Introduction: Because of the position of Polaris (the North Star) directly above the earth’s north pole, observed height of Polaris above the horizon (altitude in degrees of arc) is equal to latitude. Navigators have taken advantage of Polaris for at least 2000 years. Sailors in the Middle East and Southern Asia would measure the altitude of the North Star when they were in a certain port by using a kamal. This was made by tying a piece of wood to a string and holding it at arm’s length. The sailor would then move the wood toward or away from his eyes until the bottom just touched the horizon and the top touched the star. He would mark this distance by making a knot on the string. When sailing, he would compare the altitude of Polaris to the known measurement to determine if he was at the correct latitude or had to sail north or south to arrive at the port. Because the sizes of each kamal was different, each navigator would make one for his own use. The sun can also be used to find latitude, but since its height at noon varies north and south during the year, corrections must be made to sun observations.
Even in Columbus’ time, latitude was relatively easy to determine by observation of Polaris or the sun. There was no satisfactory method for determining longitude until the late 1700s. For this reason, many early explorers “sailed the latitudes;” that is, they sailed north or south until they reached the latitude of the desired destination, then sailed east or west until they arrived or ran into some other land, as Columbus did.
Instead of using a kamal, navigators in Columbus’ time used the astrolabe and cross-staff. Both are used to find the angle of the sun or a star above the horizon. A sextant (named for its arc, one-sixth of a circle) is used to measure angles very precisely. Until filters and reflectors were built into navigation instruments, many navigators damaged or ruined their sight in one eye through observing the sun’s altitude.
Students can measure the altitude of Polaris and find their latitude using a protractor, or very roughly using their hands, as shown in the activity Azimuth & Altitude.
What to Expect: It is great fun to use a protractor or a cross-staff to measure altitudes. Have the students practice with both in school, then try them at home on a clear night. Be sure they know how to find Polaris – it’s not an especially bright star.
Students should find a location with a horizon as distant and flat as possible – the angle measured from the tops of trees to Polaris would be too small, and from a nearby spot on the ground, the angle would be too large.
For a special occasion, invite parents and students to do some star-gazing in a large open area some evening. Learn some star names and stories or bring along someone who knows them. Have students show off their navigation skills and tools. Winter evenings are cold, but dark falls early and young students need not be up too late while enjoying the stars.
PART I: The Protractor
– Protractors (one for each student)
– Nut or other small weight
– Paper star to use for practice
1. Tie a piece of string about 15 cm long to the straight edge of the protractor.
2. Tie a small weight, such as a nut, to the end of the string to hold it vertical.
3. Hold the straight edge of the protractor near your eye and point it directly at Polaris. You can use a paper star taped high on a classroom wall for practice, or go outside use the top of the school flagpole.
4. Pinch the string against the side to mark the angle, then read the angle on the protractor.
5. Use the number that is less than 90° . Subtract this number from 90° . This is your latitude.
PART II: The Cross-Staff
– Yard stick or meter stick
– Paint stirrer
– Thick rubber band or ponytail band
– Masking tape
Cover one side of the meter stick with masking tape. This will provide a surface for marking angles and positions of the cross-piece.
Fasten a paint stirrer or similar short, straight stick to the meter stick using a rubber band. The two should be perpendicular, and the fastening should permit the smaller stick to slide along the meter stick.
Tape a paper star up on a classroom wall, or use a high point outdoors to represent Polaris.
Place the end of the meter stick close to one eye and slide the cross piece until it touches the horizon at the bottom and Polaris at the top.
One way to measure the angle you’ve observed (recommended for younger students) is to lay the cross-staff on a piece of paper and draw a straight line from the top of the cross-piece to the end of the meter stick, and another from the bottom of the cross-piece to the same point on the end of the meter stick. Remove the cross-staff and measure the angle with the protractor.
Another way to measure the angle you’ve observed is to set up the meter stick like a real cross-staff, with markings on it for each location of the cross-piece corresponding to an angle. These can be found by drawing angles and sliding the cross-piece along the meter stick until it just touches the top and bottom of the angles, then marking that place on the meter stick.
A third way is to calculate the correct distances using trigonometry. Measure the cross-piece. Half of this length will be the value of the opposite side (a) in a right triangle. Because the cross-piece extends above and below the cross-staff, you must work with half of the angle to be measured when calculating where to mark the cross-staff. For example, to find the location of the 50° mark on the cross-staff, use the tangent function for 25° . Let’s assume half the cross-piece is 20 cm long.
tan q = opposite/adjacent
tan 25° = 20/adjacent
.466 = 20/adjacent
adjacent = 20/.466 = 42.9 cm
Evaluation: Each student will demonstrate correct use of the protractor or cross staff, each student will demonstrate correct measuring and reading of the angle, and each student will identify the measured latitude on a map or globe and name some geographic point of interest on that latitude.
Extensions: Have students use an almanac to find the changing altitude of the sun during the year. How does this relate to the tropics of Cancer and Capricorn and the equator?
Resources: Online Astronomy Lesson