Problem 1st
Page 4 of 13
The Astronomy Notebook
Page 1^{1} 
From the altitude of the sun, together with the estimated Greenwich time, and the true hour of the day, apparent time, at place of observation; to compute the latitude of [the] place [of] observation. Directions  At any time when the sun is not more than one or two hours from the meridian, but the nearer to the meridian the better, take its altitude and note the time per watch, making allowance for its probable error from true apparent time.
 From the apparent altitude find the true altitude, subtract this from 90° and the remainder will be the zenith distance which call north or south according as the zenith is north or south of the body observed. [p. 2:]
 Find the true declination of the sun for the estimated Greenwich time.
 From the time at place observation find the hour angle of the sun at the time of observation.
From the above data the latitude may be computed as in the following example. Note, In all the following Forms, the capital letters signify the corresponding arches in the adjoining column, and the small letters, the sines, tangents, secants &c. of those arches respectively. When the small letter is omitted in the Form, the arch is found from the sine, tangent, &c; but when the small letter is prefixed, the sine, tangent, &c. is found from the arch. In taking out the sine, tangent, &c. of an arch from the table, or in finding the arch from the sine, tangent &c, it will be sufficient [p. 3:] to take it out to the nearest minute only. When an ambiguous sign occurs as ± or the one or the other is to be used as directed in the explanatory note to which the number in the margin refers. Example Suppose the apparent altitude of the sun's lower limb above the southern horizon = 52°17', Estimated Greenwich time May 10th 1799 about 17 hours p.m., True apparent time at place of observation = 54m p.m. Required the latitude. [p. 4:] Form I. 1 Add when zenith distance and declination are of the same name, otherwise subtract; and the latitude will be of the same name with the greater number. This rule will give the latitude in all cases where the altitude is taken near the meridian. The rule however may be made universal thus; add when latitude and declination are of the same name, and azimuth of the sun from the meridian less than 90°, also [p. 5:] when the hour angle exceeds 90°; subtract when latitude and declination are of different names, also when azimuth from meridian [is] more than 90°. Note 1st. When the declination = 0, to the secant of the hour angle add the cosine of the zenith distance, and the sum (abating 10 from the index) will be the cosine of the latitude. Note 2nd. If an observation be taken both before and after the sun comes to the meridian, when the altitudes are both nearly equal, then a mean of the latitudes computed from these two observations will be more accurate than that computed from one observation; as the errors arising from an error in the watch or time, will in this case tend to correct each other. [p. 6:] Examples for Practice
Edited by Robert N. Bergantino, 05/95, 09/04, 06/05. Funded in part by a grant from the NPS ChallengeCost Share Program.
