VI PREFACE. 



lunar distances, and for the correction of the time, were made 

 with the small sextant of five inches radius. 



In Part first, the observations for latitude previous to the 

 31st August, 1819, were made with the circle of reflection, 

 used as a sextant. After that time they were made with the 

 larger sextant, except those made while traversing the coun- 

 try in the vicinity of the Council Bluff and at the Pawnee 

 Village, and also one made at Engineer Cantonment on the 

 20th March, 1820, which were made with the smaller sextant 

 of five inches radius, it being better calculated for land ex- 

 cursions than the other. All the lunar distances in Part first 

 were taken with the larger one. 



The first and most important point in determining the lon- 

 gitude, is to have the time and the rate of the time-piece 

 well regulated. The method I pursued for this purpose was 

 by equal altitudes of the sun, and sometime? of the stars be- 

 fore and after passing the meridian. For accuracy, provided 

 your time-piece is pretty uniform in its rate of going, whe- 

 ther it should gain or lose time, this method is perhaps pre- 

 ferable to any other that can be pursued with the use of an 

 instrument of reflection, such as a sextant or quadrant. 



In determining the longitude from lunar distances the only 

 data have been the latitude of the place determined from ob- 

 servation, the observed distance of the moon from the sun or 

 stars and the correct apparent time of observation. The true 

 and apparent altitudes of the objects are inferred from the 

 apparent time and latitude. In Part first the moon's horizon- 

 tal parallax obtained from the nautical almanac is corrected 

 for the spheroidal figure of the earth, and according to the 

 latitude of the place of observation. This correction was not 

 applied in the calculations of Part second. The difference it oc- 

 casions in the result is of little moment, within what might be 

 considered as errors of observation. In computing the right 

 ascension and declination of the moon the equation of second 

 diflPerence, a correction arising from the unequal motion of 

 the moon, has in every case been applied to the proportional 

 part of the variation of right ascension and declination cor- 

 responding to the time from noon or midnight when the ob- 

 servation was made in order to get the moon's true place at 

 the given time. 



One lunar observation made at St. Louis, and three at 

 Franklin, (Missouri,) are calculated altogether by spherical 

 trigonometry, a method which I had practised under my in- 

 structor, the late venerable Professor Ellicott, of the Military 



