XLI, that the right hand figure of the argument is supposed to be dropped. But when the greatest attainable accuracy is desired, it can be retained, and a cypher conceived to be written after the numbers in the columns of Arguments in the table. In Tables L, LI, LII, and LV, the degrees will be found by referring to the head or foot of the column. (See Problem II, Note 2). Table LIV, is for the Nutation of the Moon's Longitude. Tables LV to LIX, inclusive, are for finding the Latitude of the Moon.

Tables LX to LXIII, inclusive, are for the Equatorial Parallax of the Moon.

Table LXIV, furnishes the Reductions of Parallax and of the Latitude of a Place. The reduction of parallax is for obtaining the parallax at any given place from the equatorial parallax. The reduction of latitude is for reducing the true latitude of a place, as determined by observation, to the corresponding latitude, on the supposition of the earth being a sphere. The ellipticity to which the numbers in the table correspond is

Tables LXV and LXVI. Moon's Semi-diameter, and the Augmentation of the Semi-diameter depending on the altitude.

Tables LVII to LXXXV, inclusive, are for finding the Hourly Motions of the Moon in Longitude and Latitude.

Table LXXXVI. Mean New Moons, and the Arguments for the Equations for New and Full Moon in January. The time of mean new moon in January of each year has been diminished by 15 hours, which has been added to the equations in Table LXXXIX. Thus, 4h. 20m. has been added to equation I; 10h. 10m. to equation II; 10m. to equation III; and 20m. to equation IV.

Tables LXXXVII and LXXXVIII, are used with the preceding in finding the Approximate Time of Mean New or Full Moon in any given month of the year.

Table LXXXIX, furnishes the Equations for finding the Approximate Time of New or Full Moon.

Table XC, contains the Mean Right Ascensions and Declinations of 50 principal Fixed Stars, for the beginning of the year 1840, with their Annual Variations.

Table XCI, is for finding the Aberration and Nutation of the Stars in the preceding catalogue.

Table XCII, contains the Mean Longitudes and Latitudes of some of the principal Fixed Stars, for the beginning of the year 1840, with their Annual Variations.

Tables XCIII, XCIV, XCV. Second, Third, and Fourth Differences. These tables are useful for finding, from the Nautical Almanac, the moon's longitude or latitude for any time between noon and midnight.

Table XCVI. Logistical Logarithms. This table is convenient in working proportions when the terms are minutes and seconds, or degrees and minutes, or hours and minutes,-especially when the first term is 1h. or 60m.

To find the logistical logarithm of a number composed of minutes and seconds, or degrees and minutes of an arc; or of minutes and seconds, or hours and minutes of time.

1. If the number consists of minutes and seconds, at the top or bottom of the table seek for the minutes, and in the same column opposite the seconds in the left-hand column will be found the logistical logarithm.

2. If the number is composed of hours and minutes, the hours must be used as if they were minutes, and the minutes as if they were seconds.

3. If the number is composed of degrees and minutes, the degrees must be used as if they were minutes, and the minutes as if they were seconds.

To find the logistical logarithm of a number less than 3600. Seek in the second line of the table from the top the number next less than the given number, and the remainder, or the complement to the given number, in the first column on the left : then, in the column of the first number, and opposite the complement, will be found the logistical logarithm of the sum. Thus, to obtain the logarithm of 1531, we seek for the column of 1500, and opposite 31 we find 3713.

PROBLEM I.

To work, by logistical logarithms, a proportion the terms of which are degrees and minutes, or minutes and seconds of an arc; or hours and minutes, or minutes and seconds of time.

With the degrees or minutes at the top, and minutes or seconds at the side, or if a term consists of hours and minutes, or minutes and seconds, with the hours or minutes at the top, and minutes or seconds at the side, take from Table XCVI the logistical logarithms of the three given terms; add together the logistical logarithms of the second and third terms and the arithmetical complement of that of the first term, rejecting 10 from the index.* The result will be the logistical logarithm of the fourth term, with which take it from the table.

Note 1. The logistical logarithm of 60' is 0.

Note 2. If the second or third term contains tenths of seconds, (or tenths of minutes, when it consists of degrees and minutes), and is less than 6', or 6°, multiply it by 10, and employ the logarithm of the product in place of that of the term itself. The result obtained by the table divided by 10 will be the fourth term of the proportion, and will be exact to tenths.

Note 3. If none of the terms contain tenths of minutes or seconds, and it is desired to obtain a result exact to tenths, diminish the index of the logistical of the fourth term by 1, and cut off the right-hand figure of the number found from the table, for tenths.

Exam. 1. When the moon's hourly motion is 30' 12", what is its motion in 16m. 24s.?

* Instead of adding the arithmetical complement of the logarithm of the first term, the logarithm itself may be subtracted from the sum of the logarithms of the other two terms.

2. If the moon's declination change 1° 31' in 12 hours, what will be the change in 7h. 42m.?

3. When the moon's hourly motion in latitude is 2' 26".8, what is its motion in 36m. 22s.?

4. When the sun's hourly motion in longitude is 2′ 28", what is its motion 49m. 11s.?

Ans. 2' 1".

5. If the sun's declination changes 16' 33" in 24 hours, what will be the change in 14h. 18m.?

Ans. 9' 52".

6. If the moon's declination change 54".7 in one hour, what will be the change in 52m. 18s.?

Ans. 47".7.

PROBLEM II.

To take from a table the quantity corresponding to a given value of the argument, or to given values of the arguments of the table.

Case 1. When quantities are given in the table for each sign and degree of the argument.

With the signs of the given argument at the top or bottom, and the degrees at the side, (at the left side, if the signs are found at the top; at the right side, if they are found at the bottom), take out the corresponding quantity. Also take the difference between this quantity and the next following one in the table, and say, 60': this difference: : odd minutes and seconds of

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given argument: a fourth term. This fourth term, added to the quantity taken out, when the quantities in the table are increasing, but subtracted when they are decreasing, will give the required quantity.

Note 1. When the quantities change but little from degree to degree, the required quantity may frequently be estimated without the trouble of making a proportion.

Note 2. In some of the tables the degrees or signs of the quantity sought, are to be had by referring to the head or foot of the column in which the minutes and seconds are found. (See Tables L, LI, LII, and LV.) The degrees there found are to be taken, if no horizontal mark intervenes; otherwise, they are to be increased or diminished by 1°, or 2°, according as one or two marks intervene. They are to be increased, or diminished, according as their number is less or greater than the number of degrees at the other end of the column.

Note 3. If, as is the case with some of the tables, the quantities in the table have an algebraic sign prefixed to them, neglect the consideration of the sign in determining the correction to be applied to the quantity first taken out, and proceed according to the rule above given. The result will have the sign of the quantity first taken out. It is to be observed, however, that if the two consecutive quantities chance to have opposite signs, their numerical sum is to be taken instead of their difference; also that the quantity sought will, in every such instance, be the numerical difference between the correction and the quantity first taken out, and, according as the correction is less or greater than this quantity, is to be affected with the same or the opposite sign.

Exam. 1. Given the argument 7° 6° 24' 36", to find the corresponding quantity in Table L.

7° 6° gives 0° 43′ 17".4.

The difference between 0° 43' 17".4 and the next following quantity in the table is 1' 7".3.

* The student can work the proportion either by common arithmetic, or by logistical logarithms, as he may prefer. In working this and all similar propor. tions by the arithmetical method, the seconds of the argument may be converted into the equivalent decimal part of a minute by means of Table XVII. It will be sufficient to take the fraction to the nearest tenth.