UNIT II

Where Are We?

J. Fey
Office in 3113 Mathematics Building
Phone: x53151 (voice mail)
E-Mail: JF7@umail.umd.edu

In the search for scientific theories that explain our observations and experiences of the world around us, few things have sparked as much interest as the simple question, "Where are we?" Throughout recorded history (and probably long before), people in all places and cultures have puzzled over the size and shape of the planet and the location of our Earth among the planets and stars of the heavens. What do you know or believe about:

As fascinating and important as are many facts of life on Earth and in the solar system, it is equally intriguing to ask how and when we came to current understandings of our place in the universe. What do you know or believe about the timing and methods of thought used in discovery that:

The answers to these fundamental scientific questions are generally considered part of the discipline called astronomy. However, the methods used by astronomers rely heavily on numerical and geometric ideas from mathematics. This unit explores scientific and historical connections of mathematics and astronomy in four investigations titled:

Latitude and Longitude
 Days and Nights, Years and Seasons
 Shape and Size of the Earth
 Where in the World is Christopher Columbus?

The goal is to explore the basic ideas and processes that have been involved in the long and fruitful search for mathematical models of our Earth and the solar system.

2.1 Latitude and Longitude

If you had to tell someone how to locate College Park, Maryland, it's unlikely that you would say, "It's at 77 degree west longitude and 39 degree north latitude." But as an international scheme for describing positions on earth, the (longitude, latitude) coordinate system has proven itself as an invaluable tool for navigation on land, sea, or air. It's a system that we take for granted, but

It's hard to recreate the human experience of exploration and discovery that led to our current global positioning system. But by analyzing the way that lines of longitude and latitude are drawn on a model globe, we can make some guesses at the origins of this important mapping system. To answer the following questions, and those of subsequent investigations, you'll probably need a globe map of the Earth, several feet of string, and rulers for length and angles.

1. To get started, use the globe to find approximate map coordinates for the following major cities around the Earth: 

 
a)   Chicago, Illinois	b)   Honolulu, Hawaii	c)   Tokyo, Japan
d)   Sydney, Australia	e)   Kinshasa, Zaire	f)   Sao Paulo, Brazil
g)   London, England	h)   Cairo, Egypt	i)    Moscow, Russia
2. One natural question about map coordinates is whether you can tell the distance between two points from the difference of their coordinates. Collect some data from your globe to explore this question:
  1. Describe the pattern of data relating latitude and distance from the Equator. How, if at all, does it allow you to predict north-south distancebetween two points from the latitude coordinates?

  2. Locate the two special lines of latitude in the northern hemisphere--the Tropic of Cancer and the Arctic Circle.
    1. Measure the distance on your globe from the Equator to each of those two latitude lines.
    2. Use the pattern of (latitude, distance) data from your measurements in (a) to estimate the latitude of the two special lines.
    3. If you see any interesting connection between the Tropic of Cancer and Arctic Circle latitude numbers, see if you can come up with an explanation of why that relationship occurs. If you don't see anything noteworthy right now, keep on the lookout in the investigations ahead.

3. You know, or have noticed from the globe, that latitude is reported as a measurement in degrees. That suggests that angles are involved in establishing the lines of latitude. What angles do you believe are being measured?

  1. Think again about the possible angles being measured when lines of latitude are given their degree measures. Record your ideas
  2. Any more thoughts about the Tropic of Cancer and the Arctic Circle latitudes?
4. The circles on the globe that pass through both North and South Poles are called lines of longitude. Those lines are also measured in degrees.
  1. How do you think degree measures are assigned to lines of longitude?
  2. Do some experimentation to see if you can tell the east-west distance between two points on the globe from their longitude coordinates.
5. Having studied the relation between measures of latitude and longitude and distance on your globe, scale up your findings to facts or principles about distance on the whole earth.

  1. If one point is directly north or south of the another, how can you predict the distance from one to the other, based on knowledge of the latitudes of each?
  2. If one point is directly west or east of another, how can you predict the distance from one to the other, based on knowledge of the longitudes of each?
6. A family in the northern hemisphere sets out from their home and walks 3 miles due south, then 2 miles due west, and then returns home by walking 3 miles due north. Along the way they see a bear.
  1. What color was the bear?
  2. Could a family in the southern hemisphere make a similar trip?

Conclusions and Connections--Now that you've reviewed the structure of our familiar latitude and longitude coordinate system for locating positions on the Earth, think about the ideas that had to be developed before the current system could be organized and what problems remained for navigation even with the notion of latitude and longitude developed.

1. What big ideas about the size and shape of our Earth seem prerequisites for defining longitude and latitude, and in what order do you think those ideas were discovered?

2. The system of latitude and longitude coordinates is only one of many locator systems used in a whole variety of tasks. Think about and describe several other coordinate locator systems that you are familiar with. Then compare them with the basic features of Earth longitude and latitude coordinates.

3. What is it about 90o, 180o , and 360o that makes them common in angle measurement?

4. The Equator is the line (circle) of latitude with measure 0; the prime meridian is the line (circle) of longitude with measure 0. Which of these two great circles pretty much has to have a coordinate of 0o and which has that coordinate pretty much by historical accident?

5. How do you think satellite dishes for television are "tuned in"?

6. The U. S. Defense Department now has a global positioning system that uses orbiting satellites and a portable piece of apparatus on the ground that can be carried by an individual soldier. It will tell the soldier his or her location on the Earth to within 1 meter of his true location! How do you suppose it does that?

2.2 Days and Nights ... Years and Seasons

In the search for mathematical models of our solar system, one of the toughest problems was figuring out how (if at all) the Earth, the Moon, the Sun, the other planets, and the stars are moving.

These are very hard questions, but scientists (and lay people also) conjured many theories and had heated debates about them.

You can get an understanding of the present-day theories by simulating the Earth and Sun with a globe and a flashlight. Of course, those crude simulation tools only suggest what is going on. You have to imagine the experiments scaled up many times to get a picture of what is really happening.

1. To get a rough idea of the kind of scale model that would be needed to do accurate Earth-Sun simulations, consider these data:

  1. If your globe is 1 foot in diameter, what diameter would you need for a model Sun?
  2. How far from your Earth model should your Sun model be placed to have the model be proportional to the real thing?
2. Your answers to question (1) might convince you that a globe and a flashlight will tell nothing about the real Earth-Sun relations. But with some care and thought, you'll see illustrations of many important facts of the solar system. Hold your flashlight several feet from the globe and experiment with movement of those Sun-Earth models to simulate various times of day and night and various seasons of the year.

It was not until the late 1950's that we were able to observe the solar system from orbits far above our Earth's surface! Think about how an earlier Earth-bound observer might reasonably get the following ideas about solar system phenomena:
  1. That the Sun is rotating around the Earth
  2. That the Earth itself is rotating on an axis
  3. That the Earth is rotating around the sun
  4. That the Earth moves around the Sun with its center in a fixed plane
  5. That the Earth's axis is tilted at an angle to the orbital plane?

3. Although it took thousands of years for scientists to realize (believe?) that the Earth rotates around the Sun, not the other way around, let's assume some modern knowledge of Earth-Sun relations to figure out why days and nights and years and seasons behave the way they do. The Earth moves in an elliptic orbit and the plane of that orbit cuts through the center of the Sun. At any point in the orbit (any time of year), the Earth's axis of rotation is parallel to its position at any other point in the orbit.

Use this model of Earth-Sun relations and your flashlight-globe simulation to study the following questions about familiar seasonal phenomena. Move the globe around to different positions in relation to the flashlight "Sun" and observe differences in the way the Sun illuminates the Earth as it rotates on its own axis. Record your ideas with sketches that help to describe events and their explanations.
  1. At noon on March 21 and September 21 (the spring and fall equinoxes), the Sun is directly overhead at points on the Equator. Where is the Earth in its orbit around the Sun on those dates, and why do all points on the Earth experience equal amounts of daylight and darkness on those days?

  2. At noon on June 21 (our summer solstice in the northern hemisphere), the Sun is directly overhead at points on the Tropic of Cancer. Where is the Earth in its orbit around the Sun on that date and why do some points on the Earth experience 24 hours of sunlight and others 24 hours of darkness?

  3. At noon on December 21 (our winter solstice in the northern hemisphere), the Sun is directly overhead at points on the Tropic of Capricorn. Where is the Earth in its orbit around the Sun on that date and why do some points on the Earth experience 24 hours of sunlight and others 24 hours of darkness?

  4. Why is it that in the northern hemisphere, June, July, and August are generally the warmest months, while December, January, and February are coldest?

  5. How are the latitude measurements of the Tropic of Cancer and the Tropic of Capricorn related to each other and to the tilt of the Earth's axis, and what connection does this relationship have to seasonal variations in daylight and darkness on the Earth? What is the comparable answer for the Tropic of Capricorn and the Antarctic Circle?

Conclusions and Connections --The flashlight and globe experiments use a physical model to study the solar system. It gives a general qualitative idea of how the system works. To describe things more precisely and to make predictions, it helps to have a model of the Sun and planets that can be described with geometric shapes and equations.

1. What geometric shapes and measurements play key roles in a solar system model and in explaining days and nights and years and seasons?

2. Because the Sun is so much larger than the Earth and so far away from the Earth, it is hard to make true scale model drawings of the relationship between them. However, the following drawing (inaccurate as it is in some respects) can be used to describe some important aspects of the relationship. Imagine that you are looking at a side view of the Sun and Earth with your eye directed along the orbital plane.

  1. On sketches like this, draw in the Earth's axis and Equator as you would see them from such a side view at each of the four key points in the year: The summer solstice, the fall equinox, the winter solstice, and the spring equinox.

  2. Explain how the sketches demonstrate the different sunlight hours at different seasons. In particular, how parts of the northern hemisphere have 24 hours of sunlight at the summer solstice and 24 hours of darkness at the winter solstice.

  3. Explain how the sketches show the special properties of the Tropics of Cancer and Capricorn and their relation to the Equator and the Arctic and Antarctic Circles.

2.3 Shape and Size of the Earth

The Earth is round. Are you sure? Why do you think so? Why would a resident of Nebraska (one of our flatter states) think so? When and how do you think astronomers, geographers, and explorers came to believe that the Earth is round?

When Columbus proposed his westward trip from Spain to China, via the Canary Islands off the coast of North Africa, his estimate of the distance was not very accurate and there was a large land mass blocking his planned route. However, he was right that the Earth was round. Isabella and Ferdinand of Spain financed the voyage, over the objections of their royal advisors who believed the Earth was round but thought Columbus was mistaken about the distance...which he was!

As early as 240 B. C. it was commonly believed that the Earth was round and various people had given estimates, or perhaps their guesses, as to its size. In fact, in 240 B. C. Eratosthenes, the librarian of Alexandria produced a well reasoned estimate placing the Earth's circumference at about 25,000 miles (using more modern units).

Eratosthenes' Bright Idea

The city of Alexandria in Egypt was founded in 332 B. C. by Alexander the Great. After Alexander's death nine years later, Ptolemy I emerged from the conflict among potential successors to establish control in Egypt with a government based in Alexandria. A succession of Ptolemys built Alexandria into an enormous cultural center. A center of learning was established where, for example, Euclid taught mathematics. Eratosthenes was the librarian responsible for building a collection of the best thought from around the world. Among the information he collected were some facts that led to his impressive deduction about the size of the Earth.

The questions that follow give a series of challenges and hints that allow you to trace Eratosthenes' reasoning about the size of the Earth and then to move farther back in history to see how one might first reckon the Earth to be round.

1. Imagine that the year is 240 B. C. and you live in Alexandria. You accept the general idea that the Earth is round, so you can imagine it marked up with lines of latitude and longitude, but you don't know its size. First bright idea: If you knew the connection between change in latitude and distance on the earth's surface, you could estimate the circumference of the earth.

If a one-degree change in latitude corresponds to a known distance d, how would you calculate the circumference of the Earth?

2. Imagine next that someone phones you from a place due south of your home and says "Whew, is it hot! I just went outside and the Sun is directly overhead." You hang up and go outside to find that the Sun is not directly overhead where you live.

  1. How would your friend know that the Sun was directly overhead?
  2. How would you know that the Sun was not directly overhead where you live?
  3. How could you use the observations by you and your friend to estimate the change in latitude between your homes?
This is a hard problem ... it's the key to Eratosthenes discovery. However, if you think about your flashlight Sun shining on a globe and make some sketches of the situation, you might be able to guess (if not prove) the answer. A good guess about the truth is usually the essential first step in finding a logical argument.

3. One observation that might help your reasoning is the fact that, because the Sun is so large and so far from Earth, its light comes in a beam that can be thought of as parallel light rays.

How, if at all, does this assumption change your thinking about using Sun rays to measure change in latitude between two spots on the Earth?

4. The following sketch shows a side view of the Earth with Sun rays falling at two places, one directly north of the other. The Sun is directly overhead at point A, but not directly overhead at point B. Suppose that it is you standing outside in the Sun at point B and your friend at point A.

  1. Do you see anything in the sketch that you could measure to find the change in latitude from point A to point B?

  2. The next sketch shows two points, neither of which lies directly under the sun's rays. Is it still possible to find the change in latitude from point A to point B by suitable clever measurement in this case?

5. Eratosthenes did not receive a phone call! However, he knew some facts that allowed him to reason with much the same information that your phone call from a friend could provide.
  1. Locate the cities of Alexandria and Aswan in Egypt on a map or globe. In 240 BC. the city of Syene was located at the site of present-day Aswan. Residents of Syene noticed that on June 21 of every year at about mid-day the Sun's rays were reflected from the bottom of a deep well. What does that tell you about the latitude of Syene?

  2. On June 21 at mid-day in Alexandria, Eratosthenes measured the angle between the Sun's rays and "straight up" by looking at a triangle formed by the Sun's shadow on a tall building.

    He found the angle to be about 7f(1,5) o . What does that tell you about the latitude of Alexandria, in relation to Syene?

  3. Eratosthenes then learned that the distance on land from Alexandria to Syene was a 50 day camel trip and that a camel could cover about 10 miles (in modern units) per day. What does this information imply about the distance from Alexandria to Syene and the distance around the Earth?

6. Now turn to the even older question of determining the shape of the Earth. How could people have become convinced that it is a ball and not a flat disk?
  1. What happens in a lunar or solar eclipse and how would the images seen in such an event suggest that the Earth, Moon, and Sun are round?

  2. Perhaps you are not a stargazer ... or maybe you are. People spent more time outdoors two thousand years ago. They were fascinated by star patterns. Using your twentieth century knowledge, what might observant stargazers have noticed as they traveled? How would those observations suggest something about the shape of the Earth?

  3. Imagine a ship with a tall mast sailing directly outward from land over a relatively calm sea. If you watch for a long time from sea level, what would you notice? How would this observation suggest a round Earth to people living thousands of years ago?

Conclusions and Connections -- The questions of this investigation have been designed to help you follow the course of discovery that led to good estimates of the shape and size of our Earth. Test your grasp of the main ideas by answering these questions:

1. The latitude and longitude of Washington, DC are about 39o N and 77o W; the location of Nassau in the Bahamas is about 25o N and 77o W. How far apart (in miles) are the two cities?

2. Chicago, Illinois is located at about 42o N and 87o W while Managua, Nicaragua is at about 12s (o) N and 87o W. How far apart (in miles) are those two cities?

3. The city of Madison, Wisconsin is located at about 89o west longitude and New Orleans, Louisiana at about 90o west longitude. How could you get friends in those two cities to collaborate with you to make the measurements needed in Eratosthenes method of estimating the circumference of the Earth, using the fact that it is about 1000 miles on land from Madison to New Orleans?

4. Without access to a modern clock or telephones to communicate, how could observers at Syene and Alexandria know that their Sun shadow observations were occurring at the same time on the same day?

5. Without knowledge of the magnetic compass, how could people have known about directions of North, South, East, and West?

6. On the following sketch of parallel lines cut by a transversal, what pairs of angles are congruent and how do those facts play a critical role in Eratosthenes' method for estimating the circumference of the earth?

2.4 Where in the World is Christopher Columbus (Going)?

A lot happened between 240 B. C. and 1492 A. D. During the Renaissance, the various nations of western Europe, particularly Portugal, Spain, France, and (the city states of) Italy, found trading with China and India desirable. However, between those European nations and their prospective Asian trading partners lay several other countries inclined to block the trade. So, since the Earth is round, why not go west instead of east to reach the same spot?

Of course, there were other possibilities. Working out a friendly and mutually beneficial relationship with the other countries did not seem to be considered. Perhaps it would be possible to sail around the southern tip of Africa and then up through the Indian Ocean. At the beginning of the fifteenth century Europeans were not sure that there was a southern tip of Africa. (If this sounds silly, remember that explorers were later to spend much effort trying to find a Northern Passage over the top of North America. It was perfectly reasonable to think that Africa might extend into the southern polar regions.) But the tip was reached in 1488, and then Vasco de Gama successfully reached India by that route in 1498. Thus it was de Gama who succeeded in doing what was actually intended. Nonetheless, Columbus, in error about the size of the Earth and believing that he was off the coast of Asia when he was really in the Caribbean, changed history. Such is often the way of major discovery and change.

Accounts vary as to exactly how Eratosthenes' excellent estimate of the size of the Earth got reduced to 18,000 miles by Columbus. Columbus also planned to start in the Canary Islands to sail due west at the latitude of about 30o--making the projected trip considerably shorter than it would have been at the Equator. Further, he thought that the eastward distance to the coast of China was longer than it actually was, and this led him to a corresponding reduction in the westward distance estimate. At any rate, Columbus had it figured that Japan was about 2700 miles west of the Canary Islands. Even if, as many Europeans believed, Japan was a fictional place made up by Marco Polo, it wouldn't be much farther to China.

Columbus' proposed trip was controversial. He had been turned down by Portugal, France, and England when he turned to Ferdinand and Isabella in Spain. The royal advisors in Spain were opposed to the trip. Remember, the issue was whether this was a practical way to get to China and India. No one had yet thought of plundering a whole new continent.

Advise the King and Queen

Imagine that you are a Royal Advisor to Their Majesties Queen Isabella and King Ferdinand of Spain. The Queen calls you in one day and requests that you design an experiment to discover the true size of the Earth.

1. How could you reproduce the experiment of Eratosthenes without leaving Spain?
Use whatever current globe or atlas information you need and write up a careful explanation of your method and conclusions.

2. Use your result from (1) and further reasoning to figure the distance around the Earth at a latitude of 30o. The following sketch might be a helpful guide in calculating the circumference of that circle of latitude.

Again, write up a careful explanation of your method and results.
HTML coding by Shuwen Wang, April, 1995.