NATIONAL UNIVERSITY OF SINGAPORE

DEPARTMENT OF MATHEMATICS

SEMESTER 2 EXAMINATION 2008–2009

GEK1506 Heavenly Mathematics: Cultural Astronomy

April/May 2009 — Time allowed: 2 hours

Matriculation Number:

INSTRUCTIONS TO CANDIDATES

(i) This examination paper contains five questions and comprises nine pages including thispage and a blank page at the end.

(ii) Write your matriculation number on the front page.

(iii) Write your answers on this question paper. You may also write on the back of the pages.There is a blank page at the end if you need more space. Please indicate clearly if youcontinue a question on the last page.

(iv) Explain and justify your answers carefully.

(v) Answer all questions.

(vi) You may use a calculator.

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Figure 1: Armillary sphere from the book “Marifetname” (Book of Gnosis) by the Turkish andSufi philosopher Ibrahim Hakki Erzurumi (1703 - 1780)

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1. [10 marks] Figure 1 shows a picture of an armillary sphere from the book “Marifetname”(Book of Gnosis) by the Turkish and Sufi philosopher Ibrahim Hakki Erzurumi (1703 -1780).

Match the numbers in the figure with the following objects.

• Earth

• Meridian

• Horizon

• Ecliptic

• Equinoctial colure (great circle through the equinoxes and the poles)

• Arctic circle

• Antarctic circle

• Tropic of Capricorn

• Tropic of Cancer

• Equator

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2. [20 marks] On the day of the December solstice, the Sun will cross the meridian in Oslo,Norway at 12:15 at an altitude of 6.5◦ in the South. The equation of time is about 2minutes on the December solstice and Norway lies in the UTC + 1 time zone and doesnot use daylight saving time in December. What is the latitude and longitude of Oslo?

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3. (a) [10 marks] The Hijra is the emigration of Muhammad and his followers to the cityof Medina in 622 CE and marks the first year of the Islamic calendar. Explain howyou can use this to give an estimate for what year it is in the Islamic calendar atthe moment. (It is enough to give an estimate. You do not need to use a calculatorto get an exact figure.)

(b) [5 marks] What time of the day does the Muslim day start?

(c) [15 marks] Assume for simplicity that a lunar crescent can be seen 24 hours afterthe new Moon. Assume that you live on the equator, that we ignore the equation oftime, that your longitude equals the longitude of the centre of your time zone andthat you do not use daylight saving time. Which days of the Chinese lunar monthcan the first day of the Muslim month correspond to? Estimate the probability thatit corresponds to the various days.

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4. In the Chinese lunisolar calendar, the solar year is divided into 24 solar terms or jieqı (��). They are a generalization of the solstices and equinoxes. Let us consider any one ofthe jieqı (��), for instance Qıngmıng (��).

(i) [5 marks] How will Qıngmıng move in the Gregorian calendar from one year to thenext?

(ii) [10 marks] Estimate the probability that Qıngmıng will fall on the same date in theGregorian calendar for four years.

(iii) [5 marks] How will Qıngmıng move in the Chinese calendar from one year to thenext?

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5. (a) [5 marks] In India they use both solar and lunisolar calendars. The Indian calen-dar most commonly used in Singapore is the Tamil calendar. How can you usethe calendar in Figure 2 to determine if the Tamil calendar is a solar or lunisolarcalendar?

(b) In the Wikipedia article about the Bengali Calendar, it says “In this calendar ... theyear number ... is divided by 39. If after the division the remainder becomes zeroor could be divided by 4, the year is then designated as a leap year and contains 366days ... . There are 10 leap years in every 37 years ... .”

(i) [5 marks] What is inconsistent in this statement?(ii) [5 marks] What is the leap year rule in the Gregorian calendar?(iii) [5 marks] Why is there a difference between the Gregorian and Bengali leap

year rules?

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