S EXPLANATION OF TABLES 
The limits of the table take in all values of latitude, declination, and altitude which are likely to 
be required. In its employment, care must be taken to enter the table at a place where the declina- 
tion is appropriately named (of the same or opposite name to the latitude) ; it should also be noted 
that at the bottom of the last three pages values are given for the variation of a body at lower transit, 
which can only be observed when the declination and latitude are of the same name, and in which 
case the reduction to the meridian is subtractive; the limitations in this case are stated at the foot 
of the page, and apply to all values below the heavy rules. 
TABLE 30.—CHANGE OF ALTITUDE IN GIVEN TIME FROM MERIDIAN. 
This table gives the product of the variation in altitude in one minute of a heavenly body near 
the meridian, by the square of the number of minutes. Values are given in are for every 5’ from 0° 
to 7°, or in time for every 20s from 0™ to 28™, and for all variations likely to be employed in the 
method of ‘‘reduction to the meridian.” 
The formula for computing is: 
Red.=aX#, 
where a=variation in one minute (Table 29), and 
t=number of minutes (in units and tenths) from time of meridian passage. 
The table is entered in the column of the nearest interval of time or arc from meridian, and the 
value taken out corresponding to the value of a found from Table 29: The units and tenths are 
picked out separately and combined, each being corrected by interpolation for intermediate intervals 
of time or are. 
The resuit in minutes and tenths of arc is the amount to be applied to the observed altitude to 
reduce it to the meridian altitude, which is always to be added for upper transits and subtracted 
for lower. 
TABLE 31.—NATURAL TRIGONOMETRIC FUNCTIONS. 
This table and the explanation thereto, have been prepared and copyrighted by Lyman M. 
Kells, Willis F. Kern, and James R. Bland, who have supplied them to the Hydrographie Office for 
use in its publications. Neither the table nor any new feature embodied therein, may be reproduced 
in any form without the permission of the copyright owners. 
Table of natural values of trigonometric functions.—Table 31 contains the numerical values 
of the sines, cosines, tangents, and cotangents of angles from 0° to 90° at intervals of 1’. In the 
case of an angle in the range from 0° to 45°, the number of degrees in the angle and the names of 
the functions are found at the top of the page and the left-hand minute column applies; in the case 
of angles in the range from 45° to 90°, the number of degrees in the angle and the names of the func- 
tions are found at the bottom of the page and the right-hand minute column applies. Interpolation 
must be carried out without the aid of difference columns or tables of proportional parts. 
The following examples illustrate the method of using the tables. 
Exampie 1: Find sin 68°28’. 
Solution —We first find the page at the bottom of which 68° appears and then find the row of 
the 68° block containing 28’ in the right-hand minute column. In this row and in the column 
having sin at its foot we find 020 to which we must prefix 0.93 to obtain sin 68°28’ =0.93020. 
Exame.e 2: Find sin 38°38/27’’. 
Solution.—Using the tables and computing differences, we find the values exhibited in the fol- 
lowing form: 
sin 38°38/00’’ ” = 0.62433 
sin 38°38/27/"J2" 60” =? T\og 
sin 38°39/00’’ = 0.62456 
Hence 
9 
att or z= (34 )23= 10 (nearly). 
Therefore 
sin 38°38’27’’ =0.62433-+ 0.00010=0.62443. Ans. 
EXampLeE 3: If cot 6=0.37806, find @. 
Solution.— Using the tables and computing differences, we find the values exhibited in the fol- 
lowing form: 
cot Oeil Sheena 
cot ? 60=0.37806 33 
cot 69°18’00’’ =0.37787 
Hence 
a= a or n= 35 (60) 25’’ (nearly), and 6=69°17'25’’. Ans. 
ae Binge cot @ is positive in the third quadrant, we may also write an answer 180°+69°17/25/’= 
(Pa, 
