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The Mathematics of Time
A/P Helmer AslaksenDept. of MathematicsNational University of
www.math.nus.edu.sg/aslaksen/
What is the goal of this talk?
Understanding the Heavens was among the major problems in all civilizations
Appreciate some of the problems involved and see how different cultures attempted to solve them
Appreciate the centrality of mathematics to human civilization
What is mathematics?
In ancient times, there was no divisions between mathematics, physics and astronomy
Until around 1850, astronomy was part of applied mathematics before it became astrophysics
What is time? Year Month Day Hour Second What is not in the above list? Week Minute
Year Use heliacal rising (or setting) of stars to trace
the backward motion of the Sun relative to the stars along the ecliptic
Revolution of the Earth around the Sun Precession of the equinox Sidereal year (365.256 days) vs. tropical year
(365.242 days) Hipparchos ca 150 BCE and Yú Xǐ (虞喜 ) ca
320 CE Why do Indian astronomers use sidereal year?
Month Sidereal month (27.32 days) vs. synodic
month (29.53 days) Synodic month varies between about
29.27 and 29.84 days. Lunar months can start with the new
Moon (China and South India), the first visibility of the lunar crescent (Islam) or the full Moon (North India)
Lunar months are 29 or 30 days
Lunar visibility First visibility of the lunar crescent
is an “unsolvable” scientific problem
Different Islamic countries tries to solve it in different ways
Islamic criteria tend to be “optimistic”
Types of calendars Solar calendar Lunar calendar Lunisolar calendar
Solar calendars Gregorian calendar Basic unit is the day Approximates the tropical year by
adding leap days Ignores the Moon The year is 365 or 366 days
Leap days Since the tropical year is a bit less
than 365.25 days, we should add a leap day a bit less than ever fourth year
The Julian calendar had leap years every four years
Year n is a leap year if n is divisible by 4, but not by 100, or n is divisible by 400
Lunar calendar Islamic calendar Basic unit is the month Ignores the Sun The year is 12 months or 354 (sometimes
353 or 355) days Muslim holidays are not tied to the
seasons No Muslims use the arithmetical Islamic
calendar!
Lunisolar calendar Chinese and Jewish calendars Basic unit is lunar month Approximates the tropical year by
adding leap months The year is 12 or 13 months. A 12-
month year is 354 (sometimes 353 or 355) days. A 13-month year is 384 (sometimes 383 or 385) days
Holidays are tied to the seasons within a month
Leap months A lunar consisting of 12 lunar months is
about 12 x 29.5 = 354 days A lunar years is about 365-354=11 days
shorter than a tropical year Lunisolar calendars have leap years
about every third year The Metonic or zhāng ( 章 ) cycle: 19
solar years is almost exactly 235 lunar months
Seven leap months in 19 years
Chinese New Year Beginning of spring, lì chūn (立春 ), on
February 4 is halfway between winter solstice and spring equinox
Chinese New Year is meant to approximate the beginning of spring
New Moon closest to Feb 4 January 21 to February 21 Subtract about 11 days, but if that
would take us before January 21, add 19 days
The calendar in Chinese culture Why were there more than 100
calendar reforms? Foreign talent The current Chinese calendar was
designed by a German Jesuit No linear year count in the Chinese
calendar Needham dismissed calendrical
studies
Day Nychthemeron is the 24 hour day NOT the revolution period of the Earth! Sidereal day (23h 56m 4s) vs. solar day True solar day can be from 22s shorter
to 30s longer than 24hr Mean solar day Equation of time Analemma
Analemma picture
Rising and setting analemma
Sunrise and sunset in Singapore Singapore lies almost on the equator, so we would
expect the Sun to rise at the same time each day of the year
Sunrise varies between 6.46am and 7.17am, with the earliest on November 1 and the latest on February 9
Sunset varies between 6.50pm and 7.21pm, with the earliest on November 5 and the latest on February 13
The difference between the earliest and latest sunrise in 30 minutes, but the difference between the longest and shortest day is only 8 minutes
Ptolemy predicted, but could not measure this
Eccentricity
Why does obliquity matter?
Obliquity
Combined effect
Analemma
Globe
Globe with analemma
Hours The Egyptians used temporal (unequal)
hours that divided the day and night into 12 hours, whose lengths depends on the season
Hipparchos introduced equinoctial hours of equal length
Equal hours did not catch on until the fourteenth century
O’clock refers to equinoctial hours
Seconds Tidal friction slows down the Earth’s
rotation speed The changes in the Earth’s rotations
speed are not fully understood and cannot yet be predicted properly
Leap seconds are occasionally added at the end of the year to compensate
