The correction for reducing a star's true altitude to its apparent, is obtained in the same manner, omitting what relates to parallax. Thus, if the true altitude of a star be 8 degrees, and the corresponding refraction 6:29", their sum, viz., 8:6:29" will be the augmented altitude; the refraction answering to this is 6:24", which, therefore, is the reduction of the true to the apparent altitude of the star. The correction for reducing the true altitude of the moon to the apparent, is found by diminishing the true altitude by the difference between the parallax and refraction answering thereto; then the difference between the parallax and refraction corresponding to the altitude so diminished, will be the reduction of the true to the apparent altitude. As thus :— Let the true altitude of the moon's centre be 10 degrees, and her horizontal parallax 57 minutes; required the reduction to apparent altitude? Since the solution of the Problem for finding the longitude at sea, by celestial observation, is very considerably abridged by the introduction of an auxiliary angle into the operation, the true central distance being hence readily determined to the nearest second of a degree by the simple addition of five natural versed sines; this Table has, therefore, been computed; and to render it as convenient as possible, it is extended to every tenth minute of the moon's apparent altitude, and to each minute of her horizontal parallax; with proportional parts adapted to the intermediate minutes of altitude, and to the seconds of horizontal parallax. This Table was calculated in the following manner :— To the moon's apparent altitude apply the correction from Table XVIII., and the sum will be her true altitude; from the log. cosine of which (the index being augmented by 10) subtract the log. cosine of her apparent altitude, and the remainder will be a log., which, being diminished by the constant log. .300910,* will give the logarithmic cosine of the auxiliary angle. Example. Let the moon's apparent altitude be 4 degrees, and her horizontal parallax 55 minutes; required the corresponding auxiliary angle? Moon's apparent altitude. 4: 0 0 Log, cosine.. 9.998941 The correction of the auxiliary angle for the sun's or star's apparent altitude, given at the bottom of each page of the Table, was computed by the following rule-viz. From the log. cosine of the sun's or star's true altitude subtract the log. cosine of the apparent altitude, and find the difference between the remainder and the constant log. .000120. † Now, this difference being subtracted from the log. cosine of 60 degrees, will leave the log. cosine of an arch; the difference between which and 60 degrees will be the correction of the auxiliary angle depending on the apparent altitude of the sun or star. This is the log. secant, less radius, of 60 degrees diminished by .000120, the difference between the log. cosines of a star's true and apparent altitude betwixt 30 and 90 degrees. This is the difference between the log. ccsines of a star's true and apparent altitude between 30 and 90 degrees. Example. Let the sun's or star's apparent altitude be 3 degrees; required the correction of the auxiliary angle? required correction of the auxiliary angle. In this Table the auxiliary angle is given to every tenth minute of the moon's apparent altitude (as has been before observed) from the horizon to the zenith, and to each minute of horizontal parallax. The proportional part for the excess of the given, above the next less tabular altitude is contained in the right-hand column of each page; and that answering to the seconds of parallax is given in the intermediate part of the Table. The correction depending on the sun's or star's apparent altitude is placed at the bottom of the Table in each page. As the size of the paper would not admit of the complete insertion of the auxiliary angle, except in the first vertical column of each page under or over 53; therefore, in the eight following columns, it is only the excess of the auxiliary angle above 60 degrees that is given: hence, in taking out the auxiliary angle from those columns, it is always to be prefixed with 60 degrees. The auxiliary angle is to be taken out of the Table, as thus : Enter the Table with the moon's apparent altitude in the left-hand column of the page, or the altitude next less if there be any odd minutes, opposite to which and under the minutes of the moon's horizontal parallax at top, will be found the approximate auxiliary angle. Enter the compartment of the "Proportional parts to seconds of parallax," abreast of the approximate auxiliary angle, with the tenths of seconds of the moon's horizontal parallax in the vertical column, and the units at the top; in the angle of meeting will be found a correction, which place under the approximate auxiliary angle; then enter the last or right-hand column of the page abreast of where the approximate auxiliary angle was found, or nearly so, and find the proportional part corresponding to the odd minutes of altitude, which place under the former. To these three let the correction, at the bottom of the Table, answering to the sun's or star's apparent altitude, be applied, and the sum will be the correct auxiliary angle. Example. Let the moon's apparent altitude be 25:37, the sun's apparent altitude 5820, and the moon's horizontal parallax 59:47"; required the corresponding auxiliary angle ? Aux. angle ans. to moon's app. alt. 25:30, and hor. par. 59: is 60:13:47" Auxiliary angle, as required 12 4 4 60:14: 7: TABLE XXI. Correction of the Auxiliary Angle when the Moon's Distance from a Planet is observed. The arguments of this Table are, a planet's apparent altitude in the left or right hand column, and its horizontal parallax at top; in the angle of meeting stands the correction, which is always to be applied by addition to the auxiliary angle deduced from the preceding Table: hence, if the apparent altitude of a planet be 26 degrees, and its horizontal parallax 23 seconds, the correction of the auxiliary angle will be 6 seconds, additive. This Table was calculated by a modification of the rule (page 43) for computing the correction of the auxiliary angle, answering to the sun's or star's apparent altitude; as thus:— To the logarithmic secant of the planet's apparent altitude, add the logarithmic cosine of its true altitude, and the constant logarithm 9.698850;* and the sum (abating 20 in the index) will be the logarithmic cosine of an arch; the difference between which and 60 degrees will be the required correction. This is the log.cosine of 60 degrees diminished by .000120, the difference between the log, cosines of the true and apparent altitude of a fixed star between 30 and 90 degrees. Example Let the apparent altitude of a planet be 30 degrees, and its horizontal parallax 23 seconds: required the correction of the auxiliary angle? Error arising from a Deviation of one Minute in the Parallelism of the Surfaces of the Central Mirror of the Circular Instrument of Reflection. This Table contains the error of observation arising from a deviation of one minute in the parallelism of the surfaces of the central mirror of the reflecting circle, the axis of the telescope being supposed to make an angle of 80 degrees with the horizon mirror; it is very useful in finding the verification of the parallelism of the surfaces of the central mirror in the reflecting circle, or of the index-glass in the sextant; as thus: : Let the instrument be carefully adjusted, and then take four or five observations of the angular distance between two well-defined objects, whose distance is not less than 100 degrees; the sum of these, divided by their number, will be the mean observation. Then, Take out the central mirror, and turn it so that the edge which was before uppermost may now be downwards, or next the plane of the instrument; rectify its position, and take an equal number of observations of the angular distance between the same two objects, and find their mean, as before now, half the difference between the mean of these and that of the former, will be the error of the mirror answering to the observed angle. If the first mean exceeds the second, the error is subtractive; otherwise additive the mirror being in its first or natural position. Hence, if the |