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An Experiment in the Determination of Latitude
by Bob Graham

In his book, Drake's Bay, Unravelling California's Greatest Maritime Mystery, regarding Francis Drake's ability to determine his latitude for those positions taken on land, Brian Kelleher has shown by statistical analysis that "...the mean deviation between the reported position and the actual latitude is about minus two minutes within a range of minus fifteen minutes to plus fifteen minutes and the standard deviation is about eleven and a half minutes." This in not only remarkable accuracy for the late sixteenth century, but it provides powerful evidence in the search for the actual landing site that Drake made on the coast of California in 1579. Brian Kelleher's has concluded that Drake did not land at what is now called Drake's Bay, but rather at Campbell Cove on Bodega Head.

Subsequent to the publication of Brian's book, and after hearing Brian speak on the subject, I embarked on a study of sixteenth century navigation which resulted in the paper DETERMINATION OF LATITUDE BY FRANCIS DRAKE ON THE COAST OF CALIFORNIA IN 1579 which is posted go here. That paper not only supports Brian's conclusion that Drake landed at Campbell Cove, but it explains the reasons for, and quantifies the errors that Drake unavoidably made which show up in Brian's statistical examination of the reported latitudes. On September 2, 1999, the following experiment was conducted to test the conclusions drawn in the paper.

An astrolabe of 9 inches diameter was constructed of heavy card stock. The lower right quadrant was divided into 90 degrees. The alhidada, also constructed of card stock, has two vanes, or sights, one inch square, each pierced in the center by a common paper punch. Like sixteenth century astrolabes, this astrolabe is suspended by a double articulation that allows it to hang vertical in both vertical planes, but does not allow it to swing in the horizontal plane. It is accurately made, but of course, the materials are totally unsuitable for wind and weather.

Examination for index error by back sighting showed that it was smaller than I was able to detect. As explained in the article referred to, with an astrolabe there is no correction necessary for dip of the horizon (it is its own horizon, referrenced by gravity) or semi-diameter of the sun (it takes the center of the sun). The corrections forparallax and refraction are too small to matter in this case with this or similar instruments).

About a quarter to noon on September 2 (with noon occurring at approximately 12:56 A.M. PDT in my longitude (119 West), I began to track the sun. The alhidada was moved to keep the light passing through the fore sight centered in the hole of the back sight. The holes in the two, being of equal size, left a halo of light around the edge of the hole of the back sight. This halo of light, and the same on the vane itself around the shadow of the front vane, were of great assistance in keeping the alhidada constantly aligned. Note the spot of light on the back sight in the photo at the bottom of the page.

As noon approached, less frequent correction was necessary. Then, for several minutes (at this resolution time is not too critical), the sun seemed to stop its ascent and remain stationary. Gradually it became obvious that the sun was descending and that noon had indeed occurred and that further movement of the alhidada was unnecessary.

The index on the alhidada was found to be indicating 59 1/2 degrees, the observed altitude of the sun, which I called 59° 30'. Referring to an online site which provides solar declinations, I found the declination for noon at Greenwich for the day to be 8° 09'. This subtracted from the observed altitude gives 51° 21' for the true altitude of the sun. Since Sacramento is about 8 hours (1/3 of the way around the earth) behind Greenwich, I needed to correct for longitude 121° W. The difference in declination between September 2nd and September 3rd is about 18', so 1/3 of that was added to the true altitude, so that it became 51° 27'. Subtracting 51° 27' from the 90°, the result was latitude 38° 33' North. The actual latitude of my back yard in Sacramento is in fact 38° 33' 06".

What would Francis Drake have determined had he been here on this date in the 20th century with his instruments and William Bourne's A Regiment for the Sea?

Drake would have read 59 1/2 degrees just as I did. He would have called it 59° 30' as I did. He would have referred to Bourne's Tables of Declination and found the figure for the date (23rd July, old calander) to be 8° 18' declination. Not quite right, as explained in the original article, because of errors in tables of the time. Drake would have subtracted the declination from his observed altitude and found the true altitude to be 51° 48'.

Drake could not correct the declination for longitude. We know that he did not, and could not, because when he had returned to England after the circumnavigation, he stated that the day was, "...Monday in the just and ordinary reckoning of those that had stayed at home in one place or country, but in our computation was the Lord's Day, or Sunday." Therefore, he would have subtracted his true altitude from the 90° of the celestial pole, and determined the latitude to be 38° 12'.

In spite of the accuracy of Drake's instruments, and his ability to use them, because of errors in the best declination tables of his time, and his inability to correct for longitude, his determination of latitude in this particular case would have been, on this day of the year, at this longitude, 11' too low.

I have since made a new mariner's astrolabe of more substantial materials (a plastic bread board). I find that it is possible to estimate 1/4, 1/3, 1/2, 2/3, 3/4 degrees quite accurately. That gives readings of 15', 20', 30', 40', 45, and anything just less than or more than a degree can be called 1/8, or 07' 30". Depending on the day, my determinations during September and early October have all been within 5 minutes of latitude, but that is the extreme error, and most are within 3 minutes or less (about 3 miles).

I have also experimented with determinations from the pole star.There are four positions during the 24 hours that Drake could have used for the best results--two 0 correction times and two maximum correction times. The 0 correction positions are when a line between ursa major an cassiopea is horizontal to the horizon; the max correction is with this line vertical, or perpendicular to the horizon.The 0 correction times seem like a good bet, but I find, so far, that it is much more difficult to shoot the north star and center it. This is eye to star thru the holes in the sights. Holding the instrument above eye level is also dfficult. I think it is +/- 15 minutes, but I don't really have a good sky for observing. The other problem with the 0 correction time is that the rate of declination change to either side is rapid.

At the positions of maximum correction the declination change is the slowest, but there is the extra error from the various Sixteenth Century miscalculations of the distance of polaris from the celestial pole. These distances ranged from 4 degrees 9 minutes to 3 degrees 30 minutes. Just after Drake's voyage, the number was refined by Edward Wright to 2 degrees 52 minutes.

The pole star has since moved closer to the pole, and the maximum correction today is about 1 degree added or subtracted from the observed altitude of polaris. Drake would have known from comparisons of determinations that the best he could expect to do with the pole star would be a 30 mile fix compared to a 5 or 10 mile fix by the sun. Indeed, William Bourne, in a REGIMENT FOR THE SEA, cautioned against the use of Polaris (See previous article). Polaris would be very useful at sea though, for a fix several times during the night!

In working through recorded polaris sights taken by John Charles Frémont in the Sierra Nevada in the winter of 1844, I have found that he chose to make his shots at the published times of greatest correction. This meant rising at very inconvenient hours, but the apparent change in position is slowest at these times; important because he was having problems with his chronometer maintining its rate, so he did nor have his time precisely. The correction needed was often nearly 1 1/2 degrees.

It is easier and more accurate to measure the sun with the astrolabe because; (1) it is a projection, and (2) the position from which the reading is taken is comfortable and steady. I find that a seated position, with the elbows on the knees, and the astrolabe hanging just above ground level is the best position.

go For an actual on-site determination of latitude at Campbell Cove, see A Day at the Cove.

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©1999, 2007
Bob Graham