In taking up the study of calendar work the first thing that the student observes is the irregularity of motion of the various members. Every other portion of a clock has for its main object the attainment of the nicest regularity of motion, while the calendar must necessarily have irregular motion. The hand of the day of the month proceeds around its dial regularly from 1 to 28 and then jumps to 1 in February of some years, while it continues to 29 in others; sometimes it revolves regularly from 1 to 31 for several revolutions and then jumps from 30 to 1. What is the reason of this?

If the moon’s phases are shown they do not agree with the changes of the month wheels, but keep gaining on them, while if an “equation of time” is shown, we have a hand that moves irregularly back and forth from the Figure XII at the center of its dial. What is the cause of this gaining and losing?

In order to understand this mechanism properly we shall have to first know what it is intended to show and this brings us to the study of the various kinds of calendar.

The earth revolves about its axis with a circular motion; it revolves about the sun with an elliptical motion. This means that the earth will move through a greater angular distance, measured from the sun’s center, in a given time at some portions of its journey than it will do at others; at times the sun describes an arc of 57 minutes of the ecliptic; at other times an arc of 61 minutes in a day; hence the sun will be directly over a given meridian of the earth (noon) a little sooner at some periods than at others. Now the time at which the sun is directly over the given meridian is apparent noon, or solar noon. As before stated, this is irregular, while the motion of our clocks is regular, consequently the sun crosses the meridian a little before or a little after twelve by the clock each day, varying from 15 minutes before twelve to 15 minutes after twelve by the clock. The best we can do under these circumstances is to divide these differences of gaining or losing, take the average or mean of them and regulate the clock to keep mean time. Here then we have two times—the irregular apparent time and the regular mean apparent time. The amount to be added to or subtracted from the mean in order to get the solar or actual apparent time is called the equation of time and this is shown by the equation hand on an astronomical or perpetual calendar clock.

The moon revolves on its axis with a circular motion and it revolves about the earth with an elliptical motion, the earth being at one focus of the ellipse; as this course does not agree with that of the sun, but is shorter, it keeps gaining so that the lunar months do not agree with the solar.

Certain stars are so far away that they apparently have no motion of their own and are called fixed; hence in observing them the only motion we can discern is the circular motion of the earth. We can set our clocks by watching such stars and a complete revolution of the earth, measured by such a star, is called an astronomical or sidereal day. This is the one used in computing all our time. It is shorter than the mean solar day by 3 minutes 56 seconds.

A year is defined as the period of one complete revolution of the earth about the sun, returning to the same starting point in the heavens. By taking different starting points we are led to different kinds of years. The point generally taken is the vernal equinoctial point, and when measured thus it is called the tropical year, which gives us the seasons. It is 20 minutes shorter than the siderial year.

A siderial year is the period of a complete revolution of the earth about the sun. This period is very approximately 365 days, 6 hours, 9 minutes, 9.5 seconds of mean time. Here we see an important difference between the siderial and the civil year of 365 days, and it is this difference, which must be accounted for somehow, that causes the irregularities in our calendar work.

For ordinary and business purposes the public demands that the year shall contain an exact number of days and that it should bear a simple relation to the recurrence of the seasons. For this reason the civil year has been introduced. The Roman emperor, Julius Caesar, ordered that three successive years should have 365 days each and the fourth year should have 366 days.

The fourth year, containing 366 days, is called a leap year, because it leaps over, or gains, the difference between the civil and siderial time of the preceding three years. For convenience the leap year was designated as any year whose number is exactly divisible by 4. This is called the Julian calendar.

But as a siderial year is 365 days, 6 hours, 9 minutes, 9.5 seconds of mean time, the addition of one day of twenty-four hours would not exactly balance the two calendars; therefore Pope Gregory XIII., in 1582, ordered that every year whose number is a multiple of 100 shall be a year of 365 days, unless the number of the year is divisible by 400, when it shall be a leap year of 366 days.

The calendar constructed in this way is called the Gregorian calendar, and is the one in common use. Its error is very small and will amount to only 1 day, 5 hours, 30 minutes in 4,000 years.

The revolution of the moon around the earth in relation to the stars, takes place in 27 days, 7 hours and 43 minutes this is called a siderial month. But during this period the earth has advanced along the plane of its path about the sun and the moon must make up this distance in order to return to the same point in relation to the sun. This period is called a synodic month. Its average length is 29 days, 12 hours, 44 minutes, 2.9 seconds.

Having now understood these differences we shall be able to intelligently examine the various calendar mechanisms on the market and understand the reasons for their apparent departures from regular mechanical progression, as the equation of time gives us the difference between real and mean apparent, or solar time; we regulate our clocks by means of siderial time; the irregular procession of 30 and 31 days makes the civil calendar agree with the seasons, or the tropical year, and the remainder of the discrepancy between civil and siderial time is made up in February at the period when it is of the least consequence.

Simple Calendar Work.—Fig. 111 shows the American method of making a simple calendar, the example shown being drawn from a movement of the Waterbury Clock Company as a typical example. No attempt is made here to show the day of the week or the month. The days of the month are shown by a series of numbers from 1 to 31, arranged concentrically with the time dial and the current day is indicated by a hand of different color, carried on a pipe outside the pipe of the hour hand on the center arbor.

In order to accomplish this the motion work for the hands is mounted inside the frames, the hour pipe and center arbor being suitably lengthened. In the Figure A is the cannon pinion; B, the minute wheel; C, the minute pinion; D, the hour wheel at the rear end of the hour pipe; this pipe projects through the frame and forms a bearing in the frame for the center arbor. Friction-tight on the hour pipe, in front of the front plate, is the pinion E, which drives a wheel F of twice as many teeth. This wheel F is mounted loosely on a stud and has a pin which meshes with the teeth of a ratchet wheel G. G is carried at the bottom end of a pipe which fits loosely on the hour pipe and carries the calendar hand H under the hour hand and close to the dial. The pinion on the hour pipe revolves once in twelve hours. The wheel E has twice as many teeth and will therefore revolve once in twenty-four hours. It moves the ratchet G one tooth at each revolution; therefore the hand H moves one space every twenty-four hours. There are 31 teeth, so that the hand must be set forward every time it reaches the 28th and 29th of February and the 30th of April, June, September and November. This is the simplest and cheapest of all the calendars, occupies the least space and is frequently attached to nickel alarm clocks for that reason.

A simple calendar work often met with in old clocks of European origin is shown in Fig. 112. Gearing with the hour wheel is a wheel, A, having twice its number of teeth, and turning therefore once in twenty-four hours. A three-armed lever is planted just above this wheel; the lower arm is slotted and the wheel carries a pin which works in this slot, so that the lever vibrates to and fro once every twenty-four hours. The three upper wheels, B, C and D in the drawing, represent three star wheels. B has seven teeth, corresponding to the days of the week; C has 31 teeth, for the days of the month; and D has 12 teeth, for the months of the year. Each carries a hand in the center of a dial on the other side of the plate. Every time the upper arms of the lever vibrate they move forward the day of the week, B, and the day of the month, C, wheels each one tooth. The extremities of the two upper levers are jointed so as to yield on the return vibration, and are brought into position again by a weak spring. There is a pin in the wheel, C, which, by pressing on a lever once every revolution, actuates the month of the year wheel, D. This last lever is also jointed, and is pressed on by a spring so as to return to its original position. Each of the star wheels has a click kept in contact by means of a spring. For months with less than 31 days, the day of the month hand has to be shifted forward.

Perpetual Calendar Work.—Figs. 113, 114, 115, show a perpetual calendar which gives the day of the week, day of the month and the month, making all changes automatically at midnight, and showing the 31 days on a dial beneath the time dial, by means of a hand, and the days of the week and the month by means of cylinders operating behind slots in the dial on each side of the center. This is also a Waterbury movement.

A pinion on the hour pipe engages a wheel, A, having twice the number of teeth and mounted on an arbor which projects through both plates. The rear end of this arbor carries a cam, B, on which rides the end of a lever, C, which is pivoted to the rear frame. The lever is attached to a wire, D, which operates a sliding piece, E, which is weighted at its lower end. The cam, which, of course, revolves once in twenty-four hours, drops its lever at midnight and the weight on E pulls it down. E bears a spring pawl, F, which on its way down, raises the spring actuated retaining click, H, and then moves the 31-toothed wheel G one notch. This wheel is mounted on the arbor which carries the hand and, of course, advances the hand.

Lying on top of the wheel, G, is a cam, I, pivoted to G near its circumference and having an arm reaching toward the months cylinder and another reaching towards the right leg of the pawl, H, while it is cut away in the center, so as to clear the center arbor carrying the hand. Trace this cam, I, carefully in Figs. 113 and 114, as its action is vital. The lower arm of this cam is shown more clearly in Fig. 114. It projects above the wheel and engages the long teeth, J, and the cam, K, mounted on the year cylinder arbor; where the lower arm of I strikes one of these teeth it shoves the upper arm outward, so that it strikes the retaining end of the pawl, H, and holds it up, and the descending pawl, F, may then push the wheel, G, forward for more than one tooth. The upper end of I is broad enough to cover three teeth of the wheel, G, when pushed outward, and the slot in E is long enough so that F may descend far enough to push G forward three teeth at once, unless it is stopped by H falling into a tooth, so that the position of I, when it is holding up H and the extra drop thus given to E serve to operate the jumps of 30 to 1, 28 to 1 and 29 to 1 of the hand on the dial. The teeth, J, Fig. 114, operate for two notches, thus making the changes from 30 to 1. The wide tooth, M, and cam, K, acting together, make the change for February from 28 to 31. The 29th day is added by the movement of the cam, K, narrowing the acting surface once in four years, as follows:

Looking at Fig. 114 we see an ordinary stop works finger, mounted on the months arbor and engaging a four-armed maltese cross on the wheel. Behind the wheel is a circular cam (shown dotted in) with one-fourth of its circumference cut away; the pivot holds the cam and cross rigidly together while permitting them to revolve loosely in the wheel. The cam, K, lies close to the wheel and is pressed against the cam on the cross by a spring, so that ordinarily the full width of M and K act as one piece on the end of the cam, I, which thus is pressed against the retaining pawl, H, during the passage of three teeth, making the jump from 28 to 1 each of these three years.

The fourth revolution of the maltese cross brings the cut portion of its cam to operate on K and allows K to move behind M, thus narrowing the acting surface so that I only covers two teeth (30 and 31) for every fourth revolution of the month’s cylinder, thus making the leap year every fourth year.

The months cylinder is kept in position by the two-armed pawl, N, engaging the teeth, L, which stand at 90 degrees from the wheel, as shown in Fig. 113. Attached to the bearing for the week cylinder (not shown) is one revolution of a screw track, or worm, surrounding the arbor for the hand. Attached to the arbor is a finger, O, held taut by a spring and engaging the track, P. The revolution of the arbor raises O on P until it slips off, when O, drawn downward by its spring, raises the pawl, N, drops on one of the teeth, L, and revolves the cylinder one notch.

Q is a shifter for raising the pawl, H, and allowing the hand to be set.

Fig. 115 shows the inner end of the cylinder for the days of the week. There are two sets of these and fourteen teeth on the sprocket, R, so as to get the two cylinders approximately the same size (there being 14 days and 12 months on the respective cylinders). S is a pawl whose upper end is forked so as to embrace a tooth and hold the cylinder in position. T is a hook, carried on the sliding piece, E, which swings outward in its upward passage as E is raised and on its downward course raises the pawl, S, and revolves the sprocket, R, one tooth, thus changing the day of the week at the same time the hand is advanced.

To set the calendar, raise the pawl, N, and revolve the year cylinder until M and K are at their narrowest width; that is, a leap year. Then give the year cylinder as many additional turns as there are years since the last leap year, stopping on the current month of the current year. For instance, if it is two years and four months since the 29th of February last occurred, give the cylinder 2 and 4/12 turns which should bring you to the current month, raise the shifter, Q, and set the hand to the current day. Then raise the pawl, S, and set the week cylinder to the current day. Place the hour hand on the movement so that the cam will drop E at midnight.

Fig. 116 shows the dial of Brocot’s calendar work, which, with or without the equation of time and the lunations, is to be met with in many grandfather, hall and astronomical clocks. We will assume that all of these features are present, in order to completely cover the subject. It consists of two circular plates of which the front plate is the dial and the rear plate carries the movement, arranged on both sides of it. All centers are therefore concentric and we have marked them all with the same letters for better identification in the various views as the inner plate is turned about to show the reverse side, thus reversing the position of right to left in one view of the inner plate.

Fig. 117 shows the wheel for the phases of the moon, which is mounted on the outside of the inner plate immediately behind the opening in the dial. The dark circles have the same color as the sky of the dial and the rest is gilt, white or cream color to show the moon as in Fig. 116. The position of this plate is also shown in Fig. 120. By the dotted circles, about the center D.

The inner side containing the mechanism for indicating the days of the week and the days of the month is shown in Fig. 118. The calendar is actuated by means of a pin, C, fixed to a wheel of the movement which turns once in twenty-four hours in the manner previously described with Fig. 113. Two clicks, G and H, are pivoted to the lever, M. G, by means of its weighted end, see Fig. 119, is kept in contact with a ratchet wheel of 31 teeth, and H with a ratchet wheel of 7 teeth. As a part of these clicks and wheels is concealed in Fig. 118, they are shown separately in Fig. 119.

When the lever, M, is moved to the left as far as it will go by the pin, e, the clicks, G and H, slip under the teeth; their beaks pass on to the following tooth; when e has moved out of contact the lever, M, falls quickly by its own weight, and makes each click leap a tooth of the respective wheels, B of 7 and A of 31 teeth. The arbors of these wheels pass through the dial (Fig. 116), and have each an index which, at every leap of its own wheel, indicates on its special dial the day of the week and the day of the month. A roll, or click, kept in position by a sufficient spring, keeps each wheel in its place during the interval of time which separates two consecutive leaps.

This motion clearly provides for the indication of the day of the week, and would be also sufficient for the days of the month if the index were shifted by hand at the end of the short months.

To secure the proper registration of the months of 30 days, for February of 28 during three years, and of 29 in leap year, we have the following provision: The arbor, A, of the day of the month wheel goes through the circular plate, and on the other side is fixed (see Fig. 120) a pinion of 10 leaves. This pinion, by means of an intermediate wheel, I, works another wheel (centered at C) of 120 teeth, and consequently turning once in a year. The arbor of this last wheel bears an index indicating the name of the month, G, Fig. 116. The arbor, C, goes through the plate, and at the other end, C, Fig. 118, is fixed a little wheel gearing with a wheel having four times as many teeth, and which is centered on a stud in the plate at F. This wheel is partly concealed in Fig. 118 by a disc V, which is fixed to it, and with the wheel makes one turn in four years. On this disc, V, are made 20 notches, of which the 16 shallowest correspond to the months of 30 days; a deeper notch corresponds to the month of February of leap year, and the last three deepest to the month of February common years in each quarternary period. The uncut portions of the disc correspond to the months of 31 days in the same period. The wheel, A, of 31 teeth, has a pin (i) placed before the tooth which corresponds to the 28th of the month. On the lever, M, is pivoted freely a bell-crank lever (N), having at the extremity of the lower arm a pin (o) which leans its own weight upon the edge of the disc, V, or upon the bottom of one of the notches, according to the position of the month, and the upper arm of N is therefore higher or lower according to the position of the pin, o, upon the disc.

It will be easy to see that when the pin, o, rests on the contour of the disc the upper arm, N, of the bell-crank lever is as high as possible, and out of contact with the pin as it is dotted in the figure, and then the 31 teeth of the month wheel will each leap successively one division by the action of the click, G, as the lever, M, falls backward till the 31st day. But when the pin, o, is in one of the shallow notches of the plate, V, corresponding to the months of 30 days, the upper arm, N, of the bell-crank lever will take a lower position, and the inclination that it will have by the forward movement of the lever, M, will on the 30th bring the pin, i, in contact with the bottom of the notch, just as the lever, M, has accomplished two-thirds of its forward movement, so the last third will be employed to make the wheel 31 advance one tooth, and the hand of the dial by consequence marks the 31st, the quick return of the lever, M, as it falls putting this hand to the 1st by the action of the click, G. If we suppose the pin, o, is placed in the shallowest of the four deep notches, that one for February of leap year, the upper end of the arm, N, will take a position lower still, and on the 29th the pin, i, will be met by the bottom of the notch, just as the lever has made one-third of its forward course, so the other two-thirds of the forward movement will serve to make two teeth of the wheel of 31 jump. Then the hand of the dial, A, Figs. 116 and 118, will indicate 31, and the ordinary quick return of the lever, M, with its detent, G, will put it to the 1st. Lastly, if, as it is represented in the figure, the pin, o, is in one of the three deepest notches, corresponding to the months of February in ordinary years, the pin will be in the bottom of the notch on the 28th just at the moment the lever begins its movement, and three teeth will pass before the return of the lever makes the hand leap from the 31st to the 1st.

The pin, o, easily gets out of the shallow notches, which, as will be seen, are sloped away to facilitate its doing so. To help it out of the deeper notches there is a weighted finger (j) on the arbor of the annual wheel. This finger, having an angular movement much larger than the one of the disc, V, puts the pin, o, out of the notch before the notch has sensibly changed its position.

Phases of the Moon.—The phases of the moon are obtained by a pinion of 10, Fig. 120, on the arbor, B, which gears with the wheel of 84 teeth, fixed on another of 75, which last gears with a wheel of 113, making one revolution in three lunations. By this means there is an error only of .00008 day per lunation. On the wheel of 113 is fixed a plate on which are three discs colored blue, having between them a distance equal to their diameter, as shown in Fig. 117, these discs slipping under a circular aperture made in the dial, produce the successive appearance of the phases of the moon.

Equation of Time.—On the arbor of the annual wheel, C, Figs. 116, 118, 120, is fixed a brass cam, Y, on the edge of which leans the pin, s, fixed to a circular rack, R. This rack gears with the central wheel, K, which carries the hand for the equation. That hand faces XII the 15th of April, 14th of June, 1st of September and the 25th of December. At those dates the pin, s, is in the position of the four dots marked on the cam, Y. The shape of the cam, Y, must be such as will lead the hand to indicate the difference between solar and mean time, as given in the table of the Nautical almanac.

To set the calendar first see that the return of the lever, M, be made at the moment of midnight. To adjust the hand of the days of the week, B, look at an almanac and see what day before the actual date there was a full or new moon. If it was new moon on Thursday, it would be necessary, by means of a small button fixed at the back, on the arbor of the hand of the wheel, B, of the week, to make as many returns as requisite to obtain a new moon, this hand pointing to a Thursday; afterward bring back the hand to the actual date, passing the number of divisions corresponding to the days elapsed since the new moon. To adjust the hand of the day of the month, A, see if the pin, o, is in the proper notch. If for the leap year, it is in the month of February in the shallowest of the four deep notches (o); if for the same month of the first year after leap year, then the pin should be, of course, in the notch, I, and so on.