10 EXPLANATION OF TABLES 
The increment 12.6 was rounded off to 13 because we are not justified in writing more than five 
cimal places in the mantissa. : 
corns ene of this procedure is embodied in the following statement. To find the logarithm of a 
number composed of five significant figures, first find the logarithm corresponding to the first four figures 
and to it add one-tenth of the tabular difference multiplied by the fifth digit. rags 
To shorten the process of interpolation, 10° times each tabular difference occurring in the table 
has been multiplied by 0.1, 0.2, . . . 0.9, and the results have been tabulated on the right-hand 
sides of the pages on which these differences occur. The abbreviation Prop. Parts written at the 
top of the page over these small tables abbreviates the words proportional parts. To interpolate in 
the example just solved, note the tabular difference 18, locate the Prop. Parts table headed 18 and 
find opposite 7 in its left-hand column the entry 13. In general, this difference should not be com- 
puted but should be obtained from the number opposite the fifth digit in the appropriate table of 
proportional parts. y ’ i ; 
To find the number corresponding to a given logarithm.—If log N =, the number N is called 
the antilogarithm of L. The sequence of digits of a number N corresponding to a given logarithm L 
is found from its mantissa, and the decimal point is then placed in accordance with the italicized 
rules stated above. 
Exampie: Given log N=1.92955, find N. 
Solution.—The mantissa .92955 lies between the entries .92952 and .92957 of Table 32. Using 
the table and computing the differences indicated, we write the following form: 
oases =log ie ; # 
1.92955 J" )5=log N 10. 
1.92957 =log 85.030 
Assuming that changes in the logarithm are proportional to the corresponding changes in the number, 
we write 5 - 
x a =i) — 
5-10 or ==10(2)—6. 
Hence 
N=85.026. 
The essence of the process of interpolation is indicated in the foregoing procedure. However, 
in practice, the student should always interpolate by using the table of proportional parts. The fifth 
figure 6 should have been obtained from the table of porportional parts. In the small Prop. Parts 
table corresponding to the tabular difference 5, we read either 5 or 6 in the left-hand column opposite 
the entry 3. However, the 6 must be chosen; for in case there is a choice between two or more 
entries one of which is opposite a number printed in boldface, give preference to the entry opposite 
the bold-faced figure. 
Rue: Whenever a number lying exactly half way between two entries is under consideration or is 
the same as two or more adjacent entries, give preference to that character which has a bold-faced 
part nearest the entry. 
TABLE 33.—LOGARITHMS OF 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 Hydrographic Office for use in 
its publications. Neither the table nor any new features embodied therein, may be reproduced in 
any form without the permission of the copyright owners. 
Table of logarithms of trigonometric functions.—Table 33 gives the logarithms of the sines, 
cosines, tangents, cotangents, secants, and cosecants of angles at intervals of 1’ from 0° to 90°. 
The names of the functions written at the top of any page apply to angles having the number of 
degrees written at the top of the page, and the function names written at the bottom apply to angles 
having the number of degrees written at the bottom. The left-hand or the right-hand minute 
column applies according as the number of degrees in the angle is written on the left side or on the 
right side of the block of numbers under consideration. One of the arrowheads attached to each 
number representing degrees points toward the column of minutes to be used in connection with 
an angle involving that number of degrees, the other points toward the row of names to be considered. 
For example, to find log sin 32°46’, we find the page at the top of which 32° appears, find the 
row containing 46 in the left-hand minute column, and read 9.73387 in this row and in the column 
headed sin. The part 9.73 was found above the 46’ entry or it could have been found lower down in 
the column, and 10 is to be subtracted from every logarithm in the table. Again, to find log tan 
142°36’, find the page at the top of which 142° appears, find the row containing 86 in the right- 
hand minute column, and read 9.88341 in this row and in the column headed tan. Hence log 
tan 142°36’=(—) 9.88341—10. The minus sign in parentheses before the log indicates that a 
negative number is under consideration. The 9.88 was found three spaces higher in the column, 
or it could have been found lower in the column. 
Given the angle to find the logarithm of a trigonometric function—Interpolation.—The prin- 
ciples involved here are the same as those involved in finding logarithms and antilogarithms of 
numbers. Interpolation for seconds is accomplished by direct interpolation or by using the columns 
headed ‘‘diff.”” The following example will illustrate the use of the difference columns. 
