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EXPLANATION OF THE TABLES
(c) When the number contains three significant figures.
In the column headed No., find the number. On a line with it, and in the column having o at the top, is the mantissa. characteristic (Rules I. and II.).
(d) When the number contains four significant figures.
In the column headed No., find the first three significant figures. On a line with these, and in the column having at the top the fourth significant figure, is the mantissa. Prefix the proper characteristic (Rules I. and II.).
log 9275 3.96731;
(e) When the number contains more than four significant figures. Suppose the logarithm of 62543 is required. Since the number lies between 62540 and 62550, its logarithm lies between their logarithms. In the column headed No., find the first three figures. On a line with these, and in the columns having 4 and 5 at the top, are the mantissæ .79616 and .79623. Prefixing the proper characteristic, we have
log 62550 4.79623
Here we see that while the number increases from 62540 to 62550, the logarithm increases .00007. Now our number, 62543, is of the way from 62540 to 62550; hence, if to the logarithm
of 62540 we add
of .00007, a nearly correct logarithm of 62543
log 62540 4.79616 correction= .00002 .. log 62543 4.79618
We have here assumed that the differences of logarithms are proportional to the differences of their corresponding numbers, which gives us results that are approximately correct. For greater accuracy we must use tables of more places.
To find the number corresponding to a logarithm.
(a) When the given mantissa can be found in the table.
The first three figures of the number are in the column headed No., and on a line with the mantissa; the fourth figure is at the top of the column containing the mantissa.
Find the number whose logarithm is 2.93202.
The mantissa, found on page 17, corresponds to the number 8551. As the characteristic is 2, the required number is 855.1.
(b) When the given mantissa cannot be found in the table. Find the number whose logarithm is 8.82252 10 or 2.82252. As the exact mantissa is not in the table, take out the next larger, .82256, and the next smaller, .82249, and retain the characteristic in arranging the work.
Thus, the number corresponding to 2.82256 is .06646 and the number corresponding to 2.82249 is .06645
Now the given logarithm, 2.82252, is .00003 greater than the smaller of the two logarithms, and the difference in logarithms of .00007 corresponds to a difference in numbers of .00001; therefore we should increase the number corresponding to the logarithm .00003 2.82249 by 3 of .00001. or 2 .00007 7
Thus, the number corresponding to 2.82249 = .06645
and the correction (of .00001)
.. the number corresponding to
This table contains the logarithmic sine, tangent, cotangent, and cosine for every ten seconds from o° to 2°, and for every minute from 1° to 89°.
To find the logarithmic sine, tangent, cotangent, or cosine of an angle less than 90°.
When the angle is less than 45°, use the column headings at the top of the page and the left-hand minute column. When the angle is greater than 45°, use the column headings at the bottom of the page and the right-hand minute column.
Find log sin 20° 25' 12".
On page 43, we find
log sin 20° 25' = 9.54263.
This logarithm must be increased by of the difference between it and log sin 20° 26'.
.. log sin 20° 25' 12" 9.54263 + of .00034 = 9.54270.
Find log tan 52° 17' 10".
log tan 52° 17′ = 10.11162.
This value must be increased by 18 of the difference between it and log tan 52° 18'.
.. log tan 52° 17′ 10" = 10.11162 + of .00026 = 10.11166.
NOTE. In finding log sin or log tan we add the correction, but subtract in finding log cos or log cot.
For closer work, larger tables, such as those of Vega, should be employed.
To find an angle less than 90°, having given its logarithmic sine, tangent, cotangent, or cosine.
Find the angle for which log cos = On page 48, we find the next smaller logarithm, 9.94062 10, which corresponds to the angle 29° 17', and the next larger logarithm, which corresponds to the angle 29° 16'.
The given logarithm is .00003 larger than the smaller of these logarithms; therefore we have a correction of to make.
or & of 60"
Thus, log cos 9.94065 10 corresponds to angle 29° 17' of 60" 29° 16' 26".
Find the angle for which log tan = 0.15782.
0.15800 55° 12'
0.15773 = 55° 11'
The given logarithm is .00009 larger than the smaller of these fogarithms; therefore we have a correction of or of 60" to make.
Thus, log tan o.15782 corresponds to angle 55° 11' + of 60" = 55° 11' 20".
NOTE. In finding the angle corresponding to log sin or log tan we add the correction, but subtract for log cos or log cot.
TABLE III (a), (b), (c), (d)
These tables contain the natural trigonometric functions from o° to 90° at intervals of 6'.