Before going further with the mechanism of our clocks we will now consider the means by which the various members are held in their positions, namely, the plates. Like most other parts of the clock these have undergone various changes. They have been made of wood, iron and brass and have varied in shapes and sizes so much that a great deal may be told concerning the age of a clock by examining the plates.

Most of the wooden clocks had wooden plates. The English and American movements were simply boards of oak, maple or pear with the holes drilled and bushed with brass tubes—full plates. The Schwarzwald movements were generally made with top and bottom boards and stanchions, mortised in between them to carry the trains, which were always straight-line trains. The rear stanchions were glued in position and the front ones fitted friction-tight, so that they could be removed in taking down the clock. This gave a certain convenience in repairing, as, for instance, the center (time) train could be taken down without disturbing the hour or quarter trains, or vice versa. Various attempts have been made since to retain their convenience with brass plates, but it has always added so much to the cost of manufacture that it had to be abandoned.

The older plates were cast, smoothed and then hammered to compact the metal. The modern plate is rolled much harder and stiffer and it may consequently be much thinner than was formerly necessary. The proper thickness of a plate depends entirely upon its use. Where the movement rests upon a seat board in the case and carries the weight of a heavy pendulum attached to one of the plates they must be made stiff enough to furnish a rigid support for the pendulum, and we find them thick, heavy and with large pillars, well supported at the corners, so as to be very stiff and solid. An example of this may be seen in that class of regulators which carry the pendulum on the movement. Where the pendulum is light the plates may therefore be thin, as the only other reason necessary for thickness is that they may provide a proper length of bearing for the pivots, plus the necessary countersinking to retain the oil.

In heavy machinery it is unusual to provide a length of box or journal bearing of more than three times the diameter of the journal. In most cases a length of twice the diameter is more than sufficient; in clock and other light work a “square” bearing is enough; that is one in which the length is equal to the diameter. In clocks the pivots are of various sizes and so an average must be found. This is accomplished by using a plate thick enough to furnish a proper bearing for the larger pivots and countersinking the pivot holes for the smaller pivots until a square bearing is obtained. This countersinking is shaped in such a manner as to retain the oil and as more of it is done on the smaller and faster moving pivots, where there is the greatest need of lubrication, the arrangement works out very nicely, and it will be seen that with all the lighter clocks very thin plates may be employed while still retaining a proper length of bearing in the pivot holes.

The side shake for pivots should be from .002 to .004 of an inch; the latter figure is seldom exceeded except in cuckoos and other clocks having exposed weights and pendulums. Here much greater freedom is necessary as the movement is exposed to dust which enters freely at the holes for pendulum and weight chains, so that such a clock would stop if given the ordinary amount of side shake.

We are afraid that many manufacturers of the ordinary American clock aim to use as thin brass as possible for plates without paying too much attention to the length of bearing. If a hole is countersunk it will retain the oil when a flat surface will not. The idea of countersinking to obtain a shorter bearing will apply better to the fine clocks than to the ordinary. In ordinary clocks the pivots must be longer than the thickness of the plates for the reason that freight is handled so roughly that short pivots will pop out of the plates and cause a lot of damage, provided the springs are wound when the rough handling occurs.

It will be seen by reference to Chapter VII (the mechanical elements of gearing), Figs. 21 to 25, that a wheel and pinion are merely a collection of levers adapted to continuous work, that the teeth may be regarded as separate levers coming into contact with each other in succession; this brings up two points. The first is necessarily the relative proportions of those levers, as upon these will depend the power and speed of the motion produced by their action. The second is the shapes and sizes of the ends of our levers so that they shall perform their work with as little friction and loss of power as possible.

To Get Center Distances.—As the radii and circumferences of circles are proportional, it follows that the lengths of our radii are merely the lengths of our levers (See Fig. 24), and that the two combined (the radius of the wheel, plus that of the pinion) will be the distance at which we must pivot our levers (our staffs or arbors of our wheels) in order to maintain the desired proportions of their revolution. Consequently we can work this rule backwards or forwards.

For instance if we have a wheel and pinion which must work together in the proportion of 7½ to 1; then 7½ + 1 = 8½ and if we divide the space between centers into 8½ spaces we will have one of these spaces for the radius of the pitch circle of the pinion and 7½ for the pitch circle of the wheel, Fig. 65. This is independent of the number of teeth so long as the proportions be observed; thus our pinion may have eight teeth and the wheel sixty, 60 ÷ 8 = 7.5, or 75 ÷ 10 = 7.5, or 90 ÷ 12 = 7.5, or any other combination of teeth which will make the correct proportion between them and the center distances. The reason is that the teeth are added to the wheel to prevent slipping, and if they did not agree with each other and also with the proportionate distance between centers there would be trouble, because the desired proportion could not be maintained.

Now we can also work this rule backwards. Say we have a wheel of 80 teeth and the pinion has 10 leaves but they do not work together well in the clock. Tried in the depthing tool they work smoothly. 80 ÷ 10 = 8, consequently our center distance must be as 8 and 1. 8 + 1 = 9; the wheel must have 8 parts and the pinion 1 part of the radius of the pitch circle of the wheel. Measure carefully the diameter of the pitch circle of the wheel; half of that is the pitch radius, and nine-eighths of the pitch radius is the proper center distance for that wheel and pinion.

Say we have lost a wheel; the pinion has 12 teeth and we know the arbor should go seven and one-half times to one of the missing wheel; we have our center distances established by the pivot holes which are not worn; what size should the wheel be and how many teeth should it have? 12 × 7.5 = 90, the number of teeth necessary to contain the teeth of the pinion 7.5 times. 7.5 + 1 = 8.5, the sum of the center distances; the pitch radius of the pinion can be closely measured; then 7.5 times that is the pitch radius of the missing wheel of 90 teeth. Other illustrations with other proportions could be added indefinitely but we have, we think, said enough to make this point clear.

Conversion of Numbers.—There is one other point which sometimes troubles the student who attempts to follow the expositions of this subject by learned writers and that is the fact that a mathematician will take a totally different set of numbers for his examples, without explaining why. If you don’t know why you get confused and fail to follow him. It is done to avoid the use of cumbersome fractions. To use a homely illustration: Say we have one foot, six inches for our wheel radius and 4.5 inches for our pinion radius. If we turn the foot into inches we have 18 inches. 18 ÷ 4.5 = 4, which is simpler to work with. Now the same thing can be done with fractions. In the above instance we got rid of our larger unit (the foot) by turning it into smaller units (inches) so that we had only one kind of units to work with. The same thing can be done with fractions; for instance, in the previous example we can get rid of our mixed numbers by turning everything into fractions. Eighteen inches equals 36 halves and 4.5 equals 9 halves; then 36 ÷ 9 = 4. This is called the conversion of numbers and is done to simplify operations. For instance in watch work we may find it convenient to turn all our figures into thousands of a millimeter, if we are using a millimeter gauge. Say we have the proportions of 7.5 to 1 to maintain, then turning all into halves, 7½ × 2 = 15 and 1 × 2 = 2. 15 + 2 = 17 parts for our center distance, of which the pitch radius of the pinion takes 2 parts and that of the wheel 15.

The Shapes of the Teeth.—The second part of our problem, as stated above, is the shapes of the ends of our levers or the teeth of our wheels, and here the first consideration which strikes us is that the teeth of the wheels approach each other until they meet; roll or slide upon each other until they pass the line of centers and then are drawn apart. A moment’s consideration will show that as the teeth are longer than the distance between centers and are securely held from slipping at their centers, the outer ends must either roll or slide after they come in contact and that this action will be much more severe while they are being driven towards each other than when they are being drawn apart after passing the line of centers. This is why the engaging friction is more damaging than the disengaging friction and it is this butting action which uses up the power if our teeth are not properly shaped or the center distances not right. Generally speaking this butting causes serious loss of power and cutting of the teeth when the pivot holes are worn or the pivots cut, so that there is a side shake of half the diameter of the pivots, and bushing or closing the holes, or new and larger pivots are then necessary. This is for common work. For fine work the center distances should be restored long before the wear has reached this point.

If we take two circular pieces of any material of different diameters and arrange them so that each can revolve around its center with their edges in contact, then apply power to the larger of the two, we find that as it revolves its motion is imparted to the other, which revolves in the opposite direction, and, if there is no slipping between the two surfaces, with a velocity as much greater than that of the larger disc as its diameter is exceeded by that of the larger one. We have, then, an illustration of the action of a wheel and pinion as used in timepieces and other mechanisms. It would be impossible, however, to prevent slipping of these smooth surfaces on each other so that power (or motion) would be transmitted by them very irregularly. They simply represent the “pitch” circles or circles of contact of these two mobiles. If now we divide these two discs into teeth so spaced that the teeth of one will pass freely into the spaces of the other and add such an amount to the diameter of the larger that the points of its teeth extend inside the pitch circle of the smaller, a distance equal to about 1⅛ times the width of one of its teeth, and to the smaller so that its teeth extend inside the larger one-half the width of a tooth, the ends of the teeth being rounded so as not to catch on each other and the centers of revolution being kept the same distance apart, on applying power to the larger of the two it will be set in motion and this motion will be imparted to the smaller one. Both will continue to move with the same relative velocity as long as sufficient power is applied. Other pairs of mobiles may be added to these to infinity, each addition requiring the application of increased power to keep it in motion.

These pairs of mobiles as applied to the construction of timepieces are usually very unequal in size and the larger is designated as a “wheel” while the smaller, if having less than 20 teeth, is called a “pinion” and its teeth “leaves.” Now while we have established the principle of a train of wheels as used in various mechanisms, our gearing is very defective, for while continuous motion may be transmitted through such a train, we will find that to do so requires the application of an impelling force far in excess of what should be required to overcome the inertia of the mobiles, and the amount of friction unavoidable in a mechanism where some of the parts move in contact with others.

This excess of power is used in overcoming a friction caused by improperly shaped teeth, or when formed thus the teeth of the wheel come in contact with those of the pinion and begin driving at a point in front of what is known as the “line of centers,” i. e., a line drawn through the centers of revolution of both mobiles, and as their motion continues the driven tooth slides on the one impelling it toward the center of the wheel. When this line is reached the action is reversed and the point of the driving tooth begins sliding on the pinion leaf in a direction away from the center of the pinion, which action is continued until a point is reached where the straight face of the leaf is on a line tangential to the circumference of the wheel at the point of the tooth. It then slips off the tooth, and the driving is taken up on another leaf by the next succeeding tooth. The sliding action which takes place in front of the line of centers is called “engaging,” that after this line has been passed “disengaging” friction.

Now we know that in the construction of timepieces, friction and excessive motive power are two of the most potent factors in producing disturbances in the rate, and that, while some friction is unavoidable in any mechanism, that which we have just described may be almost entirely done away with. Let us examine carefully the action of a wheel and pinion, and we will see that only that part of the wheel tooth is used, which is outside the pitch circle, while the portion of the pinion leaf on which it acts is the straight face lying inside this circle, therefore it is to giving a correct shape to these parts we must devote our attention. If we form our pinion leaves so that the portion of the leaf inside the pitch circle is a straight line pointing to the center, and give that portion of the wheel tooth lying outside the pitch circle (called the addenda, or ogive of the tooth) such a degree of curvature that during its entire action the straight face of the leaf will form a tangent to that point of the curve which it touches, no sliding action whatever will take place after the line of centers is passed, and if our pinion has ten or more leaves, the “addenda” of the wheel is of proper height, and the leaves of the pinion are not too thick, there will be no contact in front of the line of centers. With such a depth the only friction would be from a slight adhesion of the surfaces in contact, a factor too small to be taken into consideration.

Here, then, we have an ideal depth. How shall we obtain the same results in practice? It is comparatively an easy matter to so shape our cutters that the straight faces of our pinion leaves will be straight lines pointing to the center, but to secure just the proper curve for the addenda of our wheel teeth requires rather a more complicated manipulation. This curve does not form a segment of a circle, for it has no two radii of equal length, and if continued would form, not a circle, but a spiral. To generate this curve, we will cut from cardboard, wood, or sheet metal, a segment of a circle having a radius equal to that of our wheel, on the pitch circle, and a smaller circle whose diameter is equal to the radius of the pinion, on the pitch circle. To the edge of the small circle we will attach a pencil or metal point so that it will trace a fine mark. Now we lay our segment flat on a piece of drawing paper, or sheet metal and cause the small circle to revolve around its edge without slipping. We find that the point in the edge of the small circle has traced a series of curves around the edge of the segment.

These curves are called “epicycloids,” and have the peculiar property that if a line be drawn through the generating point and the point of contact of the two circles, this will always be at right angles to a tangent of the curve at its point of intersection. It is this property to which it owes its value as a shape for the acting surface of a wheel tooth, for it is owing to this that a tooth whose acting surface is bounded by such a curve can impel a pinion leaf through the entire lead with little sliding action between the two surfaces. This, then, is the curve on which we will form the addenda of our wheel teeth.

In Fig. 66, the wheel has a radius of fifteen inches and the pinion a radius of one and one-half, and these two measurements are to be added together to find the distance apart of the two wheels; 16.5 inches is then the distance that the centers of revolution are apart of the wheels. Now, the teeth and leaves jointly act on one another to maintain a sure and equable relative revolution of the pair.

In Fig. 66, the pinion has its leaves radial to the center, inside of the pitch line D, and the ends of the leaves, or those parts outside of the pitch line, are a half circle, and serve no purpose until the depthings are changed by wear, as they never come in contact with the wheel; the wheel teeth only touch the radial part of the pinion and that occurs wholly within the pitch line. So in all pinions above 10 leaves in number the addendum or curve is a thing of no moment, except as it may be too large or too long. In many large pieces of machinery the pinions, or small driven wheels, have no addendum or extension beyond their pitch diameter and they serve every end just as well. In watches there is so much space or shake allowed between the teeth and pinions that the end of a leaf becomes a necessity to guard against the pinion’s recoiling out of time and striking its sharp corner against the wheel teeth and so marring or cutting them. In a similar pair of wheels in machinery there are very close fits used and the shake between teeth is very slight and does not allow of recoil, butting, or “running out of time.”

Running out of time is the sudden stopping and setting back of a pinion against the opposite tooth from the one just in contact or propelling. This, with pinions of suppressed ends, is a fault and it is averted by maintaining the ends.

The wheel tooth drives the pinion by coming in contact with the straight flank of the leaf at the line of centers, that is a line drawn through the centers of the two wheels; centers of revolution.

The curve or end of the wheel tooth outside of the pitch line is the only part of the tooth that ever touches the pinion and it is the part under friction from pressure and slipping. At the first point of contact the tooth drives the pinion with the greatest force, as it is then using the shortest leverage it has and is pressing on the longest lever of the leaf. As this action proceeds, the tooth is acted on by the pinion leaf farther out on the curve of the wheel tooth, thus lengthening the lever of the wheel and at the same time the tooth thus acts nearer to the center of the pinion by touching the leaf nearer its center of revolution.

By these joint actions it will appear that the wheel first drives with the greatest force and then as its own leverage lengthens and its force consequently decreases, it acts on a shorter leverage of the pinion, as the end of a tooth is nearer to the center of the pinion, or on the shortest pinion leverage, just as the tooth is about ceasing to act.

The action is thus shown from the above to be a variable one, which starts with a maximum of force and ends with a minimum. Practically the variable force in a train is not recognized in the escapement, as the other wheels and pinions making up the train are also in the same relations of maximum and minimum forces at the same time, and thus this theoretical and virtual variability of train force is to a great extent neutralized at the active or escaping end of the movement.

There is another action between the tooth and leaf that is not easy to explain without somewhat elaborate sketches of the acting parts, and as this is not consistent with such an article, we may dismiss it, and merely state that it is the one of maintaining the relative angular velocities of the two wheels at all times during their joint revolutions.

In Fig. 66 will be seen the teeth of the wheel, their heights, widths and spacing, and the epicycloidal curves. Also the same features of the pinion’s construction. The curve on the end of the wheel teeth is the only curve in action during the rotation between wheel and pinion. Each flank (both teeth and leaves) is a straight line to the center of each. A tooth is composed of two members—the pillar or body of the tooth inside of the pitch line and the cycloid or curve, wholly outside of this line. The pinion also has two members, the radial flank wholly inside of the pitch line, and its addendum or circle outside of this line.

In Fig. 66 will be seen a tooth on the line of centers A B, just coming in action against the pinion’s flank and also one just ceasing action. It will be seen that the tooth just entering is in contact at the joint pitches, or radii, of the two wheels, and that when the tooth has run its course and ceased to act, that it will be represented by tooth 2. Then the exit contact will be at the dotted line o o. From this may be seen just how far the tooth has, in its excursion, shoved along the leaf of the pinion and by the distance the line o o, is from the wheel’s pitch line G, at this tooth. No. 2, is shown the extent of contact of the wheel tooth. By these dotted lines, then, it may be seen that the tooth has been under friction for nearly its whole curve’s length, while the pinion’s flank will have been under friction contact for less than half this distance. In brief, the tooth has moved about ⁸⁰⁄₁₀₀ of its curved surface along the straight flank .35 of the surface of the pinion leaf. From this relative frictional surface may be seen the reason why a pinion is apt to be pitted by the wheel teeth and cut away. In any case it shows the relation between the two friction surfaces. In part a wheel tooth rolls as well as slides along the leaf, but whatever rolling there may be, the pinion is also equally favored by the same action, which leaves the proportions of individual friction still the same.

In Fig. 66 may be seen the spaces of the teeth and pinion. The teeth are apart, equal to their own width and the depths of the spaces are the same measurement of their width—that is, the tooth (inside of the pitch line) is a pillar as wide as it is high and a space between two teeth is of like proportions and extent of surface. The depth of a space between two teeth is only for clearance and may be made much less, as may be seen by the pinion leaf, as the end of the circle does not come half way to the bottom of a space.

The dotted line, o o, shows the point at which the tooth comes out of action and the pointed end outside of this line might be cut off without interfering with any function of the tooth. They generally are rounded off in common clock work.

The pinion is 3 inches diameter and is divided into twelve spaces and twelve leaves; each leaf is two-fifths of the width of a space and tooth. That is one-twelfth of the circumference of the pinion is divided into five equal parts and the leaf occupies two and a space three of these parts. The space must be greater than the width of a leaf, or the end of a leaf would come in contact with a tooth before the line of centers and cause a jamming and butting action. Also the space is needed for dirt clearance. As watch trains actuated by a spring do not have any reserve force there must be allowance made for obstructions between the teeth of a train and so a large latitude is allowed in this respect, more than in any machinery of large caliber. As will be seen by Fig. 66, the spans between the leaves are deep, much more so than is really necessary, and a space at O C shows the bottom of a space, cut on a circle which strengthens a leaf at its root and is the best practice.

Having determined the form of our curve, our next step will be to get the proper proportions. Saunier recommends that in all cases tooth and space should be of equal width, but a more modern practice is to make the space slightly wider, say one-tenth where the curve is epicycloidal. When the teeth are cut with the ordinary Swiss cutters, which, of course, cannot be epicycloidal, it is best to make the spaces one-seventh wider than the tooth. This proportion will be correct except in the case of a ten-leaf pinion, when, if we wish to be sure the driving will begin on the line of centers, the teeth must be as wide as the spaces; but in this case the pinion leaf is made proportionately thinner, so that the requisite freedom is thus obtained.

The height of the addenda of the wheel teeth above the pitch circle is usually given as one and one-eighth times the width of a tooth. While this is approximately correct, it is not entirely so, for the reason that as we use a circle whose diameter is equal to the pitch radius of the pinion for generating the curve, the height of the addenda would be different on the same wheel for each different numbered pinion. So that if a wheel of 60 were cut to drive a pinion of 8, the curve of this tooth would be found too flat if used to drive a pinion of 10. Now, since the pitch diameter of the pinion is to the pitch diameter of the wheel as the number of leaves in the pinion are to the number of teeth in the wheel, in order to secure perfect teeth: we must adopt for the height of the addenda a certain proportion of the radius or diameter of the pinion it is to drive, this proportion depending on the number of leaves in the pinion.

A careful study of the experiments on this subject with models of depths constructed on a large scale, shows that the proportions given below come the nearest to perfection.

When the pinion has six leaves the spaces should be twice the width of the leaves and the depth of the space a little more than one-half the total radius of the pinion. The addenda of the pinion should be rounded, and should extend outside the pitch circle a distance equal to about one-half the width of a leaf. The addenda of the wheel teeth should be epicycloidal in form and should extend outside the pitch circle a distance equal to five-twelfths of the pitch radius of the pinion.

With these proportions, the tooth will begin driving when one-half the thickness of a leaf is in front of the line of centers, and there will be engaging friction from this point until the line of centers is reached.

This cannot be avoided with low numbered pinions without introducing a train of evils more productive of faulty action than the one we are trying to overcome. There will be no disengaging friction.

When a pinion of seven is used, the spaces of the pinion should be twice the width of the leaves, and the depth of a space about three-fifths of the total radius of the pinion. The addenda of the pinion leaves should be rounded, and should extend outside the pitch circle about one-half the width of a leaf. The addenda of the wheel teeth should be epicycloidal, and the height of each tooth above the pitch circle equal to two-fifths of the pitch radius of the pinion.

There is less engaging friction when a pinion of seven is used than with one of six, as the driving does not begin until two-thirds of the leaf is past the line of centers. There is no disengaging friction.

With an eight-leaf pinion the space should be twice as wide as the leaf, and the depth of a space about one-half the total radius of the pinion. The addenda of the pinion leaves should be rounded and about one-half the width of a leaf outside the pitch circle. The addenda of the wheel teeth should be epicycloidal, and the height of each tooth above the pitch circle equal to seven-twentieths of the pitch radius of the pinion.

With a pinion of eight there is still less engaging friction than with one of seven, as three-quarters of the width of a leaf is past the line of centers when the driving begins. As there is no disengaging friction, a pinion of this number makes a very satisfactory depth.

A pinion with nine leaves is sometimes, though seldom, used. It should have the spaces twice the width of the leaves, and the depth of a space one-half the total radius. The addenda should be rounded, and its height above the pitch circle equal to one-half the width of the leaf. The addenda of the wheel teeth should be epicycloidal, and the height of each tooth above the pitch circle equal to three-sevenths of the total radius of the pinion. With this pinion the driving begins very near the line of centers, only about one-fifth of the width of a leaf being in front of the line.

A pinion of ten leaves is the lowest number with which we can entirely eliminate engaging friction, and to do so in this case the proper proportions must be rigidly adhered to. The spaces on the pinion must be a little more than twice as wide as a leaf; a leaf and space will occupy 36° of arc; of this 11° should be taken for the leaf and 25° for the space. The addenda should be rounded and should extend about half the width of a leaf outside the pitch circle. The depth of a space should be equal to about one-half the total radius. For the wheel, the teeth should be equal in width to the spaces, the addenda epicycloidal in form, and the height of each tooth above the pitch circle, equal to two-fifths the pitch radius of the pinion.

A pinion having eleven leaves would give a better depth, theoretically, than one of ten, as the leaves need not be made quite so thin to ensure its not coming in action in front of the line of centers. It is seldom seen in watch or clock work, but if needed the same proportions should be used as with one of ten, except that the leaves may be made a little thicker in proportion to the spaces.

A pinion having twelve leaves is the lowest number with which we can secure a theoretically perfect action, without sacrificing the strength of the leaves or the requisite freedom in the depths. In this pinion, the leaf should be to the space as two to three, that is, we divide the arc of the circumference needed for a leaf and space into five equal parts, and take two of these parts for the leaf, and three for the space; depth of the space should be about one-half the total radius. The addenda of the wheel teeth should be epicycloidal, and the height of each tooth above the pitch line equal to two-sevenths the pitch radius of the pinion.

As the number of leaves is increased up to twenty, the width of the space should be decreased, until when this number is reached the space should be one-seventh wider than the leaf. As these numbers are used chiefly for winding wheels in watches, where considerable strength is required, the bottoms of the spaces of both mobiles should be rounded.

Circular Pitch. Diametral Pitch.—In large machinery it is usual to take the circumference and divide by the number of teeth; this is called the circular pitch, or distance from point to point of the teeth, and is useful for describing teeth to be cut out as patterns for casting.

But for all small wheels it is more convenient to take the diameter and divide by the number of teeth. This is called the diametral pitch, and when the diameter of a wheel or pinion which is intended to work into it is desired, such diameter bears the same ratio or proportion as the number required. Both diameters are for their pitch circles. As the teeth of each wheel project from the pitch circle and enter into the other, an addition of corresponding amount is made to each wheel; this is called the addendum. As the size of a tooth of the wheel and of a tooth of the pinion are the same, the amount of the addendum is equal for both; consequently the outside diameter of the smaller wheel or pinion will be greater than the arithmetical proportion between the pitch circles. As the diameters are measured presumably in inches or parts of an inch, the number of a wheel of given size is divided by the diameter, which gives the number of teeth to each inch of diameter, and is called the diametral pitch. In all newly-designed machinery a whole number is used and the sizes of the wheels calculated accordingly, but when, as in repairing, a wheel of any size has any number of teeth, the diametral number may have an additional fraction, which does not affect the principle but gives a little more trouble in calculation. Take for example a clock main wheel and center pinion: Assuming the wheel to be exactly three inches in diameter at the pitch line, and to have ninety-six teeth, the result will be 96 ÷ 3 = 32, or 32 teeth to each inch of diameter, and would be called 32 pitch. A pinion of 8 to gear with this wheel would have a diameter at the pitch line of 8 of these thirty-seconds of an inch or ⁸⁄₃₂ of an inch. But possibly the wheel might not be of such an easily manageable size. It might, say, be 3.25 inches, in which case, 96 being the number of the wheel and 8 of the pinion, the ratio is ⁸⁄₉₆ or ¹⁄₁₂, so ¹⁄₁₂ of 3.25 = 0.270, the pitch diameter of the pinion. These two examples are given to indicate alternative methods, the most convenient of which may be used. After arriving at the true pitch diameters the matter of the addendum arises, and it is for this that the diametral number is specially useful, as in every case when figuring by this system, whatever the number of a wheel or pinion, two of the pitch numbers are to be added. Thus with the 32 pitch, the outside diameter of the wheel will be 3 in. + ²⁄₃₂, and if the pinion ⁸⁄₃₂ + ²⁄₃₂ = ¹⁰⁄₃₂. With the other method the same exactness is more difficult of attainment, but for practical purposes it will be near enough if we use ²⁄₃₀ of an inch for the addendum, when the result will be 3.25 + ²⁄₃₀ or 3¼ + ²⁄₃₀ = 3⅓; in. nearly and the pinion 0.270 + ²⁄₃₀ = 0.270 + .0666 = 0.3366; or to work by ⅓ of an inch is near enough, giving the outside diameter of the pinion a small amount less than the theoretical, which is always advisable for pinions which are to be driven.

We represent by Figs. 67 to 71 a wheel of sixty teeth gearing with a pinion of six leaves. The wheel, whose pitch diameter is represented by the line mm is the same in each figure. The pinion, which has for its pitch diameter the line kk, is in Fig. 67, of a size proportioned to that of the wheel, and its center is placed at the proper distance; that is to say, the two pitch diameters are tangential.

In Fig. 68 the same pinion, of the proper size, has its center too far off; the depthing is too shallow. In Fig. 69 it is too deep. Figs. 70 and 71 represent gearing in which the pitch circles are in contact, as the theory requires, but the size of the pinions is incorrect. If the wheels and pinion actuated each other by simple contact the velocity of the pinion with reference to that of the wheel would not be absolutely the same; but the ratio of the teeth being the same, the same ratio of motion obtains in practice, and there is necessarily bad working of the teeth with the leaves.

We will observe what passes in each of these cases, and refer to the suitable remedies for obtaining a passable depthing and a comparatively good rate, without the necessity of repairs at a cost out of all proportion with the value of the article repaired.

Fig. 67 represents gearing of which the wheel and pinion are well proportioned and at the proper distance from each other. Its movement is smooth, but it has little drop or none at all. By examining the teeth h, h′, of the wheel, it is seen that they are larger than the interval between them. With a cutter FF, introduced between the teeth, they are reduced at d, d′, which gives the necessary drop without changing the functions, since the pitch circles mm and kk have not been modified. The drop, the play between the tooth d′ and the leaf a, is sufficiently increased for the working of the gearing with safety.

We have the same pair in Fig. 68, but here their pitch circles do not touch; the depthing is too shallow. The drop is too great and butting is produced between the tooth h and the leaf r, which can be readily felt. The remedy is in changing the center distance, by closing the holes, if worn, or moving one nearer the other. But in an ordinary clock this wheel may be replaced with a larger one, whose pitch circle reaches to e. The proportions of the pair are modified, but not sufficiently to produce inconvenience.

Too great depthing, Fig. 69, can generally be recognized by the lack of drop. When the teeth of the wheel are narrow, the drop may appear to be sufficient. When the train is put in action the depthing that is too great produces scratching or butting and the ’scape wheel trembles. This results from the fact that the points of the teeth of the wheel touch the core of the pinion and cause it to butt against the leaf following the one engaged, as is visible at r in Fig. 69. It should be noticed that in this figure the pitch circles mm and kk overlap each other, instead of being tangential.

To correct this gearing, the cutter should act only on the addenda of the teeth of the wheel, so as to diminish them and bring the pitch circle mm to n. The dots in the teeth d, d′, show the corrected gearing. It is seen that there will be, after this change, the necessary drop, and that the end of the tooth d′ will not touch the leaf r.

In the two preceding cases we have considered wheels and pinions of accurate proportion, and the defects of the gearing proceeding from the wrong center distances. We will not speak of the gearing in which the pinion is too small. The only theoretic remedy in this case, as in that of too large a pinion, is to replace the defective piece; but in practice, when time and money are to be saved, advantage must be taken, one way or another, of what is in existence.

The buzzing produced when the train runs in a gearing with too small a pinion proceeds from the fact that each tooth has a slight drop before engaging with the corresponding leaf. If we examine Fig. 70, it will be easy to see how this drop is produced. The wheel revolving in the direction indicated by the arrow, it can be seen that when the tooth h leaves the leaf r, the following tooth, p, does not engage with the corresponding leaf, s; this tooth will therefore have some drop before reaching the leaf. A friction may even be produced at the end or addendum of the tooth p against the following leaf v.

To obtain a fair depthing without replacing the pinion, the wheels can be passed to the rounding-up machine, having a cutter which will take off only the points of the teeth, as is indicated in the figure; the result may be observed by the dotted lines. The tooth h being shorter, it will leave the leaf r of the pinion when the latter is in the dotted position; that is to say, a little sooner. At this moment the tooth p is in contact with the leaf s, and there is no risk of friction against the leaf v. Care must be taken to touch only the addendum of the tooth so as not to weaken the teeth. The circumference i will be that of a pinion of accurate size, and if the pinion is replaced, it will be necessary to diminish the wheel so that its pitch circle shall be tangential with i.

With too small a pinion a passable gearing can generally be produced. In any case stoppage can be prevented. This is not so easy when the pinion is too large. In Fig. 71, the pinion has as its pitch circle the line k, instead of i, which would be nearer the size with reference to that of the wheel. This is purposely drawn a little small for clearness of illustration. The essential defect of such a gearing can be seen; the butting produced between the tooth p and the leaf s will cause stoppage. How shall this defect be corrected without replacing the pinion?

To remedy the butting as far as possible, some watchmakers slope the teeth of the wheel by decentering the cutter on the rounding-up machine. At FF the cutter is seen working between the teeth d and d′. It is evident that when the wheel becomes smaller it is necessary to stretch it out, and to make use of the cutter afterwards. However, the most rational method is to leave the teeth straight, and to give them the slenderest form possible, after having enlarged the wheel or having replaced it with another. The motive force of the wheel being sufficiently weak, the size of the teeth may be reduced without fear. The essential thing is to suppress the butting. Success will be the easiest when the teeth are thinner.

In conclusion, we recommend verification of all suspected gearings by the depthing tool, which is easier and surer than by the clock itself. One can see better by the tool the working of the teeth with the leaves, and can form a better idea of the defect to be corrected. With the aid of the illustrations that have been given it can be readily noticed whether the depthing is too deep or too shallow, or the pinion too large or too small.

The defects mentioned are of less consequence in a pinion of seven leaves, and they are corrected more readily. With pinions of higher numbers the depthings will be smoother, provided sufficient care has been taken in the choice of the rounding-up cutters.

Rounding-Up Wheels.—It is frequently observed that young watchmakers, and (regretfully be it said) some of the older and more experienced ones, are rather careless when fitting wheels on pinions. In many cases the wheel is simply held in the fingers and the hole opened with a broach, and in doing this no special care is taken to keep the hole truly central and of correct size to fit the pinion snugly, and should it be opened a little too large it is riveted on the pinion whether concentric or not. Many suppose the rounding-up tool will then make it correct without further trouble and without sufficient thought of the irregularities ensuing when using the tool.

To make the subject perfectly clear the subjoined but rather exaggerated sketch is shown, Fig. 72. Of course, it is seldom required to round-up a wheel of twelve teeth, and the eccentricity of the wheel would be hardly as great as shown; nevertheless, assuming such a case to occur the drawing will exactly indicate the imperfections arising from the use of a rounding-up tool.