Since the same difference is common to the sine and cosecant, to the tangent and cotangent, in this arrangement, then, it must be particularly borne in mind, that the first "Diff." column (from the left) belongs to the first column of logarithms on the left hand of the page and to the first column on the right of the page; that the second column of "Diff." belongs the second column of logarithms from the right or left of the page; and that the third column of "Diff." belongs to the third column from either the right or the left.

102. In the use of these Tables, as in that of the natural sines, two questions present themselves:-First, having given the angle in degrees, minutes, and seconds, required the log. sine, log. cosine, &c. Second, having given the log. sine, log. cosine, &c., required the value of the angle in degrees, minutes, and seconds.

103. When an angle is presented in degrees and minutes only, the tabular logarithm of its sine, tangent, &c., will be found simply by inspection, according to the following:

RULE XXXVIII.

1o. If the angle or arc is less than 45°. Find the degrees at the top of the page, and the minutes in the left-hand marginal column, then opposite the minutes, and in the column which is marked at the top with the name of the ratio, will be found the logarithm sought.

2o. If the angle be greater than 45°. Look for the degrees at the bottom of the page, and for the minutes in the right-hand column; the logarithm of the proposed function of the angle will be found opposite the minutes in the column marked at the foot with the name of the ratio whose logarithm is sought.

Ex. I. Find the log. sine of 37° 47'.

EXAMPLES.

As the arc is less than 45°, by looking at the top of the table for the degrees (37°), and in the first column on the left for the minutes (47′), we find in the column having at its top the word sine, the figures 9.787232, which is the log. sine of the arc required.

Ex. 2. Find the log. tang. of 75° 34'.

Here, as the arc is greater than 45°, looking at the bottom of the tables for the degrees (75°), and in the last or right hand column for the minutes (34), we find in the column having tang. at the bottom 10:589431, which is the log. tangent of 75° 34'.

EXAMPLES FOR PRACTICE.

Take out the logarithms of the following trigonometrical ratios.

104. If the value of the angle be given in degrees, minutes, and seconds, we proceed by

RULE XXXIX.

1°. Find from the table the sine, tangent, secant, cosine, &c., which corresponds to the degrees and minutes; also take out the number in the contiguous column headed "Diff." on the same line (See Nos. 101 and 102, page 72.)

2°. Multiply the tabular difference ("Diff.") by the seconds, reject the last two figures of the product for the division by 100, and the remaining figures will furnish the proper correction for seconds.

NOTE 1.-If the value of the two figures cut off is not less than fifty, one must be added to the first right hand figure left.

3°. If the required quantity be a sine, tangent, or secant, add the result to the last figures obtained in 1°; if it be a cosine, cotangent, or cosecant subtract.*

The result will be the required sine, tangent, secant, cosine, &c.

NOTE 2.-The process above is sufficiently accurate unless for the sines and tangents of very small angles, and for the tangents and secants of angles very near 90°. When an angle of degrees, minutes, and seconds, and of less magnitude than 3°, occurs in calculation, neither the logarithmic sine nor the logarithmic tangent will be found very accurately from the ordinary Tables. In some books, as Hutton's "Mathematical Tables," a special Table is given, containing the logarithmic sines and tangents to every second in the first two degrees of the quadrant. By that Table we should find the correct log. tang. of 1° 25′ 45′′ to be 2 3970503, whereas, by using the tab. diff. for 1° 25′ and 1o 26′ in the ordinary Table, we should get the less accurate result, 23970448, because for such small angles, the successive tabular differences for one minute shows too rapidly a wide departure from equality. When an angle of degrees, minutes, and seconds, and within less than 3° of 90° occurs in calculation, we cannot, for the reason just stated, obtain very accurately from the ordinary Tables either the logarithmic or the natural tangent. Thus, the true log. tang. of 88° 4′ 15′′ is 1.6029497; but by the ordinary Tables we would get for the last three figures 552. Norie gives the log. sin. and log. tang. to every ten seconds of the first two degrees of the quadrant, and Raper gives the log. sines to every second up to 1° 30', and to every ten seconds up to 4° 30'.

*In some tables, these differences are those due to 1 minute, or 60 seconds, and are got by simply subtracting the greater of the logarithms from the less. The difference d, due to any smaller number (a) of seconds is found from such tables by the proportion 60: a :: D: d, so that d= But as before observed the differences usually given in the tables are those due not to 60 seconds but to 100 seconds, so that in these tables, d= ; and thus d is found somewhat more readily.

Da
60

Da
100

EXAMPLES.

1. Find the log. sine of 6° 36′ 27′′.

Here the given number of degrees (6°) being less than 45°, look for them in the head line at the top of the page, turning over the leaves till the proper page is found, then in that page look in the second line for the name of the column wanted, viz., the sine; and in the left hand vertical column marked M at the top, find the number of minutes (36'); having found the minutes, then in the same line and under sine is found 9'060460, which is the log. sine corresponding to 6° 36'. Now this log. being found in the first column on the left, the tabular difference must be taken out of the first "diff." column from the left. It will be noticed that there is no diff. exactly opposite to 36' but between 36' and 37' will be found the diff. 1817, which multiplied by the seconds (27) gives 49059, and rejecting the two last figures from this product (for the division by 100) gives quotient 490, which being increased by 1, since the figures cut off exceed 50 (see Note 1, page 74) gives 491 as the correction of the logarithm for the seconds. The work will stand thus:

The log. cosine of 13° 5′ is 9.988578, and the tabular difference corresponding to the log. cosine of the given degrees and minutes is 50; this being multiplied by 32 (the given number of seconds), and pointing off two figures to the right, is 16 to be subtracted, because the cosine is a decreasing log.; therefore

The log. tangent of 72° 59′ is 10'514209, and the tab. diff. corresponding to the given degrees and minutes is 753; this being multiplied by 8 (the number of seconds), and pointing off two figures to the right is 60, which is additive; thus:

The log. cotangent of 73° 21′ is 9'475763, and the tab. diff. corresponding to the cotangent of the given degrees and minutes is 767; this being multiplied by 7 (the given number of seconds), and pointing off two figures to the right is 54; which is to be subtracted in this instance, being a colog.

Log. cotang. 73° 21' 0" 9'475763
Parts for
7"=- 54

Log. cotang. 73° 21′ 7′′ 9°475709

(Tab. diff. 767) X7 = 53,69

100

or 54

The parts for the seconds are subtracted in this instance being a colog. (See Rule XXXIX, 3°.

Here the angle whose log. sine is sought being less than 22, it must, therefore, be taken out of the special part of the Table (see Table XXV, page 107, NORIE). The next less angle to be found in the Table is 1° 5' 30", the log. sine of which 8.279941, and the corresponding tabular, "Diff" (for 10" in this part of the Table) is 1104, which multiplied by 4, the seconds over 30, gives 4416, and cutting off one figure from the right, for the division by 10 gives the correction 442, to be added to the logarithm taken out of the Table; thus the work stands as follows:

105. For the functions* of an angle between 90° and 180° we may take the same functions of its supplement; hence to find the logarithm of a trigonometrical ratio of an angle greater than 90°, i.e., of an obtuse angle, we have the following

RULE XL.

Subtract the angle from 180° and look for the remainder, which is called its supplement in the Tables.

Look for the log. sine of remainder (namely 69° 36′), which is 9'971870; or log. sine 110° 24′ 9971870.

Ex. 2.

=

Find the log. secant of 95° 43'; also the log. cosecant of the same.

Look for the log. secant of 84° 17', which is 11001701; .. log. secant of 95° 43' is

11.001701.

Again, look for the cosecant of 84° 17', which is 10-002165; .. log. cosecant of 95° 43′ is 10'002165.

*

By the functions of angles, sometimes called their trigonometrical or gonoimetrical functions, are meant their sines, tangents, secants, versed sines, and chords; the word function signifying any quantity that is dependent on another, changing as it changes.

.. Log. tangent 128° 55′ 47′′ = 10'092720, Ans.

106. But a readier way, and the better practical method, is to proceed as follows::

RULE XLI.

Diminish the given angle by 90°, and look out the remainder in the tables, observing that if the trigonometrical ratio have "co" prefixed to drop the "co," but if it have not "co," prefix it, then find the logarithm corresponding to the new ratio. Or,

If A denote any angle less than 90°, then

For sine ... •(90+ A) take out ... ... cosine A

This rule may easily be remembered by observing that to the sine, tangent, and secant, co is prefixed, while from the cosine, cosecant, and cotangent the co is dropped, and in each case the excess of 90° of the angle is used.

Find the log. cosine of 110°.

EXAMPLES.

To find the log. cosine of 110°, or log. cosine (90+20), take out the log. sine 20°, which is 9'534052.

To find the log. secant of 160° 12', take out the cosecant 70° 12' which is 10026465.

In this instance "co" is prefixed to the given trigonometrical ratio, then, according to rule, "co" is dropped, and the log. corresponding to the new ratio is taken out for the remainder resulting from the given angle when diminished by 90°.