given in any line, they are to be taken from the first line above containing them, unless the last three are preceded by a star *, in which case they are to be taken from the line immediately below.

Thus, (p. 13) log 6615 = 3.82053, log 67.36 = 1.82840, log 6.764 = 0.83020.

and

(3) For a number of more than four figures.

To find log 2845.672.

We find from the table on p. 5, as in (2),

log 2845 = 3.45408

log 2846 = 3.45423

diff. for 1 = 0.00015

Thus, for an increase of 1 in the number there is an increase of .00015 in the logarithm.

Hence, assuming that the increase of the logarithm is pro portional to the increase of the number, then an increase in the number of .672 will correspond to an increase in the logarithm of .672 x .00015.00010, to the nearest fifth decimal place.

Hence

NOTE 3.

log 2845 = 3.45408
diff. for .672 = .00010

.. log 2845.672 = 3.45418

We assumed in this method that the increase in a logarithm is proportional to the increase in the number. Although this is not strictly true, yet in most cases it is sufficiently exact for practical purposes.

8. From the above work we have the following rule for a number of more than four figures:

Find the tabular mantissa of the first four significant figures of the number; subtract this mantissa from the next greater tabular mantissa; multiply the difference thus found by the remaining figures of the number, as a decimal; add the product to the mantissa of the first four figures, and prefix the proper characteristic.

NOTE 4. The difference between any mantissa in the table and the mantissa of the next higher number of four figures, is called the tabular difference; and the corresponding proportional parts are placed in the column headed P.P. By means of this column of proportional parts the above multiplication is facilitated.

It will be seen that this difference between the logarithms of two consecutive numbers is not always the same; for instance, those in the upper part of p. 5 differ by .00018, while those in the

middle and the lower parts differ by .00016 and .00014. In the column with the heading 15 we see the difference 9 corresponding to the figure 6, which implies that when the difference between the logarithms of two consecutive numbers is .00015, the increase in the logarithm corresponding to an increase of .6 in the number is .00009.

Thus, the mantissæ of the logarithms of 2845 and 2846 differ by .00015; therefore 15 is the tabular difference. Then in the proportional table for 15, we find

or 10 to the nearest integer, which agrees with the value above.

9. To find the number corresponding to a given logarithm.

By reversing the above operations, the number corresponding to a given logarithm may be found, as will be seen by the following example:

Find the number whose logarithm is 3.47384.

We find that this mantissa does not occur exactly in the table. We therefore take out the next smaller mantissa, .47378 (on p. 5), whose corresponding number is 2977, and the next greater mantissa .47392, whose corresponding number is 2978.

The difference between these two mantissæ

=.00014.

The difference between the smaller and given mantissæ=.00006. Thus, for an increase of 1 in the number, there is an increase of .00014 in the mantissa; hence for an increase of .00006 in the mantissa there will be an increase of of 1 in the number = .43.

Hence, the number corresponding = 2977.43.

From the above work we have the following rule:

Find the tabular mantissa next less than the given mantissa, and the corresponding number of four figures; divide the difference of these mantissæ by the tabular difference, annex the quotient to the first four figures of the number, and point off the result according to

NOTE. The labor of division may be saved by using the table of proportional parts for 14, as follows:

10. Arithmetic Complement. By the arithmetic complement of the logarithm of a number, or briefly, the cologarithm of the number, is meant the remainder found by subtracting the logarithm from 10. To subtract one logarithm, b, from another, a, is the same as to add the cologarithm, 10 b, and then subtract 10 from the result.

Thus,

a-ba +(10-6)-10.

When one logarithm is to be subtracted from the sum of several others, it is more convenient to add its cologarithm to the sum, and reject 10. The advantage of using the cologarithm is that it enables us to exhibit the work in a more compact form. The cologarithm is easily taken from the table mentally by subtracting the last significant figure on the right from 10, and all the others from 9.

TABLE II. LOGARITHMS OF SINES, TANGENTS, ETC.

11. This table (pages 21-82) contains the logarithms of the sines, tangents, cotangents, and cosines of all angles from 0° to 90°.

If the angle is less than 45°, we look for the name of the function and the number of degrees in the angle at the top of the page, and the minutes in the left-hand column.

If the angle is between 45° and 90°, we look for the name of the function and the number of degrees at the bottom of the page, and the minutes in the right-hand column. In each case the horizontal rows at the top of the pages go with the degrees at the top, and the horizontal rows at the bottom go with the degrees at the bottom.

On pp. 21-33 the minutes and each ten seconds are given in columns at the left and right, and the odd seconds are given in a horizontal row at the top and bottom of each page. On

pp. 34-82 the minutes are given in columns at the left and rightand on pp. 34-43 each ten seconds is given in a horizontal row at the top and bottom of each page.

It is sufficient to have tables which give the functions of angles only in the first quadrant, since the functions of all angles of whatever size can be reduced to functions of angles less than 90° (Art. 35).

12. Since the sines and cosines of all angles, the tangents of angles less than 45°, and the cotangents of angles greater than 45°, are less than unity, the logarithms of these functions are negative. To avoid the inconvenience of using negative characteristics, 10 is added to the logarithms of all these functions before they are entered in the table. The logarithms so increased are called the tabular logarithms of the sine, tangent, etc.

Thus,

log sin 27° 48' = 9.66875;

log tan 27° 48' = 9.72201;

log cot 70° 5' = 9.55910;

log cos 27° 48' = 9.94674.

13. To find the logarithmic sine, tangent, etc., of a given angle. (1) When the angle contains only degrees and minutes.

In this case the logarithm is given immediately in the table. Thus we find (pp. 56 and 57), the following:

log sin 18° 38':
log cot 19° 23'

= 9.50449.

= 0.45367.

log sin 71° 13' = 9.97623.

log tan 70° 51' = 0.45935.

(2) When the angle contains degrees, minutes, and seconds.

In this case we take out the logarithmic function for the degrees and minutes, as in (1); the correction for the seconds has to be calculated in the same manner as for the logarithms of numbers (Art. 7). For this purpose, on pp. 44-82, the differences of the logarithmic sines and cosines for 1' are given in the columns headed d. (difference), and those of the logarithmic tangents and cotangents in the columns headed c. d. (common difference). In the case of tangent and cotangent, only one column of differences is necessary for both functions.

Ex. 1. To find log sin 18° 25' 35".

Page 56, log sin 18° 25'

=

9.49958. Tabular difference
22

=

9.49980.

= 38.

From this work, we have the following rule:

Find from the table the logarithmic function for the degrees and minutes, and the corresponding tabular difference; divide this difference by 60, multiply the quotient by the number of seconds, and add this correction, if the function is a sine or tangent; or subtract it, if the function is a cosine or cotangent.

NOTE. The labor of multiplication and division may be saved by means of the tables of proportional parts given in the right margin, the use of which is similar to those given in the table of logarithms of numbers. The proportional parts between 0° and 1°, and 890 and 90°, are given for 1, 2, 3, etc., tenths of a second; and between 10 and 89° they are given for 1, 2, 3, etc., seconds.

Ex. 2. To find log tan 64° 35′ 18′′.

Page 63,

log tan 64° 35' = 0.32313. Tab. diff. = 33.

Under diff. 33, P.P. for 10"=

5.5

NOTE.

.. log cos 35° 44′ 49" 9.90935.

Since the cosine diminishes as the angle increases, we subtract the proportional parts.

14. To find the angle corresponding to a given logarithmic sine or tangent.

By reversing the above operations, the angle corresponding to a given logarithmic function may be found, as will be seen by the following examples:

= 9.81510.

Ex. 1. Find the angle whose log sin We find that this mantissa does not occur exactly in the column of logarithmic sines. We therefore take out the next smaller logarithmic sine, 9.81505 (on p. 78), whose corresponding angle is 40° 47', and the tabular difference 14. The difference between this logarithm and the given one is 5. The difference for 1" is 14 ÷ 60 or .23; hence for an increase of 5 in the mantissa there will be an increase of seconds in the angle = 21". Hence, the angle corresponding = 40° 47′ 21′′.

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