When contact ensues between the tooth _D_ and pallet _C_ the tooth _D_ will attack the pallet at the point where the radial line _A v_ crosses the tooth face. We have now explained how we can delineate a tooth or pallet at any point of its angular motion, and will next explain how to apply this knowledge in actual practice.

PRACTICAL PROBLEMS IN THE LEVER ESCAPEMENT.

To delineate our entrance pallet after one-half of the engaged tooth has pa.s.sed the inner angle of the entrance pallet, we proceed, as in former ill.u.s.trations, to establish the escape-wheel center at _A_, and from it sweep the arc _b_, to represent the pitch circle. We next sweep the short arcs _p s_, to represent the arcs through which the inner and outer angles of the entrance pallet move. Now, to comply with our statement as above, we must draw the tooth as if half of it has pa.s.sed the arc _s_.

To do this we draw from _A_ as a center the radial line _A j_, pa.s.sing through the point _s_, said point _s_ being located at the intersection of the arcs _s_ and _b_. The tooth _D_ is to be shown as if one half of it has pa.s.sed the point _s_; and, consequently, if we lay off three degrees on each side of the point _s_ and establish the points _d m_, we have located on the arc _b_ the angular extent of the tooth to be drawn.

To aid in our delineations we draw from the center _A_ the radial lines _A d'_ and _A m'_, pa.s.sing through the points _d m_. The arc _a_ is next drawn as in former instructions and establishes the length of the addendum of the escape-wheel teeth, the outer angle of our escape-wheel tooth being located at the intersection of the arc _a_ with the radial line _A d'_.

As shown in Fig. 92, the impulse planes of the tooth _D_ and pallet _C_ are in contact and, consequently, in parallel planes, as mentioned on page 91. It is not an easy matter to determine at exactly what degree of angular motion of the escape wheel such condition takes place; because to determine such relation mathematically requires a knowledge of higher mathematics, which would require more study than most practical men would care to bestow, especially as they would have but very little use for such knowledge except for this problem and a few others in dealing with epicycloidal curves for the teeth of wheels.

For all practical purposes it will make no difference whether such parallelism takes place after eight or nine degrees of angular motion of the escape wheel subsequent to the locking action. The great point, as far as practical results go, is to determine if it takes place at or near the time the escape wheel meets the greatest resistance from the hairspring. We find by a.n.a.lysis of our drawing that parallelism takes place about the time when the tooth has three degrees of angular motion to make, and the pallet lacks about two degrees of angular movement for the tooth to escape. It is thus evident that the relations, as shown in our drawing, are in favor of the train or mainspring power over hairspring resistance as three is to two, while the average is only as eleven to ten; that is, the escape wheel in its entire effort pa.s.ses through eleven degrees of angular motion, while the pallets and fork move through ten degrees. The student will thus see we have arranged to give the train-power an advantage where it is most needed to overcome the opposing influence of the hairspring.

[Ill.u.s.tration: Fig. 92]

As regards the exalted adhesion of the parallel surfaces, we fancy there is more harm feared than really exists, because, to take the worst view of the situation, such parallelism only exists for the briefest duration, in a practical sense, because theoretically these surfaces never slide on each other as parallel planes. Mathematically considered, the theoretical plane represented by the impulse face of the tooth approaches parallelism with the plane represented by the impulse face of the pallet, arrives at parallelism and instantly pa.s.ses away from such parallelism.

TO DRAW A PALLET IN ANY POSITION.

As delineated in Fig. 92, the impulse planes of the tooth and pallet are in contact; but we have it in our power to delineate the pallet at any point we choose between the arcs _p s_. To describe and ill.u.s.trate the above remark, we say the lines _B e_ and _B f_ embrace five degrees of angular motion of the pallet. Now, the impulse plane of the pallet occupies four of these five degrees. We do not draw a radial line from _B_ inside of the line _B e_ to show where the outer angle of the impulse plane commences, but the reader will see that the impulse plane is drawn one degree on the arc _p_ below the line _B e_. We continue the line _h h_ to represent the impulse face of the tooth, and measure the angle _B n h_ and find it to be twenty-seven degrees. Now suppose we wish to delineate the entrance pallet as if not in contact with the escape-wheel tooth--for ill.u.s.tration, say, we wish the inner angle of the pallet to be at the point _v_ on the arc _s_. We draw the radial line _B l_ through _v_; and if we draw another line so it pa.s.ses through the point _v_ at an angle of twenty-seven degrees to _B l_, and continue said line so it crosses the arc _p_, we delineate the impulse face of our pallet.

We measure the angle _i n B_, Fig. 92, and find it to be seventy-four degrees; we draw the line _v t_ to the same angle with _v B_, and we define the inner face of our pallet in the new position. We draw a line parallel with _v t_ from the intersection of the line _v y_ with the arc _p_, and we define our locking face. If now we revolve the lines we have just drawn on the center _B_ until the line _l B_ coincides with the line _f B_, we will find the line _y y_ to coincide with _h h_, and the line _v v'_ with _n i_.

HIGHER MATHEMATICS APPLIED TO THE LEVER ESCAPEMENT.

We have now instructed the reader how to delineate either tooth or pallet in any conceivable position in which they can be related to each other. Probably nothing has afforded more efficient aid to practical mechanics than has been afforded by the graphic solution of abstruce mathematical problems; and if we add to this the means of correction by mathematical calculations which do not involve the highest mathematical acquirements, we have approached pretty close to the actual requirements of the practical watchmaker.

[Ill.u.s.tration: Fig. 93]

To better explain what we mean, we refer the reader to Fig. 93, where we show preliminary drawings for delineating a lever escapement. We wish to ascertain by the graphic method the distance between the centers of action of the escape wheel and the pallet staff. We make our drawing very carefully to a given scale, as, for instance, the radius of the arc _a_ is 5". After the drawing is in the condition shown at Fig. 93 we measure the distance on the line _b_ between the points (centers) _A B_, and we thus by graphic means obtain a measure of the distance between _A B_. Now, by the use of trigonometry, we have the length of the line _A f_ (radius of the arc _a_) and all the angles given, to find the length of _f B_, or _A B_, or both _f B_ and _A B_. By adopting this policy we can verify the measurements taken from our drawings. Suppose we find by the graphic method that the distance between the points _A B_ is 5.78", and by trigonometrical computation find the distance to be 5.7762". We know from this that there is .0038" to be accounted for somewhere; but for all practical purposes either measurement should be satisfactory, because our drawing is about thirty-eight times the actual size of the escape wheel of an eighteen-size movement.

HOW THE BASIS FOR CLOSE MEASUREMENTS IS OBTAINED.

Let us further suppose the diameter of our actual escape wheel to be .26", and we were constructing a watch after the lines of our drawing.

By "lines," in this case, we mean in the same general form and ratio of parts; as, for ill.u.s.tration, if the distance from the intersection of the arc _a_ with the line _b_ to the point _B_ was one-fifteenth of the diameter of the escape wheel, this ratio would hold good in the actual watch, that is, it would be the one-fifteenth part of .26". Again, suppose the diameter of the escape wheel in the large drawing is 10" and the distance between the centers _A B_ is 5.78"; to obtain the actual distance for the watch with the escape wheel .26" diameter, we make a statement in proportion, thus: 10 : 5.78 :: .26 to the actual distance between the pivot holes of the watch. By computation we find the distance to be .15". These proportions will hold good in every part of actual construction.

All parts--thickness of the pallet stones, length of pallet arms, etc.--bear the same ratio of proportion. We measure the thickness of the entrance pallet stone on the large drawing and find it to be .47"; we make a similar statement to the one above, thus: 10 : .47 :: .26 to the actual thickness of the real pallet stone. By computation we find it to be .0122". All angular relations are alike, whether in the large drawing or the small pallets to match the actual escape wheel .26" in diameter.

Thus, in the pallet _D_, Fig. 93, the impulse face, as reckoned from _B_ as a center, would occupy four degrees.

MAKE A LARGE ESCAPEMENT MODEL.

Reason would suggest the idea of having the theoretical keep pace and touch with the practical. It has been a grave fault with many writers on horological matters that they did not make and measure the abstractions which they delineated on paper. We do not mean by this to endorse the cavil we so often hear--"Oh, that is all right in theory, but it will not work in practice." If theory is right, practice must conform to it.

The trouble with many theories is, they do not contain all the elements or factors of the problem.

[Ill.u.s.tration: Fig. 94]

Near the beginning of this treatise we advised our readers to make a large model, and described in detail the complete parts for such a model. What we propose now is to make adjustable the pallets and fork to such a model, in order that we can set them both right and wrong, and thus practically demonstrate a perfect action and also the various faults to which the lever escapement is subject. The pallet arms are shaped as shown at _A_, Fig. 94. The pallets _B B'_ can be made of steel or stone, and for all practical purposes those made of steel answer quite as well, and have the advantage of being cheaper. A plate of sheet bra.s.s should be obtained, shaped as shown at _C_, Fig. 95. This plate is of thin bra.s.s, about No. 18, and on it are outlined the pallet arms shown at Fig. 94.

[Ill.u.s.tration: Fig. 95]

[Ill.u.s.tration: Fig. 96]

[Ill.u.s.tration: Fig. 97]

[Ill.u.s.tration: Fig. 98]

To make the pallets adjustable, they are set in thick disks of sheet bra.s.s, as shown at _D_, Figs. 95, 96 and 97. At the center of the plate _C_ is placed a bra.s.s disk _E_, Fig. 98, which serves to support the lever shown at Fig. 99. This disk _E_ is permanently attached to the plate _C_. The lever shown at Fig. 99 is attached to the disk _E_ by two screws, which pa.s.s through the holes _h h_. If we now place the bra.s.s pieces _D D'_ on the plate _C_ in such a way that the pallets set in them correspond exactly to the pallets as outlined on the plate _C_, we will find the action of the pallets to be precisely the same as if the pallet arms _A A'_, Fig. 94, were employed.

[Ill.u.s.tration: Fig. 99]

To enable us to practically experiment with and to fully demonstrate all the problems of lock, draw, drop, etc., we make quite a large hole in _C_ where the screws _b_ come. To explain, if the screws _b b_ were tapped directly into _C_, as they are shown at Fig. 95, we could only turn the disk _D_ on the screw _b_; but if we enlarge the screw hole in _C_ to three or four times the natural diameter, and then place the nut _e_ under _C_ to receive the screw _b_, we can then set the disks _D D'_ and pallets _B B'_ in almost any relation we choose to the escape wheel, and clamp the pallets fast and try the action. We show at Fig. 97 a view of the pallet _B'_, disk _D'_ and plate _C_ (seen in the direction of the arrow _c_) as shown in Fig. 95.

PRACTICAL LESSONS WITH FORK AND PALLET ACTION.

It will be noticed in Fig. 99 that the hole _g_ for the pallet staff in the lever is oblong; this is to allow the lever to be shifted back and forth as relates to roller and fork action. We will not bother about this now, and only call attention to the capabilities of such adjustments when required. At the outset we will conceive the fork _F_ attached to the piece _E_ by two screws pa.s.sing through the holes _h h_, Fig. 99. Such an arrangement will insure the fork and roller action keeping right if they are put right at first. Fig. 100 will do much to aid in conveying a clear impression to the reader.

The idea of the adjustable features of our escapement model is to show the effects of setting the pallets wrong or having them of bad form. For ill.u.s.tration, we make use of a pallet with the angle too acute, as shown at _B'''_, Fig. 101. The problem in hand is to find out by mechanical experiments and tests the consequences of such a change. It is evident that the angular motion of the pallet staff will be increased, and that we shall have to open one of the banking pins to allow the engaging tooth to escape. To trace out _all_ the consequences of this one little change would require a considerable amount of study, and many drawings would have to be made to ill.u.s.trate the effects which would naturally follow only one such slight change.

[Ill.u.s.tration: Fig. 100]

[Ill.u.s.tration: Fig. 101]

Suppose, for ill.u.s.tration, we should make such a change in the pallet stone of the entrance pallet; we have increased the angle between the lines _k l_ by (say) one and a half degrees; by so doing we would increase the lock on the exit pallet to three degrees, provided we were working on a basis of one and a half degrees lock; and if we pushed back the exit pallet so as to have the proper degree of lock (one and a half) on it, the tooth which would next engage the entrance pallet would not lock at all, but would strike the pallet on the impulse instead of on the locking face. Again, such a change might cause the jewel pin to strike the horn of the fork, as indicated at the dotted line _m_, Fig.

99.

Dealing with such and similar abstractions by mental process requires the closest kind of reasoning; and if we attempt to delineate all the complications which follow even such a small change, we will find the job a lengthy one. But with a large model having adjustable parts we provide ourselves with the means for the very best practical solution, and the workman who makes and manipulates such a model will soon master the lever escapement.

QUIZ PROBLEMS IN THE DETACHED LEVER ESCAPEMENT.

Some years ago a young watchmaker friend of the writer made, at his suggestion, a model of the lever escapement similar to the one described, which he used to "play with," as he termed it--that is, he would set the fork and pallets (which were adjustable) in all sorts of ways, right ways and wrong ways, so he could watch the results. A favorite pastime was to set every part for the best results, which was determined by the arc of vibration of the balance. By this sort of training he soon reached that degree of proficiency where one could no more puzzle him with a bad lever escapement than you could spoil a meal for him by disarranging his knife, fork and spoon.

Let us, as a practical example, take up the consideration of a short fork. To represent this in our model we take a lever as shown at Fig.

99, with the elongated slot for the pallet staff at _g_. To facilitate the description we reproduce at Fig. 102 the figure just mentioned, and also employ the same letters of reference. We fancy everybody who has any knowledge of the lever escapement has an idea of exactly what a "short fork" is, and at the same time it would perhaps puzzle them a good deal to explain the difference between a short fork and a roller too small.

[Ill.u.s.tration: Fig. 102]

[Ill.u.s.tration: Fig. 103]

In our practical problems, as solved on a large escapement model, say we first fit our fork of the proper length, and then by the slot _g_ move the lever back a little, leaving the bankings precisely as they were.

What are the consequences of this slight change? One of the first results which would display itself would be discovered by the guard pin failing to perform its proper functions. For instance, the guard pin pushed inward against the roller would cause the engaged tooth to pa.s.s off the locking face of the pallet, and the fork, instead of returning against the banking, would cause the guard pin to "ride the roller"

during the entire excursion of the jewel pin. This fault produces a sc.r.a.ping sound in a watch. Suppose we attempt to remedy the fault by bending forward the guard pin _b_, as indicated by the dotted outline _b'_ in Fig. 103, said figure being a side view of Fig. 102 seen in the direction of the arrow _a_. This policy would prevent the engaged pallet from pa.s.sing off of the locking face of the pallet, but would be followed by the jewel pin not pa.s.sing fully into the fork, but striking the inside face of the p.r.o.ng of the fork at about the point indicated by the dotted line _m_. We can see that if the p.r.o.ng of the fork was extended to about the length indicated by the outline at _c_, the action would be as it should be.

To practically investigate this matter to the best advantage, we need some arrangement by which we can determine the angular motion of the lever and also of the roller and escape wheel. To do this, we provide ourselves with a device which has already been described, but of smaller size, for measuring fork and pallet action. The device to which we allude is shown at Figs. 104, 105 and 106. Fig. 104 shows only the index hand, which is made of steel about 1/20" thick and shaped as shown. The jaws _B''_ are intended to grasp the pallet staff by the notches _e_, and hold by friction. The p.r.o.ngs _l l_ are only to guard the staff so it will readily enter the notch _e_. The circle _d_ is only to enable us to better hold the hand _B_ flat.