1. The Mathematics of Time A/P Helmer Aslaksen Dept. of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/

2. 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

3. 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

4. What is time? Year Month Day Hour Second What is not in the above list? Week Minute

5. 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 X ǐ ( 虞喜 ) ca 320 CE Why do Indian astronomers use sidereal year?

6. 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

7. 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”

8. Types of calendars Solar calendar Lunar calendar Lunisolar calendar

9. 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

10. 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

11. 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!

12. 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

13. 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

14. 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

15. 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

16. 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

17. Analemma picture

18. Rising and setting analemma

19. 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

20. Eccentricity

21. Why does obliquity matter?

22. Obliquity

23. Combined effect

24. Analemma

25. Globe

26. Globe with analemma

27. 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

28. 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