Let the pair of wheels be the same for
each example, the teeth simply being
different; let the number of teeth in the
smaller wheel be 16 and in the larger 56.
Also take the same pitch for all, and
equal to two inches.
1st. For epicycloidal teeth with radial
flanks.  For this we must use table I,
which requires the ratio of the radii of
the wheels; or, what is the same thing,
the ratio of the number of teeth in the
wheels.
For the larger wheel the required ratio is
- =3  and N=56.
Running down the column headed -
we find 3~ comes midway between 3 and
4; and opposite them, under " 56," corresponding to N, the numbers.473 and.409. Taking the mean for the interpo



100
lated value, we have.441 as the multiplier for the pitch. Hence.441 X 2 inches pitch —.882 inches,
which is the index number, or the number to look out on the scale A D B, Fig.
4, which point must be brought to coinicide with the tangent D E. Then, placing the curve C F tangent to the tangent
line as shown in the figure, the tooth
face curve may be drawn.
For the smaller wheel we have
=.286 +, and,-= 16.
Rlunning down the column headed
r
we find no ratio =.286, but.25 0
and.333.  Interpolating by the wellknown method of proportional parts, we
find, in column headed " 16," for the
multiplier the number.433. Hence.433 X 2=.866 inches.
which is the setting or index number required for the small wheel.




101
2d. For the rack, suppose the large
wheel is made infinite and the pitch
line straight. Then the pinion will have
16 teeth, and the multiplier is.513.
Hence.513X2=1.023 inches,
which is the index number for setting the
odontograph to the straight line of the
rack.
For the pinion, we find the multiplier
to be.443. Hence.443 X 2 —.886 inches,
which is the required number for the
pinion.
3d: For the case of;nvoblte teeth, the
multiplier for the larger wheel is.847,
table II, and for the smaller it is.443, by
aid of which the odontograph is set a
tangent to the base circle as above explained.
4th. For the case of internal gearing
we have, for obtaining the external involute flanks,
R_ 3, and N=56.




102
Not finding 56 in the line of numbers
of teeth, table III, we interpolate between 40 and 60 for the ratio 3. This
gives for N  =  56 the multiplier.649.
Similarly for ratio 4, we find multiplier.490. Taking the mean of these for the
ratio 31, and we get.5t;9 as the desired
multiplier. Hence.569 X 2 inches pitch = 1.1 38 inches.
for the index number.
For the faces of the wheel we find,
from the lower line of table III,.841 as
the required multiplier, and 1.682 as the
index number.
To set the odontograph.  Observing
the foot-note of table III, the tangent
line E D, Fig. 4, to which the instiument is set, is drawn from the middle of the space, as though the space in
this case, outside the pitch line, were a
tooth.
For the faces of the pinion teeth the
multiplier is.443, as in lower line of table
III, &c., &c.




TABLE III.
FOR INTERNAL GEARING, PINION FLANKS RADIAL.
V,,l ue of' rin i7lches: or, number by which to multiply the pitch, measured in inches, to obtain
the index number for setting the odontograph.
R
R or
qr                             Wheel: Number of teeth, or N.'t    6       8          16        24         40         60         100        150
2             r.400.556.818     1.082      1.514      1.887.,     l. ~.210.317.497.687      1 017      1.322
4      Faces ~.232.370.520.767      1.027
6.268.387.555.687
12     _                     _.282.387.447
Flanks of wheel and faces of pinion.
1.323.443   1.551   1.718   I.872         1.084   I 1.270
N. B.-In laying out the flatnks for the iwheel, the tangent should be drawn from
the center of the space, and not center of tooth.




104
CA1SE OF CIIANGE WHEELS, AND cc SETS "
OF EPICYCLOIDAL GEARING.
The demonstrations for, and the alpplications of, the new odontograph have,
thus far, proceeded upon the supposition
that the flanks of the teeth laid out are
radial. Though this form is the most
convenient to delineate, and possesses
greatest freedom in action as shown in
Part I, yet some may prefer to construct
wheels so that all, having a given pitch,
may be used in  any interchangeable
manner; or, it may be desired in a pair
of gears that the teeth have thick bases,
such as obtained by using small generating circles, describing curved hypocycloidal flanks.
The preceding demonstrations and tables cover this case. The equations may
N             R
have -- replaced by -, in which r has
n             r)
been regarded as the radius of the wheel
gearling with the one sought, and equal
to the diameter of the generating circle.
But we readily see that it is only necessary to consider r as the diameter of the




105
generating circle in all cases, due regard
being given only to using the same value
of r in laying out the other wheel, outside of one and inside the other, and conversely; or, for a given "set," the same
throughout, as pointed out in Part I.
The new odontograph is therefore of
universal application  in  epicycloidal
gearing, giving  any such flare, and
thickness of tooth base, as may be demanded by the most fastidious draughtsmall.
-For describing the faces of the teeth
now considered Table I is sufficient, the
diameter of the generating circle always
forming the denominator of the fraction
-.  For the flanks, the first part of Table
III would apply, though with less approxilnative results than those obtained
by aid of the latter part of Table IV,
prepared expressly for this.
The odontograph is found to osculate
best with the hypocycloid when it is located by aid of a tangent line E D, Fig.
4, drawn from A, instead of E; or, from




106
the sides instead of middle of tooth; and
hence, to lay out flanks other than radial,
draw tangents to the pitch line at each
side of the tooth, and on them, locate
the instrument by aid of second part of
Table IV, as on any other tangent, observing that the back edge of the odontograph should now be turned toward
the center of the wheel, instead of from
it, as in drawing faces.
This portion of this table was computed by aid of equations (13), (14) and
(7), after suppressing the last term of
(14); which term, it is seen, is a b, Fig.
7, now made zero by reducing A C, Fig.
7, to zero, as above explained.
From the fact that the curve C D B,
and C F H, Fig. 4, are one and the
same, except as to length, the instrument
may, if preferred, be located by a radius
instead of a tangent line. Suppose, for
instance, D F, be a radius, drawn at
one side of a tooth, with the pitch line
intersecting it at F. By laying off F
D, outward from the circle, and equal,
in inches, to the proper index number,




107
then F H, when the instrument is properly located is the required flank.
For this mode of locating the instrunment the index number obtained from
the table should be divided by two, because always 2 C D-C F.
EXAMPLE.
Take number of teeth 16 and 56, and
pitch 2 inches, as in previous examples,
and a four-inch generating circle. The
pitch circle radii are 5.1 and 17.8 inches,
and hence, for wheel, "radius of wheel
sought divided by diameter of describing
circle" — 4.45. For this, and for N.56, TABLE IV, referring to TABLE I, gives.89, which, multiplied by pitch, gives.78, the index number for wheel faces.
Interpolating in TABLE IV the numbers 4.45 and 56 give.52, and multiplied
by pitch 1.04, for the index number for
the wheel flanks. Similarly, for the pinion, we get the index numbers.628 and
2.90.




Radius of Wheel                -
soug-ht divided
by diamete of  
describing circle.
5   -;_____        F      CD      l      V  
z                c*  z  -
C
_    cD    H
0 -
C-5"=                 t801




For Flanks.
1.5.77.5     1.177     1.540     1.958     2.607     3.229     3.875
2.440.632.805     1.094     1.391  i 1.786      2.217.454.65,0.860     1.190     1.585
4.276      410.592  1.836      1.126
l6.200.324.440.603.724
12                        1.322.417.470
N. B.-For the faces  the tangent D E, Fig. 4, should be drawn from the
middle of a tooth; and for the  ffl,~k i tie tangent should be drawn from the sides
of a tooth.




110
MODIFIED FORM OF THE WILLIS ODONTOGRAPH.
The odontograph now under consideration is termed a modified form of the
Willis odontograph, not because it bears
any resemblance to the Willis odontograph, but because it is based upon the
theory of Prof. Willis for the forming
of teeth of wheels by circle arcs, and
also because it gives centers to tooth arcs
as does the Willis odontograph, instead
of forming in itself a template for the
curves.
By constructing the instrument so
that it canll be set to various angles, instead of being fixed at 750; and by preparing suitable tables to accompany it,
we may be able with it to lay out teeth
with straight flanks for all wheels, the
faces of the teeth being circular arcs
whose centers are given by the instrument.
The analysis by which these facts are
demonstrated is essentially that of Prof.
Willis, and given in his Principles qf
lJrechaduismn, 2d ed., pp. 132, 136.




111
According to the Willis theory for
forming each tooth with foul circular
arcs, whose centers are on the line of
action, if certain centers be removed to
an infinite distance, the arcs struck from
such centers will be straight lines, perpendicular to the line of action. This is
supposed to be done in the present case
for the flank arcs, and when the line of
action is suitably inclined, these perpendiculars become radii on the pinion and
inclined lines on the wheel.
With these conditions imposed at the
outset, the analysis becomes extremely
simple, as follows: In Fig. 9 let A and
13 represent the pitch circles, C the pitch
point, and E F the line of action. Draw
A E and B F perpendicular to E F as
shown. Let cc = angle C A E = CB F,
AC = R, BC = r, andCG = D. Now
Prof. Willis' diagram, p. 135, becomes
Fig. 9 when A k ancd B K are made parallel to line of action as in Fig. 9. I
will simply explain Fig. 9, referring
those who wish to study the diagram
further to Prof. Willis' book. G is a




112
center for describing a tooth face of A,
anld H a center for a face of a tooth of
B
K
\   rig.9    /
1). The center for the flank against
which the face, struck from G, works,
being on B K at an infinite distance, the
flank is perpendicular to E F. The
point serving as the origin for this face




113
and flank may be assumed at any position * on the line E F, but should generally be not far from F, and for radial
flanks for B, must be at F.  Similarly
for E if the flanks for E are to be radial.
To render the odontograph convenient
in practical use, and to keep the radii
of the tooth faces within reasonable
limits for all cases, such as when the pair
of wheels are of equal size and also of
very different sizes, it is found advisable
to make the flanks of the smaller of the
pair of wheels or pinion radial in all
cases; and for the larger wheel, of such
inclination that the same radius for the
tooth faces of the wheel will be right for
the pinion. To this end C F is taken,
arbitrarily equal to one-third of the
pitch, so that the radius of the faces for
both wheels is G F =G C +     pitch.
The point G is given by the odontograph
and it is easy to lay off a third of the
pitch C F. Then we will have, C F B
being always preserved a right angle,; This fact is pointed out by Prof. Willis.




114
C F= p=  sin. cc
P    27r
whr  =number3 of teeth in pinio
where n=number of teeth in pinion.
Again
CF: CH: R+r: R
C H=D=CF R   R P,-Il=D=CFR F
R+ —R+' 3
But
C  F: CE      I',: 1
and
CG: CE: r: R+r
4r        r R    PC P.. C G=C E R+ C FR+
or
C G=C H.
This shows that it is only necessary to
set the odontograph once to obtain either
C G or C H. Taking the distance from
C to the origin of tooth arcs the same




115
for both wheels gives equal radii for the
faces of both wheels, and the arcs and
flanks are easily struck in as shown in
Fig. 9. To draw all other arcs the centers may be found, which lie in circles
concentric with the pitch lines.  The
drawing of the non-radial flanks of A
will require a little attention that they be
drawn at the proper angle with the radius, or rather simply perpendicular to
E F at the proper point.
The quantities cx and p are given in
terms of n and  r, respectively, in the
following table for a number of values,
which will aid in the practical use of the
instrument, the necessary setting values
being thus obtained beforehand. When
the data fall between values in the table,
interpolations will be necessary, though
readily made.
THE INSTRUMENT, AND MODE OF USING IT.
The modified odontograph is shown
full size in Fig. 10, with its scales, &c.,




116
ready for use, in delineating teeth of six
inches pitch.
TABLE  VT,
FOR THE MODIFIED WILLIS ODONTOGRAPH.
D
Values of the setting angle C(, and of p for position of arc cen-ter, for tooth face for radial
flanks for pinion and for inclined flanks for the
wheel.
Setting   Larger ra- Tooth center
Number  anrgle for dius divid- distance diof teeth   odonto- I ed by       vided by
in pinion,  graph,   smaller,      pitch,
}=?.  -= -0  _R               _D
Deg. Min.
10      12  05        1.16
15       8  02       1.5.200
20       6  01        2.222
30       4  00        3.250
40       3  00        4.267
60       2  00        5.278
80       1  30        6.286
100       1  12        7.292
130       (  55        8.296
160       0  45        9.300
200       0  36       10.303
260       0  28       11.306
300       0  24       12.308




117
N. B.-The two parts of this table are
entirely independent of each other, the
first giving the setting angle ~o, without
regard to where the line is in second part
of table giving P.
In using the instrument the point C
is to be brought to the pitch point C,
Fig. 9. Then the angle OC obtained
from the table is to be looked out on
the sectoral scale AB, and that point
placed upon the radius A C, Fig. 9. The
instrument is then in position. A line
should then be drawn along E C and
produced in the direction of F, perpendicular to which the flanks are to be
drawn. Then the distance D, obtained
by multiplying D of the last table by P,
is to be set off on the scale C E, Fig. 10,
which point is the center G, Fig. 9, from
which to strike the tooth face. Then
C F, equal I P as explained above, is set
off on line E F as the origin of a tooth
face and flank as shown in Fig. 9. We
then have the length G F for the radius




118
Fig. 10.
E          I   I|   I     I    I; I    I   I     I     I    I   i    -  I-  C
2                        I
B /
/110~~ A
Modified Willis Odontograph.
Full size.




119
of a face of A, which is also the radius
for a face of B to be used at the centers H
and G respectively. The flanks are per
pendicular to E F and tangent to the
tooth arcs as shown, and each face works
with its tangent flank, as shown in Fig. 9.
By repeating these faces and flanks the
teeth become fully drawn, as in Fig. 13.
The flanks of the smaller wheel are always
radial, but those of the larger wheel never,
except in case both wheels become equal,
when the flanks of both are radial.
It will hardly be necessary to follow
this description with an example.
ODONTOGRAPH TEETH COMPARED.
Fig. 11 is an accurate drawing of a
part of the teeth of a pair of wheels as
laid out by the well known  Willis
Odontograph.
Fig. 12 is likewise a carefully executed
drawing of a portion of the same pair of
gears as laid out by the Xew  Odontograph.
Fig. 13 shows the form of teeth of the




120
same pair of wheels as laid out by the
Modified  Willis Instrioment.
The smaller of the wheels has twelve
teeth, and the larger has 144 teeth.
These drawings will enable the reader
to form a very correct opinion as to the
relative merits of the results of the three
instruments. This is not difficult to such
as are familiar with the proper form of
gear teeth. While this may be true,
still it remains doubtful which to adopt
for practical use. Such doubts however
may be removed on taking a direct comparative view of the respective difficulties, and conveniences of application of
the three instruments, which, for the
purpose of furthering this object is given
below.
PRiACrICE  AVITH TIHE THREE  ODONT0GiRAPHS COMPARED.
The well known Willis Odontogr~aph/,
in laying out one side of a tooth requires:
1st, that the pitch be laid off on the
pitch line; 2d, that this pitch be bisect



121
ed to obtain the origin of arcs; 3d, that
a radius be drawn at one end of the pitch
and the instrument placed on it; 4th,
a quantity taken from a table by which
the center point is found on the odontograph scale and point noted on drawing;
5th, that a radius be drawn at the other
end of the pitch, and the instrument placed on it; 6th, a second
quantity taken from the table by which
a second center point is found and noted
on drawing for a second arc center; 7th,
that a pair of dividers be placed at one
center, the face or flank struck; 8th,
that similarly a flank or face be struck
from the second center; 9th, and finally
that caution be exercised to point off
the right center at the two settings of
instrument, and also that the proper
part, flank or face, be drawn from the
centers.
The New Odontograph in laying out
the side of a tooth requires: 1st, that a
tangent line be drawn to the pitch circle;
2d, that the half thickness of tooth be
laid off from the point of tangency; 3d,




122
a quantity looked out from a table; 4th,
that this be multiplied by the pitch by
which the instrument is properly set on the
tangent line; 5th, that a scribe be drawn
along edge of instrument for the face;
6th, and lastly, that a radial flank be
drawn from the intersection of face with
the pitch line.
The Modified Willis Odontograph requires: 1st, that a radius be drawn; 2d,
that a setting angle be found from a
table by which the instrument is set;
3d, that a second quantity be taken from
a table; 4th, that this quantity be multiplied by the pitch, by which the center
point for the face is found and noted;
5th, that a line be drawn along the edge
of the odontograph, perpendicular to
which the flank is afterwards drawn;
6th, that the one-third pitch be laid off
from the end of the radius upon which
the instrument was set; 7th, that by aid
of a pair of dividers the face arc be
struck; 8th, and finally, that the flank
be drawn perpendicular to the line traced
for the purpose, as above described.




123
COMPARATIVE VIEW OF THE INSTRUMENT;S
AND TABLES.
As to the tables for the three instruments, those for the modified Willis
odontograph are the most compact of
all, having only four vertical columns
and fourteen horizontal, or fifty-six
blocks, including everything. The table
for the Willis odontograph, as published
in his book,* has nine vertical columns
and twenty-eight horizontal, or 250
blocks for figures. As printed on the
card which accompanies the brass instrument, it has fourteen vertical columns
and thirty horizontal ones, or 420 blocks
for figures. The tables for the new instrument, that for radial flanks and epicycloidal teeth, has fourteen vertical
columns and twenty-six horizontal ones,
or 304 blocks for figures; or, adding
those for table for internal gearing, we
get 353. It is believed also, that table
I might be reduced by cutting out about
half of the horizontal columns, and still
Willis, p. 137.




124
the gaps to be filled, in practice, by interpolation be no greater than in the
tables for the Willis instrument. This
would reduce the tables for the new odontograph, for epicycloidal and rack and
pinion gearing to 182 blocks for figures.
Then adding the seventy-three blocks
for figures in tables II and III, we have
the tables for applying the new instrument to all ordinary forms of gears occupying 255 blocks for figures, a number
of places much less than is found in the
tables on the printed card accompanying
the Willis instrument, and but few more
than given in the tables published in
Professor Willis' book. Taking the
table that accompanies the instrument, as
the one which it is fair to compare
with, we find the tables for the new instrumnent most conlpact.
Comparing the instruments themselves
for compactness, we see, by reference to
the figures, that the modified Willis instrument is a little ahead, though either
this, or the new one, may be carried
under the tuck of a common pocket



125
book. In this regard, either of the new
instruments possess a degree of portability and ready usefulness, not common
to an instrument 6 X 14 inches, as in the
older form of odontograph. In fact the
new and modified instruments might be
treated as mere " pocket pieces," the accompanying tables being recorded in a
note-book. This matter of compactness
of tables and instruments is, however,
only of secondary importance when
eounterposed with desirable results and
facility of obtaining them.
These comparisons indicate that the
modified Willis odontograph has advantages in the minor points of compactness
of instrument and tables. But in the
more essential qualities of simplicity of
application, the new instrument, as above
pointed out, appears to be superior in
the ratio of six to eight or nine; and of
form of tooth, superior in the almost mathematical correctness of the curves for
faces, and in having radial flanks, as indicated by Figs. 1, 2, 11, 12, and 13.
Designers of gearing, familiar with




126
properly formed gear teeth, however
much they may admire the short-cuts
secured by odontographs, will undoubtedly deprecate the angularity of junction
of flanks and faces found in the Willis
tooth, and the rude approximation to
correct tooth curves inseparable from the
circle are. The excessively thick base
of tooth found in Fig. 1, 11, and 13 are
undesirable features.
Fig. 12, as compared with Figs. 11
and 13, shows superiority in form of
tooth, which, by those who advocate the
common form of radial flanks at least,
cannot be questioned. This superiority
of form of tooth, added to the compactness of instrument and especially to the
important advantage possessed by it of
great facility of application, will, it is
believed, merit for the new odontograph
decided preferences.




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