How Columbus and Apollo Astronauts Navigated

The animation above depicts the means Columbus and mariners of the fifteenth century determined their location on the sea. They used simple arithmetic and a technique called Dead Reckoning Navigation. (Actually, Columbus judged the ship's speed through his own version of dead reckoning. Knowing the time elapsed between changes in his vessel's speed and direction of travel, Columbus entered his estimate of distances in a log. Rather than using a float to determine speed, Columbus, from his "feel" of the keel moving through the water along with his sense of wind and sail behavior, "knew" fairly accurately what the ship's speed was.)

DEAD RECKONING NAVIGATION

Here's how it worked:
First of all, the navigator had to collect some articles to make a system which could "dead-reckon". Among these items were: a long rope (the yellow line in the above animation), an hour glass (upper left of the animation), and a piece of wood or log (the orange square block above). The navigator also needed a compass and a log book to record his navigational findings. The rope was knotted every 4 to six feet along its length. The log (piece of wood) was tossed into the sea after it had been tied to the end of the rope. The rest of the rope was piled on the ship's deck.

Items Above Comprised Dead Reckoning Navigation

The Process
From arithmetic, we know that distance traveled in a selected direction is determined by multiplying speed by the time the vehicle travels at the chosen speed. If a car has a steady speed of 30 mile per hour for a period of two hours, it has covered a distance of 60 miles (30 miles per hour multiplied by two hours comes to a 60 mile distance) in the direction the driver chose to drive. In order to determine the car's position after a number of hours, the driver would need to record on paper the speed, direction, and time the vehicle traveled at a steady speed prior to changing speed or direction. Each time a change in direction or speed occurred, the driver must record the new speed, direction, and time in his log.

This is the process Columbus used to navigate. A compass determined the direction the ship moved in the water. Speed was measured by throwing the wood float attached to a long rope overboard. At the time the float entered the water, an hour glass was turned so that the sand began dropping into the empty lower portion of the hour glass. Knowing the length of the rope which trailed behind the ship during the time it took for the hour-glass's sand to empty into the bottom portion of the timer enabled the navigator to measure both the speed and distance traveled. While not very precise compared to a car's speedometer, the process was done in a regular fashion so that the recorded measurements often achieved up to 90 percent accuracy.

Questions:
1. What reasons would make "dead reckoning" navigation inaccurate?

2. Would it be easier to use this technique during the day or the night? Explain.

3. What would the effect of storms have on dead reckoning navigating?

4. What purpose did the knots in the rope serve?

5. Columbus had a rope of 100 knots length. The distance between each knot was about 6 feet (a man's height). When Columbus cast the float overboard, he turned his hour-glass so that the sand poured into the bottom chamber. The sand completely emptied from the top chamber filling the bottom portion of the hour-glass after 50 knots had been pulled from the deck into the ocean. If the deposit of sand into the glass's bottom took two minutes, what was the Santa Maria's speed? How far would the vessel travel in a day's sail at this speed? If the journey to the New World was a distance of 4,000 miles, how many days would it take Columbus's fleet to reach the West Indies at this speed?

6. If the ship in the animation above is 45 feet long, how long would it take to travel 4,000 miles to the New World. Use dead reckoning navigation to estimate an answer? (Hint: Use a school clock or watch with a second hand. Lay a ruler on the image of the ship and compare it to the length of the yellow rope at the time the sand has completely entered the hour-glass's bottom chamber.)

HOW APOLLO 17 NAVIGATED

Celestial Navigation

The above picture represents the science of celestial navigation used by Apollo astronauts to reach the Moon. Notice the spaceship rocketing through the cosmos (the moving flash of light) with the field of star markers in the background. The lunar landing program employed 37 stars as guides for spacecraft navigation. Their prescribed position is so predictable and constant that a spaceship could journey hundreds of thousands of miles and land within hundreds of feet of a desired lunar landmark.


Apollo Astronaut Sighting Star with Space Sextant

Despite the accomplishments of space age navigation, many of the principles and instruments used date back to centuries prior to Columbus's journey to the New World. These primitive instruments employed the stars, planets, Moon, and the Sun in the same fashion that Gene Cernan and his crew did.

Medieval Celestial Navigation (How It Worked)

The animated sketch above shows a way Columbus could have determined his latitude in degrees north of the equator. By using an instrument (A cross staff is shown above. ) capable of measuring the angle between the horizon and Polaris (North Star), Columbus could determine the latitude of his vessel on Earth's surface. While dead reckoning provided a measure of both latitude (distance north and south) and longitude ( distance east and west), the celestial navigation technique shown above yielded a measure of latitude without the need for a log and the dead reckoning process.

In the device above, a vertical slide member moves along a horizontal rule such that the tangent of the angle determined by sighting along the hypotenuse (longest side of the triangle formed by the measuring device) can be determined. The horizontal and vertical sides of the device could have ruled markings to calculate the angle's tangent (tangent ratio of an angle = length of the side opposite the angle / length of the side adjacent the angle ), hence angle calculation. Since the vertical member of the cross-staff is fixed in length, the markings on that member could be ignored. The markings on the horizontal member simply were the angle whose tangent was the ratio of the two members.

Notice in the animation the correlation between the angle and the position of the ship's marker on Earth's latitude lines. As the angle increases, the location of the marker ascends northward. As the angle decreases, the location of the latitude marker descends southward toward the equator. The first sighting in the animation is approximately 30 degrees North latitude which was the latitude of the Canary Islands from which Columbus set sail for the New World.

Of course, to begin the process, the mariner must be able to locate the North Star in the heavens. This is done by finding the Big Dipper as shown below. Note: The North Star is an extension of the line formed by the vertical side of the Big Dipper opposite the handle.

Difficulties with Determining Latitude at Sea

The above technique has obvious handicaps when employed at sea. Since the movement of the ship is affected by waves, the line of sight is constantly changing. Only in very calm water can this sighting be made with any hope of accuracy. Land based sightings are much more accurate. For this reason, a mariner would be wise to find an island for making such a measurement. Additionally, a haze about the ship would make sighting the North Star unlikely. At night, sighting the horizon as well as the North Star was not easy. Obviously, at large angles corresponding to latitudes far north, sighting both the North Star and horizon simultaneously would be difficult with a cross staff. Finally, even on land, the horizontal member must be aligned with the horizon. To assure this alignment, some instruments employed a weighted plumb line. Gravity assured the plumb line was perpendicular to the horizon. Knowing this, the operator made certain the horizontal member was at an angle of 90 degrees with the direction of the plumb line. The following discussion deals with celestial navigational instruments using the process shown in the animation above.

THE LONGITUDE PROBLEM
Knowing East-West Position Accurately

"The Discovery of the Longitude is of such Consequence to Great Britain for the safety of the Navy and Merchant Ships as well as for the improvement of Trade that for want thereof many Ships have been retarded in their voyages, and many lost..."

If you dear friend can solve this problem, yours will be the award of several millions of dollars!

The above quoted passage reached the Parliament of England in March of 1704. The second passage is the reward offered by Parliament in 1714 as it might have appeared in a newspaper of the year 2000. Knowing such a reward is offered, one would, indeed, wonder, "What is this longitude problem?"

The above discussions dealt with dead reckoning and celestial navigation. Each resulted in considerable error due to measurement technique and infrequency of determination. These errors led to countless ship wrecks, loss of precious cargo and lives. While dead reckoning was suited to measuring location north and south or east and west, it proved useless over large distances where precision of several miles was needed. Though the use of coarse celestial navigation proved useful in measuring latitude (north-south mapping), it offered little in measuring longitude (east-west mapping). These factors led to the unique reward offered for a longitude measuring solution.

Based on the knowledge of the era (1704), how might one have begun to win the contest? Since dead reckoning was longitude and latitude independent, perhaps, an improved means of performing the process might achieve the prize. If one could devise a very precise flow meter capable of measuring water speeds as accurately as a twentieth century automobile odometer, might that solve the problem? All that would be required would be a paddlewheel mechanism geared to a clock-like movement which notched up rotations of the paddle-wheel in a mechanical counter fashion. Yet, even such a device could not contend with the flow of currents which had little to do with ship's speed relative to the water, i.e., the frame of reference was also moving.

Somehow a means independent of the movement of the wind, waves, and ship had to be devised. This solution had to be something like celestial navigation in the sense that the invariant predictable movements of the Earth's rotation, the Sun, and the Moon, planets, stars and other celestial bodies would yield the solution.

Among these changeless facets of planetary bodies is the rotation of the planets. Since the problem was knowing east-west position, might not the answer relate to Earth's east-west rotation on its axis. For any known latitude, the time of sun-up or sun-down might offer a solution. These times at any longitude might be compared to a standard time determined at a reference point on the planet. The time at the reference point would be time zero at zero longitude

Here's how it might work:(See Above Animation) Virtually every school child knows the Earth's rotation of 360 degrees takes twenty-four hours given a few seconds. It follows that the school room moves one twenty-fourth of the 360 degrees of one day's rotation in an hour. Dividing 360 degrees by 24 yields a rotation of 15 degrees east in an hour's time increment.

Now if the Sun comes up at 6:00 AM at the 0 degrees mark called longitude 0 degrees and the child's school is at the same latitude only 15 degrees west, the same clock will show 7:00 AM at sunrise over the school house. Instead of measuring longitude, all is needed is a device to measure the time of day when the sun rises. Knowing this time gives a measure of longitude west of the reference mark of 0 degrees longitude.

The concept is so very simple, but for several difficulties. The accuracy of the longitude determination is no better than the clock accuracy or the positioning accuracy of the sun locator device. The circumference of the Earth is known very accurately. Simply dividing the circumference (approximately 25,000 miles at the Equator) by the unit of longitude (degree) gives a measure of approximately 1,000 miles at the Equator per 15 degrees of Earth's rotation. Each degree relates to the time required for the Earth to rotate past the sun's illuminating rays.

The above animation is especially created to depict the path of Columbus's first voyage. The red marker on the map of Earth is the position of Columbus's Santa Maria west of Palos, Spain, the port of embarkation for the New World. Imagine the Admiral of the Ocean Sea possessed a clock sufficiently accurate to keep time within a fraction of a second after being at sea several months. Besides the precise time piece, on board the Santa Maria is a device capable of noting the exact moment the Sun is directly overhead (midday), i.e., noon. Based on the technique described above, Columbus checks his clock at noon the day he sets sail for the New World. He assigns Palos, Spain the longitude zero degrees. Each successive check of his clock's time at midday (noon) will provide a means of measuring longitude (distance west of his starting point).

Before continuing in these suppositions, it must be said that neither Columbus nor anyone else in 1492 knew the extent of the Earth's circumference. Again, for this exercise, assume Columbus knows the Earth to be about 25,000 miles around at the equator.

After Columbus departs the Canary Islands, a distance of approximately 1000 miles west of Palos, Spain, he notes the time when the Sun is directly overhead. He remembers not to reset his clock as some might to keep noon conveniently at twelve o'clock (12:00 PM). Instead, he keeps his longitude clock unchanged. Knowing the Earth rotates about 1000 miles eastward every hour (at the Equator), Columbus reads his longitude clock once more at the exact moment the Sun is directly overhead. Since the clock's time shows exactly 1:00 PM, he is aware his position is about 15 degrees of longitude west of Palos his starting point. (360 degrees divided by 24) At the Equator this would equate to approximately 1,000 miles west of Palos. Because degrees of longitude vary in width, being a maximum at the Equator and quite small in magnitude near the poles, locations of islands, reefs, shorelines, and other sailing landmarks are given in degrees of longitude and latitude. However, it is a simple geometry exercise to convert these degree readings to actual north-south east-west distances from a given location.

The process is repeated as shown in the above animation. Midway along the path of his journey West, late September of 1492, another noon day (Sun directly overhead) clock time is recorded: three o'clock. Now he knows his fleet to be more than 3,000 miles west of Palos at 45 degrees west longitude based on his standard of 0 degrees assigned Palos, Spain. Several days before, a dove was sighted. Land was thought to be near. A cloud had appeared to be over a distant island. It proved to be a false landfall.

The crew is becoming restless, wanting to turn back. Once more, Captain Columbus consults his on board longitude clock as the Sun beats directly down on the Santa Maria. It reads four o'clock (4:00 PM). October 4, 1492: He must be more than 4,000 miles west of Spain. "Surely, the Orient must be just over the horizon," is his thought. Seeing seaweed and many land birds comforts the crew. The voyage continues.

It is the 36th day since leaving the Canary Islands. The crew is near mutiny. Again, as shown in the above animation, a noon reading of the clock is recorded: almost five o'clock, about five thousand miles west of Palos with longitude about 70 degrees west of Palos. Later that night Columbus cries out, "I see a light!" About four hours later, in the darkness (2:00 AM)of the 12th of October, 1492, the 37th day since departing the Canary Islands landfall is confirmed.

Though eighteenth century celestial instruments accurately positioned the Sun in the sky, mariner's clocks kept poor time. Clocks of the era utilized pendulum based movements. Pitch, roll, and ship yaw caused by wind and wave corrupted free-swinging pendulums. Certainly, the hour glass could not achieve the needed accuracy of a few seconds over months of time keeping. Such a clock must, effectively, hold the same time as the time piece maintained at the zero degree reference longitude. Simply reading this on board clock after having determined the Sun's location at the given time would provide the needed accuracy.


Copyright - Dava Sobel. New York: Walker & Co., 1995.

The 1996 best selling book Longitude by Dava Sobel shown above is an exciting story of the invention which won the 20,000 pound prize offered by the English Parliament in 1714.

It was a man named John Harrison (pictured on the above cover) whose genius contributed the solution, a time piece known as the chronometer. This extremely accurate mechanical clock (shown on the cover photo above ) operated independently of the motions of the sailing vessel. So accurate was Harrison's clock that sailing ships found the chronometer useful for longitude measurement well into the twentieth century.

The Instruments of Celestial Navigation

Though Columbus, for the most part depended on the technique of dead reckoning described about, he also commented in his log that he performed celestial sightings with a quadrant. The art and science of celestial navigation, then as now, required an instrument to determine the angle a chosen navigational celestial body made with the horizon of the navigator position on land or sea. Below are examples of some of these angle determining devices:

THE QUADRANT

A mariner of the 15th century might have at his disposal a quadrant. Later derivative instruments for navigation were the Octant or Sextant. Of course, the name Quadrant describes it characteristic. A quadrant is one quarter of a circle. Like the astrolabe, it measures vertical angles like the height of the North Star. This gave the mariner a measure of Latitude. On November 2 of 1492, Columbus noted in his log that he made a quadrant sighting of the North Star. Apparently, the Admiral combined his version of dead reckoning along with celestial navigation to judge his location.

An octant is simply a quadrant which is folded making it lighter and easier touse, however, it accuracy is lessened by the fact its scale marking are twice as close as a quadrant with an identical radius.

THE SEXTANT

A sextant (a sixth of a circle or 60 degrees of arc) is as easy to use as a quadrant. It is almost as accurate as well. Additionally, it has a light weight. Both the octant and sextant were invented after the quadrant.

Questions
1. Could Columbus use the North Star if his ship was at latitude 20 degrees south of the equator?

2. If the angle found in the latitude sighting above was 37 degrees, how far north was the ship with respect to the United States, i.e., what geography was due west of the ship?

3. If Columbus's ship was off the east coast of Cuba, what was the angle of the device with the North Star?

4. Columbus battled a terrible storm as he returned to Spain. Would he have been able to use the celestial device above to find his latitude? Why or why not?

5. What is the latitude of your school? Suggest how you might make a device like the mariner uses above from card-board, a ruler, scissors, tape, a string, a pencil, paper clip, and a small paper weight. Sketch your device. Will it work during the school day? Why or why not? Use it to find the latitude of your school. How accurate was it?

6. Examine the accuracy of modern wrist watches. Many advertise having a quartz crystal for maintaining accuracy. How accurate are such watches? In terms of determining longitude after a year at sea, how accurate would a mariner be able to determine his ship's longitude position with a quartz watch?

7. Parliament's prize of 20,000 pounds ($2,000,000 in today's dollars) required the mariner's clock yield an accuracy of a half degree of longitude after a voyage to the West Indies. Based on the knowledge that Columbus's first voyage required 37 days for a passage from the Canary Islands to its destination in the West Indies, what would the error have been in time per day if Harrison's chronometer exactly met the terms of Parliament's award? Harrison's clock yielded an accuracy of 1/5 of a second per day. What would the longitude error have been for a 37 day trip to the West Indies? Answer in degrees and miles (assume a degree of longitude equals 70 miles). Why do you think the specification of a "voyage to the West Indies" is not specific enough for determining the winner of the contest?


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