In describing the method for drawing the cylinder escapement we shall make a radical departure from the systems usually laid down in text-books, and seek to simplify the formulas which have heretofore been given for such delineations. In considering the cylinder escapement we shall pursue an a.n.a.lytical course and strive to build up from the underlying principles. In the drawings for this purpose we shall commence with one having an escape wheel of 10" radius, and our first effort will be the primary drawing shown at Fig. 129. Here we establish the point _A_ for the center of our escape wheel, and from this center sweep the short arc _a a_ with a 10" radius, to represent the circ.u.mference of our escape wheel. From _A_ we draw the vertical line _A B_, and from the intersection of said line with the arc _a a_ we lay off twelve degree s.p.a.ces on each side of the line _A B_ on said arc _a_ and establish the points _b c_. From _A_ as a center we draw through the points _b c_ the radial lines _b' c'_.

To define the face of the incline to the teeth we set our dividers to the radius of any of the convenient arcs of sixty degrees which we have provided, and sweep the arc _t t_. From the intersection of said arc with the line _A b'_ we lay off on said arc sixty-four degrees and establish the point _g_ and draw the line _b g_. Why we take sixty-four degrees for the angle _A b g_ will be explained later on, when we are discussing the angular motion of the cylinder. By dividing the eleventh degree from the point _b_ on the arc _a a_ into thirds and taking two of them, we establish the point _y_ and draw the radial line _A y'_. Where this line _A y'_ intersects the line _b g_ we name the point _n_, and in it is located the point of the escape-wheel tooth. That portion of the line _b g_ which lies between the points _b_ and _n_ represents the measure of the inner diameter of the cylinder, and also the length of the chord of the arc which rounds the impulse face of the tooth. We divide the s.p.a.ce _b n_ into two equal portions and establish the point _e_, which locates the position of the center of the cylinder. From _A_ as a center and through the point _e_ we sweep the arc _e' e'_, and it is on this line that the points establishing the center of the cylinder will in every instance be located. From _A_ as a center, through the point _n_ we sweep the arc _k_, and on this line we locate the points of the escape-wheel teeth. For delineating the curved impulse faces of the escape-wheel teeth we draw from the point _e_ and at right angles to the line _b g_ the line _e o_. We next take in our dividers the radius of the arc _k_, and setting one leg at either of the points _b_ or _n_, establish with the other leg the point _p'_ on the line _e o_, and from the point _p'_ as a center we sweep the arc _b v n_, which defines the curve of the impulse faces of the teeth. From _A_ as a center through the point _p'_ we sweep the arc _p_, and in all instances where we desire to delineate the curved face of a tooth we locate either the position of the point or the heel of such tooth, and setting one leg of our dividers at such point, the other leg resting on the arc _p_, we establish the center from which to sweep the arc defining the face of said tooth.

ADVANTAGES GAINED IN SHAPING.

The reason for giving a curved form to the impulse face of the teeth of cylinder escape wheels are somewhat intricate, and the problem involves several factors. That there are advantages in so shaping the incline or impulse face is conceded, we believe, by all recent manufacturers. The chief benefit derived from such curved impulse faces will be evident after a little thought and study of the situation and relation of parts as shown in Fig. 129. It will be seen on inspection that the angular motion imparted to the cylinder by the impulse face of the tooth when curved as shown, is greater during the first half of the twelve degrees of escape-wheel action than during the last half, thus giving the escape wheel the advantage at the time the balance spring increases its resistance to the pa.s.sage of the escape-wheel tooth across the lip of the cylinder. Or, in other words, as the ratio of resistance of the balance spring increases, in a like ratio the curved form of the impulse face of the tooth gives greater power to the escape-wheel action in proportion to the angular motion of the escape wheel. Hence, in actual service it is found that cylinder watches with curved impulse planes to the escape-wheel teeth are less liable to set in the pocket than the teeth having straight impulse faces.

THE OUTER DIAMETER OF THE CYLINDER.

[Ill.u.s.tration: Fig. 129]

To define the remainder of the form of our escape-wheel tooth we will next delineate the heel. To do this we first define the outer diameter of our cylinder, which is the extent from the point _n_ to _c_, and after drawing the line _n c_ we halve the s.p.a.ce and establish the point _x_, from which point as a center we sweep the circle _w w_, which defines the outer circ.u.mference of our cylinder. With our dividers set to embrace the extent from the point _n_ to the point _c_ we set one leg at the point _b_, and with the other leg establish on the arc _k_ the point _h_. We next draw the line _b h_, and from the point _b_ draw the line _b f_ at right angle to the line _b h_. Our object for drawing these lines is to define the heel of our escape-wheel tooth by a right angle line tangent to the circle _w_, from the point _b_; which circle _w_ represents the curve of the outer circ.u.mference of the cylinder. We shape the point of the tooth as shown to give it the proper stability, and draw the full line _j_ to a curve from the center _A_. We have now defined the form of the upper face of the tooth. How to delineate the U arms will be taken up later on, as, in the present case, the necessary lines would confuse our drawing.

We would here take the opportunity to say that there is a great lat.i.tude taken by makers as regards the extent of angular impulse given to the cylinder, or, as it is termed, the "actual lift." This lat.i.tude governs to a great extent the angle _A b g_, which we gave as sixty-four degrees in our drawing. It is well to understand that the use of sixty-four degrees is based on no hard-and-fast rules, but varies back and forth, according as a greater or lesser angle of impulse or lift is employed.

In practical workshop usage the impulse angle is probably more easily estimated by the ratio between the diameter of the cylinder and the measured (by lineal measure) height of the impulse plane. Or, to be more explicit, we measure the radial extent from the center _A_ between the arcs _a k_ on the line _A b_, and use this for comparison with the outer diameter of the cylinder.

We can readily see that as we increase the height of the heel of the impulse face of our tooth we must also increase the angle of impulse imparted to the cylinder. With the advantages of accurate micrometer calipers now possessed by the horological student it is an easy matter to get at the angular extent of the real lift of any cylinder. The advantage of such measuring instruments is also made manifest in determining when the proper proportion of the cylinder is cut away for the half sh.e.l.l.

[Ill.u.s.tration: Fig. 130]

In the older methods of watchmaking it was a very common rule to say, let the height of the incline of the tooth be one-seventh of the outer diameter of the cylinder, and at the same time the trade was furnished with no tools except a clumsy douzieme gage; but with micrometer calipers which read to one-thousandths of an inch such rules can be definitely carried into effect and not left to guess work. Let us compare the old method with the new: Suppose we have a new cylinder to put in; we have the old escape wheel, but the former cylinder is gone.

The old-style workman would take a round broach and calculate the size of the cylinder by finding a place where the broach would just go between the teeth, and the size of the broach at this point was supposed to be the outer diameter of the cylinder. By our method we measure the diameter of the escape wheel in thousandths of an inch, and from this size calculate exactly what the diameter of the new cylinder should be in thousandths of an inch. Suppose, to further carry out our comparison, the escape wheel which is in the watch has teeth which have been stoned off to permit the use of a cylinder which was too small inside, or, in fact, of a cylinder too small for the watch: in this case the broach system would only add to the trouble and give us a cylinder which would permit too much inside drop.

DRAWING A CYLINDER.

We have already instructed the pupil how to delineate a cylinder escape wheel tooth and we will next describe how to draw a cylinder. As already stated, the center of the cylinder is placed to coincide with the center of the chord of the arc which defines the impulse face of the tooth.

Consequently, if we design a cylinder escape wheel tooth as previously described, and setting one leg of our compa.s.ses at the point _e_ which is situated at the center of the chord of the arc which defines the impulse face of the tooth and through the points _d_ and _b_ we define the inside of our cylinder. We next divide the chord _d b_ into eight parts and set our dividers to five of these parts, and from _e_ as a center sweep the circle _h_ and define the outside of our cylinder. From _A_ as a center we draw the radial line _A e'_. At right angles to the line _A e'_ and through the point _e_ we draw the line from _e_ as a center, and with our dividers set to the radius of any of the convenient arcs which we have divided into sixty degrees, we sweep the arc _i_.

Where this arc intersects the line _f_ we term the point _k_, and from this point we lay off on the arc _i_ 220 degrees, and draw the line _l e l'_, which we see coincides with the chord of the impulse face of the tooth. We set our dividers to the same radius by which we sweep the arc _i_ and set one leg at the point _b_ for a center and sweep the arc _j'_. If we measure this arc from the point _j'_ to intersection of said arc _j'_ with the line _l_ we will find it to be sixty-four degrees, which accounts for our taking this number of degrees when we defined the face of our escape-wheel tooth, Fig. 129.

There is no reason why we should take twenty-degrees for the angle _k e l_ except that the practical construction of the larger sizes of cylinder watches has established the fact that this is about the right angle to employ, while in smaller watches it frequently runs up as high as twenty-five. Although the cylinder is seemingly a very simple escapement, it is really a very abstruce one to follow out so as to become familiar with all of its actions.

THE CYLINDER PROPER CONSIDERED.

[Ill.u.s.tration: Fig. 131]

We will now proceed and consider the cylinder proper, and to aid us in understanding the position and relation of the parts we refer to Fig.

131, where we repeat the circles _d_ and _h_, shown in Fig. 130, which represents the inside and outside of the cylinder. We have here also repeated the line _f_ of Fig. 130 as it cuts the cylinder in half, that is, divides it into two segments of 180 degrees each. If we conceive of a cylinder in which just one-half is cut away, that is, the lips are bounded by straight radial lines, we can also conceive of the relation and position of the parts shown in Fig. 130. The first position of which we should take cognizance is, the tooth _D_ is moved back to the left so as to rest on the outside of our cylinder. The cylinder is also supposed to stand so that the lips correspond to the line _f_. On pressing the tooth _D_ forward the incline of the tooth would attack the entrance lip of the cylinder at just about the center of the curved impulse face, imparting to the cylinder twenty degrees of angular motion, but the point of the tooth at _d_ would exactly encounter the inner angle of the exit lip, and of course the cylinder would afford no rest for the tooth; hence, we see the importance of not cutting away too much of the half sh.e.l.l of the cylinder.

But before we further consider the action of the tooth _D_ in its action as it pa.s.ses the exit lip of the cylinder we must finish with the action of the tooth on the entrance lip. A very little thought and study of Fig. 130 will convince us that the incline of the tooth as it enters the cylinder will commence at _t_, Fig. 130, but at the close of the action the tooth parts from the lip on the inner angle. Now it is evident that it would require greater force to propel the cylinder by its inner angle than by the outer one. To compensate for this we round the edge of the entrance lip so that the action of the tooth instead of commencing on the outer angle commences on the center of the edge of the entrance lip and also ends its action on the center of the entrance lip. To give angular extent enough to the sh.e.l.l of the cylinder to allow for rounding and also to afford a secure rest for the tooth inside the cylinder, we add six degrees to the angular extent of the entrance lip of the cylinder sh.e.l.l, as indicated on the arc _o'_, Fig. 131, three of these degrees being absorbed for rounding and three to insure a dead rest for the tooth when it enters the cylinder.

WHY THE ANGULAR EXTENT IS INCREASED.

Without rounding the exit lip the action of the tooth on its exit would be entirely on the inner angle of the sh.e.l.l. To obviate this it is the usual practice to increase the angular extent of the cylinder ten degrees, as shown on the arc _o'_ between the lines _f_ and _p_, Fig.

131. Why we should allow ten degrees on the exit lip and but six degrees on the entrance lip will be understood by observing Fig. 130, where the radial lines _s_ and _r_ show the extent of angular motion of the cylinder, which would be lost if the tooth commenced to act on the inner angle and ended on the outer angle of the exit lip. This arc is a little over six degrees, and if we add a trifle over three degrees for rounding we would account for the ten degrees between the lines _f_ and _p_, Fig.

131. It will now be seen that the angular extent is 196 degrees. If we draw the line _w_ we can see in what proportion the measurement should be made between the outer diameter of the cylinder and the measure of the half sh.e.l.l. It will be seen on measurement that the distance between the center _e_ and the line _w_ is about one-fifteenth part of the outer diameter of the cylinder and consequently with a cylinder which measures 45/1000 of an inch in diameter, now the half sh.e.l.l should measure half of the entire diameter of the cylinder plus one-fifteenth part of such diameter, or 25 thousandths of an inch.

After these proportions are understood and the drawing made, the eye will get accustomed to judging pretty near what is required; but much the safer plan is to measure, where we have the proper tools for doing so. Most workmen have an idea that the depth or distance at which the cylinder is set from the escape wheel is a matter of adjustment; while this is true to a certain extent, still there is really only one position for the center of the cylinder, and that is so that the center of the pivot hole coincides exactly with the center of the chord to the curve of the impulse face of the tooth or the point _e_, Fig. 130. Any adjustment or moving back and forth of the chariot to change the depth could only be demanded where there was some fault existing in the cylinder or where it had been moved out of its proper place by some genius as an experiment in cylinder depths. It will be evident on observing the drawing at Fig. 131 that when the cylinder is performing an arc of vibration, as soon as the entrance lip has pa.s.sed the point indicated by the radial line _e x_ the point of the escape-wheel tooth will commence to act on the cylinder lip and continue to do so through an arc of forty degrees, or from the lines _x_ to _l_.

MAKING A WORKING MODEL.

To practically study the action of the cylinder escapement it is well to make a working model. It is not necessary that such a model should contain an entire escape wheel; all that is really required is two teeth cut out of bra.s.s of the proper forms and proportions and attached to the end of an arm 4-7/8" long with studs riveted to the U arms to support the teeth. This U arm is attached to the long arm we have just mentioned. A flat ring of heavy sheet bra.s.s is shaped to represent a short transverse section of a cylinder. This segment is mounted on a yoke which turns on pivots. In making such a model we can employ all the proportions and exact forms of the larger drawings made on a ten-inch radius. Such a model becomes of great service in learning the importance of properly shaping the lips of the cylinder. And right here we beg to call attention to the fact that in the ordinary repair shop the proper shape of cylinder lips is entirely neglected.

PROPER SHAPE OF CYLINDER LIPS.

The workman buys a cylinder and whether the proper amount is cut away from the half sh.e.l.l, or the lips, the correct form is entirely ignored, and still careful attention to the form of the cylinder lips adds full ten per cent. to the efficiency of the motive force as applied to the cylinder. In making study drawings of the cylinder escapement it is not necessary to employ paper so large that we can establish upon it the center of the arc which represents the periphery of our escape wheel, as we have at our disposal two plans by which this can be obviated. First, placing a bit of bristol board on our drawing-board in which we can set one leg of our dividers or compa.s.ses when we sweep the peripheral arc which we use in our delineations; second, making three arcs in bra.s.s or other sheet metal, viz.: the periphery of the escape wheel, the arc pa.s.sing through the center of the chord of the arc of the impulse face of the tooth, and the arc pa.s.sing through the point of the escape-wheel tooth. Of these plans we favor the one of sticking a bit of cardboard on the drawing board outside of the paper on which we are making our drawing.

[Ill.u.s.tration: Fig. 132]

At Fig. 132 we show the position and relation of the several parts just as the tooth pa.s.ses into the sh.e.l.l of the cylinder, leaving the lip of the cylinder just as the tooth parted with it. The half sh.e.l.l of the cylinder as shown occupies 196 degrees or the larger arc embraced between the radial lines _k_ and _l_. In drawing the entrance lip the acting face is made almost identical with a radial line except to round the corners for about one-third the thickness of the cylinder sh.e.l.l. No portion, however, of the lip can be considered as a straight line, but might be described as a flattened curve.

[Ill.u.s.tration: Fig. 133]

A little study of what would be required to get the best results after making such a drawing will aid the pupil in arriving at the proper shape, especially when he remembers that the thickness of the cylinder sh.e.l.l of a twelve-line watch is only about five one-thousandths of an inch. But because the parts are small we should not shirk the problem of getting the most we possibly can out of a cylinder watch.

The extent of arc between the radial lines _k f_, as shown in Fig. 132, is four degrees. Although in former drawings we showed the angular extent added as six degrees, as we show the lip _m_ in Fig. 132, two degrees are lost in rounding. The s.p.a.ce _k f_ on the egress or exit side is intended to be about four degrees, which shows the extent of lock. We show at Fig. 133 the tooth _D_ just having pa.s.sed out of the cylinder, having parted with the exit lip _p_.

In making this drawing we proceed as with Fig. 132 by establishing a center for our radius of 10" outside of our drawing paper and drawing the line _A A_ to such center and sweeping the arcs _a b c_. We establish the point _e_, which represents the center of our cylinder, as before. We take the s.p.a.ce to represent the radial extent of the outside of our cylinder in our dividers and from _e_ as a center sweep a fine pencil line, represented by the dotted line _t_ in our drawing; and where this circle intersects the arc _a_ we name it the point _s_; and it is at this point the heel of our escape-wheel tooth must part with the exit lip of the cylinder. From _e_ as a center and through the point _s_ we draw the line _e l''_. With our dividers set to the radius of any convenient arc which we have divided into degrees, we sweep the short arc _d'_. The intersection of this arc with the line _e l''_ we name the point _u_; and from _e_ as a center we draw the radial line _e u f'_. We place the letter _f''_ in connection with this line because it (the line) bears the same relations to the half sh.e.l.l of the cylinder shown in Fig. 133 that the line _f_ does to the half sh.e.l.l (_D_) shown in Fig.

132. We draw the line _f'' f'''_, Fig. 133, which divides the cylinder into two segments of 180 degrees each. We take the same s.p.a.ce in our dividers with which we swept the interior of the cylinder in Fig. 132 and sweep the circle _v_, Fig. 133. From _e_ as a center we sweep the short arc _d''_, Fig. 133, and from its intersection of the line _f''_ we lay off six degrees on said arc _d''_ and draw the line _e' k''_, which defines the angular extent of our entrance lip to the half sh.e.l.l of the cylinder in Fig. 133. We draw the full lines of the cylinder as shown.

We next delineate the heel of the tooth which has just pa.s.sed out of the cylinder, as shown at _D'_, Fig. 133. We now have a drawing showing the position of the half sh.e.l.l of the cylinder just as the tooth has pa.s.sed the exit lip. This drawing also represents the position of the half sh.e.l.l of the cylinder when the tooth rests against it on the outside. If we should make a drawing of an escape-wheel tooth shaped exactly as the one shown at Fig. 132 and the point of the tooth resting at _x_, we would show the position of a tooth encountering the cylinder after a tooth which has been engaged in the inside of the sh.e.l.l has pa.s.sed out.

By following the instructions now given, we can delineate a tooth in any of its relations with the cylinder sh.e.l.l.

DELINEATING AN ESCAPE-WHEEL TOOTH WHILE IN ACTION.

We will now go through the operation of delineating an escape-wheel tooth while in action. The position we shall a.s.sume is the one in which the cylinder and escape-wheel tooth are in the relation of the pa.s.sage of half the impulse face of the tooth into the cylinder. To do this is simple enough: We first produce the arcs _a b c_, Fig. 133, as directed, and then proceed to delineate a tooth as in previous instances. To delineate our cylinder in the position we have a.s.sumed above, we take the s.p.a.ce between the points _e d_ in our dividers and setting one leg at _d_ establish the point _g_, to represent the center of our cylinder.

If we then sweep the circle _h_ from the center of _g_ we define the inner surface of the sh.e.l.l of our cylinder.

Strictly speaking, we have not a.s.sumed the position we stated, that is, the impulse face of the tooth as pa.s.sing half way into the cylinder. To comply strictly with our statement, we divide the chord of the impulse face of the tooth _A_ into eight equal s.p.a.ces, as shown. Now as each of these s.p.a.ces represent the thickness of the cylinder, if we take in our dividers four of these s.p.a.ces and half of another, we have the radius of a circle pa.s.sing the center of the cylinder sh.e.l.l. Consequently, if with this s.p.a.ce in our dividers we set the leg at _d_, we establish on the arc _b_ the point _i_. We locate the center of our cylinder when one-half of an entering tooth has pa.s.sed into the cylinder. If now from the new center with our dividers set at four of the s.p.a.ces into which we have divided the line _e f_ we can sweep a circle representing the inner surface of the cylinder sh.e.l.l, and by setting our dividers to five of these s.p.a.ces we can, from _i_ as a center, sweep an arc representing the outside of the cylinder sh.e.l.l. For all purposes of practical study the delineation we show at Fig. 133 is to be preferred, because, if we carry out all the details we have described, the lines would become confused.

We set our dividers at five of the s.p.a.ces on the line _e f_ and from _g_ as a center sweep the circle _j_, which delineates the outer surface of our cylinder sh.e.l.l.

Let us now, as we directed in our former instructions, draw a flattened curve to represent the acting surface of the entrance lip of our cylinder as if it were in direct contact with the impulse face of the tooth. To delineate the exit lip we draw from the center _g_, Fig. 134, to the radial line _g k_, said line pa.s.sing through the point of contact between the tooth and entrance lip of the cylinder. Let us next continue this line on the opposite side of the point _g_, as shown at _g k'_, and we thus bisect the cylinder sh.e.l.l into two equal parts of 180 degrees each. As we previously explained, the entire extent of the cylinder half sh.e.l.l is 196 degrees. We now set our dividers to the radius of any convenient arc which we have divided into degrees, and from _g_ as a center sweep the short arc _l l_, and from the intersection of this arc with the line _g k'_ we lay off sixteen degrees on the said arc _l_ and establish the point _n_, from _g_ as a center draw the radial line _g n'_. Take ten degrees from the same parent arc and establish the point _m_, then draw the line _g m'_. Now the arc on the circles _h j_ between the lines _g n'_ and _g m_ limits the extent of the exit lip of the cylinder and the arc between the lines _g k'_ and _g m'_ represents the locking surface of the cylinder sh.e.l.l.

[Ill.u.s.tration: Fig. 134]

To delineate the U arms we refer to Fig. 135. Here, again, we draw the arc _a b c_ and delineate a tooth as before. From the point _e_ located at the heel of the tooth we draw the radial line _e e'_. From the point _e_ we lay off on the arc _a_ five degrees and establish the point _p_; we halve this s.p.a.ce and draw the short radial line _p' s'_ and _p s_.

From the point _e_ on the arc _A_ we lay off twenty-four degrees and establish the point _t_, which locates the heel of the next tooth in advance of _A_. At two and a half degrees to the right of the point _t_ we locate the point _r_ and draw the short radial line _r s_. On the arc _b_ and half way between the lines _p s_ and _r s_, we establish the point _u_, and from it as a center we sweep the arc _v_ defining the curve of the U arms.

We have now given minute instructions for drawing a cylinder escapement in all its details except the extent of the banking slot of the cylinder, which is usually made to embrace an angular extent of 270 degrees; consequently, the pillar of the cylinder will not measure more than ninety degrees of angular extent.

There is no escapement constructed where carefully-made drawings tend more to perfect knowledge of the action than the cylinder. But it is necessary with the pupil to inst.i.tute a careful a.n.a.lysis of the actions involved. In writing on a subject of this kind it is extremely perplexing to know when to stop; not that there is so much danger of saying too much as there is not having the words read with attention.