To Subtract quantities having regard to their signs. Suppose the sign of the quantity, which is to be subtracted, to be changed, that is, if it is affirmative, suppose it to be negative, or if it is negative, suppose it to be affirmative. Then proceed as in the above rule for adding quantities.

When one arc is to subtracted from another and the latter is the less of the two, we may increase it by 360°.

To find the Logarithmic Sine, Cosine, Tangent, or Cotangent of an arc, with its proper Sign, from Tables that extend only to each minute of the quadrant.

When the given arc does not exceed 180°. With the given arc, or when it exceeds 90°, with its supplement to 180°, take out from the table the required Sine or Tangent, &c. When there are seconds, take out the quantity corresponding to the given degrees and minutes; also take the difference between this quantity and the next following one, in the table. Then 60" the odd seconds of the given arc the difference: a fourth term. This fourth term, added to the quantity taken out, when it is increasing, but subtracted when it is decreasing, will give the required quantity.

When the given arc exceeds 180°. Subtract 180° from it, and proceed as before. When the arc exceeds 270°, it is more convenient, and amounts to the same, to subtract it from 360°.

To determine the Sine of the quantity. Call the arc from 0° to 90°, the first quadrant; from 90° to 180°, the second quadrant; from 180° to 270°, the third quadrant; and from 270° to 360°, the fourth quadrant. Then,

The Sine of an affirmative arc is affirmative for the first and second quadrants; and negative for the third and fourth. For a negative arc it is just the reverse; the sine being negative in the first and second quadrants and affirmative in the third and fourth.

The Cosine of an affirmative arc is affirmative for the first and fourth quadrants, and negative for the second and third. It is the same for a negative arc.

The Tangent or Cotangent of an affirmative arc is affirmative for the first and third quadrants, and negative for the second and fourth. For a negative arc it is just the reverse; the tangent and cotangent being negative in the first and third quadrants, and affirmative in the second and fourth.

Note. Negative logarithms or logarithmic sines &c., are frequently designated by a small n, placed at the right hand, instead of the sign, before them.

By attending to the preceding rules, the student will easily find the Sine, Cosine &c. of an arc in either quadrant, with its appropriate sign, as exemplified in the following table:

The logarithmic Sine, Cosine, Tangent, or Cotangent of an arc being

given, to find the arc.

When the given quantity can be found in the table, under or over its name, take out the corresponding arc. When the given quantity is not found exactly in the table, and the arc is required to seconds, take out the degrees and minutes corresponding to the next less quantity, when that quantity is increasing; but to the next greater when it is decreasing. Take the difference between the quantity corresponding to the degrees taken out, and the next following one in the table; also, take the difference between the same quantity and the given one. Then, the first difference : the second :: 60′′: the number of seconds which is to be annexed to the degrees and minutes. Then,

For a Sine. When it is affirmative, the required affirmative arc will be, either the arc found in the table, or its supplement to 180°. When the

sine is negative, the required arc will be, either the arc found in the table, increased by 180°, or its supplement to 360°.

For a Cosine. When it is affirmative, the required affirmative arc will be, either the arc found in the table, or its supplement to 360°. When the cosine is negative, the required are will be, either the supplement of the arc found in the table, to 180°, or that arc increased by 180°.

For a Tangent or Cotangent. When it is affirmative, the required affirmative arc will be, either the arc found in the table, or that arc, increased by 180°. When the tangent or cotangent is negative, the required arc will be, either the supplement of the arc found in the table, to 180°, or its supplement to 360°.

When the required arc comes out more than 180°, the equivalent negative arc is frequently taken.

These rules are exemplified by the quantities in the following table:

Note. Tables which extend only to five decimals, will give the arc, for a tangent or cotangent, true to the nearest second, for a few degrees, near to 0°, 90°, 180°, or 270°; for a sine, near to 0° or 180°; and for a cosine near to 90° or 270°. In other cases they cannot be depended on to give the seconds accurately. They are, however, sufficient for many calculations; particularly, when the nature of the problem does not make it necessary that the required arc or angle should be determined with great accuracy.

As most mathematical students are furnished with a set of such tables, and as an example worked by them will serve as well to illustrate a rule, as if worked by those which are more extensive, they will generally be used, in working the examples and questions in the following problems.

Observations relative to the Signs and Indices of Logarithms. A logarithm is affirmative when the natural number is affirmative, and negative when it is negative.

When several logarithms, or logarithms and the arithmetical complements of logarithms, are added together, if they are all affirmative or if there is an even number of negative ones, the resulting logarithm will be affirmative; but if there is an odd number of negative ones, the resulting logarithm will be negative.

Instead of the negative index of the logarithm of a decimal number, the index increased by 10, is frequently used. Thus, when there is no cipher

on.

between the decimal point and first significant figure, 9 is put for the index; when there is one cipher between them, 8; when there is two, 7; and so When this is done and the resulting logarithm of a computation is the logarithm of a natural number, if the index is 9, the number will be a decimal without any cipher between the decimal point and first significant figure; if it is 8, there must be one cipher between them; if it is 7, there must be two; and so on. If the index is near to 0, the resulting number is generally integral.

Rejection of the tens in the index of the sum of logarithms. In working the following problems, when several logarithms or logarithms and the arithmetical complements of logarithms are added together, the tens in the index of the sum are to be rejected. When, however, the sum is the tangent or cotangent of an arc, and a table of tangents is used in which the 10 in the index has not been rejected, one 10 should be retained in the index of the sum, if its rejection would reduce this index below 5.

Note. By the term Sine, Cosine, &c. in the rules for working the following problems, the logarithmic Sine, Cosine &c. is to be understood.

PROBLEMS FOR MAKING VARIOUS ASTRONOMICAL

CALCULATIONS.

PROBLEM I.

To work, by logistical logarithms, a proportion, the terms of which are minutes and seconds of a degree, or of time, or hours and minutes. With the minutes at the top and seconds at the side, or if a term con. sists of hours and minutes, with the hours at the top and minutes at the side, take from table LXXX, the logistical logarithms of the three given terms, and proceed in the usual manner of working a proportion by logarithms. The quantity, in the table, corresponding to the resulting logarithm, will be the fourth term.

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

2. The student will easily perceive that proportions that are worked by logistical logarithms, may also be worked by the common rule in arithmetic.

EXAM. 1. When the moon's hourly motion is 31′ 57′′, what is its motion in 39m. 22sec.? Ans, 20′ 58′′.

2. If the moon's declination change 2° 29′ in 12 hours, what will be the change in 8h. 21m.? Ans. 1° 44'.

3. When the sun's hourly motion is 2' 31", what is its motion in 17m. 18sec.? Ans. 0' 44".

4. When the sun's declination changes 22′ 14′′ in 24 hours, what is its change in 19h. 25m.? Ans. 17′ 59′′.

PROBLEM II.

From a table in which quantities are given, for each Sign and Degree of the circle, to find the quantity corresponding to Signs, Degrees, Minutes, and Seconds.

Take out, from the table, the quantity corresponding to the given signs and degrees; also take the difference between this quantity and the next following one. Then 60' odd minutes and seconds :: this difference : 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. The given quantity with which a quantity is taken from a table, is called the Argument.

Note 3. In many tables, the argument is given in parts of the circle, supposed to be divided into 100, 1000, or 10,000, &c. parts. The method of taking quantities from such tables is the same as is given in the above rule; except that when the argument changes by 10, the first term of the proportion must be 10, and the second, the odd units; when the argument changes by 100, the first term must be 100, and the second, the odd parts between hundreds; and so on.

EXAM. 1. Given the argument 1' 9° 31' 26", to find the corresponding quantity in table XLIV. Ans. 11° 13′ 57′′.

1. 9° gives 11° 11′ 15′′

The difference between 11° 11' 15" and the next following quantity in the table is 5' 9".