Chapter I: The Detached Lever Escapement (4)

The proportion in size between the two rollers in a double-roller escapement is an open question, or, at least, makers seldom agree on it. Grossmann shows, in his work on the lever escapement, two sizes: (1) Half the diameter of the acting roller; (2) two-thirds of the size of the acting roller. The chief fault urged against a smaller safety roller is, that it necessitates longer horns to the fork to carry out the safety action. Longer horns mean more metal in the lever, and it is the conceded policy of all recent makers to have the fork and pallets as light as possible. Another fault pertaining to long horns is, when the horn does have to act as safety action, a greater friction ensues.

In all soundly-constructed lever escapements the safety action is only called into use in exceptional cases, and if the watch was lying still would theoretically never be required. Where fork and pallets are poised on their arbor, pocket motion (except torsional) should but very little affect the fork and pallet action of a watch, and torsional motion is something seldom brought to act on a watch to an extent to make it worthy of much consideration. In the double-roller action which we shall consider, we shall adopt three-fifths of the pitch diameter of the jewel-pin action as the proper size. Not but what the proportions given by Grossmann will do good service; but we adopt the proportions named because it enables us to use a light fork, and still the friction of the guard point on the roller is but little more than where a guard roller of half the diameter of the acting roller is employed.

The fork action we shall consider at present is ten degrees, but subsequently we shall consider a double-roller action in which the fork and pallet action is reduced to eight degrees. We shall conceive the play between the guard point and the safety roller as one degree, which will leave half a degree of lock remaining in action on the engaged pallet.

THEORETICAL ACTION OF DOUBLE ROLLER CONSIDERED.

In the drawing at Fig. 81 we show a diagram of the action of the double-roller escapement. The small circle at _A_ represents the center of the pallet staff, and the one at _B_ the center of the balance staff. The radial lines _A d_ and _A d'_ represent the arc of angular motion of fork action. The circle _b b_ represents the pitch circle of the jewel pin, and the circle at _c c_ the periphery of the guard or safety roller. The points established on the circle _c c_ by intersection of the radial lines _A d_ and _A d'_ we will denominate the points _h_ and _h'_. It is at these points the end of the guard point of the fork will terminate. In construction, or in delineating for construction, we show the guard enough short of the points _h h'_ to allow the fork an angular motion of one degree, from _A_ as a center, before said point would come in contact with the safety roller.

We draw through the points _h h'_, from _B_ as a center, the radial lines _B g_ and _B g'_. We measure this angle by sweeping the short arc _i_ with any of the radii we have used for arc measurement in former delineations, and find it to be a trifle over sixty degrees. To give ourselves a practical object lesson, let us imagine that a real guard point rests on the circle _c_ at _h_. Suppose we make a notch in the guard roller represented by the circle _c_, to admit such imaginary guard point, and then commence to revolve the circle _c_ in the direction of the arrow _j_, letting the guard point rest constantly in such notch. When the notch _n_ in _c_ has been carried through thirty degrees of arc, counting from _B_ as a center, the guard point, as relates to _A_ as a center, would only have passed through an arc of five degrees. We show such a guard point and notch at _o n_. In fact, if a jewel pin was set to engage the fork on the pitch circle _b a_, the escapement would lock. To obviate such lock we widen the notch _n_ to the extent indicated by the dotted lines _n'_, allowing the guard point to fall back, so to speak, into the notch _n_, which really represents the passing hollow. It is not to be understood that the extended notch at _n_ is correctly drawn as regards position, because when the guard point was on the line _A f_ the point _o_ would be in the center of the extended notch, or passing hollow. We shall next give the details of drawing the double roller, but before doing so we deemed it important to explain the action of such guard points more fully than has been done heretofore.

HOW TO DESIGN A DOUBLE-ROLLER ESCAPEMENT.

We have already given very desirable forms for the parts of a double-roller escapement, consequently we shall now deal chiefly with acting principles as regards the rollers, but will give, at Fig. 82, a very well proportioned and practical form of fork. The pitch circle of the jewel pin is indicated by the dotted circle _a_, and the jewel pin of the usual cylindrical form, with two-fifths cut away. The safety roller is three-fifths of the diameter of the pitch diameter of the jewel-pin action, as indicated by the dotted circle _a_.

The safety roller is shown in full outline at _B'_, and the passing hollow at _E_. It will be seen that the arc of intersection embraced between the radial lines _B c_ and _B d_ is about sixty-one and a half degrees for the roller, but the angular extent of the passing hollow is only a little over thirty-two degrees. The passing hollow _E_ is located and defined by drawing the radial line _B c_ from the center _B_ through the intersection of radial line _A i_ with the dotted arc _b_, which represents the pitch circle of the safety roller. We will name this intersection the point _l_. Now the end of the guard point _C_ terminates at the point _l_, and the passing hollow _E_ extends on _b_ sixteen degrees on each side of the radial line _B c_.

The roller action is supposed to continue through thirty degrees of angular motion of the balance staff, and is embraced on the circle _a_ between the radial line _B k_ and _B o_. To delineate the inner face of the horn _p_ of the fork _F_ we draw the short arc _g_, from _A_ as a center, and on said arc locate at two degrees from the center at _B_ the point _f_. We will designate the upper angle of the outer face of the jewel pin _D_ as the point _s_ and, from _A_ as a center, sweep through this point _s_ the short arc _n n_. Parallel with the line _A i_ and at the distance of half the diameter of the jewel pin _D_, we draw the short lines _t t'_, which define the inner faces of the fork.

The intersection of the short line _t_ with the arc _n_ we will designate the point _r_. With our dividers set to embrace the space between the point _r_ and the point _f_, we sweep the arc which defines the inner face of the prong of the fork. The space we just made use of is practically the same as the radius of the circle _a_, and consequently of the same curvature. Practically, the length of the guard point _C'_ is made as long as will, with certainty, clear the safety roller _B_ in all positions. While we set the point _f_ at two degrees from the center _B_, still, in a well-constructed escapement, one and a half degrees should be sufficient, but the extra half degree will do no harm. If the roller _B'_ is accurately made and the guard point _C'_ properly fitted, the fork will not have half a degree of play.

The reader will remember that in the escapement model we described we cut down the drop to one degree, being less by half a degree than advised by Grossmann and Saunier. We also advised only one degree of lock. In the perfected lever escapement, which we shall describe and give working drawings for the construction of, we shall describe a detached lever escapement with only eight degrees fork and pallet action, with only three-fourths of a degree drop and three-fourths of a degree lock, which we can assure our readers is easily within the limits of practical construction by modern machinery.

HOW THE GUARD POINT IS MADE.

The guard point _C'_, as shown at Fig. 82, is of extremely simple construction. Back of the slot of the fork, which is three-fifths of the diameter of the jewel pin in depth, is made a square hole, as shown at _u_, and the back end of the guard point _C_ is fitted to this hole so that it is rigid in position. This manner of fastening the guard point is equally efficient as that of attaching it with a screw, and much lighter--a matter of the highest importance in escapement construction, as we have already urged. About the best material for such guard points is either aluminum or phosphor bronze, as such material is lighter than gold and very rigid and strong. At Fig. 83 we show a side view of the essential parts depicted in Fig. 82, as if seen in the direction of the arrow _v_, but we have added the piece which holds the jewel pin _D_. A careful study of the cut shown at Fig. 82 will soon give the horological student an excellent idea of the double-roller action.

We will now take up and consider at length why Saunier draws his entrance pallet with fifteen degrees draw and his exit pallet with only twelve degrees draw. To make ourselves more conversant with Saunier's method of delineating the lever escapement, we reproduce the essential features of his drawing, Fig. 1, plate VIII, of his "Modern Horology," in which he makes the draw of the locking face of the entrance pallet fifteen degrees and his exit pallet twelve degrees. In the cut shown at Fig. 84 we use the same letters of reference as he employs. We do not quote his description or directions for delineation because he refers to so much matter which he has previously given in the book just referred to. Besides we cannot entirely endorse his methods of delineations for many reasons, one of which appears in the drawing at Fig. 84.

MORE ABOUT TANGENTIAL LOCKINGS.

Most writers endorse the idea of tangential lockings, and Saunier speaks of the escapement as shown at Fig. 84 as having such tangential lockings, which is not the case. He defines the position of the pallet staff from the circle _t_, which represents the extreme length of the teeth; drawing the radial lines _A D_ and _A E_ to embrace an arc of sixty degrees, and establishing the center of his pallet staff _C_ at the intersection of the lines _D C_ and _E C_, which are drawn at right angles to the radial lines _A D_ and _A E_, and tangential to the circle _t_.

Here is an error; the lines defining the center of the pallet staff should have been drawn tangent to the circle _s_, which represents the locking angle of the teeth. This would have placed the center of the pallet staff farther in, or closer to the wheel. Any person can see at a glance that the pallets as delineated are not tangential in a true sense.

We have previously considered engaging friction and also repeatedly have spoken of tangential lockings, but will repeat the idea of tangential lockings at Fig. 85. A tangential locking is neutral, or nearly so, as regards engaging friction. For illustration we refer to Fig. 85, where _A_ represents the center of an escape wheel. We draw the radial lines _A y_ and _A z_ so that they embrace sixty degrees of the arcs _s_ or _t_, which correspond to similar circles in Fig. 84, and represent the extreme extent of the teeth and likewise the locking angle of such teeth. In fact, with the club-tooth escapement all that part of a tooth which extends beyond the line _s_ should be considered the same as the addendum in gear wheels. Consequently, a tangential locking made to coincide with the center of the impulse plane, as recommended by Saunier, would require the pallet staff to be located at _C'_ instead of _C_, as he draws it. If the angle _k'_ of the tooth _k_ in Fig. 84 was extended outward from the center _A_ so it would engage or rest on the locking face of the entrance pallet as shown at Fig. 84, then the draw of the locking angle would not be quite fifteen degrees; but it is evident no lock can take place until the angle _a_ of the entrance pallet has passed inside the circle _s_. We would say here that we have added the letters _s_ and _t_ to the original drawings, as we have frequently to refer to these circles, and without letters had no means of designation. Before the locking angle _k'_ of the tooth can engage the pallet, as shown in Fig. 84, the pallet must turn on the center _C_ through an angular movement of at least four degrees. We show the situation in the diagram at Fig. 86, using the same letters of reference for similar parts as in Fig. 84.

As drawn in Fig. 84 the angle of draft _G a I_ is equal to fifteen degrees, but when brought in a position to act as shown at _G a' I'_, Fig. 86, the draw is less even than twelve degrees. The angle _C a I_ remains constant, as shown at _C a' I'_, but the relation to the radial _A G_ changes when the pallet moves through the angle _w C w'_, as it must when locked. A tangential locking in the true sense of the meaning of the phrase is a locking set so that a pallet with its face coinciding with a radial line like _A G_ would be neutral, and the thrust of the tooth would be tangent to the circle described by the locking angle of the tooth. Thus the center _C_, Fig. 86, is placed on the line _w'_ which is tangent to the circle _s_; said line _w'_ also being at right angles to the radial line _A G_.

The facts are, the problems relating to the club-tooth lever escapement are very intricate and require very careful analysis, and without such care the horological student can very readily be misled. Faulty drawings, when studying such problems, lead to no end of errors, and practical men who make imperfect drawings lead to the popular phrase, "Oh, such a matter may be all right in theory, but will not work in practice." We should always bear in mind that _theory, if right, must lead practice_.

CORRECT DRAWING REQUIRED.

If we delineate our entrance pallet to have a draw of twelve degrees when in actual contact with the tooth, and then construct in exact conformity with such drawings, we will find our lever to "hug the banks" in every instance. It is inattention to such details which produces the errors of makers complained of by Saunier in section 696 of his "Modern Horology," and which he attempts to correct by drawing the locking face at fifteen degrees draw.

We shall show that neither _C_ nor _C'_, Fig. 85, is the theoretically correct position for the pallet center for a tangential locking.

We will now take up the consideration of a club-tooth lever escapement with circular pallets and tangential lockings; but previous to making the drawings we must decide several points, among which are the thickness of the pallet arms, which establishes the angular motion of the escape wheel utilized by such pallet arms, and also the angular motion imparted to the pallets by the impulse faces of the teeth. We will, for the present, accept the thickness of the arms as being equivalent to five degrees of angular extent of the pitch circle of the escape wheel.

In making our drawings we commence, as on former occasions, by establishing the center of our escape wheel at _A_, Fig. 87, and sweeping the arc _a a_ to represent the pitch circle of such wheel. Through the center _A_ we draw the vertical line _A B_, which is supposed to also pass through the center of the pallet staff. The intersection of the line _A B_ with the arc _a_ we term the point _d_, and from this point we lay off on said arc _a_ thirty degrees each side of said intersection, and thus establish the points _c b_. From _A_, through the point _c_, we draw the line _A c c'_. On the arc _a a_ and two and a half degrees to the left of the point _c_ we establish the point _f_, which space represents half of the thickness of the entrance pallet. From _A_ we draw through the point _f_ the line _A f f'_. From _f_, and at right angles to said line _A f_, we draw the line _f e_ until it crosses the line _A B_.

Now this line _f e_ is tangent to the arc _a_ from the point _f_, and consequently a locking placed at the point _f_ is a true tangential locking; and if the resting or locking face of a pallet was made to coincide with the line _A f'_, such locking face would be strictly "dead" or neutral. The intersection of the line _f e_ with the line _A B_ we call the point _C_, and locate at this point the center of our pallet staff. According to the method of delineating the lever escapement by Moritz Grossmann the tangent line for locating the center of the pallet staff is drawn from the point _c_, which would locate the center of the pallet staff at the point _h_ on the line _A B_.

Grossmann, in delineating his locking face for the draw, shows such face at an angle of twelve degrees to the radial line _A f'_, when he should have drawn it twelve degrees to an imaginary line shown at _f i_, which is at right angles to the line _f h_. To the writer's mind this is not just as it should be, and may lead to misunderstanding and bad construction. We should always bear in mind the fact that the basis of a locking face is a neutral plane placed at right angles to the line of thrust, and the "draw" comes from a locking face placed at an angle to such neutral plane. A careful study of the diagram at Fig. 88 will give the reader correct ideas. If a tooth locks at the point _c_, the tangential thrust would be on the line _c h'_, and a neutral locking face would be on the line _A c_.

NEUTRAL LOCKINGS.

To aid in explanation, let us remove the pallet center to _D_; then the line of thrust would be _c D_ and a neutral locking face would coincide with the line _m m_, which is at right angles to the line _c D_. If we should now make a locking face with a "draw" and at an angle to the line _c D_, say, for illustration, to correspond to the line _c c'_ (leaving the pallet center at _D_), we would have a strong draw and also a cruel engaging friction.

If, however, we removed the engaging tooth, which we have just conceived to be at _c_, to the point _k_ on the arc _a' a'_, Fig. 88, the pallet center _D_ would then represent a tangential locking, and a neutral pallet face would coincide with the radial line _A k'_; and a locking face with twelve degrees draw would coincide nearly with the line _l_. Let us next analyze what the effect would be if we changed the pallet center to _h'_, Fig. 88, leaving the engaging tooth still at _k_. In this instance the line _l l_ would then coincide with a neutral locking face, and to obtain the proper draw we should delineate the locking face to correspond to the line _k n_, which we assume to be twelve degrees from _k l_.

It is not to be understood that we insist on precisely twelve degrees draw from a neutral plane for locking faces for lever pallets. What we do insist upon, however, is a "safe and sure draw" for a lever pallet which will hold a fork to the banks and will also return it to such banks if by accident the fork is moved away. We are well aware that it takes lots of patient, hard study to master the complications of the club-tooth lever escapement, but it is every watchmaker's duty to conquer the problem. The definition of "lock," in the detached lever escapement, is the stoppage or arrest of the escape wheel of a watch while the balance is left free or detached to perform the greater portion of its arc of vibration. "Draw" is a function of the locking parts to preserve the fork in the proper position to receive and act on the jewel pin of the balance.

It should be borne in mind in connection with "lock" and "draw," that the line of thrust as projected from the locked tooth of the escape wheel should be as near tangential as practicable. This maxim applies particularly to the entrance pallet. We would beg to add that practically it will make but little odds whether we plant the center of our pallet staff at _C_ or _h_, Fig. 87, provided we modify the locking and impulse angles of our pallets to conform to such pallet center. But it will not do to arrange the parts for one center and then change to another.

PRACTICAL HINTS FOR LEVER ESCAPEMENTS.

Apparently there seems to be a belief with very many watchmakers that there is a set of shorthand rules for setting an escapement, especially in American watches, which, if once acquired, conquers all imperfections. Now we wish to disabuse the minds of our readers of any such notions. Although the lever escapement, as adopted by our American factories, is constructed on certain "lines," still these lines are subject to modifications, such as may be demanded for certain defects of construction. If we could duplicate every part of a watch movement perfectly, then we could have certain rules to go by, and fixed templets could be used for setting pallet stones and correcting other escapement faults.

Let us now make an analysis of the action of a lever escapement. We show at Fig. 89 an ordinary eighteen-size full-plate lever with fork and pallets. The dotted lines _a b_ are supposed to represent an angular movement of ten degrees. Now, it is the function of the fork to carry the power of the train to the balance. How well the fork performs its office we will consider subsequently; for the present we are dealing with the power as conveyed to the fork by the pallets as shown at Fig. 89.

The angular motion between the lines _a c_ (which represents the lock) is not only absolutely lost--wasted--but during this movement the train has to retrograde; that is, the dynamic force stored in the momentum of the balance has to actually turn the train backward and against the force of the mainspring. True, it is only through a very short arc, but the necessary force to effect this has to be discounted from the power stored in the balance from a former impulse. For this reason we should make the angular motion of unlocking as brief as possible. Grossmann, in his essay, endorses one and a half degrees as the proper lock.

In the description which we employed in describing the large model for illustrating the action of the detached lever escapement, we cut the lock to one degree, and in the description of the up-to-date lever escapement, which we shall hereafter give, we shall cut the lock down to three-quarters of a degree, a perfection easily to be attained by modern tools and appliances. We shall also cut the drop down to three-quarters of a degree. By these two economies we more than make up for the power lost in unlocking. With highly polished ruby or sapphire pallets ten degrees of draw is ample. But such draw must positively be ten degrees from a neutral locking face, not an escapement drawn on paper and called ten degrees, but when actually measured would only show eight and a half or nine degrees.

THE PERFECTED LEVER ESCAPEMENT.

With ten degrees angular motion of the lever and one and a half degrees lock, we should have eight and a half degrees impulse. The pith of the problem, as regards pallet action, for the practical workman can be embodied in the following question: What proportion of the power derived from the twelve degrees of angular motion of the escape wheel is really conveyed to the fork? The great leak of power as transmitted by the lever escapement to the balance is to be found in the pallet action, and we shall devote special attention to finding and stopping such leaks.

WHEN POWER IS LOST IN THE LEVER ESCAPEMENT.

If we use a ratchet-tooth escape wheel we must allow at least one and a half degrees drop to free the back of the tooth; but with a club-tooth escape wheel made as can be constructed by proper skill and care, the drop can be cut down to three-quarters of a degree, or one-half of the loss with the ratchet tooth. We do not wish our readers to imagine that such a condition exists in most of the so-called fine watches, because if we take the trouble to measure the actual drop with one of the little instruments we have described, it will be found that the drop is seldom less than two, or even three degrees.

If we measure the angular movement of the fork while locked, it will seldom be found less than two or three degrees. Now, we can all understand that the friction of the locking surface has to be counted as well as the recoil of the draw. Locking friction is seldom looked after as carefully as the situation demands. Our factories make the impulse face of the pallets rounded, but leave the locking face flat. We are aware this condition is, in a degree, necessary from the use of exposed pallets. In many of the English lever watches with ratchet teeth, the locking faces are made cylindrical, but with such watches the pallet stones, as far as the writer has seen, are set "close"; that is, with steel pallet arms extending above and below the stone.

There is another feature of the club-tooth lever escapement that next demands our attention which we have never seen discussed. We refer to arranging and disposing of the impulse of the escape wheel to meet the resistance of the hairspring. Let us imagine the dotted line _A d_, Fig. 89, to represent the center of action of the fork. We can readily see that the fork in a state of rest would stand half way between the two banks from the action of the hairspring, and in the pallet action the force of the escape wheel, one tooth of which rests on the impulse face of a pallet, would be exerted against the elastic force of the hairspring. If the force of the mainspring, as represented by the escape-wheel tooth, is superior to the power of the hairspring, the watch starts itself. The phases of this important part of the detached lever escapement will be fully discussed.

ABOUT THE CLUB-TOOTH ESCAPEMENT.

We will now take up a study of the detached lever escapement as relates to pallet action, with the point specially in view of constructing an escapement which cannot "set" in the pocket, or, in other words, an escapement which will start after winding (if run down) without shaking or any force other than that supplied by the train as impelled by the mainspring. In the drawing at Fig. 90 we propose to utilize eleven degrees of escape-wheel action, against ten and a half, as laid down by Grossmann. Of this eleven degrees we propose to divide the impulse arc of the escape wheel in six and five degrees, six to be derived from the impulse face of the club tooth and five from the impulse plane of the pallet.

The pallet action we divide into five and four, with one degree of lock. Five degrees of pallet action is derived from the impulse face of the tooth and four from the impulse face of the pallet. The reader will please bear in mind that we do not give these proportions as imperative, because we propose to give the fullest evidence into the reader's hands and enable him to judge for himself, as we do not believe in laying down imperious laws that the reader must accept on our assertion as being correct. Our idea is rather to furnish the proper facts and put him in a situation to know for himself.

The reader is urged to make the drawings for himself on a large scale, say, an escape wheel 10" pitch diameter. Such drawings will enable him to realize small errors which have been tolerated too much in drawings of this kind. The drawings, as they appear in the cut, are one-fourth the size recommended, and many of the lines fail to show points we desire to call attention to. As for instance, the pallet center at _B_ is tangential to the pitch circle _a_ from the point of tooth contact at _f_. To establish this point we draw the radial lines _A c_ and _A d_ from the escape-wheel center _A_, as shown, by laying off thirty degrees on each side of the intersection of the vertical line _i_ (passing through the centers _A B_) with the arc _a_, and then laying off two and a half degrees on _a_ and establishing the point _f_, and through _f_ from the center _A_ draw the radial line _A f'_. Through the point _f_ we draw the tangent line _b' b b''_, and at the intersection of the line _b_ with _i_ we establish the center of our pallet staff at _B_. At two and a half degrees from the point _c_ we lay off two and a half degrees to the right of said point and establish the point _n_, and draw the radial line _A n n'_, which establishes the extent of the arc of angular motion of the escape wheel utilized by the pallet arm.

We have now come to the point where we must exercise our reasoning powers a little. We know the locking angle of the escape-wheel tooth passes on the arc _a_, and if we utilize the impulse face of the tooth for five degrees of pallet or lever motion we must shape it to this end. We draw the short arc _k_ through the point _n_, knowing that the inner angle of the pallet stone must rest on this arc wherever it is situated. As, for instance, when the locking face of the pallet is engaged, the inner angle of the pallet stone must rest somewhere on this arc (_k_) inside of _a_, and the extreme outer angle of the impulse face of the tooth must part with the pallet on this arc _k_.

HOW TO LOCATE THE PALLET ACTION.

With the parts related to each other as shown in the cut, to establish where the inner angle of the pallet stone is located in the drawing, we measure down on the arc _k_ five degrees from its intersection with _a_, and establish the point _s_. The line _B b_, Fig. 90, as the reader will see, does not coincide with the intersection of the arcs _a_ and _k_, and to conveniently get at the proper location for the inner angle of our pallet stone, we draw the line _B b'_, which passes through the point _n_ located at the intersection of the arc _a_ with the arc _k_. From _B_ as a center we sweep the short arc _j_ with any convenient radius of which we have a sixty-degree scale, and from the intersection of _B b'_ with _j_ we lay off five degrees and draw the line _B s'_, which establishes the point _s_ on the arc _k_. As stated above, we allow one degree for lock, which we establish on the arc _o_ by laying off one degree on the arc _j_ below its intersection with the line _B b_. We do not show this line in the drawing, from the fact that it comes so near to _B b'_ that it would confuse the reader. Above the arc _a_ on the arc _k_ at five degrees from the point _n_ we establish the point _l_, by laying off five degrees on the arc _j_ above the intersection of the line _B b_ with _j_.

The point _l_, Fig. 90, establishes where the outer angle of the tooth will pass the arc _k_ to give five degrees of angular motion to the lever. From _A_ as a center we sweep the arc _m_, passing through the point _l_. The intersection of the arc _m_ with the line _A h_ we call the point _r_, and by drawing the right line _r f_ we delineate the impulse face of the tooth. On the arc _o_ and one degree below its intersection with the line _B b_ we establish the point _t_, and by drawing a right line from _t_ to _s_ we delineate the impulse face of our entrance pallet.

"ACTION" DRAWINGS.

One great fault with most of our text books on horology lies in the fact that when dealing with the detached lever escapement the drawings show only the position of the pallets when locked, and many of the conditions assumed are arrived at by mental processes, without making the proper drawings to show the actual relation of the parts at the time such conditions exist. For illustration, it is often urged that there is a time in the action of the club-tooth lever escapement action when the incline on the tooth and the incline on the pallet present parallel surfaces, and consequently endure excessive friction, especially if the oil is a little thickened.

We propose to make drawings to show the exact position and relation of the entrance pallet and tooth at three intervals viz: (1) Locked; (2) the position of the parts when the lever has performed one-half of its angular motion; (3) when half of the impulse face of the tooth has passed the pallet. The position of the entrance pallet when locked is sufficiently well shown in Fig. 90 to give a correct idea of the relations with the entrance pallet; and to conform to statement (2), as above. We will now delineate the entrance pallet, not in actual contact, however, with the pallet, because if we did so the lines we employed would become confused. The methods we use are such that _we can delineate with absolute correctness either a pallet or tooth at any point in its angular motion_.

We have previously given instructions for drawing the pallet locked; and to delineate the pallet after five degrees of angular motion, we have only to conceive that we substitute the line _s'_ for the line _b'_. All angular motions and measurements for pallet actions are from the center of the pallet staff at _B_. As we desire to now delineate the entrance pallet, it has passed through five degrees of angular motion and the inner angle _s_ now lies on the pitch circle of the escape wheel, the angular space between the lines _b' s'_ being five degrees, the line _b''_ reducing the impulse face to four degrees.

DRAWING AN ESCAPEMENT TO SHOW ANGULAR MOTION.

To delineate our locking face we draw a line at right angles to the line _B b''_ from the point _t_, said point being located at the intersection of the arc _o_ with the line _B b''_. To draw a line perpendicular to _B b''_ from the point _t_, we take a convenient space in our dividers and establish on the line _B b''_ the points _x x'_ at equal distances from the point _t_. We open the dividers a little (no special distance) and sweep the short arcs _x'' x'''_, as shown at Fig. 91. Through the intersection of the short arcs _x'' x'''_ and to the point _t_ we draw the line _t y_. The reader will see from our former explanations that the line _t y_ represents the neutral plane of the locking face, and that to have the proper draw we must delineate the locking face of our pallet at twelve degrees. To do this we draw the line _t x'_ at twelve degrees to the line _t y_, and proceed to outline our pallet faces as shown. We can now understand, after a moment's thought, that we can delineate the impulse face of a tooth at any point or place we choose by laying off six degrees on the arc _m_, and drawing radial lines from _A_ to embrace such arc. To illustrate, suppose we draw the radial lines _w' w''_ to embrace six degrees on the arc _a_. We make these lines contiguous to the entrance pallet _C_ for convenience only. To delineate the impulse face of the tooth, we draw a line extending from the intersection of the radial line _A' w'_ with the arc _m_ to the intersection of the arc _a_ with the radial line _A w''_.

We next desire to know where contact will take place between the wheel-tooth _D_ and pallet _C_. To determine this we sweep, with our dividers set so one leg rests at the escape-wheel center _A_ and the other at the outer angle _t_ of the entrance pallet, the short arc _t' w_. Where this arc intersects the line _w_ (which represents the impulse face of the tooth) is where the outer angle _t_ of the entrance pallet _C_ will touch the impulse face of the tooth. To prove this we draw the radial line _A v_ through the point where the short arc _t t'_ passes through the impulse face _w_ of the tooth _D_. Then we continue the line _w_ to _n_, to represent the impulse face of the tooth, and then measure the angle _A w n_ between the lines _w n_ and _v A_, and find it to be approximately sixty-four degrees. We then, by a similar process, measure the angle _A t s'_ and find it to be approximately sixty-six degrees. 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 passed the inner angle of the entrance pallet, we proceed, as in former illustrations, 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 passed the arc _s_.

To do this we draw from _A_ as a center the radial line _A j_, passing 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 passed 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'_, passing 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 analysis 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 passes 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.

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 passes 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 illustrate 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 illustration, 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 passes 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.

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 illustration, 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.

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 brass should be obtained, shaped as shown at _C_, Fig. 95. This plate is of thin brass, about No. 18, and on it are outlined the pallet arms shown at Fig. 94.

To make the pallets adjustable, they are set in thick disks of sheet brass, as shown at _D_, Figs. 95, 96 and 97. At the center of the plate _C_ is placed a brass 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 pass through the holes _h h_. If we now place the brass 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.

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 passing 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.