2. •8

6. 96

6. 6

Given the nat. sines, to find the angle.

I. 898002

3. 444 4. 20389 Given the nat. cosine, to find the angle.

I. *448807 2. 948397 3. 514841 4. '974822 5. 999000

TABLES OF LOGARITHMS OF TRIGONOMETRICAL RATIOS.

67. The Trigonometrical Ratios being numbers, have logarithms that correspond to them. In practice the logarithmic are generally far more useful than the natural sines, &c., though the latter are often necessary, or in some simple kinds of calculation, preferable.

68. As the sines and cosines of all angles, and the tangents of angles less than 45°, are less than radius or unity, being proper fractions, the logarithms of the value of these quantities, properly, have negative characteristics. In order to avoid the inconvenience of printing negative logarithms, and for other reasons, 10 is added to the characteristic before it is registered in the table of logarithmic sines, &c., so that we find the characteristic 9 instead of ī, 8 instead of 2, &c.

Thus, on referring to the Table of Natural Sines (Table XXVI, NORIE), we find natural sine of 16' 275637. If we calculate the logarithm of 275637, we find its value is I'440338; if to this 10 is added we find that

To preserve uniformity, the characteristics of the logarithms of all the other ratios, namely, of the log. tangents, cotangents, secants, and cosecants are increased by 10. In trigonometrical operations this is convenient, but principally because the extraction of roots very seldom occurs.

It may be observed here that the uniform addition of 10 to the characteristic gives the logarithm of 10000 million times the natural number.

Thus, 9'599327 is the log. of 3979486000, and this latter number is the natural sine corresponding to a radius of 10000 millions, instead of a radius of unity.

69. Usual arrangement of Tables of Logarithmic Sines, Cosines, &c.— The table of logarithmie sines, cosines, tangents, cotangents, secants, and cosecants, contain all ares from zero (0°) through all magnitudes up to 90°, the log. of radius, as just stated, being 10. At the top of the page is placed the number of degrees, and in the left-hand column each minute of the degree, opposite to which are arranged the numerical values of the log. sine, cosine, &c., of the corresponding angle in those columns, at the top of which those terms are placed. The headings of the columns run along the top, thus, as far as 44°. The degrees from 45° to 90° are placed at the bottom of the page, and the minutes of the degree arranged in a right-hand column, so that the angles read off on the right-hand side are complemental to those read off at the points exactly opposite on the left-hand side, the values of the sines, cosines, tangents, &c., being found in the columns at the bottom of which those terms are found. This arrangement is rendered practicable by the circumstance of every angle between 45° and 90° being the complement of another between 45° and o°, every sine of an angle less than 45° is the cosine of another greater than 45°, every tangent is a cotangent, &c.; the sines, tangents, &c., of angles being respectively equal to the cosines, cotangents, &c., of the complements of the same angle. (Art. 44.)

The following shows the usual arrangement of such tables:

Besides the columns headed "sine, tangent," &c., are three smaller columne headed "D." or "Diff." They contain, in most tables, the differences between the values of the consecutive logarithms in the contiguous columns on either side, but corresponding to a change of 100" in the arc (not the difference corresponding to 60" of arc or angle); and it must be kept in mind that the same difference is common to the sine and cosecant, to the tangent and cotangent, and to the secant and cosine. They are inserted for the convenience of finding the values of the sines and cosines, &c., of angles which are expressed in degrees, minutes, and seconds.

70. The above, as just stated, is the usual arrangement of most tables, but in the earlier editions of NORIE and some other works the arrangement is somewhat different.

The columns are arranged thus:

M Sine. Diff. Cosine. Diff. Tangent. Diff. Cotangent Secant. Cosecant. M

M Cosine. Diff. Sine. Diff. Cotangent Diff. Tangent. Cosecant. Secant. M

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

H

of logarithms on the left hand of the page, and is also the "Diff." for the first column on the right of the page; that the second column of "Diff." (from the left) belongs to the second column of logarithms from either 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, which may be otherwise expressed, thus:-A cosecant takes a sine "diff."; a secant takes a cosine "diff."; and cotangent takes a tangent "diff.”

71. 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.

72. When an angle is presented in degrees and minutes only, the tabular logarithm of its sine, tangent, &c., will be found (Table XXV, NORIE, or Table 68, RAPER) simply by inspection, according to the following:

RULE XVIII.

1o. If the angle or arc is less than 45°. Find the page having the given degrees at the top, and in the left hand marginal column find the minutes, 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 page having the given degrees at the bottom, and find 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 table 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'.

Take out the logarithms of the following trigonometrical ratios.

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

RULE XIX.

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. 69 and 70, page 49).

2°. Multiply the tabular difference ("Diff.") by the seconds, reject the last two figures (always two) 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, '3970448, 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. sine 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 1o 30', and to every ten seconds up to 4° 30°.

EXAMPLES.

Ex. 1. Find the log. sine of 6° 36′ 27′′.,

Here, as the arc is less than 45°, by looking at the top of the Table for the degrees (6o), and in the first column on the left, marked M at the top, for the minutes (36'), we find in the column having at its top the word sine the figures 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 multipled 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

• 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 small 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 dis found somewhat more readily.

Da
60

Da

1001

and thus

by 1, since the figures out off exceed 50 (see Note 1, page 51) gives 491 as the correction of the logarithm for the seconds. The work will stand thus:

The log. cosine of 13o 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 parts for the seconds are subtracted in this instance, being a colog. XIX, 3°).

Ex. 3. Find the log. tangent 72° 59′ 8′′.

(See Rule

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:

Log, tang, 72° 59' 0" 10'514209
Parts for
8"=+ 60

Log. tang. 72° 59′ 8′′ = 10°514269

Ex. 4. Find the log. cotangent of 73° 21′ 7′′.

Tab. diff. 753

60,24

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, beng a colog.

Log. cotang. 73° 21′ 0′′ = 9°475763
Parts for
7" ==

54

100

(Tab. diff. 767) × 7
= 53,69
or 54

Log. cotang. 73° 21′ 7′′ = 9°475709

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

Ex. 5. Take out log. sine 1° 5' 34".

Here the angle whose log. sine is sought being less than 2o, 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: