EXPLANATION OF THE TABLES. 
229 
dec. part of the log. due to 70' 137", or 72' 17", is 8962, the prop. log. 
required is 1 * 3962. 
So the prop. log. of an arc, under 1' 48" may be found to the hundredth 
of a second by multiplying by 100. 
To find the arc or time to the tenth of a second to a given prop. log. 
exceeding 1 * 0000. Look in the Table till the decimal part again occurs, 
and divide the arc by 10. 
Example .—Find the time to the prop. log. 2 * 5106. Look for 1 * 5106 ; 
the nearest found is 1*5110, against 5m. 33s., or 333s.; hence the time 
required is 33s. * 3. 
Four places are enough for common purposes; but since the fourth 
place ceases to change by 1 after lh. 13m., a greater time than this 
cannot be found truly to Is. So also, a time exceeding 2h. 25m. cannot 
be found truly to 2s. This defect may be avoided in some cases by 
employing the complement of the interval to 3h. 
Table XXVIII. Natural Cosines .—This table gives the natural cosines 
of angles from 0° to 90°. The several columns of cosines are headed 
by degrees, the accompanying minutes being inserted in the first column 
on the left of the page; this is equally a column of seconds, and is 
accordingly headed with the marks for minutes and seconds. The 
number of degrees and minutes of an arc or angle is found in the column 
of cosines under the degrees and in a line with the minutes found in the 
first column; if there are seconds also in the arc or angle, again refer to 
the first column for these, and in the same horizontal line with them 
in the column headed parts for,” next to the column from which the 
cosine has been extracted, will be found the correction for seconds, which 
is always to be subtracted , and the remainder will be the cosine of the 
given degrees, minutes, and seconds. When the angle or arc for which 
the cosine is required is greater than 90°, the table must be entered with 
its supplement and the corresponding cosine regarded as negative. The 
decimal points have not been inserted before each cosine; and in com¬ 
putation, the numbers may always be regarded as integers. 
Example 1. Suppose the natural cosine of 39° 22' 33" were required: 
Turning to the page containing 39 on the top, we find “ parts ” against 
33" to be 103, and the cosine against 22' to be 773103; subtracting 
103 from this, we get the cosine required, 773000. 
2. Required the cosine for 120° 18' 20": the supplement of this is 
