The Project Gutenberg eBook of An Analysis of the Lever Escapement
    
This ebook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this ebook or online
at www.gutenberg.org. If you are not located in the United States,
you will have to check the laws of the country where you are located
before using this eBook.

Title: An Analysis of the Lever Escapement

Author: H. R. Playtner

Release date: June 30, 2007 [eBook #21978]

Language: English

Credits: Produced by Sigal Alon, Fox in the Stars, Laura Wisewell
        and the Online Distributed Proofreading Team at
        http://www.pgdp.net


*** START OF THE PROJECT GUTENBERG EBOOK AN ANALYSIS OF THE LEVER ESCAPEMENT ***




Produced by Sigal Alon, Fox in the Stars, Laura Wisewell
and the Online Distributed Proofreading Team at
http://www.pgdp.net









[Illustration: THOMAS MUDGE

_The first Horologist who successfully applied the Detached Lever
Escapement to Watches._

_Born 1715--Died 1794._]




AN ANALYSIS

OF THE

LEVER ESCAPEMENT

BY H. R. PLAYTNER.

A LECTURE DELIVERED BEFORE THE CANADIAN WATCHMAKERS' AND RETAIL
JEWELERS' ASSOCIATION.

ILLUSTRATED.

CHICAGO:

HAZLITT & WALKER, PUBLISHERS.

1910.




PREFACE.


Before entering upon our subject proper, we think it advisable to
explain a few points, simple though they are, which might cause
confusion to some readers. Our experience has shown us that as soon as
we use the words "millimeter" and "degree," perplexity is the result.
"What is a millimeter?" is propounded to us very often in the course of
a year; nearly every new acquaintance is interested in having the metric
system of measurement, together with the fine gauges used, explained to
him.

The metric system of measurement originated at the time of the French
Revolution, in the latter part of the 18th century; its divisions are
decimal, just the same as the system of currency we use in this country.

A meter is the ten millionth part of an arc of the meridian of Paris,
drawn from the equator to the north pole; as compared with the English
inch there are 39+3708/10000 inches in a meter, and there are
25.4 millimeters in an inch.

The meter is sub-divided into decimeters, centimeters and millimeters;
1,000 millimeters equal one meter; the millimeter is again divided into
10ths and the 10ths into 100ths of a millimeter, which could be
continued indefinitely. The 1/100 millimeter is equal to the 1/2540 of
an inch. These are measurements with which the watchmaker is concerned.
1/100 millimeter, written .01 mm., is the side shake for a balance
pivot; multiply it by 2¼ and we obtain the thickness for the spring
detent of a pocket chronometer, which is about 1/3 the thickness of a
human hair.

The metric system of measurement is used in all the watch factories of
Switzerland, France, Germany, and the United States, and nearly all the
lathe makers number their chucks by it, and some of them cut the leading
screws on their slide rests to it.

In any modern work on horology of value, the metric system is used.
Skilled horologists use it on account of its _convenience_. The
millimeter is a unit which can be handled on the small parts of a watch,
whereas the inch must always be divided on anything smaller than the
plates.

Equally as fine gauges can be and are made for the inch as for the
metric system, and the inch is decimally divided, but we require another
decimal point to express our measurement.

Metric gauges can now be procured from the material shops; they consist
of tenth measures, verniers and micrometers; the finer ones of these
come from Glashutte, and are the ones mentioned by Grossmann in his
essay on the lever escapement. Any workman who has once used these
instruments could not be persuaded to do without them.

No one can comprehend the geometrical principles employed in escapements
without a knowledge of angles and their measurements, therefore we deem
it of sufficient importance to at least explain what a degree is, as we
know for a fact, that young workmen especially, often fail to see how to
apply it.

Every circle, no matter how large or small it may be, contains 360°; a
degree is therefore the 360th part of a circle; it is divided into
minutes, seconds, thirds, etc.

To measure the _value_ of a degree of any circle, we must multiply the
diameter of it by 3.1416, which gives us the circumference, and then
divide it by 360. It will be seen that it depends on the size of that
circle or its radius, as to the value of a degree in any _actual_
measurement. To illustrate; a degree on the earth's circumference
measures 60 geographical miles, while measured on the circumference of
an escape wheel 7.5 mm. in diameter, or as they would designate it in a
material shop, No. 7½, it would be 7.5 × 3.1416 ÷ 360 = .0655 mm., which
is equal to the breadth of an ordinary human hair; it is a degree in
both cases, but the difference is very great, therefore a degree cannot
be associated with any actual measurement until the radius of the
circle is known. Degrees are generated from the center of the circle,
and should be thought of as to ascension or direction and relative
value. Circles contain four right angles of 90° each. Degrees are
commonly measured by means of the protractor, although the ordinary
instruments of this kind leave very much to be desired. The lines can be
verified by means of the compass, which is a good practical method.

It may also be well to give an explanation of some of the terms used.

_Drop_ equals the amount of freedom which is allowed for the action of
pallets and wheel. See Z, Fig. 1.

_Primitive or Geometrical Diameter._--In the ratchet tooth or English
wheel, the primitive and real diameter are equal; in the club tooth
wheel it means across the locking corners of the teeth; in such a wheel,
therefore, the primitive is _less_ than the real diameter by the height
of two impulse planes.

_Lock_ equals the depth of locking, measured from the locking corner of
the pallet at the moment the drop has occurred.

_Run_ equals the amount of angular motion of pallets and fork to the
bankings _after_ the drop has taken place.

_Total Lock_ equals lock plus run.

A _Tangent_ is a line which _touches_ a curve, but does not intersect
it. AC and AD, Figs. 2 and 3, are tangents to the primitive circle GH at
the points of intersection of EB, AC, and GH and FB, AD and GH.

_Impulse Angle_ equals the angular connection of the impulse or ruby pin
with the lever fork; or in other words, of the balance with the
escapement.

_Impulse Radius._--From the face of the impulse jewel to the center of
motion, which is in the balance staff, most writers assume the impulse
angle and radius to be equal, and it is true that they must conform with
one another. We have made a radical change in the radius and one which
does not affect the angle. We shall prove this in due time, and also
that the wider the impulse pin the greater must the impulse radius be,
although the angle will remain unchanged.

Right here we wish to put in a word of advice to all young men, and that
is to learn to draw. No one can be a thorough watchmaker unless he can
draw, because he cannot comprehend his trade unless he can do so.

We know what it has done for us, and we have noticed the same results
with others, therefore we speak from personal experience. Attend night
schools and mechanic's institutes and improve yourselves.

The young workmen of Toronto have a great advantage in the Toronto
Technical School, but we are sorry to see that out of some 600 students,
only five watchmakers attended last year. We can account for the
majority of them, so it would seem as if the young men of the trade were
not much interested, or thought they could not apply the knowledge to be
gained there. This is a great mistake; we might almost say that
knowledge of any kind can be applied to horology. The young men who take
up these studies, will see the great advantage of them later on; one
workman will labor intelligently and the other do blind "guess" work.

We are now about to enter upon our subject and deem it well to say, we
have endeavored to make it as plain as possible. It is a deep subject
and is difficult to treat lightly; we will treat it in our own way,
paying special attention to all these points which bothered us during
the many years of painstaking study which we gave to the subject. We
especially endeavor to point out how theory can be applied to practice;
while we cannot expect that everyone will understand the subject without
study, we think we have made it comparatively easy of comprehension.

We will give our method of drafting the escapement, which happens in
some respects to differ from others. We believe in making a drawing
which we can reproduce in a watch.




AN ANALYSIS OF THE LEVER ESCAPEMENT.


The lever escapement is derived from Graham's dead-beat escapement for
clocks. Thomas Mudge was the first horologist who successfully applied
it to watches in the detached form, about 1750. The locking faces of the
pallets were arcs of circles struck from the pallet centers. Many
improvements were made upon it until to-day it is the best form of
escapement for a general purpose watch, and when made on mechanical
principles is capable of producing first rate results.

Our object will be to explain the whys and wherefores of this
escapement, and we will at once begin with the number of teeth in the
escape wheel. It is not obligatory in the lever, as in the verge, to
have an uneven number of teeth in the wheel. While nearly all have 15
teeth, we might make them of 14 or 16; occasionally we find some in
complicated watches of 12 teeth, and in old English watches, of 30,
which is a clumsy arrangement, and if the pallets embrace only three
teeth in the latter, the pallet center cannot be pitched on a tangent.

Although advisable from a timing standpoint that the teeth in the escape
wheel should divide evenly into the number of beats made per minute in a
watch with seconds hand, it is not, strictly speaking, necessary that it
should do so, as an example will show. We will take an ordinary watch,
beating 300 times per minute; we will fit an escape wheel of 16 teeth;
multiply this by 2, as there is a forward and then a return motion of
the balance and consequently two beats for each tooth, making
16 × 2 = 32 beats for each revolution of the escape wheel. 300 beats are
made per minute; divide this by the beats made on each revolution, and
we have the number of times in which the escape wheel revolves per
minute, namely, 300 ÷ 32 = 9.375. This number then is the proportion
existing for the teeth and pitch diameters of the 4th wheel and escape
pinion. We must now find a suitable number of teeth for this wheel and
pinion. Of available pinions for a watch, the only one which would
answer would be one of 8 leaves, as any other number would give a
fractional number of teeth for the 4th wheel, therefore 9.375 × 8 = 75
teeth in 4th wheel. Now as to the proof: as is well known, if we
multiply the number of teeth contained in 4th and escape wheels also by
2, for the reason previously given, and divide by the leaves in the
escape pinion, we get the number of beats made per minute; therefore
(75 × 16 × 2)/8 = 300 beats per minute.

Pallets can be made to embrace more than three teeth, but would be much
heavier and therefore the mechanical action would suffer. They can also
be made to embrace fewer teeth, but the necessary side shake in the
pivot holes would prove very detrimental to a total lifting angle of
10°, which represents the angle of movement in modern watches. Some of
the finest ones only make 8 or 9° of a movement; the smaller the angle
the greater will the effects of defective workmanship be; 10° is a
common-sense angle and gives a safe escapement capable of fine results.
Theoretically, if a timepiece could be produced in which the balance
would vibrate without being connected with an escapement, we would have
reached a step nearer the goal. Practice has shown this to be the proper
theory to work on. Hence, the smaller the pallet and impulse angles the
less will the balance and escapement be connected. The chronometer is
still more highly detached than the lever.

The pallet embracing three teeth is sound and practical, and when
applied to a 15 tooth wheel, this arrangement offers certain geometrical
and mechanical advantages in its construction, which we will notice in
due time. 15 teeth divide evenly into 360° leaving an interval of 24°
from tooth to tooth, which is also the angle at which the locking faces
of the teeth are inclined from the center, which fact will be found
convenient when we come to cut our wheel.

From locking to locking on the pallet scaping over three teeth, the
angle is 60°, which is equal to 2½ spaces of the wheel. Fig. 1
illustrates the lockings, spanning this arc. If the pallets embraced 4
teeth, the angle would be 84°; or in case of a 16 tooth wheel scaping
over three teeth, the angle would be 360 × 2.5/16 = 56¼°.

[Illustration: Fig. 1.]

Pallets may be divided into two kinds, namely: equidistant and circular.
The equidistant pallet is so-called because the lockings are an equal
distance from the center; sometimes it is also called the tangential
escapement, on account of the unlocking taking place on the intersection
of tangent AC with EB, and FB with AD, the tangents, which is the
valuable feature of this form of escapement.

[Illustration: Fig. 2.]

AC and AD, Fig. 2, are tangents to the primitive circle GH. ABE and ABF
are angles of 30° each, together therefore forming the angle FBE of
60°. The locking circle MN is struck from the pallet center A; the
interangles being equal, consequently the pallets must be equidistant.

The weak point of this pallet is that the lifting is not performed so
favorably; by examining the lifting planes MO and NP, we see that the
discharging edge, O, is closer to the center, A, than the discharging
edge, P; consequently the lifting on the engaging pallet is performed on
a shorter lever arm than on the disengaging pallet, also any inequality
in workmanship would prove more detrimental on the engaging than on the
disengaging pallet. The equidistant pallet requires fine workmanship
throughout. We have purposely shown it of a width of 10°, which is the
widest we can employ in a 15 tooth wheel, and shows the defects of this
escapement more readily than if we had used a narrow pallet. A narrower
pallet is advisable, as the difference in the discharging edges will be
less, and the lifting arms would, therefore, not show so much difference
in leverage.

[Illustration: Fig. 3.]

The circular pallet is sometimes appropriately called "the pallet with
equal lifts," as the lever arms AMO and ANP, Fig. 3, are equal lengths.
It will be noticed by examining the diagram, that the pallets are
bisected by the 30° lines EB and FB, one-half their width being placed
on each side of these lines. In this pallet we have two locking circles,
MP for the engaging pallet, and NO for the disengaging pallet. The weak
points in this escapement are that the unlocking resistance is greater
on the engaging than on the disengaging pallet, and that neither of them
lock on the tangents AC and AD, at the points of intersection with EB
and FB. The narrower the circular pallet is made, the nearer to the
tangent will the unlocking be performed. In neither the equidistant or
circular pallets can the unlocking resistance be _exactly_ the same on
each pallet, as in the engaging pallet the friction takes place before
AB, the line of centers, which is more severe than when this line has
been passed, as is the case with the disengaging pallet; this fact
proportionately increases the existing defects of the circular over the
equidistant pallet, and _vice versa_, but for the same reason, the
lifting in the equidistant is proportionately accompanied by more
friction than in the circular.

Both equidistant and circular pallets have their adherents; the finest
Swiss, French and German watches are made with equidistant escapements,
while the majority of English and American watches contain the circular.
In our opinion the English are wise in adhering to the circular form. We
think a ratchet wheel should not be employed with equidistant pallets.
By examining Fig. 2, we see an English pallet of this form. We have
shown its defects in such a wide pallet as the English (as we have
before stated), because they are more readily perceived; also, on
account of the shape of the teeth, there is danger of the discharging
edge, P, dipping so deep into the wheel, as to make considerable drop
necessary, or the pallets would touch on the backs of the teeth. In the
case of the club tooth, the latter is hollowed out, therefore, less drop
is required. We have noticed that theoretically, it is advantageous to
make the pallets narrower than the English, both for the equidistant and
circular escapements. There is an escapement, Fig. 4, which is just the
opposite to the English. The entire lift is performed by the wheel,
while in the case of the ratchet wheel, the entire lifting angle is on
the pallets; also, the pallets being as narrow as they can be made,
consistent with strength, it has the good points of both the equidistant
and circular pallets, as the unlocking can be performed on the tangent
and the lifting arms are of equal length. The wheel, however, is so much
heavier as to considerably increase the inertia; also, we have a metal
surface of quite an extent sliding over a thin jewel. For practical
reasons, therefore, it has been slightly altered in form and is only
used in cheap work, being easily made.

[Illustration: Fig. 4.]

We will now consider the drop, which is a clear loss of power, and, if
excessive, is the cause of much irregularity. It should be as small as
possible consistent with perfect freedom of action.

In so far as _angular_ measurements are concerned, no hard and fast rule
can be applied to it, the larger the escape wheel the smaller should be
the angle allowed for drop. Authorities on the subject allow 1½° drop
for the club and 2° for the ratchet tooth. It is a fact that escape
wheels are not cut perfectly true; the teeth are apt to bend slightly
from the action of the cutters. The truest wheel can be made of steel,
as each tooth can be successively ground after being hardened and
tempered. Such a wheel would require less drop than one of any other
metal. Supposing we have a wheel with a primitive diameter of 7.5 mm.,
what is the amount of drop, allowing 1½° by angular measurement?
7.5 × 3.1416 ÷ 360 × 1.5 = .0983 mm., which is sufficient; a hair could
get between the pallet and tooth, and would not stop the watch. Even
after allowing for imperfectly divided teeth, we require no greater
freedom even if the wheel is larger. Now suppose we take a wheel
with a primitive diameter of 8.5 mm. and find the amount of drop;
8.5 × 3.1416 ÷ 360 × 1.5 = .1413 mm., or .1413 - .0983 = .043 mm.,
more drop than the smaller wheel, if we take the same angle. This is a
waste of force. The angular drop should, therefore, be proportioned
according to the size of the wheel. We wish it to be understood that
common sense must always be our guide. When the horological student once
arrives at this standpoint, he can _intelligently_ apply himself to his
calling.

_The Draw._--The draw or draft angle was added to the pallets in order
to draw the fork back against the bankings and the guard point from the
roller whenever the safety action had performed its function.

[Illustration: Fig. 5.]

Pallets with draw are more difficult to unlock than those without it,
this is in the nature of a fault, but whenever there are two faults we
must choose the less. The rate of the watch will suffer less on account
of the recoil introduced than it would were the locking faces arcs of
circles struck from the pallet center, in which case the guard point
would often remain against the roller. The draw should be as light as
possible consistent with safety of action; some writers allow 15° on the
engaging and 12° on the disengaging pallet; others again allow 12° on
each, which we deem sufficient. The draw is measured from the locking
edges M and N, Fig. 5. The locking planes _when locked_ are inclined 12°
from EB, and FB. In the case of the engaging pallet it inclines toward
the center A. The draw is produced on account of MA being longer than
RA, consequently, when power is applied to the scape tooth S, the pallet
is drawn into the wheel. The disengaging pallet inclines in the same
direction but away from the center A; the reason is obvious from the
former explanation. Some people imagine that the greater the incline on
the locking edge of the escape teeth, the stronger the draw would be.
This is not the case, but it is certainly necessary that the point of
the tooth alone should touch the pallet. From this it follows that the
angle on the teeth must be greater than on the pallets; examine the
disengaging pallet in Fig. 5, as it is from this pallet that the
inclination of the teeth must be determined, as in the case of the
engaging pallet the motion is toward the line of centers AB, and
therefore _away_ from the tooth, which partially explains why some
people advocate 15° draw for this pallet. As illustrated in the case of
the disengaging pallet, however, the motion is also towards the line of
centers AB, and _towards_ the tooth as well, all of which will be seen
by the dotted circles MM2 and NN2, representing the paths of the
pallets. It will be noticed that UNF and BNB are opposite and equal
angles of 12°. For practical reasons, from a manufacturing standpoint,
the angle on the tooth is made just twice the amount, namely 24°; we
could make it a little less or a little more. If we made it less than
20° too great a surface would be in contact with the jewel, involving
greater friction in unlocking and an inefficient draw, but in the case
of an English lever with such an arrangement we could do with less
drop, which advantage would be too dearly bought; or if the angle is
made over 28°, the point or locking edge of the tooth would rapidly
become worn in case of a brass wheel. Also in an English lever more drop
would be required.

_The Lock._--What we have said in regard to drop also applies to the
lock, which should be as small as possible, consistent with perfect
safety. The greater the drop the deeper must be the lock; 1½° is the
angle generally allowed for the lock, but it is obvious that in a large
escapement it can be less.

[Illustration: Fig. 6.]

_The Run._--The run or, as it is sometimes called, "the slide," should
also be as light as possible; from ¼° to ½° is sufficient. It follows
then, the bankings should be as close together as possible, consistent
with requisite freedom for escaping. Anything more than this increases
the angular connection of the balance with the escapement, which
directly violates the theory under which it is constructed; also, a
greater amount of work will be imposed upon the balance to meet the
increased unlocking resistance, resulting in a poor motion and accurate
time will be out of the question. It will be seen that those workmen who
make a practice of opening the banks, "to give the escapement more
freedom" simply jump from the frying pan into the fire. The bankings
should be as far removed from the pallet center as possible, as the
further away they are pitched the less run we require, according to
angular measurement. Figure 6 illustrates this fact; the tooth S has
just dropped on the engaging pallet, but the fork has not yet reached
the bankings. At _a_ we have 1° of run, while if placed at _b_ we would
only have ½° of run, but still the same freedom for escaping, and less
unlocking resistance.

The bankings should be placed towards the acting end of the fork as
illustrated, as in case the watch "rebanks" there would be more strain
on the lever pivots if they were placed at the other end of the fork.

[Illustration: Fig. 7.]

_The Lift._--The lift is composed of the actual lift on the teeth and
pallets and the lock and run. We will suppose that from drop to drop we
allow 10°; if the lock is 1½° then the actual lift by means of the
inclined planes on teeth and pallets will be 8½°. We have seen that a
small lifting angle is advisable, so that the vibrations of the balance
will be as free as possible. There are other reasons as well. Fig. 7
shows two inclined planes; we desire to lift the weight 2 a distance
equal to the angle at which the planes are inclined; it will be seen at
a glance that we will have less friction by employing the smaller
incline, whereas with the larger one the motive power is employed
through a greater distance on the object to be moved. The smaller the
angle the more energetic will the movement be; the grinding of the
angles and fit of the pivots, etc., also increases in importance. An
actual lift of 8½° satisfies the conditions imposed very well. We have
before seen that both on account of the unlocking and the lifting
leverage of the pallet arms, it would be advisable to make them narrow
both in the equidistant and circular escapement. We will now study the
question from the standpoint of the lift, in so far as the wheel is
concerned.

[Illustration: Fig. 8.]

It is self-evident that a narrow pallet requires a wide tooth, and a
wide pallet a narrow or thin tooth wheel; in the ratchet wheel we have a
metal point passing over a jeweled plane. The friction is at its
minimum, because there is less adhesion than with the club tooth, but we
must emphasize the fact that we require a greater angle in proportion on
the pallets in this escapement than with the narrow pallets and wider
tooth. This seems to be a point which many do not thoroughly comprehend,
and we would advise a close study of Fig. 8, which will make it
perfectly clear, as we show both a wide and a narrow pallet. GH,
represents the primitive, which in this figure is also the real diameter
of the escape wheel. In measuring the lifting angles for the pallets,
our starting point is _always_ from the tangents AC and AD. The tangents
are straight lines, but the wheel describes the circle GH, therefore
they must deviate from one another, and the closer to the center A the
discharging edge of the engaging pallet reaches, the greater does this
difference become; and in the same manner the further the discharging
edge of the disengaging pallet is from the center A the greater it is.
This shows that the loss is greater in the equidistant than in the
circular escapement. After this we will designate this difference as
the "loss." In order to illustrate it more plainly we show the widest
pallet--the English--in equidistant form. This gives another reason why
the English lever should only be made with circular pallets, as we have
seen that the wider the pallet the greater the loss. The loss is
measured at the intersection of the path of the discharging edge OO,
with the circle G H, and is shown through AC2, which intersects these
circles at that point. In the case of the disengaging pallet, PP
illustrates the path of the discharging edge; the loss is measured as in
the preceding case where GH is intersected as shown by AD2. It amounts
to a different value on each pallet. Notice the loss between C and C2,
on the engaging, and D and D2 on the disengaging pallet; it is greater
on the engaging pallet, so much so that it amounts to 2°, which is equal
to the entire lock; therefore if 8½° of work is to be accomplished
through this pallet, the lifting plane requires an angle of 10½° struck
from AC.

Let us now consider the lifting action of the club tooth wheel. This is
decidedly a complicated action, and requires some study to comprehend.
In action with the engaging pallet the wheel moves _up_, or in the
direction of the motion of the pallets, but on the disengaging pallet it
moves _down_, and in a direction opposite to the pallets, and the heel
of the tooth moves with greater velocity than the locking edge; also in
the case of the engaging pallet, the locking edge moves with greater
velocity than the discharging edge; in the disengaging pallet the
opposite is the case, as the discharging edge moves with greater
velocity than the locking. These points involve factors which must be
considered, and the drafting of a correct action is of paramount
importance; we therefore show the lift as it is accomplished in four
different stages in a good action. Fig. 9 illustrates the engaging, and
Fig. 10 the disengaging pallet; by comparing the figures it will be
noticed that the lift takes place on the point of the tooth similar to
the English, until the discharging edge of the pallet has been passed,
when the heel gradually comes into play on the engaging, but more
quickly on the disengaging pallet.

We will also notice that during the first part of the lift the tooth
moves faster along the engaging lifting plane than on the disengaging;
on pallets 2 and 3 this difference is quite large; towards the latter
part of the lift the action becomes quicker on the disengaging pallet
and slower on the engaging.

To obviate this difficulty some fine watches, notably those of A. Lange
& Sons, have convex lifting planes on the engaging and concave on the
disengaging pallets; the lifting planes on the teeth are also curved.
See Fig. 11. This is decidedly an ingenious arrangement, and is in
strict accordance with scientific investigation. We should see many fine
watches made with such escapements if the means for producing them could
fully satisfy the requirements of the scientific principles involved.

[Illustration: Fig. 9.]

The distribution of the lift on tooth and pallet is a very important
matter; the lifting angle on the tooth must be _less_ in proportion to
its width than it is on the pallet. For the sake of making it perfectly
plain, we illustrate what should not be made; if we have 10½° for width
of tooth and pallet, and take half of it for a tooth, and the other
half for the pallet, making each of them 5¼° in width, and suppose we
have a lifting of 8½° to distribute between them, by allowing 4¼° on
each, the lift would take place as shown in Fig. 12, which is a very
unfavorable action. The edge of the engaging pallet scrapes on the
lifting plane of the tooth, yet it is astonishing to find some otherwise
very fine watches being manufactured right along which contain this
fault; such watches can be stopped with the ruby pin in the fork and the
engaging pallet in action, nor would they start when run down as soon as
the crown is touched, no matter how well they were finished and fitted.

[Illustration: Fig. 10.]

The lever lengths of the club tooth are variable, while with the ratchet
they are constant, which is in its favor; in the latter it would always
be as SB, Fig. 13. This is a shorter lever than QB, consequently more
powerful, although the greater velocity is at Q, which only comes into
action after the inertia of wheel and pallets has been overcome, and
when the greatest momentum during contact is reached. SB is the
primitive radius of the club tooth wheel, but both primitive and _real_
radius of the ratchet wheel. The distance of centers of wheel and pallet
will be alike in both cases; also the lockings will be the same distance
apart on both pallets; therefore, when horologists, even if they have
worldwide reputations, claim that the club tooth has an advantage over
the ratchet because it begins the lift with a shorter lever than the
latter, it does not make it so. We are treating the subject from a
purely horological standpoint, and neither patriotism or prejudice has
anything to do with it. We wish to sift the matter thoroughly and arrive
at a just conception of the merits and defects of each form of
escapement, and show _reasons_ for our conclusions.

[Illustration: Fig. 11.]

[Illustration: Fig. 12.]

[Illustration: Fig. 13.]

Anyone who has closely followed our deductions must see that in so far
as the wheel is concerned the ratchet or English wheel has several
points in its favor. Such a wheel is inseparable from a wide pallet; but
we have seen that a narrower pallet is advisable; also as little drop
and lock as possible; clearly, we must effect a compromise. In other
words, so far the balance of our reasoning is in favor of the club tooth
escapement and to effect an intelligent division of angles for tooth,
pallet and lift is one of the great questions which confronts the
intelligent horologist.

Anyone who has ever taken the pains to draw pallet and tooth with
different angles, through every stage of the lift, with both wide and
narrow pallets and teeth, in circular and equidistant escapements, will
have received an eye-opener. We strongly advise all our readers who are
practical workmen to try it after studying what we have said. We are
certain it will repay them.

[Illustration: Fig. 2.]

_The Center Distance of Wheel and Pallets._ The direction of pressure of
the wheel teeth should be through the pallet center by drawing the
tangents AC and AD, Fig. 2 to the primitive circle GH, at the
intersection of the angle FBE. This condition is realized in the
equidistant pallet. In the circular pallet, Fig. 3, this condition
cannot exist, as in order _to lock_ on a tangent the center distance
should be _greater_ for the engaging and _less_ for the disengaging
pallet, therefore watchmakers aim to go between the two and plant them
as before specified at A.

When planted on the tangents the unlocking resistance will be less and
the impulse transmitted under favorable conditions, especially so in
the circular, as the direction of pressure coincides (close to the
center of the lift), with the law of the parallelogram of forces.

It is _impossible_ to plant pallets on the tangents in very small
escapements, as there would not be enough room for a pallet arbor of
proper strength, nor will they be found planted on the tangents in the
medium size escapement with a long pallet arbor, nor in such a one with
a very wide tooth (see Fig. 4) as the heel would come so close to the
center A, that the solidity of pallets and arbor would suffer. We will
give an actual example. For a medium sized escape wheel with a primitive
diameter of 7.5 mm., the center distance AB is 4.33 mm. By using 3° of a
lifting angle on the teeth, the distance from the heel of the tooth to
the pallet center will be .4691 mm.; by allowing .1 mm. between wheel
and pallet and .15 mm. for stock on the pallets we find we will have a
pallet arbor as follows: .4691 - (.1 + .15) × 2 = .4382 mm. It would not
be practicable to make anything smaller.

[Illustration: Fig. 3.]

It behooves us now to see that while a narrow pallet is advisable a very
wide tooth is not; yet these two are inseparable. Here is another case
for a compromise, as, unquestionably the pallets ought to be planted on
the tangents. There is no difficulty about it in the English lever, and
we have shown in our example that a judiciously planned club tooth
escapement of medium size can be made with the center distance properly
planted.

[Illustration: Fig. 4.]

When considering the center distance we must of necessity consider the
widths of teeth and pallets and their lifting angles. We are now at a
point in which no watchmaker of intelligence would indicate one certain
division for these parts and claim it to be "the best." It is always
those who do not thoroughly understand a subject who are the first to
make such claims. We will, however, give our opinion within certain
limits. The angle to be divided for tooth and pallet is 10½°. Let us
divide it by 2, which would be the most natural thing to do, and examine
the problem. We will have 5¼° each for width of tooth and pallet. We
_must_ have a smaller lifting angle on the tooth than on the pallet, but
the wider the tooth the greater should its lifting angle be. It would
not be mechanical to make the tooth wide and the lifting angle small, as
the lifting plane on the pallets would be too steep on account of being
narrow. A lifting angle on the tooth which would be _exactly_ suitable
for a given circular, would be _too great_ for a given equidistant
pallet. It follows, therefore, taking 5¼° as a width for the tooth, that
while we could employ it in a fair sized escapement with equidistant
pallets, we could not do so with circular pallets and still have the
latter pitched on the tangents. We see the majority of escapements made
with narrower teeth than pallets, and for a very good reason.

In the example previously given, the 3° lift on the tooth is well
adapted for a width of 4½°, which would require a pallet 6° in width.
The tooth, therefore, would be ¾ the width of pallets, which is very
good indeed.

From what we have said it follows that a large number of pallets are not
planted on the tangents at all. We have never noticed this question in
print before. Writers generally seem to, in fact do, assume that no
matter how large or small the escapement may be, or how the pallets and
teeth are divided for width and lifting angle, no difficulty will be
found in locating the pallets on the tangents. Theoretically there is no
difficulty, but in practice we find there is.

_Equidistant vs. Circular._ At this stage we are able to weigh the
circular against the equidistant pallet. In beginning this essay we had
to explain the difference between them, so the reader could follow our
discussion, and not until now, are we able to sum up our conclusions.

The reader will have noticed that for such an important action as the
lift, which supplies power to the balance, the circular pallet is
favored from every point of view. This is a very strong point in its
favor. On the other hand, the unlocking resistance being less, and as
nearly alike as possible on both pallets in the equidistant, it is a
question if the total vibration of the balance will be greater with the
one than the other, although it will receive the impulse under better
conditions from the circular pallet; but it expends more force in
unlocking it. Escapement friction plays an important role in the
position and isochronal adjustments; the greater the friction
encountered the slower the vibration of the balance. The friction should
be constant. In unlocking, the equidistant comes nearer to fulfilling
this condition, while during the lift it is more nearly so in the
circular. The friction in unlocking, from a timing standpoint,
overshadows that of the impulse, and the tooth can be a little wider in
the equidistant than the circular escapement with the pallet properly
planted. Therefore for the _finest_ watches the equidistant escapement
is well adapted, but for anything less than that the circular should be
our choice.

_The Fork and Roller Action._ While the lifting action of the lever
escapement corresponds to that of the cylinder, the fork and roller
action corresponds to the impulse action in the chronometer and duplex
escapements.

Our experience leads us to believe that the action now under
consideration is but imperfectly understood by many workmen. It is a
complicated action, and when out of order is the cause of many annoying
stoppages, often characterized by the watch starting when taken from the
pocket.

The action is very important and is generally divided into impulse and
safety action, although we think we ought to divide it into three,
namely, by adding that of the unlocking action. We will first of all
consider the impulse and unlocking actions, because we cannot
intelligently consider the one without the other, as the ruby pin and
the slot in the fork are utilized in each. The ruby pin, or strictly
speaking, the "impulse radius," is a lever arm, whose length is measured
from the center of the balance staff to the face of the ruby pin, and is
used, firstly, as a power or transmitting lever on the acting or
geometrical length of the fork (_i. e._, from the pallet center to the
beginning of the horn), and which at the moment is a resistance lever,
to be utilized in unlocking the pallets. After the pallets are unlocked
the conditions are reversed, and we now find the lever fork, through the
pallets, transmitting power to the balance by means of the impulse
radius. In the first part of the action we have a short lever engaging a
longer one, which is an advantage. See Fig. 14, where we have purposely
somewhat exaggerated the conditions. A'X represents the impulse radius
at present under discussion, and AW the acting length of the fork. It
will be seen that the shorter the impulse radius, or in other words, the
closer the ruby pin is to the balance staff and the longer the fork, the
easier will the unlocking of the pallets be performed, but this entails
a great impulse angle, for the law applicable to the case is, that the
angles are in the inverse ratio to the radii. In other words, the
shorter the radius, the greater is the angle, and the smaller the angle
the greater is the radius. We know, though, that we must have as small
an impulse angle as possible in order that the balance should be highly
detached. Here is one point in favor of a short impulse radius, and one
against it. Now, let us turn to the impulse action. Here we have the
long lever AW acting on a short one, A'X, which is a disadvantage. Here,
then, we ought to try and have a short lever acting on a long one, which
would point to a short fork and a great impulse radius. Suppose AP,
Fig. 14, is the length of fork, and A'P is the impulse radius; here,
then, we favor the impulse, and it is directly in accordance with the
theory of the free vibration of the balance, for, as before stated, the
longer the radius the smaller the angle. The action at P is also closer
to the line of centers than it is at W, which is another advantage.

[Illustration: Fig. 14.]

We will notice that by employing a large impulse angle, and consequently
a short radius, the intersection _m_ of the two circles _ii_ and _cc_ is
very _safe_, whereas, with the conditions reversed in favor of the
impulse action, the intersection at _k_ is more delicate. We have now
seen enough to appreciate the fact that we favor one action at the
expense of another.

By having a lifting angle on pallet and tooth of 8½°, a locking angle of
1½°, and a run of ½°, we will have an angular movement of the fork of
8½ + 1½ + ½ = 10½°.

[Illustration: Fig. 15.]

Writers generally only consider the movement of the fork from drop to
drop on the pallets, but we will be thoroughly practical in the matter.
With a total motion of the fork of 10½° (JAW, Fig. 15), one-half, or 5¼°
will be performed on each side of the line of centers. We are at liberty
to choose any impulse angle which we may prefer; 3 to 1 is a good
proportion for an ordinary well-made watch. By employing it, the angle
XA'Y would be equal to 31½°. The radius A'X Fig. 16, is also of the same
proportion, but the angle AA'X is greater because the fork angle WAA' is
greater than the same angle in Fig. 15. We will notice that the
intersection _k_ is much smaller in Fig. 15 than in Fig. 16. The action
in the latter begins much further from the line of centers than in the
former and outlines an action which should not be made.

[Illustration: Fig. 16.]

To come back to the impulse angle, some might use a proportion of 3.5, 4
or even 5 to 1, while others for the finest of watches would only use
2.75 to 1. By having a total vibration of the balance of 1½ turns, which
is equal to 540° a fork angle of 10° and a proportion of 2.75 for the
impulse angle which would be equal to 10 × 2.75 = 27.5°. The _free_
vibration of the balance, or as this is called, "the supplemental arc,"
is equal to 540° - 27.5° = 512.50°, while with a proportion of 5 to 1,
making an impulse angle of 50°, it would be equal to 490°. To sum up,
the finer the watch the lower the proportion, the closer the action to
the line of centers, the smaller the friction. On account of leverage
the more difficult the unlocking but the more energetic the impulse when
it does occur. The velocity of the ruby pin at P; Fig. 14, is much
greater than at W, consequently it will not be overtaken as soon by the
fork as at W. The velocity of the fork at the latter point is greater
than at P; the intersection of _ii_ and _cc_ is also not as great;
therefore the lower the proportion the finer and more exact must the
workmanship be.

We will notice that the unlocking action has been overruled by the
impulse. The only point so far in which the former has been favored is
in the diminished action before the line of centers, as previously
pointed out at P, Fig. 14.

We will now consider the width of the ruby pin and to get a good insight
into the question, we will study Fig. 17. A is the pallet center, A' the
balance center, the line AA' being the line of centers; the angle WAA
equals half the total motion of the fork, the other half, of course,
taking place on the opposite side of the center line. WA is the _center_
of the fork when it rests against the bank. The angle AA'X represents
half the impulse angle; the other half, the same as with the fork, is
struck on the other side of the center line. At the point of
intersection of these angles we will draw _cc_ from the pallet center A,
which equals the acting length of the fork, and from the balance center
we will draw _ii_, which equals the _theoretical_ impulse radius; some
writers use it as the _real_ radius. The wider the ruby pin the greater
will the latter be, which we will explain presently.

The ruby pin in entering the fork must have a certain amount of freedom
for action, from 1 to 1¼°. Should the watch receive a jar at the moment
the guard point enters the crescent or passing hollow in the roller, the
fork would fly against the ruby pin. It is important that the angular
freedom between the fork and ruby pin at the moment it enters into the
slot be _less_ than the total locking angle on the pallets. If we employ
a locking angle of 1½° and ½° run, we would have a total lock on the
pallets of 2°. By allowing 1¼° of freedom for the ruby pin at the moment
the guard point enters the crescent, in case the fork should strike the
face of the ruby pin, the pallets will still be locked ¾° and the fork
drawn back against the bankings through the draft angle.

We will see what this shake amounts to for a given acting length of
fork, which describes an arc of a circle, therefore the acting length is
only the radius of that circle and must be multiplied by two in order to
get the diameter. The acting length of fork = 4.5 mm., what is the
amount of shake when the ruby pin passes the acting corner?
4.5 × 2 × 3.1416 ÷ 360° = .0785 × 1.25 = .0992 mm. The shake of the ruby
pin in the slot of the fork must be as slight as possible, consistent
with perfect freedom of action. It varies from ¼° to ½°, according to
length of fork and shape of ruby pin. A square ruby pin requires more
shake than any other kind; it enters the fork and receives the impulse
in a diagonal direction on the jewel, in which position it is
illustrated at Z, Fig. 20. This ruby pin acts on a knife edge, but for
all that the engaging friction during the unlocking action is
considerable.

Our reasoning tells us it matters not if a ruby pin be wide or narrow,
it must have _the same_ freedom in passing the acting edge of the fork,
therefore, to have the impulse radius on the point of intersection of
A'X with AW, Fig. 17, we would require a _very_ narrow ruby pin. With 1°
of freedom at the edge, and ½° in the slot, we could only have a ruby
pin of a width of 1½°. Applying it to the preceding example it would
only have an actual width of .0785 × 1.5 = .1178 mm., or the size of an
ordinary balance pivot. At _n_, Fig. 17, we illustrate such a ruby pin;
the theoretical and real impulse radius coincide with one another. The
intersection of the circle _ii_ and _cc_ is very slight, while the
friction in unlocking begins within 1° of half the total movement of the
fork from the line of centers; to illustrate, if the angular motion is
11° the ruby pin under discussion will begin action 4½° before the line
of centers, being an engaging, or "uphill" friction of considerable
magnitude.

[Illustration: Fig. 17.]

[Illustration: Fig. 18.]

[Illustration: Fig. 19.]

[Illustration: Fig. 20.]

The intersection with the fork is also much less than with the wider
ruby pin, making the impulse action very delicate. On the other hand the
widest ruby pin for which there is any occasion is one beginning the
unlocking action on the line of centers, Fig. 17; this entails a width
of slot equal to the angular motion of the fork. We see here the
advantage of a wide ruby pin over a narrow one in the unlocking action.
Let us now examine the question from the standpoint of the impulse
action.

Fig. 18 illustrates the moment the impulse is transmitted; the fork has
been moved in the direction of the arrow by the ruby pin; the escapement
has been unlocked and the opposite side of the slot has just struck the
ruby pin. The exact position in which the impulse is transmitted varies
with the locking angle, the width of ruby pin, its shake in the slot,
the length of fork, its weight, and the velocity of the ruby pin, which
is determined by the vibrations of the balance and the impulse radius.

In an escapement with a total lock of 1¾° and 1¼ of shake in the slot,
theoretically, the impulse would be transmitted 2° from the bankings.
The narrow ruby pin n receives the impulse on the line _v_, which is
closer to the line of centers than the line _u_, on which the large ruby
pin receives the impulse. Here then we have an advantage of the narrow
ruby pin over a wide one; with a wider ruby pin the balance is also more
liable to rebank when it takes a long vibration. Also on account of the
greater angle at which the ruby pin stands to the slot when the impulse
takes place, the _drop_ of the fork against the jewel will amount to
more than its shake in the slot (which is measured when standing on the
line of centers). On this account some watches have slots dovetailed in
form, being wider at the bottom, others have ruby pins of this form.
They require very exact execution; we think we can do without them by
judiciously selecting a width of ruby pin between the two extremes. We
would choose a ruby pin of a width equal to half the angular motion of
the fork. There is an ingenious arrangement of fork and roller which
aims to, and partially does, overcome the difficulty of choosing between
a wide and narrow ruby pin, it is known as the Savage pin roller
escapement. We intend to describe it later.

If the face of the ruby pin were planted on the theoretical impulse
radius _ii_, Fig. 19, the impulse would end in a butting action as
shown; hence the great importance of distinguishing between the
theoretical and real impulse radius and establishing a reliable data
from which to work. We feel that these actions have never been properly
and thoroughly treated in simple language; we have tried to make them
plain so that anyone can comprehend them with a little study.

Three good forms of ruby pins are the triangular, the oval and the flat
faced; for ordinary work the latter is as good as any, but for fine work
the triangular pin with the corners slightly rounded off is preferable.

[Illustration: Fig. 21.]

[Illustration: Fig. 23.]

[Illustration: Fig. 22.]

English watches are met with having a cylindrical or round ruby pin.
Such a pin should never be put into a watch. The law of the
parallelogram of forces is completely ignored by using such a pin; the
friction during the unlocking and impulse actions is too severe, as it
is, without the addition of so unmechanical an arrangement. Fig. 21
illustrates the action of a round ruby pin; _ii_ is the path of the ruby
pin; _cc_ that of the acting length of the fork. It is shown at the
moment the impulse is transmitted. It will be seen that the impact takes
place _below_ the center of the ruby pin, whereas it should take place
at the center, as the motion of the fork is _upwards_ and that of the
ruby pin _downwards_ until the line of the centers has been reached.
The same rule applies to the flat-faced pin and it is important that the
right quantity be ground off. We find that 3/7 is approximately the
amount which should be ground away. Fig. 22 illustrates the fork
standing against the bank. The ruby pin touches the side of the slot but
has not as yet begun to act; _ri_ is the real impulse circle for which
we allow 1¼° of freedom at the acting edge of the fork; the face of the
ruby pin is therefore on this line. The next thing to do is to find the
center of the pin. From the side _n_ of the slot we construct the right
angle _o n t_; from _n_, we transmit ½ the width of the pin, and plant
the center _x_ on the line _n t_. We can have the center of the pin
slightly below this line, but in no case above it; but if we put it
below, the pin will be thinner and therefore more easily broken.

[Illustration: Fig. 14.]

_The Safety Action._ Although this action is separate from the impulse
and unlocking actions, it is still very closely connected with them,
much more so in the single than in the double roller escapement. If we
were to place the ruby pin at _X_, Fig. 14, we could have a much
smaller roller than by placing it at _P_. With the small roller the
safety action is more secure, as the intersection at _m_ is greater than
at _k_. It is not as liable to "butt" and the friction is less when the
guard point is thrown against the small roller. Suppose we take two
rollers, one with a diameter of 2.5 mm., the other just twice this
amount, of 5 mm. By having the guard radius and pressure the same in
each case, if the guard point touched the larger roller it would not
only have twice, but four times more effect than on the smaller one. We
will notice that the smaller the impulse angle the larger the roller,
because the ruby pin is necessarily placed farther from the center. The
position of the ruby pin should, therefore, govern the size of the
roller, which should be as small as possible. There should only be
enough metal left between the circumference of the roller and the face
of the jewel to allow for a crescent or passing hollow of sufficient
depth and an efficient setting for the jewel. For this reason, as well
as securing the correct impulse radius and therefore angle, when
replacing the ruby pin, and having it set securely and mechanically in
the roller, it is necessary that the pin and the hole in the roller be
of the same form, and a good fit. Fig. 23 illustrates the difference in
size of rollers. In the smaller one the conditions imposed are
satisfied, while in the larger one they are not. In the single roller
the safety action is at the mercy of the impulse and pallet angles. We
have noticed that in order to favor the impulse we require a large
roller, and for the safety action a small one, therefore escapements
made on fine principles are supplied with two rollers, one for each
action.

It may be well to say that in our opinion a proportion between the fork
and impulse angles in 10° pallets of 3 or 3½ to 1, _depending_ upon the
size of the escapement, is the lowest which should be made in single
roller. We have seen them in proportions of 2 to 1 in single roller--a
scientific principle foolishly applied--resulting in an action entirely
unsatisfactory.

When the guard point is pressed against the roller the escape tooth must
still rest on the locking face of the pallet; if the total lock is 2°, by
allowing 1¼° freedom for the guard point between the bank and the roller
the escapement will still be locked ¾°. How much this shake actually
amounts to depends upon the guard radius. Suppose this to be 4 mm.,
then the freedom would equal 4 × 2 × 3.1416 ÷ 360 × 1.25 = .0873 mm.

[Illustration: Fig. 24.]

[Illustration: Fig. 25.]

_The Crescent_ in the roller must be large and deep enough so it will be
impossible for the guard point to touch in or on the corners of it; at
the same time it must not be too large, as it would necessitate a longer
horn on the fork than is necessary.

Fig. 24 shows the slot _n_ of the fork standing at the bank. The ruby
pin _o_ touches it, but has not as yet acted on it; _s s_ illustrates a
single roller, while S2 illustrates the safety roller for a double
roller escapement. In order to find the dimensions of the crescent in
the single roller we must proceed as follows: WA is in the center of the
fork when it rests against the bank, and is, therefore, one of the sides
of the fork angle, and is drawn from the pallet center; V A W is an
angle of 1¼°, which equals the freedom between the guard point and the
roller; _g g_ represents the path of the guard pin _u_ for the single
roller, and is drawn at the intersection of VA with the roller A' A2 is
a line drawn from the balance center through that of the ruby pin, and
therefore also passes through the center of the crescent. By planting a
compass on this line, where it cuts the periphery of the roller, and
locating the point of intersection of VA with the roller, will give us
one-half the crescent, the remaining half being transferred to the
opposite side of the line A' A2. We will notice that the guard point has
entered the crescent 1¼° before the fork begins to move.

The angle of opening for the crescent in the double roller escapement is
greater than in the single, because it is placed closer to the balance
center, and the guard point or dart further from the pallet center,
causing a greater intersection; also the velocity of the guard point has
increased, while that of the safety roller has decreased. Fig. 24, at
_ff_, shows the path of the dart _h_, which also has 1¼° freedom between
bank and roller. From the balance center we draw A' _d_ touching the
center or point of the dart; from this point we construct at 5° angle
_b_ A' _d_. This is to ensure sufficient freedom for the dart when
entering the crescent. We plant a compass on the point of intersection
of A' A2 with the safety roller, S2, and locating the point where A'_b_
intersects it, have found one-half the opening for the crescent, the
remaining half being constructed on the opposite side of the line A' A2.

_The Horn_ on the fork belongs to the safety action: more horn is
required with the double than with the single roller, on account of the
greater angle of opening for the crescent.

The horn should be of such a length that when the crescent has passed
the guard point, the end of the horn should point to at least the center
of the ruby pin.

The dotted circle, _s s_, Fig. 25, represents a single roller. It will
be noticed that the corner of the crescent has passed the guard pin _u_
by a considerable angle, and although this is so, in case of an accident
the _acting edge_ of the fork would come in contact with the ruby pin;
this proves that a well made single roller escapement really requires
but little horn, only enough to ensure the safe entry of the ruby pin in
case the guard point at that moment be thrown against the roller. We
will now examine the question from the standpoint of the double roller;
S2, Fig. 25, is the safety roller; the corner of the crescent has safely
passed the dart _h_; the centers of the ruby pin _o_ and of the crescent
being on the line A' A2, we plant the compass on the pallet center and
the center of the face of the ruby pin and draw _k k_, which will be the
path described by the horn. The end of the horn is therefore planted
upon it from 1½° to 1¾° from the ruby pin; this freedom at the end of
the horn is therefore from ¼° to ½° more than we allow for the guard
point; it depends upon the size of the escapement and locking angles
which we would choose. It must in any case be less than the lock on the
pallets, so that the fork will be drawn back against the bank in case
the horn be thrown against the ruby pin.

When treating on the width of the ruby pin, we mentioned the Savage pin
roller escapement, which we illustrate in Figs. 26 and 27. This
ingenious arrangement was designed with the view of combining the
advantages of both wide and narrow pins and at the same time without any
of their disadvantages.

In Fig. 26 we show the unlocking pins _u_ beginning their action on the
line of centers--the best possible point--in unlocking the escapement.
These pins were made of gold in all which we examined, although it is
recorded that wide ruby pins and ruby rollers have been used in this
escapement, which would be preferable.

The functions of the two pins in the roller are simply to unlock the
escapement; the impulse is not transmitted to them as is the case in the
ordinary fork and roller action. In this action the guard pin _i_ also
acts as the impulse pin. We will notice that the passing hollow in this
roller is a rectangular slot the same as in the ordinary fork. When the
escapement is being unlocked the guard pin _i_ enters the hollow and
when the escape tooth comes into contact with the lifting plane of the
pallet the pin _i_, Fig. 27, transmits the impulse to the roller.

[Illustration: Fig. 26.]

[Illustration: Fig. 28.]

The impulse is transmitted closer to the line of centers than could be
done with any ruby pin. If the pin _i_ were wider the impulse would be
transmitted still closer to the line of centers, but the intersection of
it with the roller would be less. It is very delicate as it is,
therefore from a practical standpoint it ought to be made thin but
consistent with solidity. If the pin is anyway large, it should be
flattened on the sides, otherwise the friction would be similar to that
of the round ruby pin. It would also be preferable (on account of the
pin _i_ being very easily bent) to make the impulse piece narrow but of
such a length that it could be screwed to the fork, the same as the dart
in the double roller. The impulse radius is also the radius of the
roller, because the impulse is transmitted to the roller itself; for
this reason the latter is smaller in this action than in the ordinary
one having the same angles; also a shorter lever is in contact with a
longer one in the unlocking than in ordinary action of the same angles;
but for all this the pins _u u_ should be pitched close to the edge of
the roller, as the angular connection of the balance with the escapement
would be increased during the unlocking action. This escapement being
very delicate requires a 12° pallet angle and a proportion between
impulse and pallet angles of not less than 3 to 1, which would mean an
impulse angle of 36°; this, together with the first rate workmanship
required are two of the reasons why this action is not often met with.

George Savage, of London, England, invented this action. He was a
watchmaker who, in the early part of this century, did much to perfect
the lever escapement by good work and nice proportion, besides inventing
the two pin variety. He spent the early part of his life in Clerkenwell,
but in his old days emigrated to Canada, and founded a flourishing
retail business in Montreal, where he died. Some of George Savage's
descendants are still engaged at the trade in Canada at the present day.

The correct delineation of the lever escapement is a very important
matter. We illustrate one which is so delineated that it can be
practically produced. We have not noticed a draft of the lever
escapement, especially with equidistant pallets and club teeth, which
would act correctly in a watch.

We have been aggressive in our work and have sometimes found theories
propounded and elongated which of themselves were not right; this may
have something to do with it, that we so often hear workmen say, "Theory
is no use, because if you work according to it your machine will not
run." We say, "No, sir, if your theory is not right in itself, then your
work will certainly not be correct; but if your theory be correct then
your work _must_ be correct. Why? it simply cannot be otherwise." We
will give it another name; let us say, apply sense, reason, thought,
experience and study to your work, and what have you done? You have
simply applied theory.

A theorem is a proposition to be proved, not being able to prove it, we
must simply change it according as our experience dictates, this is
precisely what we have done with the escapement after having followed
the deductions of recognized authorities with the result that we can now
illustrate an escapement which has been thoroughly subjected to an
impartial analysis in every respect, and which is theoretically and
practically correct.

We will not only give instructions for drafting the escapement now under
consideration, but will also make explanations how to draft it in
different positions, also in circular pallet and single roller. We are
convinced that by so doing we will do a service to many, we also wish to
avoid what we may call "the stereotyped" process, that is, one which may
be acquired by heart, but introduce any changes and perplexity is the
result. It is really not a difficult matter to draft escapements in
different positions, as an example will show.

Before making a draft we must know exactly what we wish to produce. It
is well in drafting escapements to make them as large as possible, say
thirty to forty times larger than in the watch, in the present case the
size is immaterial, but we must have specifications for the proportions
of the angles. Our draft is to be the most difficult subject in lever
escapements; it is to be represented just as if it were working in a
watch; it is to represent a good and reliable action in every respect,
one which can be applied without special difficulty to a good watch, and
is to be "up to date" in every particular and to contain the majority
of the best points and conclusions reached in our analysis.

_Specifications for Lever Escapement_: The pallets are to be
equidistant; the wheel teeth of the "club" form; there are to be two
rollers; wheel, pallet, and balance centers are to be in straight line.
The lock is to be 1½°, the run ¼°, making a total lock of 1¾°; the
movement of pallets from drop to drop is to be 10°, while the fork is to
move through 10¼° from bank to bank; the lift on the wheel teeth is to
be 3°, while the remainder is to be the lift on the pallets as follows:
10¼ - (1¾ + 3) = 5½° for lift of pallets.

The wheel is to have 15 teeth, with pallets spanning 3 teeth or 2½
spaces, making the angle from lock to lock = 360 ÷ 15 × 2½ = 60°, the
interval from tooth to tooth is 360 ÷ 15 = 24°; divided by 2
pallets = 24 ÷ 2 = 12° for width of tooth, pallet and drop; drop is to
be 1½°, the tooth is to be ¾ the width of the pallet, making a tooth of
a width of 4½° and a pallet of 6°.

The draw is to be 12° on each pallet, while the locking faces of the
teeth are to incline 24°. The acting length of fork is to be equal to
the distance of centers of scape wheel and pallets; the impulse angle
is to be 28°; freedom from dart and safety, roller is to be 1¼°, and
for dart and corner of crescent 5°; freedom for ruby pin and acting
edge of fork is to be 1¼°; width of slot is to be ½ the total motion,
or 10¼ ÷ 2 = 5 1/8°; shake of ruby pin in slot = ¼°, leaving
5 1/8 - ¼ = 4 7/8° for width of ruby pin.

Radius of safety roller to be 4/7 of the theoretical impulse radius. The
length of horn is to be such that the end would point at least to the
center of the ruby pin when the edge of the crescent passes the dart;
space between the end of horn and ruby pin is to be 1½°.

It is well to know that the angles for width of teeth, pallets and drop
are measured from the wheel center, while the lifting and locking angles
are struck from the pallet center, the draw from the locking corners of
the pallets, and the inclination of the teeth from the locking edge.

In the fork and roller action, the angle of motion, the width of slot,
the ruby pin and its shake, the freedom between dart and roller, of ruby
pin with acting edge of fork and end of horn are all measured from the
pallet center, while the impulse angle and the crescent are measured
from the balance center. A sensible drawing board measures 17 × 24
inches, we also require a set of good drawing instruments, the finer the
instruments the better; pay special attention to the compasses, pens and
protractor; add to this a straight ruler and set square.

The best all-round drawing paper, both for India ink and colored work
has a rough surface; it must be fastened firmly and evenly to the board
by means of thumb tacks; the lines must be light and made with a hard
pencil. Use Higgins' India ink, which dries rapidly.

[Illustration]

We will begin by drawing the center line A' A B; use the point B for the
escape center; place the compass on it and strike G H, the primitive or
geometrical circle of the escape wheel; set the center of the protractor
at B and mark off an angle of 30° on each side of the line of centers;
this will give us the angles A B E and A B F together, forming the angle
F B E of 60°, which represents from lock to lock of the pallets. Since
the chord of the angle of 60° is equal to the radius of the circle, this
gives us an easy means of verifying this angle by placing the compass at
the points of intersection of F B and E B with the primitive circle G H;
this distance must be equal to the radius of the circle. At these points
we will construct right angles to E B and F B, thus forming the tangents
C A and D A to the primitive circle G H. These tangents meet on the line
of centers at A, which will be the pallet center. Place the compass at A
and draw the locking circle M N at the points of intersection of E B and
F B with the primitive circle G H. The locking edges of the pallets will
always stand on this circle no matter in what relation the pallets
stand to the wheel. Place the center of the protractor at B and draw the
angle of width of pallets of 6°; I B E being for the engaging and J B F
for the disengaging pallet. In the equidistant pallet I B is drawn on
the side towards the center, while J B is drawn further from the center.
If we were drawing a circular pallet, one-half the width of pallets
would be placed on each side of E B and F B. At the points of
intersection of I B and J B with the primitive circle G H we draw the
path O for the discharging edge of the engaging and P for that of the
disengaging pallet. The total lock being 1¾°, we construct V' A at this
angle from C A; the point of intersection of V' A with the locking
circle M N, is the position of the locking corner of the engaging
pallet. The pallet having 12° draw when locked we place the center of
the protractor on this corner and draw the angle Q M E. Q M will be the
locking face of the engaging pallet. If the face of the pallet were on
the line E B there would be no draw, and if placed to the opposite side
of E B the tooth would repel the pallet, forming what is known as the
repellant escapement.

[Illustration: Fig. 28.]

Having shown how to delineate the locking face of the engaging pallet
when locked, we will now consider how to draft both it and the
disengaging pallet in correct positions when unlocked; to do so we
direct our attention until further notice to Fig. 28. The locking faces
Q M of the engaging and S N of the disengaging pallets are shown in
dotted lines _when locked_. We must now consider the relation which the
locking faces will bear to E B in the engaging, and to F B in the
disengaging pallets when unlocked. This is a question of some
importance; it is easy enough to represent the 12° from the 30° angles
when locked; we must be certain that they would occupy exactly that
position and yet show them unlocked; we shall take pains to do so. In
due time we shall show that there is no appreciable loss of lift on the
engaging pallet in the escapement illustrated; the angle T A V
therefore shows the total lift; we have not shown the corresponding
angles on the disengaging side because the angles are somewhat
different, but the total lift is still the same. G H represents the
primitive circle of the escape wheel, and X Z that of the real, while
M N represents the circular course which the locking corners of the
pallets take in an equidistant escapement. At a convenient position we
will construct the circle C C' D from the pallet center A. Notice the
points _e_ and _c_, where V A and T A intersect this circle; the space
between _e_ and _c_ represents the extent of the motion of the pallets
at this particular distance from the center A; this being so, then let
us apply it to the engaging pallet. At the point of intersection _o_ of
the dotted line Q M (which is an extended line on which the face of the
pallet lies when locked), with the circle C C' D, we will plant our
dividers and transfer _e c_ to _o n_. By setting our dividers on _o_ M
and transferring to _n_ M', we will obtain the location of Q' M', the
locking face when unlocked. Let us now turn our attention to the
disengaging pallet. The dotted line S N represents the location of the
locking face of the disengaging pallet when locked at an angle of 12°
from F B. At the intersection of S N with the circle C C' D we obtain
the point _j_. The motion of the two pallets being equal, we transfer
the distance _e c_ with the dividers from _j_ and obtain the point _l_.
By setting the dividers on _j_ N and transferring to _l_ N' we draw the
line S' N' on which the locking face of the disengaging pallet will be
located when unlocked. It will be perfectly clear to anyone that through
these means we can correctly represent the pallets in any desired
position.

We will notice that the face Q' M' of the engaging pallet when unlocked
stands at a greater angle to E B than it did when locked, while the
opposite is the case on the disengaging pallet, in which the angle
S' N' F is much less than S N F. This shows that the _deeper_ the
engaging pallet locks, the lighter will the draw be, while the opposite
holds good with the disengaging pallet; also, that the draw increases
during the unlocking of the engaging, and decreases during the unlocking
of the disengaging pallet. These points show that the draw should be
measured with the _fork standing against the bank_; not when the locking
corner of the pallet stands on the primitive circle, as is so often
done. The recoil of the wheel (which determines the draw), is
illustrated by the difference between the locking circle M N and the
face Q M for the engaging, and S N for the disengaging pallet, and along
the _acting_ surface it is alike on each pallet, showing that the draft
angle should be the same on each pallet.

A number of years ago we constructed the escapement model which we
herewith illustrate. All the parts are adjustable; the pallets can be
moved in any direction, the draft angles can be changed at will. Through
this model we can practically demonstrate the points of which we have
spoken. Such a model can be made by workmen after studying these
papers.

[Illustration]

In both the equidistant and circular pallets the locking face S N of the
disengaging pallet deviates more from the locking circle M N than does
the locking face Q M of the engaging pallet, as will be seen in the
diagram. This is because the draft angle is struck from E B which
deviates from the locking circle in such a manner, that if the face of a
pallet were planted on it and _locked deep enough_ to show it, the
wheel would actually _repel_ the pallet, whereas with the disengaging
pallet if it were planted on F B, it would actually produce draw if
locked very deep; this is on account of the natural deviation of the 30°
lines from the locking circle. This difference is more pronounced in the
circular than in the equidistant pallet, because in the former we have
two locking circles, the larger one being for the engaging pallet, and
as an arc of a large circle does not deviate as much from a straight
line as does that of a smaller circle, it will be easily understood that
the natural difference before spoken of is only enhanced thereby. For
this reason in order to produce an _actual_ draw of 12°, the engaging
pallet may be set at a slightly greater angle from E B in the circular
escapement; the amount depends upon the width of the pallets; the
requirements are that the recoil of the wheel will be the same on each
pallet. We must, however, repeat that one of the most important points
is to measure the draw when the fork stands against the bank, thereby
_increasing_ the draw on the engaging and _decreasing_ that of the
disengaging pallet _during_ the unlocking action, thus _naturally_
balancing one fault with another.

We will again proceed with the delineation of the escapement here
illustrated. After having drawn the locking face Q M, we draw the angle
of width of teeth of 4½°, by planting the protractor on the escape
center B. We measure the angle E B K, from the locking face of the
pallet; the line E B does not touch the locking face of the pallet at
the present time of contact with the tooth, therefore a line must be
drawn from the point of contact to the center B. We did so in our
drawing but do not illustrate it, as in a reduced engraving of this kind
it would be too close to E B and would only cause confusion. We will now
draw in the lifting angle of 3° for the tooth. From the tangent C A we
draw T A at the required angle; at the point of intersection of T A with
the 30° line E B we have the real circumference of the escape wheel. It
will only be necessary to connect the locking edge of the tooth with the
line K B, where the real or outer circle intersects it. It must be drawn
in the same manner in the circular escapement; if the tooth were drawn
up to the intersection of K B with T A, the lift would be too great, as
that point is further from the center A than the points of contact are.

If the real or outer circle of the wheel intersects both the locking
circle M N and the path O of the discharging edge at the points where
T A intersects them, then there will be _no loss_ of lift on the
engaging pallet. This is precisely how it is in the diagram; but if
there is any deviation, then the angle of loss must be measured on the
_real_ diameter of the wheel and not on the primitive, as is usually
done, as the real diameter of the wheel, or in other words the heel of
the tooth, forms the last point of contact. With a wider tooth and a
greater lifting angle there will even be a _gain_ of lift on the
engaging pallet; the pallet in such a case would actually require a
smaller lifting angle, according to the amount of gain. We gave full
directions for measuring the loss when describing its effects in Fig. 8.
Whatever the loss amounts to, it is added to the lifting plane of the
pallet. In the diagram under discussion there is no loss, consequently
the lifting angle on the pallet is to be 5½°. From V' A we draw V A at
the required angle; the point of intersection of V A with the path O
will be the discharging edge O. It will now only be necessary to connect
the locking corner M with it, and we have the lifting plane of the
pallet; the discharging side of the pallet is then drawn parallel to the
locking face and made a suitable length. We will now draw the locking
edges of the tooth by placing the center of the protractor on the
locking edge M and construct the angle B M M' of 24° and draw a circle
from the scape center B, to which the line M M' will be a tangent. We
will utilize this circle in drawing in the faces of the other teeth
after having spaced them off 24° apart, by simply putting a ruler on
the locking edges and on the periphery of the circle.

We now construct W' A as a tangent to the outer circle of the wheel,
thus forming the lifting angle D A W' of 3° for the teeth; this
corresponds to the angle T A C on the engaging side. W' A touches the
outer circle of the wheel at the intersection of F B with it. We will
notice that there is considerable deviation of W' A from the circle at
the intersection of J B with it. At the intersecting of this point we
draw U A; the angle U A W' is the loss of lift. This angle must be added
to the lifting angle of the pallets; we see that in this action there is
no loss on the engaging pallet, but on the disengaging the loss amounts
to approximately 7/8° in the action illustrated. As we have allowed ¼° of
run for the pallets, the discharging edge P is removed at this angle
from U A; we do not illustrate it, as the lines would cause confusion
being so close together. The lifting angle on the pallet is measured
from the point P and amounts to 5½° + the angle of the loss; the angle
W A U embraces the above angles besides ¼° for run. If the locks are
equal on each pallet, it proves that the lifts are also equal. This
gives us a practical method of proving the correctness of the drawing;
to do so, place the dividers on the locking circle M N at the
intersection of T A and V A with it, as this is the extent of motion;
transfer this measurement to N, if the _actual_ lift is the same on each
pallet, the dividers will locate the point which the locking corner N
will occupy _when locked_; this, in the present case, will be at an
angle of 1¾° below the tangent D A. By this simple method, the
correctness of our proposition that the loss of lift should be measured
from the outside circle of the wheel, can be proven. We often see the
loss measured for the engaging pallet on the primitive circumference
G H, and on the real circumference for the disengaging; if one is right
then the other must be wrong, as there is a noticeable deviation of the
tangent C A from the primitive circle G H at the intersection of the
locking circle M N; had we added this amount to the lifting angle V' A V
of the engaging pallet, the result would have been that the discharging
edge O would be over 1° below its present location, thus showing that by
the time the lift on the engaging pallet had been completed, the locking
corner N of the disengaging pallet would be locked at an angle of 2¾°
instead of only 1¾°. Many watches contain precisely this fault. If we
wish to make a draft showing the pallets at any desired position, at the
center of motion for instance, with the fork standing on the line of
centers, we would proceed in the following manner: 10¼° being the total
motion, one-half would equal 5 1/8°; as the total lock equals 1¾°, we
deduct this amount from it which leaves 5 1/8 - 1¾ = 3 3/8°, which is the
angle at which the locking corner M should be shown above the tangent
C A. Now let us see where the locking corner N should stand; M having
moved up 5 1/8°, therefore N moved down by that amount, the lift on the
pallet being 5½° and on the tooth 3° (which is added to the tangent
D A), it follows that N should stand 5½ + 3 - 5 1/8 = 3 3/8° above D A.
We can prove it by the lock, namely: 3 3/8° + 1¾ = 5 1/8°, half the
remaining motion. This shows how simple it is to draft pallets in
various positions, remembering always to use the tangents to the
primitive circle as measuring points. We have fully explained how to
draw in the draft angle on the pallets when unlocked, and do not require
to repeat it, except to say, that most authorities draw a tangent R N to
the locking circle M N, forming in other words, the right angle R N A,
then construct an angle of 12° from R N. We have drawn ours in by our
own method, which is the correct one. While we here illustrate S N R at
an angle of 12° it is in reality _less_ than that amount; had we
constructed S N at an angle of 12° from R N, then the draw would be 12°
from F B, when the primitive circumference of the wheel is reached, but
_more_ than 12° when the fork is against the bank.

The space between the discharging edge P and the heel of the tooth forms
the angle of drop J B I of 1½°; the definition for drop is that it is
the freedom for wheel and pallet. This is not, strictly speaking,
perfectly correct, as, during the unlocking action there will be a
recoil of the wheel to the extent of the draft angle; the heel of the
tooth will therefore approach the edge P, and the discharging side of
the pallet approaches the tooth, as only the discharging edge moves on
the path P.

A good length for the teeth is 1/10 the diameter of the wheel, measured
from the primitive diameter and from the locking edge of the tooth.

The backs of the teeth are hollowed out so as not to interfere with the
pallets, and are given a nice form; likewise the rim and arms are drawn
in as light and as neat as possible, consistent with strength.

Having explained the delineation of the wheel and pallet action we will
now turn our attention to that of the fork and roller. We tried to
explain these actions in such a manner that by the time we came to
delineate them no difficulty would be found, as in our analysis we
discussed the subject sufficiently to enable any one of ordinary
intelligence to obtain a correct knowledge of them. The fork and roller
action in straight line, right, or any other angle is delineated after
the methods we are about to give.

We specified that the acting length of fork was to be equal to the
center distance of wheel and pallets; this gives a fork of a fair
length.

Having drawn the line of centers A' A we will construct an angle equal
to half the angular motion of the pallets; the latter in the case under
consideration being 10¼°, therefore 5 1/8° is spaced off on each side of
the line of centers, forming the angles _m_ A _k_ of 10¼°. Placing our
dividers on A B the center distance of 'scape wheel and pallets, we
plant them on A and construct _c c_; thus we will have the acting length
of fork and its path. We saw in our analysis that the impulse angle
should be as small as possible. We will use one of 28° in our draft of
the double roller; we might however remark that this angle should vary
with the construction of the escapements in different watches; if too
small, the balance may be stopped when the escapement is locked, while
if too great it can be stopped during the lift; both these defects are
to be avoided. The angles being respectively 10¼° and 28° it follows
they are of the following proportions: 28° ÷ 10.25 = 2.7316. The impulse
radius therefore bears this relation (but in the inverse ratio to the
angles), to the acting length of fork.

We will put it in the following proportion; let A_c_ equal acting length
of fork, and _x_ the unknown quantity; 28:10.25 :: A_c_:_x_; the answer
will be the theoretical impulse radius. Having found the required radius
we plant one jaw of our measuring instrument on the point of
intersection of _c c_ with _k_ A or _m_ A and locate the other jaw on
the line of centers; we thus obtain A' the balance center. Through the
points of intersection before designated we will draft X A' and Y A'
forming the impulse angle X A' Y of 28°. At the intersection of this
angle with the fork angle _k_ A' _m_, we draw _i i_ from the center A;
this gives us the theoretical impulse circle. The total lock being 1¾°
it follows that the angle described by the balance in unlocking
= 1¾ × 2.7316 = 4.788°. According to the specifications the width of
slot is to be 5 1/8°; placing the center of the protractor on A we
construct half of this angle on each side of _k_ A, which passes through
the center of the fork when it rests against the bank; this gives us the
angle _s_ A _n_ of 5 1/8°. If the disengaging pallet were shown locked then
_m_ A would represent the center of the fork. The slot is to be made of
sufficient depth so there will be no possibility of the ruby pin
touching the bottom of it. The ruby pin is to have 1¼° freedom in
passing the acting edge of the fork; from the center A we construct the
angle _t_ A _n_ of 1¼°; at the point of intersection of _t_ A with _c c_
the acting radius of the fork, we locate the real impulse radius and
draw the arc _ri ri_ which describes the path made by the face of the
ruby pin. The ruby pin is to have ¼° of shake in the slot; it will
therefore have a width of 4 7/8°; this width is drawn in with the ruby pin
imagined as standing over the line of centers and is then transferred to
the position which the ruby pin is to occupy in the drawing.

The radius of the safety roller was given as 4/7 of the theoretical
impulse radius. They may be made of various proportions; thus 2/3 is often
used. Remember that the smaller we make it, the less the friction during
accidental contact with the guard pin, the greater must the passing
hollow be and the horn of fork and guard point must be longer, which
increases the weight of the fork.

Having drawn in the safety roller, and having specified that the freedom
between the dart and safety roller was to be 1¼°, the dart being in the
center of the fork, consequently _k_ A is the center of it; therefore we
construct the angle _k_ A X of 1¼°. At the point of intersection of X A
with the safety roller we draw the arc _g g_; this locates the point of
the dart which we will now draw in. We will next draw _d_ A' from the
balance center and touching the point of the dart; we now construct
_b_ A' at an angle of 5° to it. This is to allow the necessary freedom
for the dart when entering the crescent; from A' we draw a line through
the center of the ruby pin. We do not show it in the drawing, as it
would be indiscernible, coming very close to A' X. This line will also
pass through the center of the crescent. At the point of intersection of
A' _b_ with the safety roller we have one of the edges of the crescent. By
placing our compass at the center of the crescent on the periphery of
the roller and on the edge which we have just found, it follows that our
compass will span the radius of the crescent. We now sweep the arc for
the latter, thus also drawing in the remaining half of the crescent on
the other side of A' X and bringing the crescent of sufficient depth
that no possibility exists of the dart touching in or on the edges of
it. We will now draw in the impulse roller and make it as light as
possible consistent with strength. A hole is shown through the impulse
roller to counterbalance the reduced weight at the crescent. When
describing Fig. 24, we gave instructions for finding the dimensions of
crescent and position of guard pin for the single roller. We will find
the length of horn; to do so we must closely follow directions given for
Fig. 25. In locating the end of the horn, we must find the location of
the center of the crescent and ruby pin _after_ the edge of the crescent
has passed the dart. From the point of intersection of A' _b_ with the
safety roller we transfer the radius of the crescent on the periphery of
the safety roller towards the side against the bank, then draw a line
from A' through the point so found. At point of intersection of this
line with the real impulse circle _r i r i_ we draw an arc radiating
from the pallet center; the end of the horn will be located on this arc.
In our drawing the arc spoken of coincides with the dart radius _g g_.
As before pointed out, we gave particulars when treating on Fig. 25,
therefore considered it unnecessary to further complicate the draft by
the addition of all the constructional lines. We specified that the
freedom between ruby pin and end of horn was to be 1½°; these lines,
(which we do not show) are drawn from the pallet center. Having
located the end of the horn on the side standing against the bank, we
place the dividers on it and on the point of intersection of _k_ A with
_g g_--which in this case is on the point of the dart,--and transfer
this measurement along _g g_ which will locate the end of the horn on
the opposite side.

We have the acting edges of the fork on _cc_ and have also found the
position of the ends of the horns; their curvature is drawn in the
following manner: We place our compasses on A and _r i_, spanning
therefore the real impulse radius; the compass is now set on the acting
edge of the fork and an arc swept with it which is then to be
intersected by another arc swept from the end of the horn, on the same
side of the fork. At the point of intersection of the arcs the compass
is planted and the curvature of the horn drawn in, the same operation is
to be repeated with the other horn. We will now draw in the sides of the
horn of such a form that should the watch rebank, the side of the ruby
pin will squarely strike the fork. If the back of the ruby pin strikes
the fork there will be a greater tendency of breaking it and injuring
the pivots on account of acting like a wedge. The fork and pallets are
now drawn in as lightly as possible and of such form as to admit of
their being readily poised. The banks are to be drawn at equal distances
from the line of centers. In delineating the fork and roller action in
any desired position, it must be remembered that the points of location
of the real impulse radius, the end of horn, the dart or guard pin and
crescent, must _all_ be obtained _when standing against the bank_, and
the arcs drawn which they describe; the parts are then located according
to the angle at which they are removed from the banks.

We think the instructions given are ample to enable any one to master
the subject. We may add that when one becomes well acquainted with the
escapement, many of the angles radiating from a common center, may be
drawn in at once. We had intended describing the mechanical construction
of the escapement, which does unmistakably present some difficulties on
account of the small dimensions of the parts, but nevertheless it can be
mechanically executed true to the principles enumerated. We have evolved
a method of so producing them that young men in a comparatively short
period have made them from their drafts (without automatic machinery)
that their watches start off when run down the moment the crown is
touched. Perhaps later on we will write up the subject. It is our
intention of doing so, as we make use of such explanations in our
regular work.







*** END OF THE PROJECT GUTENBERG EBOOK AN ANALYSIS OF THE LEVER ESCAPEMENT ***


    

Updated editions will replace the previous one—the old editions will
be renamed.

Creating the works from print editions not protected by U.S. copyright
law means that no one owns a United States copyright in these works,
so the Foundation (and you!) can copy and distribute it in the United
States without permission and without paying copyright
royalties. Special rules, set forth in the General Terms of Use part
of this license, apply to copying and distributing Project
Gutenberg™ electronic works to protect the PROJECT GUTENBERG™
concept and trademark. Project Gutenberg is a registered trademark,
and may not be used if you charge for an eBook, except by following
the terms of the trademark license, including paying royalties for use
of the Project Gutenberg trademark. If you do not charge anything for
copies of this eBook, complying with the trademark license is very
easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project
Gutenberg eBooks may be modified and printed and given away—you may
do practically ANYTHING in the United States with eBooks not protected
by U.S. copyright law. Redistribution is subject to the trademark
license, especially commercial redistribution.


START: FULL LICENSE

THE FULL PROJECT GUTENBERG LICENSE

PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase “Project
Gutenberg”), you agree to comply with all the terms of the Full
Project Gutenberg™ License available with this file or online at
www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing Project Gutenberg™
electronic works

1.A. By reading or using any part of this Project Gutenberg™
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/copyright) agreement. If you do not agree to abide by all
the terms of this agreement, you must cease using and return or
destroy all copies of Project Gutenberg™ electronic works in your
possession. If you paid a fee for obtaining a copy of or access to a
Project Gutenberg™ electronic work and you do not agree to be bound
by the terms of this agreement, you may obtain a refund from the person
or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement. There are a few
things that you can do with most Project Gutenberg™ electronic works
even without complying with the full terms of this agreement. See
paragraph 1.C below. There are a lot of things you can do with Project
Gutenberg™ electronic works if you follow the terms of this
agreement and help preserve free future access to Project Gutenberg™
electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the
Foundation” or PGLAF), owns a compilation copyright in the collection
of Project Gutenberg™ electronic works. Nearly all the individual
works in the collection are in the public domain in the United
States. If an individual work is unprotected by copyright law in the
United States and you are located in the United States, we do not
claim a right to prevent you from copying, distributing, performing,
displaying or creating derivative works based on the work as long as
all references to Project Gutenberg are removed. Of course, we hope
that you will support the Project Gutenberg™ mission of promoting
free access to electronic works by freely sharing Project Gutenberg™
works in compliance with the terms of this agreement for keeping the
Project Gutenberg™ name associated with the work. You can easily
comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg™ License when
you share it without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are
in a constant state of change. If you are outside the United States,
check the laws of your country in addition to the terms of this
agreement before downloading, copying, displaying, performing,
distributing or creating derivative works based on this work or any
other Project Gutenberg™ work. The Foundation makes no
representations concerning the copyright status of any work in any
country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other
immediate access to, the full Project Gutenberg™ License must appear
prominently whenever any copy of a Project Gutenberg™ work (any work
on which the phrase “Project Gutenberg” appears, or with which the
phrase “Project Gutenberg” is associated) is accessed, displayed,
performed, viewed, copied or distributed:

    This eBook is for the use of anyone anywhere in the United States and most
    other parts of the world at no cost and with almost no restrictions
    whatsoever. You may copy it, give it away or re-use it under the terms
    of the Project Gutenberg License included with this eBook or online
    at www.gutenberg.org. If you
    are not located in the United States, you will have to check the laws
    of the country where you are located before using this eBook.
  
1.E.2. If an individual Project Gutenberg™ electronic work is
derived from texts not protected by U.S. copyright law (does not
contain a notice indicating that it is posted with permission of the
copyright holder), the work can be copied and distributed to anyone in
the United States without paying any fees or charges. If you are
redistributing or providing access to a work with the phrase “Project
Gutenberg” associated with or appearing on the work, you must comply
either with the requirements of paragraphs 1.E.1 through 1.E.7 or
obtain permission for the use of the work and the Project Gutenberg™
trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg™ electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any
additional terms imposed by the copyright holder. Additional terms
will be linked to the Project Gutenberg™ License for all works
posted with the permission of the copyright holder found at the
beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg™
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg™.

1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg™ License.

1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including
any word processing or hypertext form. However, if you provide access
to or distribute copies of a Project Gutenberg™ work in a format
other than “Plain Vanilla ASCII” or other format used in the official
version posted on the official Project Gutenberg™ website
(www.gutenberg.org), you must, at no additional cost, fee or expense
to the user, provide a copy, a means of exporting a copy, or a means
of obtaining a copy upon request, of the work in its original “Plain
Vanilla ASCII” or other form. Any alternate format must include the
full Project Gutenberg™ License as specified in paragraph 1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg™ works
unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg™ electronic works
provided that:

    • You pay a royalty fee of 20% of the gross profits you derive from
        the use of Project Gutenberg™ works calculated using the method
        you already use to calculate your applicable taxes. The fee is owed
        to the owner of the Project Gutenberg™ trademark, but he has
        agreed to donate royalties under this paragraph to the Project
        Gutenberg Literary Archive Foundation. Royalty payments must be paid
        within 60 days following each date on which you prepare (or are
        legally required to prepare) your periodic tax returns. Royalty
        payments should be clearly marked as such and sent to the Project
        Gutenberg Literary Archive Foundation at the address specified in
        Section 4, “Information about donations to the Project Gutenberg
        Literary Archive Foundation.”
    
    • You provide a full refund of any money paid by a user who notifies
        you in writing (or by e-mail) within 30 days of receipt that s/he
        does not agree to the terms of the full Project Gutenberg™
        License. You must require such a user to return or destroy all
        copies of the works possessed in a physical medium and discontinue
        all use of and all access to other copies of Project Gutenberg™
        works.
    
    • You provide, in accordance with paragraph 1.F.3, a full refund of
        any money paid for a work or a replacement copy, if a defect in the
        electronic work is discovered and reported to you within 90 days of
        receipt of the work.
    
    • You comply with all other terms of this agreement for free
        distribution of Project Gutenberg™ works.
    

1.E.9. If you wish to charge a fee or distribute a Project
Gutenberg™ electronic work or group of works on different terms than
are set forth in this agreement, you must obtain permission in writing
from the Project Gutenberg Literary Archive Foundation, the manager of
the Project Gutenberg™ trademark. Contact the Foundation as set
forth in Section 3 below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
works not protected by U.S. copyright law in creating the Project
Gutenberg™ collection. Despite these efforts, Project Gutenberg™
electronic works, and the medium on which they may be stored, may
contain “Defects,” such as, but not limited to, incomplete, inaccurate
or corrupt data, transcription errors, a copyright or other
intellectual property infringement, a defective or damaged disk or
other medium, a computer virus, or computer codes that damage or
cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right
of Replacement or Refund” described in paragraph 1.F.3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg™ trademark, and any other party distributing a Project
Gutenberg™ electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium
with your written explanation. The person or entity that provided you
with the defective work may elect to provide a replacement copy in
lieu of a refund. If you received the work electronically, the person
or entity providing it to you may choose to give you a second
opportunity to receive the work electronically in lieu of a refund. If
the second copy is also defective, you may demand a refund in writing
without further opportunities to fix the problem.

1.F.4. Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO
OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of
damages. If any disclaimer or limitation set forth in this agreement
violates the law of the state applicable to this agreement, the
agreement shall be interpreted to make the maximum disclaimer or
limitation permitted by the applicable state law. The invalidity or
unenforceability of any provision of this agreement shall not void the
remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg™ electronic works in
accordance with this agreement, and any volunteers associated with the
production, promotion and distribution of Project Gutenberg™
electronic works, harmless from all liability, costs and expenses,
including legal fees, that arise directly or indirectly from any of
the following which you do or cause to occur: (a) distribution of this
or any Project Gutenberg™ work, (b) alteration, modification, or
additions or deletions to any Project Gutenberg™ work, and (c) any
Defect you cause.

Section 2. Information about the Mission of Project Gutenberg™

Project Gutenberg™ is synonymous with the free distribution of
electronic works in formats readable by the widest variety of
computers including obsolete, old, middle-aged and new computers. It
exists because of the efforts of hundreds of volunteers and donations
from people in all walks of life.

Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg™’s
goals and ensuring that the Project Gutenberg™ collection will
remain freely available for generations to come. In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg™ and future
generations. To learn more about the Project Gutenberg Literary
Archive Foundation and how your efforts and donations can help, see
Sections 3 and 4 and the Foundation information page at www.gutenberg.org.

Section 3. Information about the Project Gutenberg Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service. The Foundation’s EIN or federal tax identification
number is 64-6221541. Contributions to the Project Gutenberg Literary
Archive Foundation are tax deductible to the full extent permitted by
U.S. federal laws and your state’s laws.

The Foundation’s business office is located at 809 North 1500 West,
Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up
to date contact information can be found at the Foundation’s website
and official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project Gutenberg
Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread
public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine-readable form accessible by the widest
array of equipment including outdated equipment. Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States. Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements. We do not solicit donations in locations
where we have not received written confirmation of compliance. To SEND
DONATIONS or determine the status of compliance for any particular state
visit www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including checks, online payments and credit card donations. To
donate, please visit: www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg™ electronic works

Professor Michael S. Hart was the originator of the Project
Gutenberg™ concept of a library of electronic works that could be
freely shared with anyone. For forty years, he produced and
distributed Project Gutenberg™ eBooks with only a loose network of
volunteer support.

Project Gutenberg™ eBooks are often created from several printed
editions, all of which are confirmed as not protected by copyright in
the U.S. unless a copyright notice is included. Thus, we do not
necessarily keep eBooks in compliance with any particular paper
edition.

Most people start at our website which has the main PG search
facility: www.gutenberg.org.

This website includes information about Project Gutenberg™,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.