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+Project Gutenberg's An Analysis of the Lever Escapement, by H. R. Playtner
+
+This eBook is for the use of anyone anywhere 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
+
+
+Title: An Analysis of the Lever Escapement
+
+Author: H. R. Playtner
+
+Release Date: June 30, 2007 [EBook #21978]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK AN ANALYSIS OF THE LEVER ***
+
+
+
+
+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 of An Analysis of the Lever Escapement, by 
+H. R. Playtner
+
+*** END OF THIS PROJECT GUTENBERG EBOOK AN ANALYSIS OF THE LEVER ***
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