Meton's cycle

Show why 19-years cycle is the best. Calculate optimal distribution of the leap years.

- 1:2 Solar year is ~11 days longer than the lunar (see chapter 6); when discrepancy accumulates to ~30 days, year is made leap. - 4:1-2 Leap year - additional Adar; Pesach (15th of Nisan) must be not before the vernal equinox (Sun enters Aries, see 9:3) and before the next summer solstice.

Moznaim: Based on Sanhedrin 13a-b, the Ramah (cited in Kessef Mishneh on law 15): "Succos must be after the autumnal equinox"; Rambam doesn't mention this. With vernal equinox on Nissan 15th, autumnal could be on Tishrei 21st, 6 days into Succos. Ohr Sameach: Rambam did say this, printers missed it. Aruch HaShulchan: Rambam holds that that Sanhedrin isn't law; we only care about vernal equinox.

QUESTION: what are the properties of the fixed calendar in this respect (need true seasons)?

- 6:3 Lunar month (between conjunctions) = 29d12h793p. - 6:4 Lunar year (12 lunar months) = 354d8h876p = 354d8h48m40s; Leap lunar year (13 lunar months) = 383d21h589p; Solar year = 365d6h (see also 9:1, 10:6); longer than the lunar year by 10d21h204p. - 6:10 19 year cycle (12+7 leap); because the difference between 19 solar years and the cycle is less than a day: 1h485p. (see 10:1 - with a different year length, the difference is 0)

QUESTION: what other lengths of the cycle have this property?

- 6:11 Leap years: 3,6,8,11,14,17,19.

QUESTION: in what sense is this an optimal correspondence? and for different cycle lengths? It is possible that true Sun calculations are needed to answer those questions...