This document provides an overview of ancient calendars and how they were based on astronomical cycles of the sun, moon, and stars. It discusses key features of early calendars like the Egyptian, Julian, and Gregorian calendars. It also summarizes lunar calendars like the Islamic and lunisolar calendars like the Babylonian and Hebrew calendars. The document explores how calendars have evolved over time to account for the irregularities in astronomical cycles and seasonal changes.

1. Calendars in
the Sky
Looking Up
at Time
Presented by Dallas C. Kennedy
for Framingham Public Library,
McAuliffe Branch
July 2019

2. Astronomical Cycles as
the Basis for Calendars
• Days: Rotation of the Earth on its axis.
Most cultures have days named after the wandering
stars visible to the naked eye:
Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn
• Months: Originally tied to lunar cycles, or lunations
• Years: Usually tied to solar phenomena and Earth’s
orbit around Sun (equinoxes, solstices, seasons)
• Lunar calendars – hunter-gatherers, nomads
• Solar and lunisolar calendars – agricultural societies
• Calendars defined by periodic phenomena of the sky

3. Astronomical Time: The Essentials
• Solar day:
• Synodic lunar month marks phases of moon along a
line connecting Earth, Moon, and Sun →
Range = 29.18 to 29.93 solar days,
with mean lunation of 29d 12h 44m 3s
• Tropical solar year = 365.242 189 7 solar days
from one spring equinox to next = 365d 5h 48m 45s =
12.368 266 8 mean 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

4. Why Do We Have Seasons?

5. Predicting Seasons
• Equinoxes - Days are 12 hours each everywhere on Earth
• Sun crosses the celestial equator, the extension of our
equator on to celestial sphere.
• 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
of weekly noons

6. Major Astronomically Based Calendars
• Hebrew
• Hellenistic
• Indian
• Chinese
... all perfected in later geocentric, premodern astronomy
Solar
Egyptian
Julian/Gregorian
Lunar
Islamic
Lunisolar
Babylonian
Influence of
lunar theory
Influence of
lunar theory

7. Telling time the modern way
Clocks
• Mechanical (1300s)
• Electric motors (1870s)
• Electronic crystals (1890s)
• Atomic and maser/laser clocks (1950s)
… Modernity has freed us from
astronomical time.

8. Periodic Motion and
the 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: In later antiquity, large sundials
could resolve fraction of minute. Apart from rare high-quality water clocks
(clepsydra), ancients lacked decent nonastronomical clocks.
• Long baselines needed for naked-eye astronomy, to get precise angular
measurements. Elevated religious platforms (acropoleis). Stonehenge
aligned with Moon. Temple Mount in Jerusalem – eastern gate of Temple
aligned to sunrise at solar vernal equinox, just before Passover.

9. Telling time the ancient way
Sundials & gnomons – Water clocks
• Small, even pocket
• Medium:
Summit Park, Brookline … Ecotarium, Worcester
• Large: Can discriminate to a minute or better
– Supplemented by precise risings and settings of bright guide stars
– Also by high-quality water clocks precise to O(few − 10) seconds
• Key to pre-telescopic astronomy – Bigger is
better for naked-eye observatories.
Stonehenge, Central Asia, Central and South America

10. Egyptian Calendar
• 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.
• No leap year. But calendar always calibrated
observationally by with latest pre-dawn
(heliacal) rising of Sirius in early summer.

11. Egyptian Calendar
• After Sirius, Egyptians recorded 35 other stars
(decans) to mark the beginning of each week and 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. Probable origin of 12-hour half-day
adopted by Romans.

12. 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, last Greek monarch of Egypt.
• Basis for all later dating and calendars in the Roman
empire, Eastern and Western Christendom, and now
whole world. Still used by astronomers.

13. Julian Calendar
• Unlike Egyptians, Julian calendar added a day for leap
years - once every four years. Smaller but still
significant drift remaining.
• When first adopted, first day of spring was March
21st (correct). But by 16th century, spring equinox
was March 11th (wrong).
• Easter supposed to happen 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!

14. Gregorian Calendar
• To fix the problems with Julian calendar, which
was too simple … Pope Gregory XIII authorized
a new calendar in the 16th century.
– Designed by Neapolitan astronomer Aloysius Lilius
• Combines modified Julian (solar) calendar with
lunisolar calendar (epacts) for Easter = supposed to
be the Sunday in Passover.*
• One-time 10-day shift to eliminate cumulative drift
– October 4, 1582 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)

15. 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

16. Islamic Calendar
• Lunar calendar – Basis of Islamic crescent symbol
• 12 lunar months or lunations per year
(each 29 or 30 days)
• 7 days/week, with days starting at sunset
• Leap years: +1 day 11 times every 30 years
• Started with Hejira (622 CE)
Simplicity of Islamic calendar – Lunar only, mainly
observational, avoids complexity of calculated calendar.

17. How Lunar Cycles Work
• 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
• Lunar year = 12 lunar months
Mean lunar year = 354.366d
• On average, lunar year slips 10.6 days compared to
the solar year.

18. 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

19. Babylonian Calendar
• Rise of (neo)Babylon (612-540 BCE)
• Lunisolar: Months are 29 or 30 days.
• Days start at sunset.
• 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

20. Babylonian Calendar & Spread of
Old World Lunisolar System
• Neo-Babylonians (Chaldeans) were great astronomers. Tracked
lunar cycles to predict eclipses and new moons.
– First sighting of crescent new moon marked first day of month.
– Babylonian origin of system of 60 seconds, 60 minutes, and 360
degrees. Precise risings and settings of bright guide stars.
– High-quality water clock precise to O(few − 10) ḫalaqim*
• Basis of post-exilic Hebrew calendar and influence on Asian
calendars in India and China. Spread by Persians and Alexander
the Great, along with Babylonian astrology.
– Babylonian lunar theory also appears in Islamic system.
• Development of a common lunar theory, tropical solar year, and
precise lunisolar system throughout much of the Old World.
* Singular ḫeleq (hq) = 3-1/3 s. One hour (sha’ah) = 60 minutes (daqot) = 1080 ḫalaqim.
Key neo-Babylonian units, stated here in Hebrew.

21. Hebrew Calendar
• Needed to keep holidays in line with solar seasons and
with lunar cycles.
• Pesach (Passover) = first full Moon after spring equinox
• Sukkot (Booths) = sixth full Moon after Passover
• Rabbinic mean lunar month = 29d 12h 793hq
• Made official in 359 CE by conversion of a real-time, observational
calendar to a calculated calendar, which it remains today.
• 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 (year mod 19)
• Leap years add an extra, 13th (intercalated) month in late winter.
Common to add leap months or days in late winter, because beginning of spring so
often the new year. Roman/Julian February 29th leap day also late winter.

22. 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).

23. Calendars and
the Geocentric Era
• Aquinas & Dante: Revival of Aristotle
• Geocentrism in 1300s: Earth =
Bottom of the cosmos (not “center”)
• Looking up at what is “more eternal” until we
reach quintessence and the “fixed stars,” beyond
the “seventh heaven” of Saturn
• All the calendars in use today arose in the
geocentric era, as last, greatest, and most
precise applications of pre-telescopic astronomy.

24. The past is a different country ….
The ancients thought differently
about the sky
• Common ancient thought: “As above, so below”
– Christian: Lord’s Prayer (Our Father)
– Hebrew: oseh shalom bim’romav
– Stoic: macrocosmos = microcosmos
– China: mandate of heaven
• Harmony of the spheres
• Orderly heavens = Model for religion, philosophy,
and science. Profoundly important in history of
human thought and culture.

25. 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.”
Aristotle, On the Heavens (II.1)

26. Credits
• Calendrical Calculations, 3rd edition, E. M. Reingold
& N. Dershowitz (2007)
• The Sun in the Church, J. L. Heilbron (2001)
• Engineering in the Ancient World, J. G. Landels (1977/2000)
• Newton’s Clock: Chaos in the Solar System, I. Peterson (1995)
• The Exact Sciences in Antiquity, O. Neugebauer (1969)
• The Discarded Image, C. S. Lewis (1964)
• 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