For Heavenly Mathematics: Astrobox

____________________________________________________For Heavenly Mathematics: Astrobox ~ Contents

1


1


1

Issue 8 | October 2005 contents

Cover Story: Celestial Navigation 5 History of Celestial Navigation

9 Forefathers of Celestial Navigation

19 Guide to Celestial Navigation

29 Celestial Navigation Instruments

In conversation:

17 Singapore Naval Officer

reviews:

32 Book Reviews 33 IT section 37 Net Reviews

regualrs:

2 Astrobox 3 Heavenly Edit

trivia:

39 Crosswords puzzle 40 Just for a smile 41 Promotions

Trips to visit polaris, telescope, book vouchers, 2 years subscription, answer to crosswords puzzle 41 Astro-jokes 40

__________________________________________________For Heavenly Mathematics: Astrobox ~ Your Mails

2

Astrobox

Celestial Letter of the month

Your September issue was totally heavenly! It gave me a new insight on why there are different seasons, leap years, solar and lunar eclipse, etc. After reading your magazine, I am now able to answer enquiries post by my kids better. But I have one request here. Can you all have one issue to talk about astronav or celestial navigation? I have an uncle, who used to be a sailor. His story on how he found his way out in the ocean just by looking at the stars always fascinated me. I do hope to know about celestial navigation hence next time I am out in the seas with my friends I can show off the new gained skill. Stay heavenly! SKYE LAW Well, guess what, we are covering on Celestial Navigation this issue! Now understand this whole issue and start showing off your skill to all your friends! Also, your ‘accurate prediction’ mail has earned you an US$200 Sextant to assist you in your Celestial Navigation! Now go and navigate the world!

**This is so embarrassing to tell everybody that I have no idea how eclipse occurred till I read your last issue. And I always thought that leap years occurred due to some silly rules set by our ancestors! Never do I know the long history behind it! *Applause* Silly Gal Right… Now you know it! There are many more heavenly secrets that you never knew existed. Do take up courses in your educational institute to know more about the astronomy! I know of a few existing courses about astronomy in

NUS. And the lecturer for those modules is totally HEAVENLY! **I never know Singapore have such good astronomy magazine! I am sure it is good educational material for our kids nowsaday who have no idea of the UNIVERSE! I will continue my support for this magazine! AstroFanatics Right we are still currently the ONLY magazine that touches on astronomy totally. And we do need your continual support to keep us in heavenly business!

***********************************************************

Write In and win a

set of celestial

navigation books!

Blow us away with your feedback and you might stand a chance to win a range of Celestial Navigation Books courtesy of Astrobooks worth $300. With all these books in hand, I am sure you will not face with cases of lost in the oceans! Send in your comments by 15 November, and be in the running to walk away with these way-finding books! Other featured mails will walk away with a session of Celestial Navigation Lesson with our professional tutor from National University of Singapore. So do keep your mails coming in!

_________________________________________For Heavenly Mathematics: Heavenly Edit ~ the Editorial Page

3

Heavenly edit

Find your way! With the great Casino (uh sorry, ‘Integrated Resort) debate on every one’s lips currently, I am glad to know that we are not losing our way in life. With the many voices in bid to stop the social adversity from happening, I do hope that all of us will still have time to watch the recent full moon! This issue we bring you to a lesson on how to find your direction just by looking at the celestial bodies! I know this is totally heavenly! This skill was acquired by our ancestor centuries ago! With the development of technology, radio-transmission, Global Positioning System, this skill is moving towards EXTINCTION!

But this is the skill that will stay with us in the event of power failure or transmission failure! The stars will never play us out! Find your way!

Yours Heavenly,

jonathan

. For Heavenly Mathematics: History of Celestial Navigation

5


5

Even before history was recorded, navigation by means of following rivers, lakes and oceans has been widely used. As transportation was of key importance of contact with the world, navigation played a vital role that bridge historical people closer that allowed the progress of civilization.

Humans of early years had learned that one of the most convenient and efficient ways of traveling was by water. This mode of transport allowed them to ferry goods to and fro from other countries to engage in trading and travel.

This gave birth to the early mariners like the Sumerians, Cretans, Egyptians, Phoenicians and Greeks all of which were located near the Mediterranean Sea (an advantage was that they had many ports near the sea, so they possessed high accessibility of sea travel). The Scandinavians in northern Europe also had the means to conduct sea travel.

Early mariners did not have established data on navigation procedures, they mainly explored by trial and error. As nautical charts were not present and armed with

only a list of directions (known as periplus by the Romans – it gave details of landmarks, places of anchorages and water hazards like reefs and shoals), they only ventured areas not very far from the coasts (mostly just skirting the coastlines). This enabled them to know the positions of their ships by aligning themselves to prominent objects on land that could be seen from their ships.

History of Celestial Navigation

only a list of directions (know as periplus by the Romans – it gave details of landmarks, places of anchorages and water hazards like reefs and shoals), they only ventured areas not very far from the coasts (mostly just skirting the coastlines). This enabled them to know the positions of their ships by aligning themselves to prominent objects on land that could be seen from their ships. Usually they traveled in the day and came back to shore by night. Navigation advancement and development took place as the Phoenicians and Greeks


6

6

Roman Astronomers

became the first of the Mediterranean sailors who dared to venture far from the coastlines and to sail at night. Through observations of the position and motion of the Sun and the North Star (Polaris), they were able to determine directions and estimated distances from the time it took to cover them. Thus they created simple charts and knew a basic form of dead reckoning (see instruments).

Celestial Navigation

For centuries sailors have safely guided their ships across the oceans by means of observations of the Sun, Moon, stars and planets. This art of locating position is known as celestial navigation or nautical astronomy. During their long travels, the sailors observed that the celestial bodies seemed to change their paths across the sky which resulted in them rising and setting at different points along the horizon. As the earth is nearly spherical and that the positions of the celestial bodies changes with respect to time, the sailors were able to determine the location of their ship by observing the celestial bodies carefully.

Early Aids of Navigation

The magnetic properties of lodestone were known to men for centuries even before the Christian Era. However it was only around the 12th century that navigators began

the use of lodestones and this resulted in the construction of the greatest navigation aid of all time, the magnetic compass.

The Italians later developed a chart called a portolano during the 13th century that showed the outline of the coast and the inclusion of crosslines which aided in locating directions during their voyagers. The Greeks also contributed two inventions to measure the altitudes of celestial bodies known as the cross-staff and the astrolabe. They were also widely used by navigators to obtain measurements to determine the approximate latitude of the vessel as well as the approximate local time.

These early aids of navigation helped great explorers such as Christopher Columbus, and Ferdinand

Christopher Columbus and

advancement and complexity of navigations, these instruments were not efficient

in their various voyages. However through the

Astrolabe Portolando cross staff


7

and were deemed redundant in the 18th century as there were no clocks that could keep the time well enough to aid the fixation of longitude readings.

Past to Present: The Pioneers of Navigation

The development of navigation was dramatically spearheaded in the 17th century by powerhouse countries like Britain and France. This led to the establishments of astronomical observatories to provide almanacs. Similarly, mapmaking and the invention of required navigational instruments were also greatly encouraged.

One very fundamental instrument that proved to edge on the developments of navigation was the brainchild of two individuals; John Hadley and Thomas Godfrey who invented a quadrant that could calculate accurately the observations of celestial bodies during the early 18th century. This instrument was similar to the sextant that is commonly used by navigators today.

` John Hadley

John Harrison in England produced several chronometers between 1730 and 1763 which solved the longstanding problems of longitudinal readings during the early navigation years. Furthermore, Captain James Cook's voyages of discovery in the Pacific Ocean at this time proved the accuracy and reliability of these navigational instruments and techniques. Early in the 19th century, Nathaniel Bowditch of Salem devised many methods that improved the art of navigation. In 1837, Captain Thomas Sumner created a trigonometric method (Sumner lines of position) of obtaining reference lines from celestial observations. In 1875, Marcq

St. Hilaire, a Frenchman improvised Sumner's trigonometric calculations which were later used to supplement dead reckoning (method of correcting for drift using triangular calculations of velocity).

The advancement of navigation has come a long way and electronic devices such as radar and loran are widely used in navigation today. Most vessels nowadays use an automatic pilot. The United States Air Force has created a satellite system known as Global Positioning System (GPS) in 1978 that has been providing worldwide coverage for determining accurately the latitude and longitude to within about 10 meters. Following the use of GPS, the Navstar Global Positioning System was introduced in the 1980s allowing space personnel to store the course of the space vessel in a computer system, which helps to verify the location and speed of the spacecraft. Multiple computer systems are fast becoming the norm in managing navigations.

For Heavenly Mathematics: Forefathers of Celestial Navigation

9

Forefathers of Celestial Navigation

Early documented accounts provided proof that exploration by sea was most abundant in ancient civilizations. This was due to religious beliefs, conquest and the necessity of trade routes establishments as well as the search for riches.

The 14th century opened

a pathway to the age of exploration which was then recorded in history books. However exploration between Europe and Asia began much earlier as travels between Greece and India, for instance, was frequent during the reign of Alexander the Great as his vast empire included territories of both countries. China’s Han Dynasty as well as the Roman Empire also had regular trade relations. Early explorers had some idea of their final destination before they set sail. Although they were searching for a specific land or route, they were surprised at what they discovered most of the time.

The captain of the ship is heavily funded by wealthy merchants of those times and often required a huge manpower before voyaging out to sea. Most voyages during the 14th century were made in the name of the royal ruler of a particular government. Ironically, the crewmen on these long and dangerous voyages were not the most experienced seamen. They were mostly needed in aid of

manning the ship’s sails and to allow for attrition due to illness and death. The ships that the royal rulers gave were rarely new but the captain took what was given. The captains were usually inexperienced seaman, whose daily jobs were merchants, adventurers, soldiers or political figures. The desire for wealth or political status was often the reason in participating on these dangerous voyages. Under his command was the pilot also known as the first mate who were mainly in charge of the ships navigations and the crew that man sails and repairs during the voyages at sea.

Food supplies were often limited, consisting of only pickled or dried meat and ship's biscuits which were made from flour with a little water. Weevils would usually be found in these biscuits at the end of the voyages. Their staple diet also included cheese, onions, dried beans, salted fish and recently caught fresh fish. The absence of fresh fruits and vegetables caused the sailors to suffer from a fatal condition called scurvy due to the lack of vitamin C. Another major problem was water supply as

fresh water was hard to contain in barrels and wine turned sour easily.

The Beginning: Mongols and Marco Polo

The 4th and 5th centuries saw barbarians that existed in large numbers ruled over the plains of China, India as well as Europe, thus jeopardizing the links between these countries. Furthermore, the introduction of Islam within North Africa, the Eastern Mediterranean, Spain and France led to the break off of trade routes as there were hostilities between Christians and Muslims.

By the 13th century, nomadic Mongols had captured most of Eurasia, Korea, and Hungary thus opening the doors that enabled Europeans to travel to China safely. In the early 1260s, two brothers from the city of Venice, Niccolo Polo (father of Marco Polo) and Maffeo Polo (uncle of

Drawings of Mongols on expedition


10

Throne ruled the entire South Pacific and the Indian Ocean During his reign, numerous gifts and tributes for the Emperor were collected throughout a territorial region that spread over Korea and Japan to the Eastern coast of Africa. Admiral Zheng He began his first of seven voyages in 1405. The purpose of his first voyage was to hunt for the former dethroned emperor who had escaped. A fleet of 62 large junks and 255 smaller ships were commanded by Admiral Zheng He recaptured the deposed, emperor and also manage to defeat a large pirate fleet that had been terrorising the islands

Marco Polo) embarked on their first journey to China by land in which they met the Mongol Emperor, Kublai Khan and eventually returned to Venice.

In 1271, the Polos made another journey extending to about four years, this time bringing along the young Marco Polo with them. Kublai Khan was deeply impressed with his European visitors that led them to stay for 17 years and returned to Europe only in 1292. The route across Central Asia was lost when the Mongol Empire fell. Marco Polo's accounts of travel highly inspired many Europeans and thus they were determined create once again routes of

their own to the East in search of wealth.

Admiral Zheng He

By the 1st century, many historians claimed that the Chinese had some knowledge of the winds and currents of the Pacific. When weather prevented the Chinese from navigating by the sky, magnetized needles were used instead to guide them. As the rocking ship had no level surface the sailors floated the needle in a shallow bowl of water which gave rise to the modern day instrument, the compass. Between 1405 and 1433, Emperor Zhu Di's Treasure Fleet of the Dragon

Over a span of 28 years, Admiral Zheng He increased China's trading territory to 37 countries from the Vietnam coast to the Persian Gulf, the Red Sea, and eastern Africa. When Admiral Zheng He died around 1435, China's economic domination in the South Pacific came to and end.

Christopher Columbus (b. 1451 - d. 1506)

Born in Genoa, Italy, in 1451, young Columbus first went to sea by the age of 14 and eventually settled Portugal and married a woman of noble background. Columbus and his young son, Diego moved to Spain in 1485 after his wife’s death. Columbus knew the

near Sumatra region.

Route taken by Marco Polo


11

world was round and explained that as the earth was a sphere, thus a ship would eventually reach the Far East traveling in the opposite direction which led him to establish trade routes to Asia.

Drawing of Columbus

Europeans in the 15th century were not aware of the existence of the South and North American continents thus maps did not portray an accurate picture and the Pacific Ocean remain unknown. Columbus seek grant from the Portuguese king and the Spanish monarchs to explore possible trade routes to the West but was repeatedly turned down. However after a decade, Columbus managed to gain support from King Ferdinand and Queen Isabella and promised to bring back gold, spices and silks from the Far East. He was also keen on spreading Christianity thus leading him on a expedition to

China. In return, Columbus attained the hereditary title "admiral of the ocean seas" and became governor of all the lands he discovered. This was done in a series of four voyages.

Before celestial navigation was created, sailors navigated by deduced (dead) reckoning. This method was used by Columbus in his voyages and most other sailors of his era. Dead reckoning enables a navigator to plot the location of the ship at any given time. As a result, the journey is already planned out entirety before the start of the voyage. In dead reckoning, the navigator locates his position by measuring the course and distance that he has sailed from a known point which is recorded down on a chart. A pin would be used to mark any new position. At the end of each day, the ending position would be the starting point for the next day's course and distance measurement.

Quadrant, an equipment used

by Columbus

The most important tool that Columbus used during his voyages was the quadrant. This was a metal plate in the shape

of a quarter-circle. From the center of the circle hung a weight on a string that crossed the opposite edge of the circle.

Astrolabe used by Columbus

In addition to the quadrant, an astrolabe was also used by Columbus in his first voyage. The astrolabe made of metal, was a complete circle and had a moving arm or alidade that he would sight along to find the star's altitude. Columbus used a sandglass to measure time and a nocturnal was also used to determine when midnight would be which tells the time of the night by the rotation of stars around the celestial pole.

Columbus embarked on his first voyage with three ships, the Niña, the Pinta and the Santa Maria. The ship Santa Maria was nicknamed La Gallega as she was big and slow, designed mainly for storing cargo and not for exploration The Santa Maria had three masts (fore, main, and mizzen) with one large sail on each mast. The foresail and mainsail were square while the sail on the mizzen or rear mast was a triangular in shape also known as a lateen. The ship also carried a small square sail


12

on the bowsprit and had a small topsail on the mainmast above the mainsail. The Pinta, captained by Martín Alonso Pinzón was a leading mariner from the town of Moguer in Andalucia. Pinta was a caravel which is smaller, lighter and faster than the bulky Santa Maria. The Niña was the smallest of the three and it was captained by Vicente Añes Pinzón, brother of Martín. The Niña was another caravel and may have carried four masts that include a small counter-mizzen at the stern equipped with another lateen sail, thus making the Niña the best of the three ships at sailing upwind.

James Cook (b. 1728 - d. 1779)

James Cook was born on October 27 in 1728 in Marton in Cleveland, Yorkshire, England to a farmer of Scottish decent. Since young, he was taught by a family in sea navigations. In 1755, he joined Great Britain's Royal Navy and proved himself an expert navigator. Just after making officer rank, Lieutenant Cook was chosen by the Royal Society of London and made his first journey to observe the eclipse of the planet Venus as well as determining the location a southern continent in 1768. Cook was one of the first sea captains to discover the cure for scurvy. Being determined to keep his crew healthy, he ensured that there were abundant fresh fruits and

vegetables on board. Feeding his crew onions and pickled cabbage every day also kept other diseases at bay. On top of this, he made sure that his crew bathe everyday, wash their dirty clothes and to also air their bedding.

James Cook

Cook started his first voyage on August 25 in 1768. Accompanied by 94 crewmen and scientist, he departed from England aboard the Endeavour. On April 11 in 1769, Cook arrived and embarked on the shores of Tahiti, seven weeks earlier than expected. After viewing the eclipse of Venus between the earth and sun for several weeks, Cook opened a sealed envelope which he carried from the Royal Navy. His mission was to locate the rumoured southern continent and claim it for England. Early mapmakers in the 1570s assumed there were two major continents located at the end of each of the earth's poles. Since the continents had not yet been

found, the Royal Navy entrusted Cook with the task of finding the southern continent if it existed. On July 13, the Endeavour left Tahiti and headed southwest in search of the southern continent.

Upon reaching New Zealand on October 6, Cook discovered the native Maori people. They were hostile and Cook had no choice but to fire on them. The Endeavour explored New Zealand for a few months and Cook concluded that it was not part of the great southern continent. Cook would later on explore and be the first to document the location of Australia. After months of studying the coastlines, he concluded that this continent was not the great southern continent that the Royal Navy wanted him to find. The Endeavour continued to journey to Java in the East Indies. It was here that his crew became seriously ill with diseases like malaria and dysentery. Cook effort to keep his man healthy was futile as they were overwhelmed by these diseases. 30 of his crew died and this forced Cook to sign on new crew in South Africa for his return leg to England.

Cook's second voyage began on July in 1772 from Plymouth, England with two ships, the Resolution and the Adventure. His orders were to find the southern continent. His


13

plan of searching for the continent by circumnavigating the globe and this led him to the discovery of the continent. Cook discovered more islands in the South Pacific during his second voyage. After reaching the Antarctic Circle in January 1774, Cook had sailed farther south than any other explorer.

On his final voyage on July in 1776, Cook, with a crew strength of 112, he boarded the Resolution once again with a sister ship, the Discovery. The objective of his third voyage was to find the fabled Northwest Passage by taking a route from the Pacific side. Cook visited some islands in the Pacific as well as New Zealand and Tahiti.

On January 18 in 1778, Cook saw the Hawaiian Islands for the first time and named it after his friend, the Earl of Sandwhich. It was known as the Sandwich Islands. The natives there were very hospitable as they thought that Cook was a god and that his men were supernatural beings. Cook observed that the people in the Pacific had a common language with similar customs and deduced that these Polynesians had dispersed over the many islands in the Pacific. Things turned ugly as Cook returned again to the islands the following year. The natives were then unhappy about the presence of the Europeans.

Cook decided to leave the islands but was forced to return as they encountered a storm which damaged the ships. This angered the natives even more and they stole one of the ship’s boats. Cook took things into his hands and captured the Hawaiian chief as hostage. As Cook was heading back to his ship with his hostage, the angry natives surrounded him and his men. Cook then signaled his boats off shore for backup but he was stabbed in the back by one of the natives and was subsequently killed by the dozens of natives that continued to attack him.

John Harrison (b. 1693 – d. 1776)

Harrison was born in Foulby, near Wakefield, in Yorkshire in 1693. When he was a young boy, his family moved to Barrow, in Lincolnshire. He followed his father’s footsteps and became a carpenter. In 1713, he built his first longcase clock with a mechanism that was made entirely of wood at a young age

of 20. Three of Harrison's early wooden clocks are still present today and are located at the Worshipful Company of Clockmakers' Collection in Guildhall in London, the Science Museum and Nostell Priory in Yorkshire.

His first wife, Elizabeth died just eight years after their marriage in 1718 and took on a second wife who was also named Elizabeth. Working with with his younger brother James, they came up with a revolutionary turret clock that required no lubrication. This was a breakthrough as clocks in the 18th century had poor oil lubrication that caused them to fail as Harrison designed a clock which did not need it. It was this radical thinking that was deemed important in tackling the problem of designing a marine timekeeper later on.

In his later years and through numerals tries, Harrison had devised a portable clock which kept time to the same accuracy as precision regulators. The longitude problem that bothered all mariners at that time was eventually solved by Harrison as he took on the scientific and academic establishment of his time and won the longitude prize through extraordinary mechanical insight, talent and determination.

_________________________________________For Heavenly Mathematics: Place To Go ~ Zheng He Exhibition

14

Place To Go

Duration: 13 Aug 2005 – 10 Feb 2006 Venue: National Library, Level 10 Opening Hours: 10 am – 9 pm daily Closed on Public Holidays Admission: Free Organiser: National Library Board

The Zheng He & Maritime Asia Exhibition, held at the National Library (Level 10) Victoria Street, will run for 6 months from 13 August 2005 to 10 February 2006. It aims to enhance public's interest in learning about Zheng He and his influence on the maritime history of Southeast Asia.

_________________________________________For Heavenly Mathematics: Place To Go ~ Zheng He Exhibition

15

Why should we go ?

Admiral Zheng He commanded seven great expeditions to more than 30 countries between 1405 and 1433. With the available technology then, it is an amazing feat to be able to find his way from China to Africa and back. He made used mostly of Celestial Navigation along with some simple navigation equipments. In this exhibition, they showcase some of the equipments he used to navigate.

What to see there ?

Qianxingban is used by Admiral Zheng He to determine the latitude using the position of constellations. It is a measuring board made up of 12 pieces of square wood. Navigators aligned the qianxingban with the horizon and then used the length of their arms to calculate the position of marker stars. Degrees were measured with finger widths.

Who should go ? Anybody who is interested in early navigation expedition! For readers who have know so much of Celestial Navigation, it will be good for them to know more of one the greatest forefather of the Celestial Navigation. They will be able to get more in depth knowledge on how Celestial Navigation was put into good use during his expedition.

When to go ? Date: Saturday 22 October 2005. Time: 3.00 –5.00pm. Why? Because there is a discussion on “What does it mean to be an Explorer in Singapore?” The panel discussion includes Stefen Chow from the NUS Mountaineers who recently returned from Everest, Wilfred Tok who has climbed the highest peak in the Americas, Anaconda; and Dr. Chan Yew Wing, who has led and coordinated numerous expeditions to remote areas. Ask them how they navigate using the modern technology and whether they have used any Celestial Navigation during their exploration.

_____________________________________For Heavenly Mathematics:In Conversation: Singapore Naval Officer

17


17

In conversation

Singapore naval officer, 2LT Jerry Lui

As celestial navigation is the topic of the month for our magazine, we would like to know more about the extent to which celestial navigation is used in RSN. Actually, with all the advance technologies on board the ship, we do not really use celestial navigation often. The only time in which we usually

employ celestial navigation is either during training of new naval officers or when all other modern navigation instruments fails and celestial navigation is deployed as a last resort. How was the learning process like? It was really really tough

In this issue, we interview a naval officer from Republic Singapore Navy regarding celestial navigation and his experiences. Celestial navigation is one of the most widely used forms of navigation in the olden times. Today, it has been replaced by many advance technologies such as Global Positioning System (GPS). To your knowledge, what are the types of navigation used in the Republic Singapore Navy (RSN)? Oh. There are a few types of navigation methods that we employ. From my knowledge, we use coastal navigation, astro-navigation, also otherwise known as celestial navigation, pilotage and ocean navigation. On top of that, we also use GPS to find out our current location if we are out at sea.


18

(Laughs) we really had to study very hard to pass the tests as it was rather hard to grasp the concepts and to differentiate the stars. Actually I made so many mistakes while I was learning it, but we all learn from mistakes don’t we? Oh I see. I understand that in celestial navigation, stars are used as reference points. Since it is so difficult to differentiate the stars from one another, is there any software that is used to calculate the stars that will be seen on any particular day? Oh yes! Definitely. There is this software called PC Sightmaster that we use. It is actually used quite widely on board cargo ships etc. This software is really accurate. All we have to do is to key in certain values such as time and date and place, it will calculate and actually churn out a list of stars that will be visible at the specific time and date.

You previously mentioned that one of the difficulties in celestial navigation is the spotting of specific stars in the night sky. So are there any other difficulties or barriers faced while applying celestial navigation? Celestial navigation is a very tedious process as we need to refer to many charts and books at the same time just to do our calculations. Thus it takes quite some time just to get our calculations done. Not only that, locating of the predicted stars as stated by the software PC Sightmaster is not easy as the stars may be too dim to be seen by the naked eye or the sky may be too cloudy. In addition to that, if the waters are too rough, it makes the measuring of the declination of the stars really difficult. Hence, celestial navigation is not as easy as it seems. Personally, what is your opinion of celestial navigation?

Although celestial navigation is not easy to master, however I find it really interesting as an ancient art of sea-faring navigation. I feel that with proper training and sufficient practice, it is the most accurate form of navigation known to man. It is pretty amazing how celestial navigation is still being used in this time and date after enduring through decades of technological advances. In some parts of the world today, especially Indian Navy and Royal Navy, it is still being used frequently as a form of primary navigation. Last but not least, is there any memorable moments that occurred while using celestial navigation that you would like to share with our readers? Certainly! There was this particular training session on board ship in the middle of the South China Sea where we had to calculate the ship’s position using celestial navigation. After much calculation, I finally got the coordinates of the ship’s position. To my horror, the coordinates was actually a point in mainland China! (Chuckles) That really proved to me that I should buck up. (Laughs) Thank you for you time, Sir. We certainly gained many insights.

For Heavenly Mathematics: Cover Story: Guide to Celestial Navigation

19

Guide to Celestial Navigation

How did sailors manage to navigate themselves in a vast featureless ocean centuries ago? Sophiscated equipments like Global Positioning System (GPS) and radars were all not invented then. Without this modern electronic equipment, sailors still managed to make their way across the ocean safely.

Our forefathers had great knowledge on the celestial. They were able to find their way n direction through the position of the celestial bodies. Skilled navigators were recruited on board ships during ancient voyages to help ships find their

way out in the sea. They used angular measurements between the horizon and a common celestial object. During the day, when stars cannot be seen, the Sun was most often used. At night, the Moon, planets or one of the navigational stars were

used. These celestial bodies are constantly changing their positions with respect to observers. Hence, nautical almanacs were important references for the sailors.

Able to find their way and direction through the position of the celestial bodies. _____________________


20

The Nautical Almanac The Nautical Almanac is a publication describing the positions and movements of celestial bodies that are directly overhead to the Earth’s surface at any hour of the day. The Sun, Moon and planets 'move' independently and hence are specified separately. However, for the stars, only Aries is specified, with the other stars having a set angular distance from that. The almanac gives the location data for 173 stars, but from this listing 57 stars have been chosen from amongst these on account of brightness and distribution in the sky; they will suffice for the majority of observations. The location of a star is given by its sidereal hour angle (S.H.A.) and its declination (Dec.). The locations given below are rounded off to the degree of angle and are an aid in finding the star on a star chart, but for sight reduction, more precise values will be needed from the Nautical Almanac. No. Name Mag. S.H.A. Dec. No. Name Mag. S.H.A. Dec. 1 Alpheratz 2.2* 358 N.29 31 Gacrux 1.6 172# S. 57 2 Ankaa 2.4 354 S.42 32 Alioth 1.7 167 N. 56 3 Schedar 2.5* 350 N.56 33 Spica 1.2* 159# S. 11 4 Diphda 2.2 349# S.18 34 Alkaid 1.9 153 N. 49 5 Achernar 0.6 336# S.57 35 Hadar 0.9 149# S. 60 6 Hamal 2.2* 328 N.23 36 Menkent 2.3 149 S. 36 7 Acamar 3.1 316 S.40 37 Arcturus 0.2* 146# N. 19 8 Menkar 2.8 315 N. 4 38 Rigel Kentaurus 0.1 140# S. 61 9 Mirfac 1.9* 309 N.50 39 Zubenelgenubi 2.9* 138# S. 16 10 Aldebaran 1.1* 291# N.16 40 Kochab 2.2 137 N. 74 11 Rigel 0.3* 282# S. 8 41 Alpheca 2.3* 127 N. 27 12 Capella 0.2* 281 N.46 42 Antares 1.2* 113# S. 26 13 Bellatrix 1.7* 279# N. 6 43 Atria 1.9 108# S. 69 14 Elnath 1.8 279 N.29 44 Sabic 2.6 103 S. 16 15 Alnilam 1.8* 276# S. 1 45 Shaula 1.7 97# S. 37 16 Betelgeuse var.* 271# N. 7 46 Rasalhague 2.1 96 N. 13 17 Canopus -0.9 264# S.53 47 Eltanin 2.4 91 N. 51 18 Sirius -1.6* 259# S.17 48 Kaus Australis 2.0 84# S. 34 19 Adhara 1.6 256# S.29 49 Vega 0.1* 81 N. 39 20 Procyon 0.5* 245# N. 5 50 Nunki 2.1* 76# S. 26 21 Pollux 1.2* 244 N.28 51 Altair 0.9* 63# N. 9 22 Avior 1.7 234# S.59 52 Peacock 2.1 54# S. 57 23 Suhail 2.2 223 S.43 53 Deneb 1.3* 50 N. 45 24 Miaplacidus1.8 222# S.70 54 Enif 2.5 34 N. 10 25 Alphard 2.2 218# S. 9 55 Al Na'ir 2.2 28# S. 47 26 Regulus 1.3* 208# N.12 56 Fomalhaut 1.3 16# S. 30 27 Dubhe 2.0 194 N.62 57 Markab 2.6 14 N. 15 28 Denebola 2.2* 183# N.15 29 Gienah 2.8 176 S.17 30 Acrux 1.1 174# S.63 * = Stars that are prominent for observers in the Northern hemisphere. # = Stars that are prominent for observers in the Southern hemisphere. Var. = Variable star, mag. = 0.1 to 1.2 - Note that many stars are visible North and South of the equator.


21

How Celestial Navigation Works

The fundamentals of celestial navigation are as follows.

Angles The most common type

of number used by a celestial navigator is angles, which can be used to describe the position of the celestial bodies and points on the surface of the earth. A common instrument used to measure angles is the sextant. Angles are usually measured in degrees, minutes and seconds. The complete circumference has 360 degrees (360°), with each degree equivalent to 60 minutes. The seconds of arc are not used in

the celestial navigation, because the sextant is not precise enough to measure them. The smallest unit of angle used by navigators is the tenth of minute, until recently, with the emergence of GPS devices, the 1/100 of minute is added.

To make the conversions between angles n distances easier, a unit called the nautical mile, which is equivalent to 1852 m is selected. One nautical mile corresponds

to an arc of one minute on the surface of earth. Hence, angles on the surface of the Earth are equivalent to distances. However, an exception is the minute of longitude, which is equivalent to one mile only near the Earth’s Equator. Another important equivalence is between time and degrees of longitude. Since the earth goes one complete turn (360°) in 24 hours, each hour corresponds to 15° of longitude or 900 nautical miles (NM).


22

The Earth and the Celestial Sphere.

Imagine that the Earth is the center of the universe, surrounded by a larger outer circle, centered at the same point. This larger outer circle is named the Celestial Sphere. The stars are fixed on the Celestial Sphere, as if they were painted in its internal surface.

We use a system of latitude and longitude to specify a position on the Earth’s surface. Latitude is the angle measured from the Equator in the North-South direction. Longitude is the angle in the Pole between the Meridian of Greenwich and that of the considered position.

A similar system called the celestial coordinate system is used for the Celestial Sphere. The angle analogous to the latitude in the celestial sphere is declination and its is measured in the plane North-South, from the Celestial Equator. The analog to the longitude is named Right Ascension or RA, which is measured from an arbitrary Meridian, called the Vernal Equinox.

Apparent movement of the stars The stars have nearly

fixed positions in the Celestial Sphere. The Sun, Moon and planets move around during the year, but when compared to their apparent movement due to the rotation of the Earth, their movement is slow. Consider for now that the celestial objects (stars, planets, Sun and Moon) are fixed in the Celestial Sphere.

Using the Celestial Sphere model, imagine the Earth is stationary and the celestial sphere is turning around it, completing a turn every 24 hours. The Earth's and Celestial Sphere's axes of rotation are in the same line. Both equators are, therefore, in the same plane. The stars are fixed on the celestial sphere and turn around the earth. The Celestial poles, being in the

axis of rotation, remain fixed in the sky. Therefore, a star located near a Celestial pole will appear to be stationary in the sky. Hence, Polaris, a star that is near the North Celestial Pole (its declination is 89°05' N) is always in the north direction, a fact known by every navigator. Unfortunately there's no corresponding bright star near the South Celestial Pole.


23

Finding the Earth position by observing the stars Consider a line connecting the center of a star and the center of the Earth. The point where this line crosses the surface of the Earth is the Geographical Position (GP) of this star. An observer positioned in the GP of a star will see it directly vertically overhead.

Because stars move with the celestial sphere, their GPs also move. The GPs of stars move quickly along the Earth’s surface, for example, the Sun's GP travels a mile every four seconds. However, the GPs of other stars, closer to the celestial poles, move more slowly, for example, Polaris.

Fig. 4- Geographical Position of a Star Since both the Earth’s equator and Celestial equator are in the same plane, the latitude of the GP is equal to the declination of the star. The longitude of the GP is known as Greenwich Hour Angle (GHA) in a reference to the correspondence between hours and longitude.

With the use of a Nautical Almanac, the GP of a star (its GHA and declination) in any moment of time can be determined. Of great importance is the exact time of the observation. As we have seen, 4 seconds may correspond to one mile in the GP of a star. Hence,

the importance of having a watch or chronometer with the correct time for the celestial navigation.

Another important point is the Zenith, which is the point in the celestial sphere located in the vertical, over the head of the navigator. The line that connects the Zenith and the center of the Earth crosses the surface in the position of the navigator. Hence, the following correspondence between points:

Surface of Earth Celestial Sphere Geographical Position of a star Center of the star Position of the navigator Zenith

In this figure, the GP of the star is represented by X and the Zenith by Z. The distance XZ, from the GP of the star to the point Z of the navigator is called Zenith Distance and it can be expressed in miles or degrees, since it is an arc on the surface of the Earth.


24

The angle that XZ makes with the True North (i.e. the "bearing" of the star) is called Azimuth (Az ).

The stars are at a great distance from the earth hence their light rays that reach the Earth are parallel.

Therefore, as illustrated, the distance XZ, measured as an angle, is equal to the angle that the navigator observes between the star and the vertical.

However, it is difficult to determine the Zenith distance precisely due to surrounding factors and environment, for instance it is difficult to find the vertical direction in a rocking boat. It is, however, a lot easier to measure the angle between the star and the horizon, also known as the altitude (H) of the star. The altitude of a star is taken with the sextant held in the vertical plane, measuring the angle between the horizon and the star.

The Zenith Distance and the GP of a star, however, are not enough to determine our position. With this data we can only say that our position is in a big circle, with the center in the GP of the star and radius equal to the Zenith Distance. This is known as the Circle of Position. Point X in the figure is the GP of a star.

Any observer located on this circle will see the star at the

same altitude, but different azimuths. Suppose the navigator

observes the star with an altitude of 75°, the Zenith Distance


25

would be 90°-75° = 25°. To determine this distance in miles, multiply by 60, since one degree is equivalent to 60 nautical miles (NM). Hence, the Zenith Distance or the radius of our circle is 1500 NM.

If the exact direction where the GP of the star is known, it would establish where in the circle the navigator is. Unfortunately, the compass cannot be used for this purpose, as it is not precise enough, with an error of just 3°,

common when reading a compass, corresponding to 78 miles of error, which is not acceptable.

Hence, the way to find the position is to draw two or more circles for two or more celestial bodies and see where they intercept each other. However, drawing these circles would require large charts! This problem can be solved by making a guess at our position. Using this assumed position,

the expected altitude for the star at a given time can be calculated using the Nautical Almanac.

This Calculated Altitude can then be compared with the Observed Altitude (the actual altitude, measured with the sextant). The difference is the error of our assumed position (also known as Delta ) in the direction of the star.

The Geographical Position of a star, normally being thousands of miles from our position, causes the circle of position to very large. Hence, the small piece that interests us - the one near our position - may be considered a straight line, orthogonal to the Azimuth of the star. This line is called the Line of Position or LOP.

From the measured altitude of a star at a certain time and our assumed position, a line of position can be drawn. The

navigator’s actual position is somewhere along this line. To determine this point, another line for another star can be drawn.

The point were they intercept each other is the navigator’s position or Astronomical Position.

Normally, for better accuracy, the navigator should repeat this procedure for yet another star. Due to minor imprecisions during the measurement, the three lines will probably not intercept in a single point, resulting in a small triangle. The navigator’s position is within this triangle, hence, the smaller the triangle, the more accurate. It is usually assumed that the Astronomical Position is in the center of the triangle.

In the figure above, we can see how three circles of position determine 3 Lines of Position r1, r2 and r3.


26

In traditional celestial navigation, the determination of a Line of Position involves the computation of the GP of the star (GHA and declination) using the Nautical Almanac and the solution of the Position Triangle PXZ, which is formed by the terrestrial pole (P), the GP of the star (X) and the assumed position of the navigator (Z).

This solution yields the Calculated Altitude and the Azimuth of the star, with the help of tables. The difference, in minutes of degree, between the calculated altitude and the altitude of the star measured with the sextant is the distance between the line of position and our assumed position - the error Delta of our estimate. This can be away or towards the star.

The GP of a star and the triangle of Position are solved by computerized formulas in Navigator software. All that is required is the sextant reading (date, time and altitude), name of the star and the assumed position (latitude and longitude).

Determination of the Astronomical Position

It is not necessary to draw the lines of position when using Navigator software. Instead, its can be done using pencil and paper:

1. Plot your assumed position. 2. Using a parallel ruler, draw a line passing on the assumed position, in the direction of the Azimuth of the star. 3. Over this line, measure the error Delta of the estimate - in the direction of the star or contrary to it - according to the sign of the Delta. 4. Draw the line of position, orthogonal to the Azimuth, at this point.

Detailed Nautical Charts are usually only available for places near the shore. When in high seas, charts with the adequate scale to plot the navigator’s position are not available. instead, special plotting paper is used instead.

When navigating using Navigator software, the computer determines the altitude lines interceptions and calculates the astronomical position. A simplified map is drawn, showing the parallels, meridians, lines of altitude and the astronomical position.

Altitude corrections

But before we can use this apparent reading in our calculations, some corrections must be made, in order to obtain the true observed altitude. These corrections are:

1) the height of the eye

2) semi diameter of the body (only for Sun and Moon)


27

3) instrumental error

4) atmospheric refraction

5) parallax (only for the Moon).

Most of these corrections depend only on the selected celestial object and altitude, hence they are performed automatically by Navigator software. The only information required by the program is the height of the eye (Dip) and the instrumental error. The application of these corrections to the instrumental altitude give the corrected altitude, the one used in calculations.

An observer located in a high place will see a star with an altitude bigger than that at sea level, in the same location. This error is called height of the eye (Dip).

Due to a small misalignment of the scale of the sextant, there is a sextant index error (IE). To read the index error, adjust the scale to 0°00.0' and point towards the horizon. Turn the drum until the horizon forms a single line. Then the index error can be read.

The index error can be positive or negative. The index correction has opposite signal (i.e. must be subtracted from altitude if positive and vice-versa).

Since the navigator is not in the Earth's center, but at its surface, the apparent object position is below the true geocentric position, resulting in parallax error.

Parallax is only meaningful for the Moon. Since other celestial bodies are so far, their parallax is very small.

Nautical Almanac data is tabulated for the centers of the celestial objects. However, it is easier to measure the altitude of the lower part of the body, for the Sun and the Moon, as illustrated. This is known as the lower limb. A correction, called semi diameter must also be applied in order to obtain the altitude of the center of the body. Sometimes, the upper limb is also used.


28

Other Methods of Celestial Navigation

Lunar Distance This older method was refined in the 18th century. Although it is an old method, it is a sound one, and can be used when a timepiece is not available or its accuracy is suspected during a long sea voyage. The navigator precisely measures the angle between the Moon and a body like the Sun or a selected group

of stars lying along the ecliptic. That angle, after it is corrected for various errors, is the same at any place on the surface of the earth, facing the moon at a unique instant of time. The navigator could search through the almanac to find the angle measured, and thus know the time at Greenwich. The

calculation in minutes can be done using simple calculators, allowing the navigator to use other acceptable celestial bodies than the old nine listed in the almanac. Knowing Greenwich Time, the navigator can work out longitude.

The Use of Time The considerably more popular method was (and is) to use an accurate timepiece to directly indicate the time of a sextant sight. The need for accurate navigation led to the development of progressively more accurate chronometers in the 18th century. Time is measured with a chronometer, a quartz watch, or a short wave radio broadcast from an atomic clock. A quartz wristwatch normally keeps time within a

half-second per day. If it is worn constantly, keeping it near body heat, its rate of drift can be measured with the radio, and by compensating for this drift, a navigator can keep time to better than a second per month. Traditionally, a navigator set his chronometer from his sextant, at a geographic marker surveyed by a professional astronomer. This is now a rare skill, and most harbor masters cannot locate their harbor's marker.

Traditionally, three chronometers were kept in gimbals in a dry room near the center of the ship. They were used to set a watch for the actual sight, so that no chronometers were ever risked to the wind and salt water on deck. Winding the chronometers was a crucial duty of the navigator, logged as "chron. wound." for checking by line officers. Navigators also set the ship's clocks and calender.

Modern Celestial Navigation Modern celestial navigation methods, developed in the 19th century, allow navigators to measure latitude as part of a fix, by sighting 2 or 3 bodies within a few minutes of each other. When the celestial line of position concept was discovered in 1837, leading to the modern mathematical

"intercept method" of reducing sextant sights later in the 19th century, navigators were finally able to obtain both latitude and longitude at once from their celestial observations, giving them a position "fix", rather than viewing these two properties as separate entities. At the turn

of the 21st century, advanced electronic and computer systems had evolved enabling navigators to obtain automated celestial sight fixes. These rare systems are regarded mainly as potential backups to the more reliable satellite positioning systems, but could play a role in future space travel.

For Heavenly Mathematics: Instruments for Celestial Navigation

29

Instruments for Celestial Navigation

Before there were Global Positioning System and Radio-transmission, these are the equipment that kept sailors from being lost in the open sea. We will take a look at them and try to find our way into using them!

ASTROLABE: used to measured

the altitude of the sun or stars, later replaced by the sextant.

To determine latitude with an astrolabe, first a sailor would identify the constellations visible in the sky nearest the sun at daybreak. Then at noon, when the sun is at its highest point, the sailor would measure the degree difference between the sun and the horizon with the astrolabe's movable needle. Then with the help of a chart like Abraham Zacuto's Rules for the Astrolabe published in 1473, our sailor would use both the constallations and the sun's height in the sky to determine latitude.

SEXTANTS are instruments used by mariners to measure the angular distance between two objects (like the sun and the horizon).

They are more accurate than astrolabes by having a telescope, mirrors, and a graduated arm. The mirrors allow seeing both objects simultaneously. They also surpass the cross-staff and the quadrant by being able to measure angles up to 120 degrees (the quadrant only reads up to 60).

QUADRANTS used for measuring the angle between the horizon and the sun (or other celestial body). Invented in 1731, it had an arc that measured 1/8th of a circle plus two mirrors that allowed the sun or another star and the horizon to be viewed simultaneously. Quadrant can measure angles up to 90 degrees, for example, from the horizon to an object directly over head. This angle is used to determine latitude. The quadrant is the precursor to the sextant.

For Heavenly Mathematics: Instruments for Celestial Navigation

30

COMPASS, do I need to explain further? The astrolabe or sextant can help you find your latitude, but to determine the direction of your future course, the compass is a faithful friend. It was first used in marine navigation starting in the 1400's.

CHRONOMETER solved the great problem of finding longitude. The first seaworthy chronometers were created by an 18th-Century Englishman, John Harrison.

GLOBAL POSITIONING SYSTEM Watch allows you to find the exact position you are by just pressing a button. It comes with electronic compass, thermometer and barometer all in one watch!

STELLARSCOPE is a compact, handheld, easy-to-use educational guide to the stars and constellations. View the night sky for any date and time in either the Northern or Southern hemispheres by simply setting the time and date dials on the scope, point the scope towards a light source, and you will see the stars and constellations appearing in the sky on that night.

____________________________________________________For Heavenly Mathematics: Astro-Book Review

32

Astro-Book review

We have done a few review on Celestial Navigation related books. How this will help when you are thinking of investing in one!

the complete on-board Celestial Navigator

by George C. Bennett

The ancient skill of celestial navigation has reliably guided seafarers for centuries and is still a source of pride and respect among mariners. In this era of electronic navigation, it remains the perfect backup system, enabling you to determine your position if the GPS malfunctions or your boat loses electrical power.

Specifically designed for the novice or rusty celestial navigator, The Complete On-Board Celestial Navigator replaces $300 worth of guides and almanacs traditionally required by celestial navigators, and includes a star finder; a five-year nautical almanac (2003 - 2007) for determining precise star, sun, moon, or planet locations at the time of sighting; and sight reduction tables for crunching the numbers and producing a fix anywhere in the world.

"The only complete self-contained work available. When your electronic navigation fails, 'steering by the stars' will guide you safely

to your destination."--Sydney Afloat (Australia)

Norie's nautical tables

by A. Glance This famous set of mathematical tables has been modernised and this 1983 edition contains a fully revised list of Ports of the World and their geographical positions. This book is so reliable that most Naval Officers use them to do their Celestial Navigation. Definitely a must-have for everyone who wants to start on learning Celestial Navigation.

For Heavenly Mathematics: IT Section The Navigator Software

33

I T Section: The Navigator Software

In this issue, we will introduce one of the software available in the market that can provide navigation helps when we are out in the sea. We will explain the features of the Navigator.

Star Finder

The figure below shows the Navigator's star finder window. This is a polar chart of the visible sky in a given time and position. You can point a star and see its data (Name, Altitude and Azimuth). Or you can point on the spreadsheet and see the corresponding objects on the chart.

The center of the chart is the Zenith and the external circle is the horizon. The two blue concentric circles are the 30° and 60° altitude. The best stars to observe are between these two circles (altitudes between 30° and 60°).


34

Line of Position

Calculating a Line of Position is easy. Input:

• Date and GMT time of the observation. You can also input the local time and click GMT!. The program will calculate the GMT time, adding the time zone and watch error. Click Now! to fill with the current date and time. • Select the star observed. • Instrumental altitude of the star. For the Sun and Moon, use the altitude of the lower limb, as shown.

• Estimated position (Latitude and Longitude). • Index Error in minutes and Height of the eye in meters or feet.

Press Calculate. The LOP will show in the chart, as illustrated.

All input fields and buttons have hints that explain what they are and what units should be used. Place the cursor over the field to see the hint.


35

Astronomical position

After you calculate two or more Lines of Position, you can calculate your astronomical position. The figure bellow shows an astronomical position calculated with 2 LOPs. Note the blue chart, showing the LOPs diagram. The astronomical position is indicated by the white circle. Use the cursor to zoom in and out. Auto transport LOP feature makes calculating a running fix an easy task.


36

Chart Navigation

Navigator Chart Viewer can work with vector and raster charts. Vector charts can be produced with ChartMaker program (also included in the registered version). There are a couple charts available in our chart library. It can also import popular raster image formats, like GIF, JPG and BMP. None of the other proprietary chart formats is supported at this time.

In the Navigator chart viewer you can:

- View multiple charts, integrated in a whole world vector chart background. - Draw routes and calculate leg distances and courses - Connect a GPS and save your track on-line - Draw marks, with text. - Calculate rhumb lines and lines of great circle and generate routes.

Chart viewer showing a raster chart zoom animation

____________________________________________________For Heavenly Mathematics: Astro-Net Review

37

Astro-net review

Trying to search the net for help? Do not know which are the reliable ones? Hear what we have to say!

www.celestialnavigation.net

aa.usno.navy.mil U.S. Naval Observatory – Astronomical Applications Department. The name says it all. It provides a lot of information on the sun and moon. This includes providing azimuth, altitudes, moon rise and set timing, sun rise and set timing, etc just by keying in the longitude, latitude and timing that you are in. You may think that the loading of the page will take you a long time since they can provide so much information. But to your surprise they uses the most simple and least needed memory to give you all the data you need. Loading of results takes you not more than a few seconds!

This website although is simply designed but it is filled with all the necessary knowledge you need for Celestial Navigation. It provides a wide range of subjects from history of navigation to Celestial Navigation practice session. Apart from these, there are numerous good links to other website if you found this website insufficient to satisfy your thirst for knowledge. However, this website seems to be last updated on 2001. I hope the webmistress, as the author calls herself; will be updating this website again!

Crossword puzzle

Theme: Celestial Navigation

Across 1. A publication describing the positions and movements of celestial bodies. 4. A great circle passing through the two poles of the celestial sphere and the zenith of a given observer. 5. The angles in the Pole between the Meridian of Greenwich and that of the considered position. (Plural) 9. The point in the celestial sphere located in the vertical, over the head of the navigator.

Down 1. An old instrument used to measure the altitude of the sun or stars 2. The angle that a star makes with the True North 3. Norie’s Nautical _______s. 4. One of the important celestial body used for Celestial Navigation at night. 6. The longitude of the Geographical Position is known as _______wich Hour Angle 7. The 27th celestial body in the Nautical Almanac. 8. Number of voyages Admiral Zheng He made.

_______________________________________________________For Heavenly Mathematics: Just for a Smile

40

Just for a smile Cartoons

All these won’t happened if they know Celestial Navigation. Cartoons courtesy of www.cartoonstocks.com

Jokes When Mr. Leno of the Tonight Show went J-walking and asked pedestrians some science questions, he discovered some amazing new facts about the universe: Jay Leno: "Why does dew appear on plants in the morning when the Sun comes up?" A waitress: "Is it because the Sun makes them perspire?" Jay Leno: "Why does the Moon orbit the Earth?" An auto mechanic: "To get to the other side?" Jay Leno: What are magnets?" A taxi driver: "Are they the things crawling over a week-old dead cat?" Jay Leno: Which is more useful, the Sun or the Moon?" A thirteen-year old: [Pause] "I think it's the Moon because the moon shines at night when you want the light, whereas the Sun shines during the day when you don't need it." Jokes courtesy of www.jupiterscientific.org

___________________________________________________________For Heavenly Mathematics: Promotion

41

PROMOTION!

Celestial gifts of the month

FREE Trip to watch POLARIS! Sponsored by Spears Travel. This is a rare chance for you to visit the North Pole and view the Polaris. An experienced guided tour guide will be along with you. This special tour will cost you more than S$5000, but one lucky winner with the correct answer will walk away with this trip. Question: Who invented Chronometer? Email your answer along with your personal particulars to [email protected] Closing Date: 30th October 2005

5 sets of Book vouchers worth

***********************************************************

Other free gifts

350mm Telescope

Features:

• Color: Grey • Size: 17" L x 4"

diameter • Has an 80mm objective

lens with a 350mm focal length, and is equipped with 2 Kellner eyepieces.

SMS <TEL><NAME><NRIC No.><email> to 73888

E.g. TEL Celest 7912345G [email protected]

Closing Date: 30th October 2005

One lucky winner will bring this home!

Answer to Crosswords Puzzle

5 sets of Book vouchers worth S$50 each will be given to find lucky winners. Sponsored by Astrobooks. Just tell us in less than 200 words on why you want to have these book vouchers. Remember to include your personal particulars and email to [email protected] Free 2 years subscription to For Heavenly Mathematics magazines. Just send in the magazine barcodes from the last 3 months and you will stand a chance to win! Mail to Blk 38 Starry Road Astro Building Singapore 200038

Want to subscribe to our magazine? 2 years subscription for S$90. Save more than S$15! To subscribe: www.fhm.com/subs

Documents

For Heavenly Mathematics: Astrobox