EXPLANATION OF TABLES 11 
Examp.Le: Find log tan 65°42’17’’. 
Solution.—On the page at the foot of which 65° appears, read opposite the 42’ of the right-hand 
minute column 533; attach to this the 10.34 found four spaces above this entry, to obtain 10.34533. 
In the nearest difference column opposite 17’’ find 9 and add it to the last figures (33) of 10.34533 and 
finally subtract 10 from the result to obtain 
log tan 65°42/17'’=10.34542—10=0.34542 
In the process of interpolation for seconds, the difference column, headed ‘‘diff,’’ nearest to the 
column of entries involved should be used. The change for seconds is found in this column opposite a 
number in the adjacent column equal to the number of seconds in the given angle. This difference is added 
to or subtracted from the number represented by the last three digits of the entry opposite the given number 
of minutes according as the entry for the next higher number of minutes ts a greater or a lesser one. 
Interpolation by means of the columns headed “diff” involve slight errors which are negligible 
for most purposes of navigation. To avoid this error, direct interpolation may be used. Let n 
represent the number of seconds, D the difference between the entry corresponding to the given 
number of minutes and that corresponding to the next higher number of minutes, and d the required 
change to be added to or subtracted from the entry opposite the given number of minutes. Then 
eutalty 
~ 60 
Given the logarithm of a trigonometric function, to find the angle-—The following example will 
indicate the procedure necessary to find the angle when the logarithm of a trigonometric function 
of the angle is given. 
Exampte: Find @ if log cos 02=9.85391—10. 
Solution.—On the page at the top of which 44° appears, and in the column headed cos find the 
two entries 9.85399 and 9.85386 between which the given logarithm lies. Write 0=44°24’+ asso- 
ciated with the entry 9.85399. The difference between 9.85399 and the given logarithm is 0.00008; 
hence enter the adjacent column headed ‘‘diff’”’ and opposite the 8 in boldface read 39’’ in the asso- 
ciated seconds column. Hence 
d Dz. 
6=44°24'39’". 
In obtaining approximate position, observe only the two digits in boldface at the top of the page 
while leafing through the table in search of the desired page. 
Rue: Whenever, in the process of finding the appropriate number of seconds, there is a choice 
between two or more entries one of which is printed in boldface always give preference to the bold-faced entry. 
Here again direct interpolation may be used. For this purpose solve the formula written 
above, d=(n/60) D for n to obtain 
where n and D have the same meanings as above and d is the difference between the logarithm 
corresponding to the correct number of minutes and the given logarithm. 
TABLE 34.—LOGARITHMIC AND NATURAL HAVERSINES. 
The haversine is defined by the following relation: 
hav. A=% vers. A=%4(1—cos A)=sin? 4 A. 
hav. A=hay. (360°— A); thus hav. 210°=hav. 150°. 
It is a trigonometric function which simplifies the solution of many problems in nautical astron- 
omy as well as in plane trigonometry. To afford the maximum facility in carrying out the processes 
of solution, the values of the natural haversine and its logarithm are set down together in a single 
table for all values of angle ranging from 0° to 360°, expressed both in are and in time. 
TABLE 35.—THE LONGITUDE FACTOR. 
The change in longitude due to a change of 1’ in latitude, called the longitude factor, F, is given 
in this table at suitable intervals of latitude and azimuth. The quantities tabulated are computed 
from the formula— 
F=sec. Lat.Xcot. Az. 
When a time sight is solved with a dead-reckoning latitude, the resulting longitude is only true 
if the latitude be correct. This table, by setting forth the number of minutes of longitude due to 
each minute of error in latitude, gives the means of finding the correction to the longitude for any 
error that may subsequently be disclosed in the latitude used in the computation. 
Regarding the azimuth of the observed celestial body as less than 90° and as measured from 
either the North or the South point of the horizon toward East or West, the rule for determining 
whether the correction in longitude is to be applied to the eastward or to the westward will be as 
follows: If the change in latitude is of the same name as the first letter of the bearing, the change in 
longitude is of the contrary name to that of the second letter, and vice versa. 
