# Heavenly Mathematics & Cultural Astronomy (RI/RGS - NUS Advanced Mathematics Module Courses)

 Adam Schall (汤若望 [湯若望], Tāng Rùowàng, 1591-1666), Imperial Astronomer in Beijing. Designed the current Chinese calendar Astronomers at the Istanbul Observatory

## Objectives of the Module

The goal of this course is to study astronomy and mathematics in a cultural context. We will look at questions like: How is the date of Chinese New Year determined? Why do the Muslim and Chinese months start on different days? Will the Moon ever look like it does on the Singapore flag? What date of the year is the earliest sunrise in Singapore? How did ancient sailors navigate?

After taking this course you will become conscious of the motion of the Sun and the Moon and notice and question things you have earlier taken for granted. You will appreciate mankind's struggle through the ages and throughout the world to understand the mathematics of the heavens.

## Topics to be Covered

We start by discussing the motion of the Sun and the Moon. Most astronomy books are written from a high northern latitude point of view, but in this course we will take a "hemispherically correct", tropical point of view. We then look at applications with a cultural flavor. In Singapore we use the Gregorian, Chinese, Islamic and Indian calendars for determining the public holidays and you will learn the rules for these calendars. Most people expect the time of sunrise and sunset in Singapore to stay fixed throughout the year, but we will study the equation of time and the analemma to see why there is a half hour variation.

## Practical Information and Assessment

### Classes

There will be five lectures and an exam. The dates are 31 July, 14, 21, 28 August 2010, 18 and 25 September 2010 and the time is 1.30 to 4.30pm, except for August 14 when we will start at 12:30 and finish at 3:30.

I will lecture for 1:30, take a 20 minutes break, lecture for 30 minutes more and have a 40 minutes tutorial.

I use a cordless microphone and walk around in class and ask questions. But don't worry, I only ask easy questions! I also like to create physical demonstrations to illustrate the concepts, and I often need “volunteers” for this. I am not afraid of looking silly, and I hope you are not either!

## Recommended Texts

Unfortunately, there's no textbook for this course. Maybe I'll try to write one myself? The closest is The Ever-Changing Sky, A Guide to the Celestial Sphere by James B. Kaler. However, it goes into a lot more detail about astronomy than we will cover. The level of astronomy I expect you to learn is comparable to the first chapter and the appendix of The Copernican Revolution, Planetary Astronomy in the Development of Western Thought by Thomas S. Kuhn. For convenience, the main reference for the first part will be Strobel's Astronomy Notes on the web.

I have compiled a list of additional references. In addition to the books by Kaler and Kuhn, I've also placed The History and Practice of Ancient Astronomy by James Evans and Calendrical Calculations: The Millennium Edition by Nachum Dershowitz and Edward M. Reingold on RBR in the Science Library.

## Astronomical Java Applets and Animations

Together with Tey Meng Khoon and Frederick H. Willeboordse of CITA (Centre for Information Technology and Applications), I have developed interactive Java applets that I hope will help you understand the motion of the Earth, the Sun and the Moon.

You will notice that a lot of the applets are somewhat similar. They show the same thing, but we're trying to emphasis different aspects or points of view in each one.

You can get the Java Runtime Environment (JRE) at java.com. If you prefer to view the applets off-line, you can download them. Just unzip astro-applets.zip, and open astro-applets.html in your browser.

The last two applets are related to What Does the Waxing or Waning Moon Look Like in Different Parts of the World?

## Video Clips

Together with Keith Phua Kuan Wee of CITA (Centre for Information Technology and Applications), I have recorded video clips to help you understand the material. The clips are available in three formats:

• RealVideo SureStream, click the link to stream. The quality and file size of the clip is determined by the preference settings in your RealPlayer.

I have recorded video clips to explain the Java applets and Strobel's Astronomy Notes.

## Strobel's Astronomy Notes

Nick Strobel of Bakersfield College has written a wonderful set of Astronomy Notes for his introductory astronomy course. His chapter on Astronomy Without a Telescope is very relevant background for this module. I have recorded video clips with annotations to some of the sections of the first chapter of Strobel's notes.

Here is the table of content for the first chapter of Strobel's Astronomy Notes with links to my video clips.

## Stellarium Software

I recommend the wonderful freeware Stellarium Astronomy Software (or from Sourceforge) available for Windows, Linux/Unix and MacOSX. I have written a Help Page.

## Astronomy and its History

How can we use the celestial sphere to understand the movement of the Sun, Moon and stars across the sky? Observing the sky has always been an important part of human civilization. We give a summary of some basic facts from spherical astronomy. One of my students, Viduranga Yashasui Waisundara, has written a beautiful poem about this, called If I were God.

There are a lot of Myths about the Copernican Revolution. Many people whose main interest is philosophy of science, but with little knowledge of astronomy or history of science, have written extensively about it. Unfortunately, much of what they say is incorrect.

 Tycho Brahe (1546-1601)

## Calendars

How is the date for Chinese New Year, Eid ul-Fitr (Hari Raya Puasa) and Deepavali determined? Singapore is unique in that we use four different calendars for determining the date of the public holidays. Seven of our eleven public holidays move, and we study the rules for Chinese New Year, the two Eid's (Hari Raya's), Deepavali, Vesak Day and Good Friday. The exact rules are very complex, but the basic ideas are simple. We give two simple rules of thumb that determine the date for Chinese New Year with a margin of error of one day. I have written a set of lecture notes on The Mathematics of the Public Holidays of Singapore.

You will also see why both Eid ul-Fitrs (Hari Raya Puasas) in 2000 were celebrated one day earlier than they would have been if MUIS had relied on sightings or scientific criteria.

Why is it difficult to determine the Islamic prayer times in Northern Europe? This is an interesting astronomical problem, which very few people understand.

What are leap seconds? This requires understanding some delicate points in the definition of atomic time.

 Chinese astronomers determining the summer solstice

### General Calendar Info

 Chinese sundials from the Adler Planetarium

### The Chinese and other East Asian Calendars

 MoonCalc image

### Qibla - the direction to Mecca

Determining the direction to Mecca was together with determining the first visibility of the lunar crescent and computing prayer times one of the central problems in Islamic astronomy. It requires a clear understanding of spherical trigonometry and knowing how to find longitude. However, you would think that at the moment it would be a trivial computation. I was therefore quite surprised when I received e-mails from Muslims in the US and Canada who started asking me about it. I realized that it is a major controversy among Muslims in North America. All the famous Muslim astronomers and mathematicians in the past, like al-Khwarizmi (780-850), al-Battani (858-929) and al-Tusi (1201-1274) agreed that qibla should be measured by great circle. However, some people who are used to looking at Mercator maps and don't understand great circles argue that you should draw straight lines on a Mercator map, i.e., follow rhumb lines. They strongly attack people who follow the traditional method of great circles. For examples of this, see Direction of Al-Qiblah and Qiblah.us. For a scientifically correct discussion, please see moonsighting.com.

## Observing the Sun and the Moon from Different Parts of the World

This is one of my favorite topics. I have some lecture notes about this, which I'm trying to expand on. Most astronomy books are written from a “high-northern-latitude-centric” point of view. I will discuss the motion of the Sun and the Moon from a “hemispherically-correct” point of view, with special emphasis on the needs of “latitudinally-challenged” observers.

I have a number of web pages about astronomical topics from a Singaporean point of view.

Some of my students have contributed greatly to this. One of my favorites is the following e-mail from one of my students, LIEW Wen Hwee.

Subject: The theory of hanging clothes to dry
Date: Wed, 18 Dec 2002 12:44:51

Hi Sir,

Ohh my. I just discovered the theory behind my mum's “theory of hanging clothes to dry”.

My mum have been telling me since I was a kid that for half of the year that she will have to hang the clothes out in the kitchen and the other half of the year in the corridor for them to dry. I was perplexed by her “theory” for a long time. I thought “wasn't the sun suppose to rise East always?”

And your lesson cleared my doubts!!! My flat lies exactly (ermmm maybe not that exact. didn't notice tat until I used the compass. Was almost scared by a faulty compass that points “west” in the direction where the sun rises!!!) in the East- west plane. And so for half of the year, the sun be in the South-east and the other half in the North-east. MY MUM IS AN ASTRONOMER !!!(ha.).

Sir, what u have taught is of practical purpose!!! (Maybe u could tell the next batch of students abt tat and they could advise their mum on how to better dry their clothes. =))

Thanks a lot, Sir!!!

## The Equation of Time

I have a separate web page on Which Day Does the Sun Rise Earliest in Singapore? Singapore lies almost on the equator, so most people would expect the Sun to rise at more or less the same time each day of the year. In fact, the sunrise time varies between 6:46 a.m. and 7:17 a.m., with the earliest sunrise on November 1 and the latest on February 9. In the same way, the sunset time varies between 6:50 p.m. and 7:21 p.m., with the earliest sunset 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. The key to understanding this is the analemma, which is a graphical representation of the equation of time.

 Analemma picture by Dennis di Cicco

The famous picture by Sky & Telescope's Dennis di Cicco records the Sun's position in the sky at the same time of day on 45 different dates throughout the year.

This is also related to the concept of time zones. As you can see from the map, Singapore and West Malaysia are in the “wrong” time zone! For more details, please see my page Why is Singapore in the “wrong” time zone?

 Sundial in Enschede, Netherlands, 1836, photo by Fer J. de Vries.

## Lecture Notes

The first part of the course gives a quick introduction to spherical astronomy.

• I have a brief set of notes on The Celestial Sphere. They are useful for the first part of the course.

The second part of the course is about calendars.

## Tutorials

 Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314) Armillary sphere at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314)

## References

I have a separate web page with references.

 Adam Schall (1591-1666), Imperial Astronomer in Beijing Armillary sphere

## Good-bye!

Congratulations! You've made it to the end of my page! Thanks for your patience! But before you go, it's time for a little quiz. In the picture of Adam Schall above, can you name the five astronomical and three geometrical instruments in the picture? Can you tell what is wrong with one of them? Hint: Look at the picture on the right. I have included answers.

I hope you have found something of interest. That makes my efforts worthwhile. Feel free to send me a message.

And remember, if life is hard, do like the sundial: Count only the bright hours!

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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