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Columbus Found Longitude?

 

In our computer age, one may acquire some programming skills; learn some rudimentary astronomy and celestial navigation; and with a regressive ephemeris reconstruct the celestial tapestry of the past and focus on great historical events.

Let us revisit Columbus’s first voyage.  The eminent Columbus scholar Samuel Morison always characterized Columbus as a dead reckoning (DR) sailor who muffed his opportunity to determine his longitude on two occasions when observing eclipses.  Another view holds that Columbus may have relied upon celestial observations for his navigation.  This view contends that certain eclipses, conjunctions, and culminations could aid Columbus in determining his latitude and/or longitude.  Columbus had no sailing instructions to steer to a specific set of coordinates.  His quest was to find a western route to Cipangu (Japan) and Cathay and he had a perception of a smaller world which encouraged him to believe that Cipangu was 2,760 miles away as opposed to roughly 12,200 miles away.  He sailed with crude instruments and experienced the changes in magnetic variation as noted by comparing the direction of the pole star to his magnetic compass.  The magnetic variation was dominantly westerly.  Columbus’ charts showed that he held the latitude of 28°N for close to 1,400 nmi after departing the Canaries without altering course.  He was not concerned that his track would be deflected to the left as a result of the westerly magnetic variation.  The return voyage would experience the same deflection to the left of course in the dominantly westerly variation that prevailed.  His charts showed constant course segments with no evidence of changes (other than to avoid weather) leading one Columbus researcher, Admiral Robert McNitt USN (ret.), to conclude that Columbus relied exclusively on dead reckoning in his navigation.  Implicit in this conclusion was that if a celestial observation was made and it showed a difference in latitude over the DR position one would expect a course change to be made.

In Columbus’ age, dead reckoning was the dominant method used for navigation.  It required knowing one’s course as determined from the magnetic compass and speed by timing the movement of a chip log thrown over the side with reference to external markers on the side of the ship (although Morison believed that Columbus just estimated his speed).  The distance made good per hour was noted by placing a peg in a hole along a radiating line marked on a peg-board representing the course of the ship hourly.  The results of this peg-board tracking were then transferred to the map at the end of the day.  The DR position could be updated by celestial observations obtained by observing Polaris, low grazing stars, and conjunctions according to the recent paper by Arne Molander who noted correlation of Columbus’ fixes with certain celestial events.  The problem was that the astrolabe and the quadrant both capable for use to measure the elevation angle of the celestial body to yield latitude were sensitive to gravity owing to their pendulous element for establishing the vertical.  In a ship subject to the vagaries of the ocean motions, the pendulous element was almost constantly in motion.  These instruments were primarily used for observing celestial bodies from the land or the quiet waters of a harbor.  The cross staff would have improved the sighting accuracy, but it was not in use until the next century.

Depicted in Figure 1 is the diagram for obtaining latitude when a celestial body is on your meridian.  It is a cross section of the celestial sphere when the celestial triangle degenerates into the arc of a circle when a celestial body is observed on

Figure 1.  Low Grazing Star at Culmination on Your Meridian

your meridian.  Latitude can be calculated by the addition or subtraction of measured angles with known values.  The elevation of the star or Sun is measured and the declination of the body is obtained from a table.

On Columbus’ first voyage, his chart indicated adherence to 28°N after departing the Canaries.  If he were relying upon celestial observations to maintain this latitude line, he could verify his adherence to the course by observing the pole star with his quadrant or astrolabe (despite their susceptibility to error at sea).  Molander believes that Columbus may have used low grazing circumpolar stars when they appeared above the northern horizon at their culmination (on his meridian as they were at their lowest elevation) to determine his latitude to maintain adherence to his course.  He may have used a kamal in his measurements, a simple instrument held by both hands and held taught by string whose one end was clenched in the observer’s teeth and the other end bridled to the instrument.  Determining one’s latitude by observing the culmination of a low grazing star is not susceptible to verticality errors as the tangent to the arc of the measured star remains close to parallel to the horizon over a wide range (1-cosine effect for small angles) as viewed in Figure 1 .  We cite Schedar (Cassiopeia) as the star Columbus could have used on his return trip at the latitude of the Azores.  It is an example of a low grazing circumpolar star to aid in establishing latitude adherence.

In Figure 2 , we see how one obtains latitude by observing Polaris.  The pole star is at P with the horizon at (HH’).  Since its declination (angle between the celestial body and the equatorial plane QQ) is 90°, the polar distance to the horizon arc PH is equal to the latitude of the observer arc QZ.  In

Figure 2.  Determining Latitude by Observing Polaris
and/or a Low Grazing Star

this example, the elevation angle of Polaris is 37°, therefore Columbus’s latitude is 37°N latitude.  We ignore the effects of refraction and dip.  The question will be what elevation angle should Schedar be at culmination to establish that the observer is at 37°N latitude?  We assume that Schedar’s declination is 56° in the Columbus era. Determining longitude was accomplished on land by observing lunar eclipses.  Columbus had access to Ephemerides for lunar and solar eclipses.  Both the Regiomontanus’s Ephemerides and Zacuto’s Almanach Perpetuum contained the predicted times of total eclipses at Nuremberg and Salamanca.  A total lunar eclipse occurs when the Moon enters the umbra sector and ends when the Moon exits the umbra sector and enters the penumbra sector as seen in Figure 3 .

Figure 3.  A Lunar Eclipse

Occurs when a full moon enters the shadow of the Earth and the Moon is near or at one of its nodes (intersection of its plane of orbit and the plane of the ecliptic) (not to scale).

On February 29, 1504, Columbus observed a lunar eclipse from the middle of the north coast of the island of Jamaica.  A lunar eclipse can be observed by anyone within a hemisphere if the full Moon is observed.  One need only note the local time of the event and compare it to the local time at the reference location in the ephemeris.  This is an observation of simultaneity.  He concluded that the difference in time between the Isle of Cadiz in Spain and the center of Jamaica was 7 hours 15 minutes.  As the eclipse began before sunset, he based his calculation on observing the end of the eclipse when the illumination of the Moon returned.  He knew the elapsed time between the end of sunset and the end of the eclipse which was two-and-a-half hours as timed by the half-hour glass (five half-hour glasses in duration).  He obtained the altitude of Polaris as 18 degrees using his quadrant.  This was close to the correct latitude of his location presumed to be Santa Gloria (today’s St. Anne’s Bay) at 18° 27'N, 77° 14'W.  The difference of longitude between Cadiz and his location was actually 70°56' or 4 hours 44 minutes.  He incurred an error of 2 hours 31 minutes.  It appears that Columbus knew the difference in longitude between Salamanca and Cadiz.  (~ 39 arcminutes of longitude) since
his ephemeris was based on observations made at Salamanca and Nuremberg.

There are various reasons that could explain Columbus’ colossal errors in determining longitude by timing the lunar eclipse.  The one half-hour glass introduced an error.  If he backed into his estimate for the beginning of the eclipse by using the elapsed time from sunset to the end of the eclipse to establish the beginning of the eclipse was another source.  He could have exaggerated the longitude difference to establish a vaster domain under discovery.  Clearly this dependence on a half-hour glass as a basis for time reference was an error source and extrapolating the time of the beginning of the eclipse was another error source.

Let us assume for this Brain Game that our Columbus had a clearer awareness of the beginning of the lunar eclipse and concluded that the difference in time between the island of Cadiz and Santa Gloria for the beginning of the eclipse was 4 hours 30 minutes.  He also knew that the reading of the half-hour glass reference introduced an error of 1 percent of the time on the slow side.  Assume that 5.5 hours elapsed from local noon (last setting of the half-hour glass) to the time of the eclipse.  Columbus knew that his time master was slow.  We will presume that he also read the time of the eclipse using his nocturnal (an instrument used to determine time at night and not available until the next century).  It had an index error of ‑0.1 hour.  Based on his uncorrected nocturnal reading, he concluded that the longitude difference between the two sites was 4 hours 30 minutes.  He then corrected the readings of his time sources for their errors and averaged them.  What was his measurement of the difference of longitude between Cadiz and St. Anne’s Bay?

The elevation angle of Schedar and the longitude difference between Cadiz and St. Anne’s Bay was:

a.     Schedar 3°, longitude difference 4 hours 35 minutes

b.     Schedar 4°, longitude difference 4 hours 30 minutes

c.     Schedar 6°, longitude difference 4 hours 32 minutes

d.     Schedar 5°, longitude difference 4 hours 26 minutes

The answer is:

Referring to the elevation angle of Schedar to yield a latitude of 37°N latitude:

 

Regarding the longitude difference:

The half-hour glass error after 5.5 hours was -0.01 x 330 minutes = -3.3 minutes.

Correction is +3 minutes to be added to 4 hours 30 minutes
= 4 hours 33 minutes.

The nocturnal error was -0.1 hour or -6 minutes.

Correction is +6 minutes to be added to 4 hours 30 minutes
= 4 hours 36 minutes

Then the average difference in longitude is:  

This corresponds to 4.58 hour x 15°/hour = 68.7° west of Cadiz.  His actual longitude difference was 4 hours 44 minutes or 4.73 x 15°/hour = 70.95° west of Cadiz.  What we find with the ascribed nominal errors to the instruments that Columbus in this simulation would be within 2.25° of his actual longitude.  We would then conclude that the balance of the error was due to his perception of the beginning of the eclipse and other errors.  It is known that he measured his altitude of the pole star as 18° at St. Anne’s Bay which was close to the actual latitude of 18°27'.  Columbus typically had a history of determining his latitude with significant error.  He blamed this on his quadrant.  Some scholars believe he was reading his instrument high to record higher latitudes initially to keep his discovery within the bounds ascribed to the Spanish sovereignty in accordance with an agreement.