Calculations of Observations
Page 3 of 6
1. Latitude by Octant and Meridian Altitude of the Sun
alculations of celestial observations usually begin with determining the latitude of a point of observation, especially if the data were obtained from a Meridian Altitude observation of the sun. Lewis waited until June 9, the day before leaving Camp Chopunnish, to make this necessary observation and, because of the sun’s noon altitude, he had to take the observation with the octant by the "back sight method."
The angle that Lewis recorded for this observation, however, was not the sun’s altitude; it was the supplement of the double altitude of the sun’s lower limb. This means that the angle he observed had to be subtracted from 180° then divided by two to find the “observed” altitude of the sun’s lower limb.1 From the “observed” altitude it still was necessary to subtract the effect of refraction, add the effect of parallax and add the sun’s semidiameter; the “sum” of these three operations was the altitude of the sun’s center. Having mathematically determined the true altitude of the sun’s center it was then necessary to algebraically subtract the sun’s declination from it. Having completed that operation, subtracting the result from 90° gives the calculated latitude (for the calculations.
Using the index error for the octant that Lewis gave on May 22, 1804 and which he reaffirmed while at Fort Clatsop (see Fort Clatsop Miscellany), his observed angle yields a latitude of 46°09'12" N. The actual latitude of Camp Chopunnish determined from information in the courses and distances and from early sources who identified the camp’s location is 46°14'31"--a difference of 5'19". This difference of 5'19" is larger than any of the differences between the actual latitude and the latitude recalculated from observations that Lewis made in 1805. The difference might result from a changed index error or from the fact that Lewis had not taken an observation with the octant since August 1805. In any case, the difference is not much greater than for many observations taken for latitude at sea at that time.
2. Latitude from Sun’s Altitude, Declination and Hour Angle
heoretically, it is possible to calculate a latitude from the data derived from any observation in which one knows the sun’s altitude, its declination and its hour angle (that is, the difference between the Local Apparent Time of the observation and noon-- multiplied by 15).
Equal Altitudes observations normally provide all that data, but the observation for May 25 was not followed by an observation the next day, so the chronometer’s rate-of-going is uncertain. The Equal Altitudes observations for June 5 and June 6, however appear to provide the data necessary to determine the rate. At noon on June 5 the chronometer was 4h 33m 11s slow on Local Mean Time and on June 6 it was 5h 50m 22s slow--a loss of 1h 17m 11s in 24 hours! The question is, did the chronometer stop or change rate between the two observations or did it run at a steady rate?
This exceptionally high rate-of-loss, at first, doesn’t inspire much confidence in a calculation for latitude from chronometer’s time. Nevertheless, by plotting the chronometer error for June 5 and 6 and projecting its rate-of-loss of 1h17m per 24 hours back to May 25, one discovers that the line passes close to the chronometer’s error at noon on May 25--provided 12 hours are subtracted from that error (that is, at a rate-of-loss of about 1h17m per day, the chronometer had lost more than 14 hours in eleven days). This is an unbelievably high daily rate-of-loss, but the near-concordance of times suggests that the chronometer had been running at nearly a steady rate during this period of time. It would be unwise to make any calculation from the Equal Altitudes observation on May 25 because the projected line, although passing close to the chronometer’s error at noon determined from that observation, still misses by too far to warrant that calculation. It does appear reasonable, however, to attempt a calculation from the time given by PM observation on June 5 and that of the AM observation on June 6.
Unfortunately, the calculations, once made, show that Lewis took both the Equal Altitudes observations at a time when the sun was nearly east or west. Therefore, the last step of the equation for determining the latitude as given in Robert Patterson’s Astronomical Notebook, Problem 1, “falls apart.” The problem is that the arc-cosine needed to complete the calculation is too near unity to give reliable results.
Fortunately, Lewis took two observations for Magnetic Declination between the PM Equal Altitude observation on June 5 and the AM Equal Altitudes observation on June 6. Even though these observations for Magnetic Declination shortly followed the PM observation on June 5, the sun would have moved farther to the north of west away from the 0°-90°-problem. The calculations for the first Magnetic observation produce a latitude of 46°10'37"N, seeming to confirm the octant latitude. The second Magnetic observation, however, produces a latitude of 46°20'41”N. The average of these two calculations is 46°15'39”N, but can an average from such divergent results be trusted?
As noted above in the Summary on page 3, above, the latitude of Camp Chopunnish can be approximately determined on the Lewis and Clark Map of 1806 (Moulton, Atlas, map 123) at about 46°15' North.
--Robert N. Bergantino, 11/04
1. Lewis first would divide the observed angle by 2, then subtract the result from 90°. Both his method and that given above produce the same answer, though the one above is procedurally more correct.
Funded in part by the Idaho Governor's Lewis and Clark Trail Committee.