With the values: sun's Local Hour Angle = 53°40'45", altitude of sun's center = 34°05'14" and sun's declination = +12°23'31", Lewis is ready to use spherical trigonometry (possibly from an example in Kelley's Spherics) and solve the equations for his latitude. It's a lengthy calculation, so only the results are shown here.
After Lewis has solved the spherical triangle using the above data, he should have derived a latitude of 45°04'45" N. But if Lewis had made his calculation after correctly figuring out the sun's altitude (that is, using 34°09'37" instead of 34°05'14") he should have obtained a latitude of 44°50'37"N. The camp was at 44°59'36" N. Lewis, however, obtained a latitude of 44°33'50.5" N from this calculation. How?
Suppose Lewis made his calculation using the sun's declination at noon Greenwich time +12° 32' 40" instead of adjusting it to his estimated longitude, but kept the values of 53°40'45" for the Local Hour Angle and his incorrect sun's altitude of 34°05'14". What latitude would he have obtained?
Solving the spherical triangle with these data gives a latitude of 45°26'05"N. Note, however, that for the step just previous he would have obtained a value of 44°33'55"; this is the camp's co-latitude (90°—latitude). Did Lewis forget that he needed to subtract the co-latitude from 90° to get his latitude and, because 44°33'55" (or 44°33'50") seemed to fit the values derived from his meridian altitude observations, he accepted that value? Or, because of his unfamiliarity with the calculations, did he make some other error that can't be traced?
Part of Lewis's error in latitude for this observation can be found in the times recorded for the Equal Altitudes observation. If you look carefully at the times recorded for the Equal Altitudes observations for the morning of the twentieth you will see that the time interval between the first and second observations is out of step with that between the second and third. You will also note that the average time of the observation is significantly greater than the time of the sun's center. Similar errors exist in both the morning and afternoon observations for the twenty-first. In each case, the person who read the chronometer read it incorrectly by exactly one minute. Additionally, as noted above, if you calculate the altitude the sun should have been above the horizon at the corrected chronometer time, you find that either Lewis misread the sextant or mis-recorded the angle, or that the chronometer was running erratically.
Before the expedition got underway, Thomas Jefferson expressed the need to have all of the expedition's celestial observations recalculated by competent mathematicians after the expedition's safe return. However, he failed to make certain, while he was still president, that this necessary operation was done. The reasons for this "failure" belong to another part of the expedition's story. But suppose those re-calculations had been made. For example, Lewis didn't attempt to calculate the latitude from the Equal Altitudes observation of the twenty-first, but that observation could have provided a check on the latitude of Fortunate Camp, as well as upon Lewis's other calculations. What would that observation have shown?
For the observation on August 21, (a) determine the altitude of the sun's center correctly, 32°53'19"; (b) adjust the chronometer's time of the observation to reflect the chronometer's daily rate of loss, 3:40:40 PM; and (c) approximate the longitude of the observation from Clark's maps and factor in the time of observation to obtain a value for the sun's declination more nearly what it should have been, +12°03'29".
Using the above values produces a latitude of 44°55'42". This comes very close to the average recalculated latitudes from Lewis's meridian altitude observations, and less than 4' latitude (5 miles) south of the latitude of 44°59'36" for Fortunate Camp arrived at from the expedition's survey data and journal descriptions.