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Five pages of the Nautical Almanac, for the month of May, 1836.
Tables of Second, Third, and Fourth Differences; useful in finding, from the Nautical Almanac, the moon's longitude or latitude for any intermediate time between noon and midnight.
TABLES LXIX., to LXXIX., inclusive.
Approximate tables for the planet Mercury; including also a small table containing the Heliocentric Longitude, Latitude, &c., of the planet Venus at the times of Transit over the sun's disc in 1874 and 1882.
Logistical Logarithms. This table is convenient in working proportions when the terms are minutes and seconds, or degrees and minutes, or hours and minutes.
Reduction to the Meridian. (See Problem XXX.)
It is frequently convenient to regard quantities as separated into two classes; those of one class being called affirmative, and those of the other negative. Thus, a right line or an arc of a circle, taken in one direction, being regarded as affirmative, a line or are taken in the opposite direction, is regarded as negative. An affirmative quantity is denoted by having the sign +, called the affirmative or plus sign, prefixed to it, and a negative quantity by having the sign —, called the negative or minus sign, prefixed to it. Before an affirmative quantity the sign is frequently omitted, it being understood to be affirmative if neither sign is prefixed; but before a negative quantity the sign must always be expressed.
If an affirmative arc, and a negative arc, equal to the supplement of the former to 360°, both commence at the same point in the circumference of a circle, they must also both terminate at the same point. We may, therefore, denote the position of a point in the circumference with reference to a given or fixed point, either by an affirmative arc or by a negative one equal to its supplement to 360°. Thus, supposing the affirmative arc to be 294° 47', we may substitute in place of it, 65° 13'.
To add quantities, having regard to their signs. ties have the same sign, add them as in common
When all the quanti
arithmetic, and prefix
that sign to the sum. When the quantities have different signs, add the affirmative quantities into one sum, and the negative into another. Then take the difference between these two sums and prefix the sign of the greater.
When several arcs are to be added together, if the sum exceeds 360°, we may reject 360°, or any multiple of it, and regard the result as the sum of the arcs.
From 4' 11"
Rem. 3 16
From 312° 17′ 39′′
Rem. 294 25 52
65 34 8
27.5 Subt. 12.3
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 be subtracted from another, and the latter is the less of the two, we may increase it by 360°.
Add, 28.4 +75.2 +33.9
From 21° 17′ 25′′
Rem 135 36 48
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 are 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 are 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:
37° 18′ 21′′ 37 18 21 114 35 10 114 35 10 247 12 36 314 17 50
Log. tangent. Log. cotang. 9.88193 10.11807 9.88193n 10.11807n
9.61916n 10.33956n 9.66044n 9.61916n 10.33956 9.66044 9.58811n 10.37659 9.62341 9.84409 10.01065n 9.98935n
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 are. 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.
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 are 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 :
9.78252 arc 37° 18′ 21′′ or 142° 41′ 39′′
10 or 314 17 50
9.85475n arc 225 42 9.90060 arc 37 18 9.61916n arc 114 35 Log. tangent 9.88193 arc 37 18 Log. tangent 10.33956n arc 114 35 Log. cotangent 9.62341 arc 67 12 Log. cotangent 9.98935n arc 134 17 51 or 314 17 51
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
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 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 on. 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 log. tangent or log. cotangent of an arc, and a table of log. 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.