Re: Accurate Lunisolar Calendar

Dear Victor, Irv and Calendar People

Dear Calendar People,

The following would be a device useful for an observational lunisolar calendar.

The device consists of two gears that intermesh with each other or with a common gear.
The moon gear has 19 teeth.
The sun gear has 235 teeth.
One tooth on the moon gear is specially marked.
One tooth on the sun gear is specially marked.
Both gears are moved together each time there is a new moon. If this results in the specially marked teeth of both gears lining up, then a new year is started. So far, the Metonic cycle has been duplicated.

I’ve had difficulty understanding this. If moved together means moved together by one tooth, then it’d would take 235*19=4465 new moons, which is about 361 years to the two marked teeth to come together. I am aware that if the moon gear wheel were turned one revolution per lunation, the sun gear would turn about once per year and its marked tooth reaching the moon gear wheel would mark the new year.

I now think what Victor meant by moved together was that the moon gear wheel was turned one complete revolution each new moon.

An alternative (to one complete revolution on the new moon) is to move it one tooth every 2^nd, 3^rd , 5^th, 6^th,  8^th, 9^th,  11^th, 12^th,  14^th day after a new moon and likewise for the following 14 days, then one more tooth on the new moon to complete one revolution of the moon gear wheel.

Another alternative is to move the wheels one tooth according the same 14-day cycle till 26 cycles have passed then for a common year move one tooth the next (and 365^th) day to complete a year and in a leap year move one tooth the next but one (and 366^th) day to complete a year. The moon gear wheel would turn close to once per lunation.

However the lining up of the marked teeth on both the moon and sun gear wheels would occur just once every 19 years.

The Metonic cycle is a good approximation, but the year length is too long for the month length. So the following amends the above procedures:

It can be corrected by making the sun gear wheel slip one tooth against the moon gear wheel , so the sun wheel turns one tooth, the  moon gear wheel does not turn.

The moon gear has two pegs that travel around the gear. The pegs at epoch start out together, and one is taller than the other. At epoch, the shorter one sits in the cog with the mark that's used to align with the sun gear. Each time the marks on the moon and sun gears line up, the tall peg is moved one cog. The cogs have only one hole used by both pegs, so when the tall peg catches up to the short peg, they trade places (the short peg is moved to where the tall peg was when the tall peg is moved).

If this last leap-frog operation results in the short peg landing in the marked cog, then the sun gear is advanced one cog.

This operation effectively introduces a leap month every 342 Metonic cycles.

This gives a month:year ratio of:

19*342 : 235*342-1

One need not wait several thousand years for this device to be useful. The distance between the small peg and the mark is a measure of how far the year and month are drifting with respect to each other. You can extend that precision by using the long peg. Perhaps short and long hands, clock style, should be used instead, since we're familiar with that.

It seems that by each tooth of the moon gear wheel there is a hole in which a peg may be placed and the pegs are moved once every Metonic cycle so completing a cycle once every 342 Metonic cycles after which the moon gear wheel is turned one revolution backward (or one revolution is cancelled).

I think a much better idea is the move the pegs annually and after each peg cycle make the sun wheel slip a tooth against the moon wheel.

Since making the wheels slip could damage them, one could instead turn the moon wheel forward 99 revolutions (1881 teeth), then the sun wheel would turn 8 revolutions (1880 teeth) and one tooth, so achieving the same effect as slipping the sun wheel. This makes it evident that this correction truncates a Metonic cycle by 8 years to 11 years.

Karl

10(08(25