A. Latitude from Meridian Altitude Observations of the Sun
Determining the latitude of a location from a meridian observation of the sun is among the simplest celestial observations to take and to calculate. It is somewhat surprising, therefore, that Clark doesn't mention taking an observation for the latitude at that camp until October 5, and when he does, he writes: "Latitude of this place from the mean of two observations is 46°34'56.3" North." That's it. No mention of the sun's altitude they observed nor the date they took the observations, though it might be supposed that the last observation was taken on the 5th. The average latitude the captains obtained lies about 4'55" (5 Ω miles) north from the south side of the current river junction.
It is almost certain that the captains used the sextant to take the two meridian observations while at Clearwater Canoe Camp. Although the captains did not give the sun's altitude for these two observations nor did they show their calculations, they had, since the mouth of the Ohio River, recorded several observations they made with the sextant. For these observations they gave both the observed altitude1 and their calculated latitude.2 Furthermore, while at Camp Dubois, Clark left many examples of his methodology and gave his calculated latitude.
As it is likely that the captains used the essentially the same methodology throughout the course of the expedition, it is possible to approximate what they might have observed at Clearwater Canoe Camp, even if it is just from the average.
The captains persistently made a mistake in calculating a latitude from an observation with the sextant while using the artificial horizon. This mistake was that they first divided the observed double altitude by two, then subtracted the full index error of the sextant (+8'45"), making all their calculated latitudes too high by 4'22 Ω". By correcting just that one error, the average latitude of Clearwater Canoe Camp recalculated from the average of their observations turns out to be 46°30'34" N–a difference of just 29" too far north (about Ω mile), not 4'55" too far north.
The captains likely made other lesser mistakes in their calculating methodology,3 so it is not enough just to subtract 4'22 Ω" from their latitude and have the latitude they should have obtained from the observation. Without the altitudes they observed, however, one can only suppose their observations might have provided latitudes that were somewhat more accurate than what their calculated latitudes appear to show.
1. Observed altitude: This is the angle in degrees above the horizon that an observer instrumentally measures for a celestial body. With the sextant the captains usually measured the sun's upper limb; with the octant, its lower limb.
2. Calculated latitude: the observed altitude is but a starting place to determine the latitude. Many "corrections" are needed before the observed altitude is turned into a latitude. Usually the observed altitude is corrected first for the instrument's index error. When an artificial horizon is used the resultant angle must be divided by two. Next, the effect of refraction is subtracted and the effect of parallax is added. Then the semidiameter of the sun or moon is added if the lower limb was observed, or subtracted if the upper limb was observed. The declination of the celestial body at the time of the observation must be determined and algebraically subtracted. When all the above have been done, the result has to be subtracted from 90°, which, finally, is the calculated latitude. (And this is one of the simpler mathematical operations involving celestial observations.)
3. Among these, the most important might be errors in determining the sun's semidiameter and the sun's declination from estimated longitude.
Comment 1. In the calculation shown in the left-hand column, the double altitude of the sun's upper limb is what the captains should have observed to fit the index error, refraction, semidiameter, parallax, and declination given in the following steps.
Double Altitude: When taking a meridian altitude of the sun with a sextant or octant and using an artificial horizon, a ray from the sun that has been reflected from the index mirror to the horizon mirror and to the eye is matched with a ray that has been reflected from the artificial horizon through the horizon glass into the eye. This procedure gives the observer a true horizon on land, but doubles the true altitude of the object observed. This angle, therefore, must be divided by two to give the true observed altitude.
Comment 2. The calculation in the column on the right shows how the captains might have arrived at a latitude of 46°34'56" from the same "observed" double altitude. Using their method of calculating the latitude, and to fit the index error, refraction, semidiameter, parallax and declination, this would be the average of the two double altitudes actually observed.
Comment 3. In the third step in both calculations, note the phrase "refraction at 990 feet." Refraction is affected mainly by the density of the air above the observer. As one's altitude increases the air becomes less dense; thus refraction typically decreases with altitude. But temperature also controls the density of the air, and that must be factored in.
A common formula for refraction is: ((983 x barometric pressure inches) / (460 + temperature in °F) x cotangent sun's altitude). By this formula, at an altitude of 990 feet above sea level and a temperature of 60°F, the refraction would be 1'05", whereas at sea level it would be 1'07.5" For most observations using a sextant or octant it isn't worth while to re-calculate refraction for every observation. The tabulated refraction at sea level found in the Tables Requisite is adequate. Besides, Lewis and Clark had no way of measuring their altitude above sea level at any point, and would necessarily have used the Tables Requisite, which they carried in their portable library. But for me, every second counts.
Funded in part by the Idaho Governor's Lewis and Clark Trail Committee.