Heavenly Mathematics & Cultural Astronomy (GEK1506)

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

Welcome to Heavenly Mathematics & Cultural Astronomy!

Course Content

Additional Information

Objectives of the Module

The goal of this course is to study astronomy 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.

This course is one of the General Education Modules at the NUS.

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Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

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 tropical point of view. 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. Navigation has been important to people all over the world throughout history, and people at different latitudes developed fundamentally different methods. If we have time, we will look at ancient astronomical instruments, astrology, cartography, archaeoastronomy (the use of astronomy in archaeology), cultural astronomy and astronomy in nature.

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Brian Greig's astronomical instruments for sale at the Melbourne Arts Centre Sunday Market

Practical Information and Assessment

Classes

There will be three hours of lectures and one hour of large group tutorial each week. The time slots in 2007/2008 Semester 2 are Monday and Thursday 6-8 p.m. in LT26. I take a 10 minutes break in the middle, and aim to finnish by 10 to the hour.

I'm only supposed to give three hours of lectures so on Thursday I will use the last hour as a tutorial starting in Week 3. Alternatively, you can come to tutorial on Wed 4-5 p.m. in LT 22. The first two weeks I use both for lectures because there is so much practical to cover.

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!

This semester I am trying out webcasting. The goal of the webcast if to give you a chance to take another look at things you didn't catch, or to help you if arrive a couple of minutes late. I do not think it will be possible to learn much from just watching the webcasts. I'm afraid my lecturing style is not very suitable for webcast. I actually feel quite sorry for the poor person who's doing it! I move around a lot, and I rely on physical demonstrations that probably will not show up in the webcast. I'm also not sure how well the sound comes out. Webcast is great for lecturers who read aloud from their lecture notes, which is not my style.

If it turns out that many of you treat the webcast as a replacement of the lectures instead of a supplement, I will stop  the webcasts.

Assessment

The final exam counts 70% of your grade. You have to do a project that counts 10%. The projects are done in groups of four to six students. There will be also be two homework that count 10% each.

The project proposal is due Monday 11/2. The project itself is due on Monday 7/4. The Moon homework is due on Monday 24/3 and the Sun homework is due on Monday 31/3. In 2007/08 Semester 2 the exam will be on Thursday 8/5 in the morning.

This year I'm going to try something new for the group assessment. Each group can either choose to have everybody get the same grade, or you can choose to grade the others in your group, by saying how many percent of the final mark the others deserve. If you manage to agree, I will follow what you agree on! Just gather everybody and come and see me. If you cannot agree, come and see me (separately, if necessary), and I will take the average. But I will not let that average count 100%. I will give it a weight of x%, and the uniform distribution will get weight (100-x)%. I will not tell you what x is, and I may even use different x for different groups. I encourage you to discuss this in a polite, friendly, reasonable, generous and compassionate way, and make every effort to reach a unanimous decision. My weightage may not be fair, so try to sort it out yourselves. At the same time, I want to have a way to protect people from unreasonable group mates. To sum up:

• If you are OK with everybody getting the same mark, then do nothing.
• If you can all agree on a non-uniform marking scheme, then come and see me.
• If you cannot agree, then come and see me with your individual schemes and I will average and adjust.

This the first time I'm trying this, so I don't know how it will work out. I hope you don't mind being guinea pigs. It's for the benefit of future students!

Groups

Please do both homework and project in the same group of four to six people. If you are planning to do a very special topic, and you're having a hard time finding somebody interested in it, I MAY also approve individual projects or groups of two or three. The chances of me approving such requests are best if you approach me early.

Don't worry if you have a hard time finding a group, you're probably not alone! Post a message on the discussion forum and you'll probably find other people in the same situation who you can form a group with.

Project

Many topics will only be touched upon in lectures, and you may explore them further on your own in the projects. I have a list of possible topics, but I also encourage you to propose your own topics and send them to me for approval. Follow the links from the web page. See if you find something you think looks interesting. Use the discussion forum to brainstorm.

I want you to have fun! I hope you can find a topic that interests you, and that you will enjoy finding out more about. I want to give you the opportunity to show your creativity.

I prefer projects with a cultural astronomy focus, but if you are passionate about a pure astronomy project, then that's OK, too.

I have a page with links to some past projects.

The project can be a paper project or a web page. I don't have any set rules about length or scope of the project.

Proposal

The point of the proposal is to let me check that the topic is suitable for the course and that you have enough material. I would like to see what sources you have found and hear a bit about what you plan to do. The proposal should include the title of the Project, the number of your group, the names of the members of the group, a brief outline, and a list of the main references. One or two pages is enough.

Submitting

Please submit the proposal, the project and the homework in both hard copy in class and soft copy in the IVLE workbin. I prefer to read the hard copy, so if you create a web page, please print out a hard copy, too. I realize that the print out may not do full justice to your web page, but it will give me time to read the text before I look at your page. If you have animations or other things that you can't print out, please include a note where you indicate which parts of the web site I should look more closely at.

I don't want to get 50 files in the workbin called project.doc. Please use the number of your group as the first two characters of the file name, followed by the title of your project or “Proposal” , “Sun Homework” or “Moon Homework”. For example “09 Celestial Navigation.doc”, “12 Proposal.doc” or “44 Sun Homework.doc”.

If the project is a web page and you have a server to put it on, you can just submit a file with the URL. However, I would appreciate it if you could also zip the files into one file and upload to the workbin or give me the files on a CD or bring a USB drive to class so I can transfer them to my PC.

How long should the project be?

I don't have fixed limits. It depends on the topic. If the mathematical content is high, the project can be shorter. If you have interesting computer animations, the text can be shorter. It is easy to make a very long project by copying material from the web. That will not impress me!

What makes a project good or bad?

• Keep it real! A lot of times I get very sophisticated projects about things that I don't think you knew anything about before you took this class. I usually quickly find that you have copied a lot of stuff from some standard source. I am much more impressed when I see that you have picked something that is somehow related to your life or interests, and try to link what you've learned in this class to it.
• There has to be some depth to the project. This is the university, not JC.
• You have to make sure that you have enough material. You need to find some good books or web pages that you can use as sources.
• I would like to use your project in my teaching. A good project is one that I can use as a reference for my students next year. So your presentation must be very clear. Don't repeat things you don't understand. If you don't understand them, your classmates probably won't either! A good project is one that takes an interesting topic and tries to explain it in simple terms. Copying things you don't understand is plagiarism. Making sense of confusing material is scholarship.
• Make sure you define and explain technical terms that haven't been defined in class.
• The web is great, but don't rely it on exclusively. There's a lot of crap out there!
• Be critical of controversial claims. Present different points of view. Debunking pseudoscience makes for great projects. Carl Sagan said that extraordinary claims require extraordinary evidence. If your project involves extraordinary claims, I would like to see some critical thinking and presentation of alternative views.
• Beware of astrology projects! I would love to read a project about why astrologers claim that an Aries has certain traits, but I'm not interested in a project that just tells me that an Aries has certain traits. So far I am afraid I have not liked any of the astrology projects I've received. :-(
• Make sure that the parts done by different members of the group integrate well. Don't use different fonts for different parts of the project for instance. Avoid repetitions.
• Don't worry about having a conclusion. Your job is to present different theories. I don't necessarily care what your view is.
• If possible, scan your pictures, instead of pasting them into the hard copy.
• Please include a cover and/or title page in the main doc file. Put your names and the group number on the title page or on the second page or the last page if you have a fancy title page.
• Don't submit PowerPoint files or posters. They are designed for oral presentations or poster presentations, not for written explanations.
• There is nothing wrong with doing a project about something that other people have written about before, but in that case you must show that you can do better than them. If you plan to do something like Stonehenge or the Mayan calendar you will have to do something quite original to impress me.
• I am skeptical of games or quizzes that is intended to help with rote learning.
• In the past, a lot of people made models. Unfortunately, most of them were not that interesting, and people spent too much time on them. If you have some very special idea, I may be interested, but in general I discourage models.
• Plagiarism is a big problem. I want clear references and citations to indicate where you get your material from. In particular, I want to know where you get all your graphics from. I want to know if the drawings, figures, animations and applets you include are your own or taken from somewhere else.

Homework

There are three goals of the Sun homework.

• Observe the setting (or rising) position of the Sun over the course of three months
• Estimate the date of the zenith passage around September 20 or March 24
• Study the changes in the shape of the shadow path of a vertical sundial over the course of three months

There are four goals of the Moon homework.

• Determine the day of the first visibility of the lunar crescent, i.e., the first day of the Muslim month
• Determine the day of the first daytime visibility of the Moon
• Determine the tilt of the Moon when it is a crescent close to the horizon
• Follow the phases of the Moon in the course of a lunar month

Here are some highlights from past homework.

E-mail

I get a lot of e-mail, so it would be helpful if you try to observe the following.

• If you are confused about something, chances are that a lot of other people in class are confused about the same. So why not post a question on the IVLE Forum? Then the answer will benefit everybody! And I won't have to answer the same question over and over again in individual e-mails! I know that some of you may not feel comfortable about asking a “stupid” question, but trust me, most of your class mates will not think it is stupid. They will probably be very grateful to you! It's a great way to show some leadership!
• If the question involves geometrical arguments (as most of the material in this class does), it might be easier for both you and me if you ask me after class or come by my office some time.
• When people ask me about X, the problem is very often not that they don't understand X, but that there is something more basic they don't understand. When you ask me a question, try to state what you understand and what you understand. In that way we can get to the essence of your confusion more quickly.
• I don't mind “checking” your answer if you are genuinely unsure, but if you send me a long list of questions to check, including many that I'm convinced you know very well, I get the feeling that you're not really valuing my time.
• If you send me e-mail, please use the module code GEK1506 in the subject. Otherwise you may end up in my spam folder. This is especially important if you use a non-NUS e-mail address.
• I have two addresses, @math.nus and @nus. You can use either.
• Try to avoid sending me huge files. When you're submitting projects and homework, you can use the workbin, burn a CD or bring a thumb drive along. If you have your own web page you can just send me a link. If you send me a picture, maybe you can send me a low resolution version first?

Many of the questions involve geometry, and it is hard to discuss it by e-mail. If you want to make an appointment to see me, just mail me and tell me when you are free.

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Course Schedule 2007/08 Semester 2

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Royal Observatory Greenwich

IVLE Course Page

I have a course page at IVLE, the Integrated Virtual Learning Environment. It contains several useful tools.

• A discussion forum that I encourage you to use
• A workbin where you submit soft copies of project proposal, project and homework
• A class list with e-mail addresses
• The latest timetable information

Otherwise, the current page is the main source of information.

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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.

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Orrery by Brian Greig

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 Motion of the Earth Around the Sun. This is a pretty basic applet showing the Earth moving around the Sun. But pay attention to how you can determine the equinoxes and solstices by looking at the position of the Earth's axis, and how the declination changes.
• The Apparent Motion of the Sun. This shows the “spiral of circles”.
• The Apparent Motion of the Sun at Different Latitudes. Now you can see how the “spiral of circles” changes depending on your latitude.
• The Apparent Motion of the Sun at Different Times of the Year. This is one of my favorite applets, that I use over and over again in my class. If you really understand this applet, you will get a good grade in my class!
• The Motion of the Sun Along the Ecliptic. Another of my favorites. If you really understand this applet, you have arrived!
• A Vertical Sundial. One of the points of this applets, is to make you understand why you shouldn't make a vertical sundial. (Unless you are a polar bear or a penguin!)
• The Astrological Houses at Different Latitudes. Astrologers like to talk about your ascendant and other “houses”. Unfortunately, the house system depends very much on your latitude. Very few astrologers understand how this works. If you want to understand this, get the book Making Sense of Astrology by Ronny Martens, Tim Trachet and study this applet.
• What does the waxing or waning Moon look like in different parts of the world in the course of the year? This is something that confuses a lot of people! A simplified model of the tilt of the waxing or waning Moon in different parts of the world ignores the tilt of the ecliptic, and assumes that the Sun and the Moon both move along the celestial equator. You may want to start out with this applet.
• If you studied the previous applet, you are now hopefully ready for the real thing. What does the waxing or waning Moon look like in different parts of the world in the course of the year?

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

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Equatorial (left) and ecliptic (right) armillary spheres in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

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.
• RealVideo LAN/150Kbps, right-click to download.
• QuickTime, click to stream, right-click to download.

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

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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.

• Astronomy Without a Telescope.
• Celestial Sphere Defined: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Angles
• Reference Markers: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Motion of Our Star the Sun: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Coordinates: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Time and Seasons: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Time Zones and Equation of Time: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Seasons
• Sections Review
• Motions of the Moon: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• Phases and Eclipses
• Eclipse Details: Lunar Eclipse
• Planetary Motions

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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.

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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)

Multimedia

• Please check out my Astronomical Java Applets and Animations and Astronomical Video Clips.
• The Nebraska Astronomy Applet Project has a number of interesting modules. Remember to check all the boxes!
  • Basic Coordinates and Seasons Module with Seasons and Ecliptic Simulator.
  • The Rotating Sky Module with Rotating Sky Explorer.
  • Motions of the Sun Module with Paths of the Sun Simulator.
  • Lunar Phase Simulator with Lunar Phase Simulator.
• Demonstrations and Animations for Teaching Astronomy - D A T A at Department of Astronomy - University of Illinois at Urbana-Champaign.
• Models of Planetary Motion from Antiquity to the Renaissance.

Course pages

• 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.
• From Stargazers to Starships by David P. Stern.
• The course Introduction to Astronomy by Scott R. Anderson has great animations.
• The course Math & Culture, Late Renaissance Thought and the New Universe by Dorothy Wallace & Scott Sciortino of Dartmouth College is very interesting.

The Meridian Line at Greenwich

Astronomical Institutions

• For general astronomy info, my favorite is the web site of the Astronomical Applications Department of the U.S. Naval Observatory. Their Data Services and FAQ contain a lot of useful information.
• HM Nautical Almanac Observatory in the UK makes very nice lunar visibility charts. Their Websurf contains a wealth of information, including Global First Sighting of New Crescent Moon information for the last couple of years. The charts for the current month are also available at Moon Watch.
• The Royal Observatory Greenwich. Their information leaflets on Time and Timekeeping are very good.
• Museum of the History of Science, University of Oxford has many interesting exhibitions.

Skywatching

• Night Sky This Week - NightSkyInfo.com
• Earth & Sky: Skywatching
• Sky and Telescope - This Week's Sky at a Glance
• SPACE.com
• BBC - Science & Nature - Space - What's in the sky tonight?

General Astronomy Sites

• Wolfgang R. Dick's History of astronomy page.
• Starry Messenger by David Chart of the Department of History and Philosophy of Science of the University of Cambridge.
• Physics and astronomy applets by Juergen Giesen.
• Phil Plait's Bad Astronomy: The Home Page.
• Basic Celestial Phenomena by Kerry Magruder.
• Astronomy & Culture is part of Astronomy Homework Help by Gary Agranat.

Singaporean Astronomy Related Sites

• The Astronomical Society of Singapore.
• The Singapore Science Centre has a nice Skytrack site.
• NUS Astronomical Society.
• singastro.org.

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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

• I have a page on Calendars in Singapore. It contains a paper I have written about The Mathematics of the Chinese Calendar. I have also written a shorter introduction called When is Chinese New Year?. This paper won a fourth prize in the Fifth Annual Boeing Writing Contest, which is organized by the Griffith Observatory. The article will appear in the Griffith Observer. I have also written a set of lecture notes on The Mathematics of the Public Holidays of Singapore.
• Honours project on Calendars in Singapore by my student Rachel Lee Tang Hwee.
• The most authoritative reference on calendars is the book Calendrical Calculations by Nachum Dershowitz and Edward M. Reingold of the Department of Computer Science at the University of Illinois at Urbana-Champaign. If you need a calendar conversion program, the place to go is their Calendar Applet.
• Before the book by Dershowitz and Reingold, the chapter about Calendars by L. E. Doggett in the “Explanatory Supplement to the Astronomical Almanac” was the standard reference.
• The Calendar FAQ by Claus Tøndering.
• Calendars Through the Ages is part of WebExhibits.
• The Worldwide Holiday & Festival Site (http://www.holidayfestival.com/) was very comprehensive. (Broken link. Can anybody please help?)
• The Calendar Zone.
• CalendarHome.com.
• Time-reckoning in Iceland before literacy by Þorsteinn Vilhjálmsson.

Chinese sundials from the Adler Planetarium

The Chinese and other East Asian Calendars

• I have a page on The Chinese Calendar. It contains a paper I have written about The Mathematics of the Chinese Calendar.
• Undergraduate research project on The Chinese Calendar of the Later Han Period by my students Kuan Shau Hong and Teng Keat Huat.
• Undergraduate research project on Calendars, Interpolation, Gnomons and Armillary Spheres in the Work of Guo Shoujing (1231-1314) by my student Ng Say Tiong.
• Undergraduate research project on Strings of Long Months and Short Months in the Chinese Calendar by my student Zhang Jieping.
• Zhuo Meng's Make Your Own Chinese Calendar is a wonderful tool!
• The Western-Chinese Calendar Converter at www.mandarintools.com.
• Harold C. Hill, Department of East Asian Languages and Literatures, Washington and Lee University, has a very interesting page on http://www.wlu.edu/~hhill/calendar.html about The cyclical calendar. (Broken link. Can anybody please help?)
• The Lunar Calendar in Japan by Steve Renshaw and Saori Ihara is the best source for information about the Japanese calendar.
• Vietnamese lunar calendar conversion by Ho Ngoc Duc of Institute of Informatics, University of Leipzig.

MoonCalc image

The Islamic Calendar

• I have a page on The Islamic Calendar, which contains additional links. You will 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.
• Undergraduate research project on Lunar Visibility and the Islamic Calendar by my student Leong Wen Xin.
• Notes for lecture on How Does the Islamic Calendar Really Work in Different Parts of the World? at the AMS/MAA meeting in Atlanta 2005.
• Moonsighting.com by Khalid Shaukat.
• Dr. Monzur Ahmed is the author of MoonCalc, which for many years was the leading lunar visibility and Islamic calendar software. He has pages on Astronomy and Islam and Ramadhan (Ramadan) and Eid-al-Fitr.
• The web page of the Islamic Crescents' Observation Project (ICOP) has a number of interesting articles. They also host the software Accurate Times by Mohammad Odeh. It is a Windows program for the Islamic calendar, Qiblah direction, and prayer times. There are samples of the lunar visibility charts on Visibility of Ramadan Crescent. Their page on The Actual Saudi Dating System will probably surprise many.
• Islamic calendar for Singapore by Majlis Ugama Islam Singapura (MUIS), the Islamic Religious Council of Singapore. MUIS seems to move the location of their calendar around from year to year. If the link doesn't work, go to their homepage and look under "Downloads" or do a search for Takwim (or Taqwim). This will give you MUIS's prediction for first visibility of the lunar crescent. But it is based on a dubious method, so you will often not be able to see the Moon on the day when MUIS claims that you will. Please see my page on The Islamic Calendar.
• HM Nautical Almanac Observatory in the UK makes very nice lunar visibility charts. Their Websurf contains a wealth of information, including Global First Sighting of New Crescent Moon information for the last couple of years. The charts for the current month are also available at Moon Watch.
• Moon Watch.
• Crescent Moon Visibility and the Islamic Calendar from the Astronomical Applications Department of the U.S. Naval Observatory.
• Moon Research Centre (U.K). The article on Policies Adopted in Britain by Muslims in Celebrating Islamic Festivals is essential reading! They also have a copy of the RGO Astronomical Information Sheet No. 6.
• Crescent-Sighting and Islamic Calendars and Islamic Astronomy: Questions, Comments and Correspondence by David McNaughton.
• Crescent Section from the Jordanian Astronomical Society. Their section on The Actual Saudi Dating System is essential reading!
• Islamic Crescents' Observation Project.
• Predicting the First Visibility of the Lunar Crescent, The Umm al-Qura Calendar of Saudi Arabia, Islamic-Western Calendar Converter (Based on the Arithmetical or Tabular Calendar) and Internet sites over de islamitische tijdrekening (in Dutch) by Robert H. van Gent.

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.

The Indian Calendar

• I have a page on The Indian Calendar.
• I have a page on When is Deepavali (Diwali)?.
• Honours project on Indian Calendars by my student LEOW Choon Lian.
• Undergraduate research project by my student Daphne CHIA on Indian Calendars: Comparing the Surya Siddhanta and the Astronomical Ephemeris, 2001.
• Undergraduate research project by my studen Akshay REGULAGEDDA on Panchanga-Tantra: The Magic of the Indian Calendar System, 2002.
• Undergraduate research project by my studens Akhil DOEGAR and Akshay PRASAD on Indian Calendars, 2002.

Easter

• The Date of Easter from the Astronomical Applications Department of the U.S. Naval Observatory.

Time

• Leap Seconds, from the Time Service Department, U.S. Naval Observatory.

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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.

• What is “wrong” with the Singapore flag? Why is the coat of arms more correct? Find out on my page on The Mathematics and Astronomy of the Singapore Flag.
• What Does the Waxing or Waning Moon Look Like in Different Parts of the World? This is a complex question that confuses many people.
• If you look at a map with time zones, you will see that 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?
• Which Day Does the Sun Rise Earliest in Singapore? We are at one degree north, so most people think the Sun will rise the same time every day. Not!
• Where is the Geographical Origin of Coordinates of Singapore?
• Calendars in Singapore. We are not just a multiracial, multicultural society; we are a multicalendrical society!

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!!!

Sun and Moon info

• The National Environment Agency on Singapore has a number of interesting pages, like Moonrise and Moonset Times and Moonphase Information for Singapore and Sunrise & Sunset Times for Singapore.
• The Astronomical Applications Department of the U.S. Naval Observatory has a lot of interesting pages on their Data Services and FAQ, like Virtual Reality Moon Phase Pictures, Phases of the Moon, Phases of the Moon and Percent of the Moon Illuminated, Rise, Set, and Twilight Definitions, Earth's Seasons Equinoxes, Solstices, Perihelion, and Aphelion 1992-2020, Sun or Moon Altitude/Azimuth Table for One Day and Sun or Moon Rise/Set Table for One Year. You can use the last one to print out a table of rising and setting times for the Sun and the Moon in Singapore. The coordinates of Singapore are 104 east and 1 north. Our time zone is UTC + 8.
• If you need to do computations, there is a handy table of the equation of time and the Sun's declination on the Daily Sun Data at Tony Helyar's Home Page.
• Sun, Moon & Earth Applet and Sun Declination & Equation of Time from Juergen Giesen's GeoAstro Java Applet Collection.
• Moon Phases.
• Islamic calendar for Singapore by Majlis Ugama Islam Singapura (MUIS), the Islamic Religious Council of Singapore. MUIS seems to move the location of their calendar around from year to year. If the link doesn't work, go to their homepage and look under "Downloads" or do a search for Takwim (or Taqwim). This will give you MUIS's prediction for first visibility of the lunar crescent. But it is based on a dubious method, so you will often not be able to see the Moon on the day when MUIS claims that you will. Please see my page on The Islamic Calendar.

Singapore weather and tide info

• National Environment Agency on Singapore has a number of interesting pages, like Climatology of Singapore, FAQs-Meteorological Service and Tide Times For Singapore.
• World Weather Information Service - Singapore from the World Meteorological Organization.

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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.

The Equation of Time

• I have recorded a video clip that explains this picture: RealVideo SureStream, RealVideo LAN/150Kbps, QuickTime.
• I have a page on Which Day Does the Sun Rise Earliest in Singapore? Most books on astronomy are written from a “high-latitude” point of view; instead I take a tropical point of view. You will learn how to prepare a table of sunrise and sunset times for your location.
• I strongly recommend the analemma.com site.
• Length of Day and Night at the Equinoxes, The Dark Days of Winter and Comparative Lengths of Longest Day and Longest Night, and of Shortest Day and Shortest Night from the Astronomical Applications Department of the U.S. Naval Observatory.
• In the past you could order a copy of Dennis di Cicco's famous analemma picture from Sky & Telescope.
• Sky & Telescope used to have a great comic strip called SkyWise. One month SkyWise discussed Latest Sunrise, Earliest Sunset. Sky & Telescope no longer has a free archive, but you can buy the article or look it up in a library.
• NIGHT and DAY by Walter R. Steiger and George W. Bunton.
• The solar analemma - Luxorion.
• The Analemma Project.
• Astrophotography by Anthony Ayiomamitis.
• Stanford SOLAR Center -- Ask A Solar Physicist FAQs - Answer.
• For great graphics, see the site by Andrew Marsh at The School of Architecture and Fine Arts at The University of Western Australia (http://fridge.arch.uwa.edu.au/topics/thermal/sun/sunpos.html). (Broken link. Can anybody please help?)
• If you need to do computations, there is a handy table of the equation of time and the Sun's declination on the Daily Sun Data at Tony Helyar's Home Page.
• The Analemma by Jon Kahl of University of Wisconsin-Milwaukee.
• Analemma, my Analemma by Douglas Dodds. (You need to scroll down a bit.)
• Sundials on the Internet - the Equation of Time.

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Ancient Astronomical Instruments

Why do sundials look different in Singapore and Beijing? We study the theory behind sundials and show how to construct an accurate sundial for any latitude. There are several nice sundials in Singapore. There is one in the Botanic Gardens and one in Marina City Park. As part of their Science is everywhere program, The Singapore Science Centre together with the National Parks Board recently put up a series of sundials representing the planets. With Science Centre representing the sun, unique sculptures are placed at various locations around Singapore. The relative distance of these sculptures from the Science Centre represents the distances of each planet from the Sun. Please check out the map!

Four common types of sundials are the equatorial, armillary, horizontal and vertical sundials.

Equatorial sundial	Armillary Sundial at City Hall, Ruurlo, photo by Frans W. Maes.
Horizontal sundial by Harriet James	Vertical sundial by Harriet James

For telling the time at night, you can use a nocturnal.

Nocturnal	Nocturnal

The quadrant is a simple tool for determining latitude.

Quadrant

The astrolabe was made famous by Arab astronomers, and can be thought of as an analog astronomical computer. We discuss the difference between an astrolabe and a mariner's astrolabe.

Astrolabe by Norman Greene	Mariner's astrolabe by Norman Greene

There's a lovely Persian astrolabe from 17th century at the Asian Civilisations Museum in Singapore.

Brass astrolabe, 17th century, Persia, Asian Civilisations Museum's collection, Singapore

One of the most fascinating of all astronomical instruments is the Antikythera mechanism. The Antikythera Mechanism is the name given to an astronomical calculating device, measuring about 32 by 16 by 10 cm, which was discovered in 1900 in a sunken ship just off the coast of Antikythera, an island between Crete and the Greek mainland. Several kinds of evidence point incontrovertibly to around 80 B.C. for the date of the shipwreck.

Geoffrey Chaucer, the author of Canterbury Tales wrote a Treatise on the Astrolabe in 1391.

We then look at the cross-staff and the back staff.

Cross-staff	Backstaff

We will compare the relative advantages of the quadrant, mariner's astrolabe and cross-staff.

Quadrant	Mariner's astrolabe	Cross-staff

The Chinese astronomer Guō Shǒujìng (郭守敬,1231-1314) designed many outstanding instruments. We will look at some of the famous observatories in China and India.

Dengfeng Observatory, designed by Guō Shǒujìng (郭守敬,1231-1314) in 1276	Armillary spheres at the Purple Mountain Observatory in Nanjing designed by Guō Shǒujìng (郭守敬,1231-1314)

We study the armillary spheres made by the Danish astronomer Tycho Brahe (1546-1601). (My second cousin lives in a castle built by a nephew of Tycho Brahe!) You will learn the difference between an ecliptic armillary sphere and an equatorial armillary sphere. I also have a separate page about “Fake” Armillary Spheres.

Ecliptic armillary sphere	Equatorial armillary sphere

How were cathedrals used as solar observatories? We study meridian lines in European churches. I have created a separate page on meridian lines, A Guide to “The Sun in the Church” by J.L. Heilbron.

Meridian Line. S. Petronio, Bologna, Calter Photo	Solar eclipse at the Meridian line at S.Maria degli Angeli in Rome by Mario Catamo

The Ambassadors by Holbein is a famous example of Anamorphosis. Do you see the strange object on the floor? Close your left eye, put your face close to the computer screen near the right side of the picture. You will then see a skull! If you can't get it to work, you can cheat and look at a picture of it.

The reason why I include this painting on this page, is because it contains an incredible collection of astronomical and mathematical instruments! Please check out my page about The Ambassadors by Holbein.

The Ambassadors (1533), by Hans Holbein the Younger (1497/8 - 1543)

Ancient Astronomical Instruments

• Undergraduate research project on Calendars, Interpolation, Gnomons and Armillary Spheres in the Work of Guo Shoujing (1231-1314) by my student Ng Say Tiong.
• Medieval science & scientific instruments by Richard A. Paselk, Department of Chemistry, Humboldt State University.
• Astronomical Instruments.

Tycho Brahe (1546-1601)

• Tycho Brahe - official website.
• The astronomical instruments of Tycho Brahe.
• Dansk Astronomi gennem 400 år, 400 years of Danish Astronomy (in Danish).
• Brian Greig is going to make models of Tycho's armillary spheres for me.

Sundials

• Undergraduate research project on The Mathematics of Sundials by my students Liew Huay Ling and Lim Siew Yee.
• Sundials on the Internet.
• North American Sundial Society.
• The British Sundial Society.
• Sundials on the Internet - Four simple sundial projects for you to make.
• Sundials by Jack Aubert.
• Sundials by Francois Blateyron.
• A Horizontal Sundial by Robert C. Berrington.
• Queens' College Cambridge - Sundial.
• sundial links by Daniel Roth.
• Polyhedral sundial.

Analemmatic Sundials

• Of Analemmas, Mean Time, and the Analemmatic Sundial by Frederick W. Sawyer III.
• Make your own Sundial from the Hartebeesthoek Radio Astronomy Observatory.
• Analemmatic Sundials: How to build one and why they work by C.J. Budd and C.J. Sangwin of the University of Bath.

Moondials

• Make your own Moondial.

Nocturnals

• Pocket Sundial, Star Clock, star time by Keith Powell.
• Shepherd's Watch makes a nice nocturnal. They also make nice sundials, but notice that their Explorer model does not adjust for the season, only for the latitude.

Celestial globe made for the Beijing Observatory by Ferdinand Verbiest (1673)	Celestial Star Globe

Celestial Globes

• Celestial Star Globe.

Astrolabes

• astrolabes.org.
• Keith's astrolabes: a Java program.
• A Treatise on the Astrolabe Geoffrey Chaucer, 1391.
• Clockworks: Astrolabe from Britannica.com.
• Astrolabes by Norman Greene.
• Martin Brunold - Astrolabienmacher

The Antikythera mechanism

• The Antikythera Mechanism I by Tony Phillips with Java animations by Bill Casselman.
• The Antikythera Mechanism II by Tony Phillips with Java animations by Bill Casselman

The large Samyat Yantra, equatorial sundial, at Jaipur	Drawing of the small equatorial sundial showing the shadow, quadrants, gnomon, and Sun rays

Jaipur Observatory

• Jantar Mantar: The Astronomical Observatories of Jai Singh II by Barry Perlus, Department of Art at Cornell University, is an absolutely stunning spherical VR imaging project.
• Jaipur - Jantar Mantar.
• 18th Century Observatories of Maharaja Sawai Jai Singh II by C. Hartley, Director of the Ernest B. Wright Observatory at Hartwick College.

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Astrology

Does astrology make sense? I'm personally skeptical. In the past, however, there was a close relationship between astronomy and astrology, and astronomy has great historical interest. We look at the mathematical foundation of some of the key concepts in both Western and Chinese astrology.

15th century French woodcut

What do astrologers mean when they say that somebody has an Aries ascendant? In order to understand this, we study the division of the ecliptic into houses. This is a very difficult issue, especially in high latitudes.

Astrology

• Undergraduate research project on The Mathematics of Astrology: Does House Division Make Sense? by my student Kevin Heng Ser Guan.
• Your Astrology Defense Kit by Andrew Fraknoi.
• Skeptic's Dictionary astrology.
• SC/BC 1800C.03 - Analysis of Astrology FAQ Page collected and edited by M.M. De Robertis (Instructor) at York University.
• The Astrotest, A tough match for astrologers by Rob Nanninga.
• Skeptical Information Links: Astrology by Jim Lippard.
• Astrology & Science.

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Navigation

Why is it harder to find your position at sea than at land? We study how ancient sailors in different parts of the world navigated in different ways, and how technological advances opened up new possibilities for discoveries and trade.

Cross-staff used for computing lunar distance and height of buildings on the cover of Introductio geographica by Petrus Apianus, Ingolstadt, 1533

Why is it harder to determine the longitude than the latitude? In 1714 the British Parliament offered 20,000 pound to anybody who could come up with a practical method to determine the longitude at sea. That was the equivalent of US$12 million in modern money! The Astronomer Royal was only paid 100 pounds a year, and had to pay for his own instruments.

We will watch the movie Longitude, starring Jeremy Irons, which is based on the best-selling book Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time by Dava Sobel.

Why is it harder to find your position in an air plane than at sea? We look at some modern methods, including GPS. We will also have a mystery guest-lecturer.

General Navigation

• An honours project on The Mathematics of the Longitude by my student Wong Lee Nah.
• Henning Umland has a wonderful Short guide to celestial navigation.
• Celestial Navigation Net is one of my favorite sites. Very comprehensive!
• Celestial Navigation by Al Placette.

Harrison's 4th Timekeeper and Longitude prize-winner

Finding the Longitude

• NOVA Online Lost at Sea The Search for Longitude.
• John Harrison and the Longitude Problem from The Royal Observatory Greenwich.

General History of Navigation

• The Columbus Navigation Homepage by Keith A. Pickering.
• Determination of Latitude by Francis Drake on the Coast of California in 1579 by Bob Graham.
• The Viking Sun Compass by Franck Pettersen, Northern Lights Planetarium, Tromsø, Norway. From the Planetarian.
• The Mariners' Museum Age of Exploration.
• Wayfinders: A Pacific Odyssey on PBS.
• John Charles Fremont by Bob Graham.

Danish high school students working on a navigation project

On-line Navigation Tools

• Bowditch -The American Practical Navigator.
• JavaScript Navigator HomePage.
• On Line Nautical Almanac from Navigator Light Software.

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Cartography

Most capital cities have a geographical origin of coordinates, which is used when making maps for the country. All cities also have a zero point for road distances from the city center to other points. Why do capital cities need an origin of coordinates? In the past, finding coordinates, especially the longitude, was so hard that the best way was to pick a local origin, do astronomical observations at that point, and then do triangulation for the rest of the country. That's why the origin of coordinates is often at observatories.

In London the origin of coordinates is at Greenwich, while the zero point for road distances is at Trafalgar Square. In Oslo the origin of coordinates is at the Observatory, while the zero point for road distances is at Stortorvet. But what about Singapore? After a bit of detective work, and a lot of help from various people, I found that the origin of coordinates is at Empress Place and the zero point for road distances is at the old General Post Office, now the Fullerton Hotel. For more details, see my page on Where is the Geographical Origin of Coordinates of Singapore?

How big is the Earth? We look at Eratosthenes's solution to this problem.

Did people in the Middle Ages believe that the Earth was flat?

Why is there no perfect map? We define the curvature of a surface and show that the curvature of the Earth is positive while the curvature of the plane is zero.

Mercator's 1569 world map	Mercator map of the world attributed to Edward Wright (1599)	Modern Mercator map

What makes the Mercator projection so important in navigation? We study the geometry of rhumb lines. We also show how it is related to the invention of logarithms.

Cartography

• Cynthia Lanius' Lessons: The Mathematics of Cartography from Rice University.
• Map Projection Home Page from the Geography Department, Hunter College, City University of New York.
• Map Projection Overview by Peter H. Dana, Department of Geography, University of Colorado, Boulder.
• Mathematical Cartography by Hans Havlicek, Abteilung für Lineare Geometrie, Technische Universität Wien.

This is NOT a medieval woodcut!

The Flat Earth Myth

• This is NOT a medieval woodcut! by Kerry Magruder.
• The Myth of the Flat Earth by Jeffrey Burton Russell.

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Archaeoastronomy

Was Stonehenge an observatory? When were the pyramids of Egypt built? Archaeoastronomy is a very active area these days. Many ancient monuments reveal an astronomical background.

Newgrange	Stonehenge

Stonehenge

• Visit Stonehenge : English Heritage
• Science and Stonehenge from English Heritage.
• How They Rebuilt Stonehenge
• Stonehenge and Astronomy.
• Earth Mysteries Stonehenge.
• Stonehenge on the Discovery Channel.

Winter Solstice Sunrise at Newgrange	Winter Solstice Sunrise at Maeshowe

Newgrange and Maeshowe

• Newgrange Megalithic Passage Tomb - Ireland has a great animation showing how the passageway is illuminated by the sunrise at the Winter solstice.
• Winter Solstice at Newgrange (http://members.theglobe.com/newgrange2/solstice.html). (Broken link. Can anybody please help?)
• Geniet Heaven and earth contains both Geniet: Newgrange and Geniet Research on Maes Howe, Orkney, UK.
• Maeshowe and the Winter Solstice.

Pyramids

General Archaeoastronomy

• The Center for Archaeoastronomy at the University of Maryland.

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Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

Cultural Astronomy

What is “wrong” with the Singapore flag? Why is the coat of arms more correct? I have a page on The Mathematics and Astronomy of the Singapore Flag.

The Singapore Flag	The Singapore Coat of Arms

What is a Harvest Moon? What is a Blue Moon? What is Groundhog Day? There are a lot astronomical terms that people use but don't really understand.

What was the Star of Bethlehem? How can eclipses be used to date historical events?

Chaucer wrote a book on the astrolabe, and the Canterbury Tales are filled with astronomical references.

Celestial Navigation Net has a very interesting section on Dante and Celestial Navigation!

Donald Olson and others have championed a new field called forensic astronomy. A nice example of this is Astronomers turn detectives to pinpoint date of Ansel Adams image.

What is a Blue Moon?

• What's a Blue Moon? by Donald W. Olson, Richard Tresch Fienberg, and Roger W. Sinnott. The definite article from Sky & Telescope magazine. Sky & Telescope no longer has a free archive, but you can buy the article or look it up in a library.
• You can also search for the earlier articles “Once in a Blue Moon” by Philip Hiscock and “Blue-Moon Mystery Solved?” by Donald W. Olson and Roger W. Sinnott from Sky & Telescope.

What is a Harvest Moon?

• Watch Out for the Harvest Moon from SpaceScience.com.
• The Harvest Moon by Larry Gedney. From Alaska Science Forum.

The Star of Bethlehem

• What was the Star of Bethlehem? - Mysteries of the Universe- msnbc.com
• Star of Bethlehem by Donald Simanek.
• Common Errors in “Star of Bethlehem” Planetarium Shows by John Mosley, Program Supervisor, Griffith Observatory. From the Planetarian.
• Yet Another Eclipse for Herod by John P. Pratt. From the Planetarian.
• The Bimillenary of Christ's Birth: The Astronomical Evidence by William P. Bidelman, Warner and Swasey Observatory, Case Western Reserve University. From the Planetarian.

Phil the groundhog

Groundhog Day

• Groundhog Day and Chinese Astronomy by Kelley L. Ross.
• Groundhog Day History from Stormfax.
• Groundhog.org - the Official Site of the Punxsutawney Groundhog Club.

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Astronomy in Nature

I have a separate page on What Does the Waxing or Waning Moon Look Like in Different Parts of the World? This is a complex question that confuses many people.

Moonrise from Sky & Telescope

What does the orbit of the Moon around the Sun look like? Most people, even almost all mathematicians I've asked this question, tend to believe that it will have loops and look something like the picture below.

In fact it looks like this picture!

It is not a circle, but is close to a 12-gon with rounded corners. It is locally convex in the sense that it has no loops and the curvature never changes sign. I have a page that explains why The Orbit of the Moon around the Sun is Convex!

Halls Harbour, Nova Scotia, six hours apart

Why are there two high tides each day? This is very frequently asked question! I used to think that the answer was simple, but after looking around in the library and on the web, I discovered several different attempts at explaining it. Newton's equilibrium theory from 1687 used the differential of the gravitational force, but some people (especially oceanographers) also consider a centrifugal force caused by the rotation of the Earth around the Earth-Moon barycenter. Unfortunately, it seems to me that the methods using a centrifugal force are unnecessarily complicated, if not outright wrong! I will describe some of the attempted explanations I found.

How often can we see lunar and solar eclipses in Singapore? It turns out that it is fairly easy to make reasonably accurate eclipse predictions. There are several solar eclipses each year, but at a given place, they occur on average only every 360 years. Solar eclipses, Singapore lists all solar eclipses visible from Singapore between 1700 and 2100. The last total eclipse was in 1821, and there will be annular eclipses in 2019 and 2053, but I'm afraid there will not be any total eclipses in Singapore before 2100.

How is astronomy related to the ice ages? It turns out that changes in the Earth's movement is related to the ice ages.

Why does the Moon look bigger when it is near the horizon?

Can you balance an egg on the Spring equinox? This turns out to be related to the Chinese calendar!

Eclipses

In October 1995 I went with Professor John Parkinson to Sabah to watch a total solar eclipse.

• The Singapore Science Centre has a nice Skytrack site and info about eclipses in Singapore, including a list of lunar eclipses (excluding penumbral eclipses) visible in Singapore between 1995 and 2015.
• singastro.org is good for local astronomy info including eclipses.
• Solar eclipses, Singapore lists all solar eclipses visible from Singapore between 1700 and 2100.
• References on Eclipses from the Astronomical Applications Department of the U.S. Naval Observatory.
• Eclipses Online Portal.
• NASA has a nice Eclipse Home Page, which includes a very useful Javascript Lunar Eclipse Explorer. It can compute the local circumstances for every lunar eclipse visible from a city for any century from -1999 to 3000 (2000 BCE to 3000 CE).
• There is a nice animation of a lunar eclipse at NASA and a nice picture at UK Eclipse Group website.

Astronomy and Climate

• The Seasons and the Earth's Orbit - Milankovitch Cycles from the Astronomical Applications Department of the U.S. Naval Observatory.

What's Wrong With This Moon?

The Moon

• What's Wrong With This Moon? by Jim Loy.
• The Moon by Jim Loy.
• Bad Moons Rising from Paper Plate Education.

The Moon Illusion

• Moon illusion from Wikipedia.
• Explaining the Moon illusion by Lloyd Kaufman, and James H. Kaufman.

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Bad Astronomy and Pseudoscience

The truth may be out there, but there's a lot of crap, too! I will discuss some common misconceptions and hoaxes. One of my favorites is the “Mars will be as big at the Full Moon” e-mail. It has several variations, but here is a common one.

The Red Planet is about to be spectacular! This month and next, Earth is catching up with Mars in an encounter that will culminate in the closest approach between the two planets in recorded history. The next time Mars may come this close is in 2287. Due to the way Jupiter's gravity tugs on Mars and perturbs its orbit, astronomers can only be certain t! hat Mars has not come this close to Earth in the Last 5,000 years, but it may be as long as 60,000 years before it happens again.

The encounter will culminate on August 27th when Mars comes to within 34,649,589 miles of Earth and will be (next to the moon) the brightest object in the night sky. It will attain a magnitude of -2.9 and will appear 25.11 arc seconds wide. At a modest 75-power magnification Mars will look as large as the full moon to the naked eye.

Mars will be easy to spot. At the beginning of August it will rise in the east at 10p.m. and reach its azimuth at about 3 a.m. By the end of August when the two planets are closest, Mars will rise at nightfall and reach its highest point in the sky at 12:30a.m.

That's pretty convenient to see something that no human being has seen in recorded history. So, mark your calendar at the beginning of August to see Mars grow progressively brighter and brighter throughout the month. Share this with your children and grandchildren. NO ONE ALIVE TODAY WILL EVER SEE THIS AGAIN!

Does this make you think that Mars will be as big as the Full Moon in the sky? That's NOT what they're saying, but you have to read carefully to not get tricked.

Balancing Eggs on the Spring Equinox

• Standing an egg on end on the Spring Equinox from Phil Plait's Bad Astronomy site. This is related to the Chinese calendar!

Kepler's Mysterium Cosmographicum, 1596

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Lecture Notes

• I have modified the first chapter Astronomy without a telescope from Nick Strobel's Astronomy Notes.
• I have a brief set of notes on The Celestial Sphere that is meant to supplement Strobel's notes.
• I have a set of lecture notes on The Mathematics of the Public Holidays of Singapore.
• I have written a paper on Heavenly Mathematics: Observing the Sun and the Moon from Different Parts of the World.
• I have a brief set of notes on Celestial Navigation and the Equation of Time.
• I have a long paper on The Mathematics of the Chinese Calendar. This is just for optional reading, but you may enjoy looking at it, especially Section 2 and 3.

You can get all of them as a package at the LT 27 Co-op. I forgot to leave them there before I went to the US, but they will be available from Wednesday 16/1.

I will also post the slides from the lectures. They are intended as summaries and to help you get organized, not as a replacement for the lecture notes.

• Introduction
• Week 1
• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• The Mathematics of the Public Holidays of Singapore (updated and expanded)
• Week 10

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Homework and Tutorials

Homework

• Sun Homework
• Moon Homework

Tutorials

• Tutorial 1 Solutions
• Tutorial 2 Solutions
• Tutorial 3 Solutions
• Tutorial 4 Solutions
• Tutorial 5 Solutions
• Tutorial 6 Solutions
• Tutorial 7 Solutions
• Tutorial 8 Solutions
• Tutorial 9 Solutions
• Tutorial 10 Solutions

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Old Exams

• Exam 2003/04 Semester 1, Solutions
• Exam 2005/06 Semester 1, Solutions
• Exam 2005/06 Semester 2, Solutions
• Exam 2006/07 Semester 2, Solutions
• Exam 2007/08 Semester 2, Solutions

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Summary

A brief summary of some of the main points.

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)

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Past Homework

I have a separate page for past homework.

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Past Projects

I have a separate page for past projects.

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Project Topics

All the topics we talk about in lecture can be extended to projects. Here are some suggestions. I also encourage you to propose your own topics and send them to me for approval.

• Astronomy and its History
  • The appearance of the Moon at different latitudes.
  • The heliocentric system versus the geocentric system.
  • The works of Galileo.
  • The works of Ptolemy.
  • The works of Copernicus.
  • The works of Brahe.
  • The works of Kepler.
  • Spherical trigonometry in astronomy.
  • The journeys of Columbus.
• Calendars
  • Chinese Festivals.
  • Computing the date of the public holidays in Singapore?
  • How many birthdays can you have each year in Singapore?
  • The meaning of the names of the 24 jie qi in the Chinese calendar.
  • The Tibetan calendar.
  • Prediction and observation in the Islamic calendar.
  • Muslim prayer times at high latitudes.
  • Indian calendars.
  • History of the computation of Easter.
  • The Gregorian calendar reform.
  • The Jewish calendar.
  • The Mayan calendar.
  • Leap seconds.
• The Equation of Time
  • Sunrise and Sunset times in the tropics.
• Ancient Astronomical Instruments
  • History of the astrolabe.
  • Sundials.
  • Armillary spheres.
  • Cathedrals as solar observatories.
• Astrology
  • House division in Astrology.
  • Astronomy and astrology in the works of Chaucer.