# Longitude

Page 5 of 6

Rain and clouds! . . . clouds and rain! . . . day after day since November 17. Finally, on the morning of November 24 the sky was fair once more. The captains took observations of the sun for magnetic declination, then took the AM observation for Equal Altitudes of the sun to check the chronometer's error on Local Time. The sky remained clear until about 3:00 p.m. Local Time, allowing the captains to measure the angular distance between the moon and the sun, but clouds cove the sun before the PM Equal Altitudes observations could be taken. The sky clea sufficiently after dark to permit the captains to measure the angular distance between the moon and the star Markab (Pegasi), the brightest star in the constellation Pegasus, at about 8:30 p.m.

Because the captains could not complete the PM Equal Altitudes observation, and—worse!, because Clark failed to record the already-measu sun's altitude for the AM observation—the chronometer's error on Local Time and its daily rate of loss cannot be calculated from that observation.The chronometer's error at the time of observation can be determined to a satisfactory degree of accuracy, however, by calculating the time of the observation for Magnetic Declination on November 24 (see previous page Local Time).

## Table 5. Calculated Longitude

Assuming the Sun's Upper Limb was Observed for Magnetic Declination Longitude resuts Extending Line A-B-C-D to the Times of Observation:

1805 Date With Calculated Longitude Actual Longitude Error in ° ' Error in statute miles
(47.75  mi per° at 46°16')
Nov 24 Sun 122°15'W. 123°54½' W. 1°39½' E. 79
Nov 24 Markab 123°26' W. 123°54½' W. 0°28½' E. 23

Assuming the Sun's Center was Observed for Magnetic Declination:

1805 Date With Calculated Longitude Actual Longitude Error in ° ' Error in statute miles
(47.75  mi per° at 46°16')
Nov 24 Sun 121°54' W. 123°54½' W. 2°00½' E. 96
Nov 24 Markab 122°31' W. 123°54½' W. 1°23½' E. 66½

Table 5 clearly shows that the calculated longitudes are closer to the actual longitude when using the time of the sun's upper limb from the observation for magnetic declination, rather than the time of its center.

Usually, the expedition's Lunar Distance observations with the sun yield longitudes nearer to the actual longitude than those with stars. The Lunar Distance observation with Markab at Station Camp, however, yields a longitude that is more than a degree nearer to the actual longitude than the observation with the sun. Was the observation with Markab simply a much better observation than that with the sun?

On February 4, 1806, Lewis recorded that the sextant's index error was 5'45" subtractive (that is, it read 5'45" too high). Suppose the sextant's index error was NOT +8'45" but +5'45" . . . how would that affect the longitude calculated?

Using times produced by projecting the chronometer error obtained in Step 6 of the section LOCAL TIME for the Lunar Distance observations:

## Table 6. Longitudes calculated using an Index Error of +5' 45"

 For the Moon-Sun observation the resultant longitude is 123°50' W For the Mon-Markab observation the resultant longitude is 121°45' W

As can be seen from the above, using an index error of 5'45" (both for the Lunar Distance observations and the calculated time of those observations) results in a Moon-Sun longitude that is within 5 arc minutes of what it should be. On the other hand, it moves the Moon-Markab longitude much too far to the east.1

1. This reversal of calculated longitudes happens because with the moon-sun observations the angular distance between the near limbs is measu. At the time of the Moon-Markab observations, however, only the western limb of the moon was illuminated whereas Markab was east of the moon. Lewis, thus, had to measure the angular distance from Markab to the moon's far limb instead of its near limb. The smaller index error (in this case) makes the calculated longitude from the moon-sun distance farther west, but makes the moon-Markab longitude farther east.

Funded in part by a grant from the National Park Service's Challenge-Cost Share Program