2. Some Astronomically Based Calendars
• Egyptian (solar)
• Gregorian/Julian (solar)
• Islamic (lunar)
• Hebrew (lunisolar)
• Babylonian (lunisolar)
• Hellenistic (lunisolar)
• Indian (lunisolar)
• Chinese (lunisolar)
3. Astronomical Cycles as Basis for
Calendars
• Days: Most cultures have day names named after
the wandering stars visible to the naked eye:
– Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn
• Months: Originally tied to lunar phenomena
(lunar cycles, or lunations)
• Years: Usually tied to solar phenomena
(equinoxes, solstices, seasons)
• Lunar calendars – hunter-gatherers, nomads
• Solar and lunisolar calendars – agricultural societies
4. Irrationality of time
• Month/day, year/day, month/year all irrational numbers –
No exact representation as M/N : M, N = integers.
– Can be approximated by bigger M, N (ancient math) or
more decimal places (modern math).
→ ALL calculated (non-real-time) calendars contain some
non-periodic or non-cyclical (cumulative or secular) drift.
• Lost world of precise astronomical time: Large sundials, in later antiquity,
could resolve fraction of minute. Apart from the occasional high-quality
water clock (clepsydra), the ancients lacked decent nonastronomical
clocks. Modernity has freed us from astronomical time.
• Long baselines needed for naked-eye astronomy, to get precise angular
measurements. For example, Temple Mount – apparently eastern gate of
Temple aligned to sunrise at vernal equinox, just before Passover.
5. The ancient gods above and
the lights in the sky
“The ancients gave to the gods the heaven or
upper place, as being alone immortal; and our
present argument testifies that it is
indestructible and ungenerated.” (II.1)
Aristotle, On the Heavens
6. In the siddur …
• Kaddish: Oseh shalom bim’romav, hu ya’aseh
shalom aleinu ….
• M’romav = heavenly heights
• Shalom bim’romav = harmony of the celestial
spheres
• Common ancient thought: “As above, so below”
– Christians: Lord’s Prayer
– Stoics: macrocosmos = microcosmos
7. Telling time the modern way
Clocks
• Mechanical (1300s)
• Electric motors (1870s)
• Electronic crystals (1890s)
• Atomic and maser/laser clocks (1950s)
8. Telling time the ancient way
Sundials & gnomons
• Small, even pocket
• Medium: Brookline, Summit Park
• Large: Can discriminate to minute or better
• Key to pre-telescopic astronomy – Bigger is
better for naked-eye observatories like
Stonehenge, Central Asia
10. Predicting Seasons
• Sun crosses the celestial equator (extension of our equator on
to celestial sphere)
• Equinoxes - Days are 12 hours each everywhere on Earth
• Much harder to track than solstices!
• Solstices - First day of Summer and Winter
- High and low points of the analemma
• If Earth weren’t tilted (23.5 degrees),
crossover would be at equinox (first day
of spring and autumn)
• Instead crossovers are April 15th and
August 29th
Analemma
11. Egyptian Calendar
• Among the oldest in recorded history – 5,000+ years old
• Solar calendar: 365 days/year
• 12 months, with three 10-day-long weeks
• 5 days added at the end of last month to fill up
final days that didn’t fit 360 = 12 x 30.
• Egyptians used Sirius, the Dog Star, to
mark the new year at first appearance in
predawn sky.
• Brightest star in the sky
• Announced the Nile flood season in summer.
12. Egyptian Calendar
• After Sirius, Egyptians recorded 35 other stars
(decans) to mark the beginning of each week.
• Used the decans to record the hours of the night
• One might expect that half (18 decans) are visible
any given night.
• Because New Year was near summer solstice, only
12 decans subdivided the night.
• Therefore the length of the night was established to
be 12 hours.
13. Egyptian Calendar
• Equinox to equinox, the Earth goes around the
Sun in 365.242 189 7 days (tropical year).
• Egyptians had no leap year. But they were
always calibrating the calendar with latest
pre-dawn (heliacal) rising of Sirius in early
summer.
• So calendar re-set observationally every year
and didn’t drift too much over the centuries.
14. Julian Calendar
• Solar calendar: 365 days/year
• 12 months = 10 archaic Roman months, plus later
additions of July & August (Julius & Augustus Caesar)
• Days start at midnight = change of watch in Roman
army.
• Based on Egyptian pharaonic calendar
• Started January 1, 45 BCE. Brought from Egypt by
Julius Caesar, devised by astronomers at Alexandria
and presented by Cleopatra.
• Basis for all later dating and calendars in the Roman
empire, Eastern and Western Christendom, and now
whole world. Still used by astronomers.
15. Julian Calendar
• Unlike Egyptians, Julian calendar added a day for leap
years - once every four years.
• But drift still happened, most notably with Easter.
• When first adopted, first day of Spring was March
21st (correct). But by 16th century, Spring equinox
was March 11th (wrong).
• Easter happens on the 1st Sunday after the 1st full
Moon on or after the 1st day of spring.
• Without a fix, Easter would eventually slip to
summer!
16. Gregorian Calendar
• Need to fix the problems with Julian calendar,
which was too simple.
• Authorized in 16th century by Pope Gregory XIII
• Designed by Neapolitan astronomer Aloysius Lilius
• Combines modified Julian (solar) calendar with
lunisolar calendar (epacts) for Easter = supposed
to be the Sunday in Passover.*
* But Christian calendar rules not exactly the same as rabbinic.
17. Gregorian Calendar
Fixes came with new rules for leap years:
1) Extra day on years divisible by 4, except …
2) No leap day on “century” years, except …
3) Years divisible by 400 are leap years!
– 2000 was a leap year.
– 1900 was not.
– 2100 won’t be.
Net result: average solar year over four centuries:
365-1/4 days – 3/400 days = 365.242 5 days
Actual tropical year = 365.242 189 7 days
18. Gregorian Calendar
• Still needed one-time 10-day shift to eliminate
cumulative drift.
• So: October 4, 1582 was followed by October
15, 1582
• Not everyone accepted the fix immediately.
• Great Britain and British colonies in the New
World didn’t switch over until 1752!
– Familiar to history as O.S. and N.S. (old/new
style dating).
19. Islamic Calendar
• Lunar calendar –
Basis of Islamic crescent symbol
• 12 months/year (each 29 or 30 days)
• 7 days/week, with days starting at sunset
• Started with hijri (Hejira), 622 CE
Simplicity of Islamic calendar – Lunar only,
mainly observational, avoids complexity of
calculated calendar.
20. Islamic Calendar
• Lunar calendars wander!
• Lunar month = ranges 29.18 to 29.93 days
– 29.53 day average (synodical period, over same point on
Earth’s surface ≠ sidereal period, against “fixed stars”)
• Any given month, ±9 hours of potential variation
• On average, the lunar cycle slips 10.6 days compared
to the solar year.
• About 354 days per lunar year (mean = 354.366d)
• 12 lunations in a solar year
22. Islamic Calendar
• Islamic calendars require
real-time observation of the
thinnest crescent moons.
• Muslim astronomers
compete to be the first to
identify the youngest
moons possible.
• Current records:
– 11h 40m (telescope-aided)
– 15h 32m (unaided eye)
after 0h 0m = new moon
23. Babylonian Calendar
• Rise of (neo)Babylon (612-540 BCE)
• Days start at sunset.
• Lunisolar: Months are 29 or 30 days.
• Based upon the Metonic Cycle:
19 solar years ≈ 235 lunar months
– Off by 6.5 minutes/year
• Other cycles exist too. Better ones!
• 334 solar years ≈ 4131 lunar months
– off by 3 minutes/year
• Hindus take it even further: 180,000 solar years ≈ 2,226,389 lunar months
– off by 8 seconds/year
24. Babylonian & Hebrew Calendars
• Neo-Babylonians (Chaldeans) were great astronomers.
• Knew how to predict eclipses and track the lunar cycle to determine when
new moon would occur.
• First sighting of the crescent new moon marked first day of month.
Basis of post-exilic Hebrew calendar
• Like neo-Hebrew (Aramaic) letters, this calendar adopted by Judeans in exile
in city of Babylon after First Temple was destroyed. Need for systematic
calendar in Diaspora. Smooth continuation of pre-exilic months of First
Temple period, with some changes.
• New year moved from spring to fall: biblical Aviv = Nissan → Tishrei
• Standard sundials of later antiquity could distinguish fine fractions of an hour
~ O(few) minutes or daqot (dq = 60 s), possibly somewhat better.
• Precise risings and settings of bright guide stars
• High-quality water clock precise to O(few)− O(10) ḫalaqim (hq = 3-1/3 s)*
• Ḫodesh of Aviv means “first new moon of spring.”
* Singular ḫeleq. 1 hour (sha’ah) = 1080 ḫalaqim. Neo-Baylonian units.
25. Hebrew Calendar
• Ha-luach ha-ivri = Lunisolar calendar
• Needed to keep holidays in line with solar seasons
and with lunar cycles.
• Passover = first full Moon after spring equinox
• Rabbinic mean lunar month = 29d 12h 793hq
• Made official in 359 CE by Hillel II, who converted
a real-time, observational calendar to a calculated
calendar.
War, persecution, desolation, dispersal made it
difficult to maintain real-time system and
transmission network.
(Maimonides, comments to Mishneh Torah)
26. Hebrew Calendar
• Before destruction of Second Temple, the calendar
was halakhically observational: Two witnesses
needed to confirm the sliver of the new moon
(nolad, or new-born) in Temple courtyard.
• Holidays were authorized by a beit din and
announced by word of mouth and bonfire signals.
• Uncertainty (clouds, potential 18-hour variation in
lunar orbit) and delay of transmission are why we
have two Seders and other 2-day holidays in the
Diaspora.
(Exceptions: Rosh Hashanah is two days in Israel too, and
Yom Kippur is one day for everyone.)
27. Hebrew Calendar
• 12 or 13 months/year
• 353–355 days or 383–385 days
• Based on the 19-year Metonic cycle
• To correct for errors, leap years occur seven times
every 19 years: 0, 3, 6, 8, 11, 14, 17, 19 = 0.
• Leap years add an extra, 13th (intercalated)
month in late winter, Adar I. “Real” Adar is Adar II.
• Common to add leap months or days in late winter,
because beginning of spring so often the new year.
(B. Sanhedrin 1-2)
• Cf. Roman/Julian February 29th leap day also late winter
28. Hebrew (neo-Babylonian) months
• Nissan (Nisanu) 30 days • Tishrei (Tashritu) 30 days
• Iyar (Ayaru) 29 days • Cheshvan (Arakhsamna) 29 or 30 days
• Sivan (Simanu) 30 days • Kislev (Kislimu) 30 or 29 days
• Tammuz (Du‘uzu) 29 days • Tevet (Tebetu) 29 days
• Av (Abu) 30 days • Shevat (Shabatu) 30 days
• Elul (Ululu) 29 days • Adar (Adaru) 30 or 29 days
• Adar II (leap years) 29 days
29. Hebrew Calendar
Four different new years in Hebrew calendar:
• Nissan/Aviv — Kings and festivals (spring)
• Elul — Tithing (late summer)
• Tishrei — Counting years (early fall)
• Shevat — Planting and harvesting trees
(mid-winter)
(M. Rosh Hashanah 1:1)
30. Hebrew Calendar
Calendar falls into three types, depending on
when Rosh Hashanah falls:
• A day is taken from Kislev (353 or 383 days):
deficient or chaserah
• Normal year (354 or 384 days):
regular or k’sidrah
• A day is added to Cheshvan (355 or 385 days):
complete or shleimah
31. Hebrew Calendar
Rosh Hashanah is always 12th (or 13th) new moon after
the previous RH, unless one of four exceptions:
1) Need to round up for sunset (18 hours into the day).
2) If RH falls on a Sunday, Wednesday, or Friday:
– a. Keeps Yom Kippur from falling on a Friday or Sunday.
– b. Keeps Hoshanah Rabbah from falling on Saturday.
3) If adding a day would make the year 356 days, the
year is shortened to 354 instead (about every 30 years).
4) If leap year ends up with 382 days, another day is
added (about every 186 years).
32. Accuracy of Calendars
• Hebrew calendar is the most accurate of the
ancient calendars, but also the most complex:
three-cycle scheme (day, month, and year).
• Hebrew calendar — Year increases about 1 day
every 1176 years with respect to true solar year.
• Since 359 CE, a solar year drift of about 1.4 days –
Passover will be a month off after 34,727 years!
• Month about 13 times more accurate than year.
• Gregorian calendar — Modified Julian calendar
drifts about 1 day every 3223 years — Benefits
from additional centuries of careful observation
and from simpler two-cycle scheme (day + year).
33. Roads Not Taken:
Qumran Solar Calendar I
• Qumran sect = Dead Sea Scroll sect – almost certainly
• DSS sect = Essenes mentioned by Josephus and Pliny the Elder – probably
• Damascus Document (1897, Solomon Schechter)
Dead Sea Scrolls (1947-1990s)
• Submerged Sadducee (Tzeduqi) tradition ended by destruction of Second
Temple in 70, transmitted partly by Karaites. Sadducees: aristocratic party of
high priests.
• Essenes/DSS sect = Sadducees? No, but related by opposition. Dissident
group self-exiled into poverty in desert from Jerusalem after Hasmoneans
took office of high priesthood (kohen gadol) from Tzadokite line c. 160 BCE.
• DSS tell us about this sect, but also indirectly about Sadducees in Jerusalem
and the Pharisees, back into 200s BCE and earlier!
• And Boethusians (Baytusim) = related sect, or subsect – Spoofing the new
moon with false witnesses (M. Rosh Hashanah). Origins obscure.
34. Roads Not Taken:
Qumran Solar Calendar II
• Solar calendar in DSS and Book of Jubilees
• Omer count always starts on Sunday following Pesach.
Shavuot always on Sunday seven weeks later. Pesach always
on Tuesday. Year and quarter always start on Wednesday
(fourth day, creation of lights in sky).
• 364 days per year = 52 x 7 = 4 x 91 = 4 x 13 x 7
• In deficit with respect to solar year by a little less than
5 days/4 years; with respect to lunar year by about 10 days.
• DSS/Qumran/Essene sect did observe lunations as well, not
entirely clear why.
• How did these sects justify solar calendar on biblical basis?
Mystery! Books of Jubilees and Enoch provide a rationale.
35. Astronomical Time: The Essentials
• Solar day:
• Synodic lunar month: Along a line connecting Earth
and Sun, marks phases of moon → Range =
29.18 to 29.93 solar days, with mean lunation of
29.530 587 981 solar days (29d 12h 44m 3s) →
Lunar month between 29 and 30 solar days
• Solar (tropical) year = 365.242 189 7 solar days =
365d 5h 48m 45s = 12.368 266 8 mean lunar months
→ Solar year between 12 and 13 lunar months
* Earth’s rotation slowing from tidal friction with Moon
Ephemeris solar day (1889) ≡ 24h 00m 00s = 86,400s
Mean solar day (1960+) 86,400.002s (+2ms)*
True solar day ±16m over seasons
36. Credits
• Calendrical Calculations, 3rd edition, E. M. Reingold &
N. Dershowitz (2007)
• The Sun in the Church, J. L. Heilbron (2001)
• Reclaiming the Dead Sea Scrolls, L. Schiffman (1995)
• Engineering in the Ancient World, J. G. Landels (1977/2000)
• The Exact Sciences in Antiquity, O. Neugebauer (1969)
• Maimonides, Mishneh Torah, Sefer Zemanim: Kiddush
ha-Ḫodesh (epoch: Passover 1178 CE)
• B. Rosh Hashanah 1-3; B. Sanhedrin 1-2 & gemara ibid.
(late 1st century – early 6th century)
• David Tytell, former associate editor of Sky & Telescope, for
the original version of this talk (2006)
• Dennis di Cicco, Roger Sinnott, Rick Feinberg, and the staff
of Sky & Telescope magazine
• Youssef Ismail (www.organiclightphoto.com) for permission
to use the beautiful new Moon image for Islamic calendar
