minutes, seek first for the number of degrees among those which are written at the top or bottom of the pages; at the top if it is less than 45°; at the bottom if it is greater. Run the eye down the first column which goes on increasing from top to bottom, if the number of degrees is found at the top of the page; or up the last column which goes on increasing from the bottom upwards if the number of degrees is found at the bottom; run the eye, I say, through one or the other of these columns in the direction in which it increases until you have found the number of minutes given; upon the same horizontal line with the minutes thus found you will find the logarithm of the sine, cosine, tangent or cotangent which you seek. In order not to mistake the column, it is necessary to consult the title at the head of the column, if the number of degrees given is at the top of the page, but if it is at the bottom, the inferior title must be consulted.

EXAMPLE I.

Required the logarithmic sine, tangent, cosine and cotangent of 19° 55'.

I find 19° at the top of page 37, I descend the first column at the left marked м which goes on increasing downwards till I find 55'; upon the same horizontal line, and in the column entitled sine at top I find 9.532312, in the column entitled cosine 9.973215, in the column of tangents 9.559097, and in that of cotangents 10.440903; and these numbers are therefore the numbers required.

EXAMPLE II.

Required the logarithmic sine and tangent of 70° 10'.

I find 70° at the bottom of page 37, I ascend the last column marked м at bottom which goes on increasing upwards; I find 10' in that column; upon the same horizontal line I find in the column marked sine at bottom 9.973444, and in the column marked tangent at bottom 10.442879, which are the logarithms sought.

CASE II. If the given number is composed of degrees minutes and seconds, find the logarithm of the degrees and minutes as above, and then to know how much this should be increased for the given number of seconds, in case of the sine or tangent, or diminished in case of the cosine or cotangent, observe that the number in the column marked D is the increase of the logarithm for 1", and if multiplied by the given number of seconds the product will be the quantity to add to the logarithmic sine or tangent before found, or to subtract from the logarithmic cosine or cotangent.

The number in the columnn D is calculated by subtracting one of two consecutive logarithms in the table, which differ by 1' from the other, and dividing the remainder by 60, the number of seconds in a minute; the quotient is the difference of logarithms corresponding to a difference of 1" in the numbers to which they belong. A decimal point must be understood two places from the right of each number in the column This calculation depends upon the principle mentioned at art. 51, that the differences of logarithms are proportional to the differences of their corresponding numbers.

D.

The logarithmic sines and cosines have each their column of differences annexed, but the tangents and cotangents have but one between them, the reason of which will appear from the following demonstration.

By art. 37 we have

tan x cot = R2

applying logarithms to this equation, since the log. of a product the sum of the logs. of the factors, and the log. of a power = log. of the number raised to the power multiplied by the index of the power, we have

log. tan + log. cot = 2 log. R = 20

log. R being 10. Therefore having two arcs a and b, since log. tan log. cot. in both is 20 we have

log. tan a

posing

log. tan a

log. cot a = log. tan blog. cot b, or trans

that is the difference of the logarithmic tangents of two arcs is equal to the difference of their logarithmic cotangents.

EXAMPLE I.

Required the logarithmic sine of 40° 26' 28".

I find the log. sine of 40° 26' to be 9.811952; the number in the next column D is 247, which I multiply by the given number of seconds 28; the product, inserting the decimal point which is understood to cut off the 47 in the number taken from the column D, is 69.16, which added to the logarithm before found

Gives

9.811952

69.16

9.812021

rejecting the 16

which goes beyond the usual number of decimal places. The logarithmic tangent of any given number of degrees, minutes and seconds, is found in a similar manner from the column entitled tangent.

EXAMPLE II.

Required the logarithmic cosine of 3° 40' 40".

I find the cosine of 3° 40' to be 9.999110; the tabular difference in the adjoining column is 13, which being multiplied by the seconds 40, the product is 5.20; subtracting* this result from 9.999110, the remainder is 9.999104, the logarithmic cosine sought.

PROBLEM.

To find the degrees, minutes and seconds answering to any given logarithmic, sine, cosine, tangent or cotangent. The method is, of course, exactly the reverse of that just given. Look for the given logarithm in the proper column, which you will know from its title, either at the top or bottom, and if you find it exactly, the degrees will be found at

* It will be recollected that as the arc increases in the first quadrant, the cosine diminishes.

the top of the page, and the minutes on the same horizontal line with your logarithm, in the first column at the left, if the title of the column is at top, but the degrees will be found at the bottom of the page, and the minutes in the column at the right, if the title of the column which contains your logarithm is at the bottom.

If the given logarithm cannot be found, take the next less logarithm contained in the tables, subtract it from the given, annex two ciphers to the remainder, and divide by the number in the column marked D; the quotient is seconds, which add to the degrees and minutes belonging to the logarithm found in the tables, if your given logarithm be that of a sine or tangent, but which subtract from the degrees and minutes, if a cosine or cotangent.

EXAMPLE I.

Required the number of degrees, minutes and seconds, of which the logarithmic sine is 9.880054.

I find the next less logarithm in the column marked sine at bottom, to be 9.879963, which subtracted from the given logarithm, leaves 91; I annex two ciphers to this, and divide by the number 181, found in the column D adjoining; the quotient is 50, which is seconds. Taking the degrees from the bottom of the page, and the minutes from the column at the right, and in the same horizontal line with the logarithm 9.879963, I have 49° 20′ 50′′ for the number required.

EXAMPLE II.

Required the number of degrees, minutes and seconds, of which the log. cotangent is 10.008688.

I find the next less logarithm in the table, to be 10.008591, that of 44° 26', which subtracted from the given logarithm, leaves 97, to which annexing two ciphers, and dividing by the tabular difference 421, the quotient is 23", and the required number is 44° 26'23" or 44° 25' 37".

60. The secants and cosecants of arcs have not been in

serted in the table, because they may be easily computed from the cosines and sines. Thus: (Art. 33.)

log. of the square of a number equal to twice the log. of the number, (Art. 54,) and log. of R = 10.

To obtain the log. secant, therefore, we have this

To obtain log. cosec, subtract log. sine from 20.

EXAMPLE.

Required the log. secant of 48° 35' 27"

log. cos. 48° 35′ 27′′ 9.820485

log. cosec. 35° 27′ 24′′ = 20-9.763493 = 10.236507

A method of finding with greater accuracy the sine and tangent of a very small arc, or the cosine and cotangent of one near 90°, is pointed out at Art. 123.

To find the trigonometrical lines of arcs greater than 90°, observe the rule at Art. 17.