557-554TblExpl.pdf

EXPLANATION OF NAVIGATION

TABLES

Mathematical Tables

plies to the values half a line above and half a line below.

To determine the correction to apply to the value for the

smaller entering angle, multiply the difference by the num-

ber of tenths of a minute (or seconds

Table 1. Logarithms of Numbers – The first page

of this table gives the complete common logarithm

(characteristic and mantissa) of numbers 1 through 250.

Succeeding pages give the mantissa only of the common

logarithm of any number. Values are given for four sig-

nificant digits of entering values, the first three being in

the left-hand column, and the fourth at the heading of

one of the other columns. Thus, the mantissa of a three-

digit number is given in the column headed 0, on the line

with the given number; while the mantissa of a four-digit

number is given in the column headed by the fourth digit,

on the line with the first three digits. As an example, the

mantissa of 328 is 51587, while that of 3.284 is 51640.

For additional digits, interpolation should be used. The

difference between each tabulated mantissa and the next

larger tabulated mantissa is given in the “d” column to

the right of the smaller mantissa. This difference can be

used to enter the appropriate proportional parts (“Prop.

parts”) auxiliary table to interpolate for the fifth digit of

the given number. If an accuracy of more than five sig-

nificant digits is to be preserved in a computation, a table

of logarithms to additional decimal places should be

used. For a number of one or two digits, use the first page

of the table or add zeros to make three digits. That is, the

mantissa of 3, 30, and 300 is the same, 47712. Interpola-

tion on the first page of the table is not recommended.

The second part should be used for values not listed on

the first page.

60) of the entering

angle. Note whether the function is increasing or decreas-

ing, and add or subtract the correction as appropriate, so

that the interpolated value lies between the two values be-

tween which interpolation is made.

Table 3. Logarithms of Trigonometric Functions –

This table gives the common logarithms (+10) of sines,

cosecants, tangents, cotangents, secants, and cosines of

angles from 0

to 180

, at intervals of 1'. For angles be-

tween 0

use the column labels at the top and the

minutes at the left; for angles between 45

and 45

use

the column labels at the bottom and the minutes at the

right; for angles between 90

and 90

use the column la-

bels at the bottom and the minutes at the left; and for

angles between 135

and 135

use the column labels at

the top and the minutes at the right. These combinations

are indicated by the arrows accompanying the figures

representing the number of degrees. For angles between

180

and 180

and proceed as indicated

above to obtain the numerical values of the various

functions.

Differences between consecutive entries are shown in

the “Diff. 1'” columns, except that one difference column is

used for both sines and cosecants, another for both tangents

and cotangents, and a third for both secants and cosines.

These differences, given as an aid to interpolation, are one-

half line out of step with the numbers to which they apply,

as in a critical table. Each difference applies to the values

half a line above and half a line below. To determine the

correction to apply to the value for the smaller entering an-

gle, multiply the difference by the number of tenths of a

minute (or seconds

and 360

, subtract 180

Table 2. Natural Trigonometric Functions – This

table gives the values of natural sines, cosecants, tangents,

cotangents, secants, and cosines of angles from 0

to 180

at intervals of 1'. For angles between 0

use the col-

umn labels at the top and the minutes at the left; for angles

between 45

and 45

60) of the entering angle. Note wheth-

er the function is increasing or decreasing, and add or

subtract the correction as appropriate, so that the interpolat-

ed value lies between the two values between which

interpolation is made.

use the column labels at the bottom

and the minutes at the right; for angles between 90

and 90

and

135

use the column labels at the bottom and the minutes at

the left; and for angles between 135

use the col-

umn labels at the top and the minutes at the right. These

combinations are indicated by the arrows accompanying

the figures representing the number of degrees. For angles

between 180

and 180

Table 4. Traverse Table – This table can be used in

the solution of any of the sailings except great-circle and

composite. In providing the values of the difference of

latitude and departure corresponding to distances up to

600 miles and for courses for every degree of the com-

pass, Table 4 is essentially a tabulation of the solutions

of plane right triangles. Since the solutions are for inte-

gral values of the acute angle and the distance,

interpolation for intermediate values may be required.

Through appropriate interchanges of the headings of the

and proceed as indi-

cated above to obtain the numerical values of the various

functions.

Differences between consecutive entries are shown in

the “Diff. 1'” column to the right of each column of values

of a trigonometric function, as an aid to interpolation. These

differences are one-half line out of step with the numbers to

which they apply, as in a critical table. Each difference ap-

and 360

, subtract 180

557

558

columns, solutions for other than plane sailings can be

made. The interchanges of the headings of the different

columns

log e is the natural (Naperian) logarithm, using the base

e = 2.71828182846,

are

summarized

the

foot

each

table

(

)

log

2.3025851

log

0.36221569

opening.

The distance, difference of latitude, and departure col-

umns are labeled Dist., D. Lat., and Dep., respectively.

For solution of a plane right triangle, any number N in

the distance column is the hypotenuse; the number opposite

in the difference of latitude column is N times the cosine of

the acute angle; and the other number opposite in the depar-

ture column is N times the sine of the acute angle. Or, the

number in the column labeled D. Lat. is the value of the side

adjacent and the number in the column labeled Dep. is the

value of the side opposite the acute angle.

L is the latitude,

f 2

e is eccentricity of the earth, or

(log=8.087146894 –10)

–

0.0818188

298.26

f is earth’s flattening, or

----------------

0.00335278 log

(

7.4745949

–

)

Using these values,

log

7915.704468

(

log

3.8984896

)

Cartographic Tables

ae 2

Table 5. Natural and Numerical Chart Scales –

This table gives the numerical scale equivalents for various

natural or fractional chart scales. The scale of a chart is the

ratio of a given distance on the chart to the actual distance

which it represents on the earth. The scale may be ex-

pressed as a simple ratio or fraction, known as the natural

23.01336332

(

log

1.3619801

)

ae 4

0.05135291 log

(

8.28943495

–

)

---------

ae 6

--------

0.00020626 log

(

6.6855774

–

)

80000

scale . For example, 1:80,000 or

---------------

means that one unit

(such as an inch) on the chart represents 80,000 of the same

unit on the surface of the earth. The scale may also be ex-

pressed as a statement of that distance on the earth shown

as one unit (usually an inch) on the chart, or vice versa. This

is the numerical scale .

The table was computed using 72,913.39 inches per

nautical mile and 63,360 inches per statute mile.

Hence, the formula becomes

---

7915.704468 log tan

–

23.0133633

0.051353 sin 3 L

0.000206 sin 5 L ...

sin L

–

The constants used in this derivation and in the table

are based upon the World Geodetic System (WGS) ellip-

soid of 1972.

Table 6. Meridional Parts – In this table the meridi-

onal parts used in the construction of Mercator charts and in

Mercator sailing are tabulated to one decimal place for each

minute of latitude from the equator to the poles.

Table 7. Length of a Degree of Latitude and Longi-

tude – This table gives the length of one degree of latitude

and longitude at intervals of 1

from the equator to the

poles. In the case of latitude, the values given are the

lengths of the arcs extending half a degree on each side of

the tabulated latitudes. Lengths are given in nautical miles,

statute miles, feet, and meters.

The values were computed in meters, using the World

Geodetic System ellipsoid of 1972, and converted to other

units. The following formulas were used:

The table was computed using the formula:

e 4

---

ö - a

e 2

---- sin 3 L

a log e 10 log tan

sin

e 6

---- sin 5 L+...

M =111,132.92–559.82 cos 2L+1.175 cos 4L–

0.0023

cos + . . .

P=111,412.84 cos L–93.5 cos 3L+0.118 cos 5L–...

in which M is the number of meridional parts between the

equator and the given latitude, a is the equatorial radius of

the earth, expressed in minutes of arc of the equator, or

in which M is the length of 1

of the meridian (latitude), L

is the latitude, and P is the length of 1

of the parallel

(longitude).

21600

Piloting Tables

---------------

3437.74677078 log

(

3.5362739

)

Table 8. Conversion Table for Meters, Feet, and

559

Fathoms – The number of feet and fathoms corresponding

to a given number of meters, and vice versa, can be taken

directly from this table for any value of the entering argu-

ment from 1 to 120. The entering value can be multiplied by

any power of 10, including negative powers, if the corre-

sponding values of the other units are multiplied by the

same power. Thus, 420 meters are equivalent to 1378.0

feet, and 11.2 fathoms are equivalent to 20.483 meters.

The table was computed by means of the relationships:

1 meter = 39.370079 inches,

1 foot = 12 inches,

1 fathom = 6 feet.

Table 12. Distance of the Horizon – This table gives

the distance in nautical and statute miles of the visible sea

horizon for various heights of eye in feet and meters. The

actual distance varies somewhat as refraction changes.

However, the error is generally less than that introduced by

nonstandard atmospheric conditions. Also the formula used

contains an approximation which introduces a small error at

the greatest heights tabulated.

The table was computed using the formula:

2 r o h f

6076.1

-------------------------

b o

in which D is the distance to the horizon in nautical miles;

ro is the mean radius of the earth, 3440.1 nautical miles; h f

is the height of eye in feet; and

Table 9. Conversion Table for Nautical and Statute

Miles – This table gives the number of statute miles corre-

sponding to any whole number of nautical miles from 1 to

100, and the number of nautical miles corresponding to any

whole number of statute miles within the same range. The

entering value can be multiplied by any power of 10, in-

cluding negative powers, if the corresponding value of the

other unit is multiplied by the same power. Thus, 2,700

nautical miles are equivalent to 3,107.1 statute miles, and

0.3 statute mile is equivalent to 0.2607 nautical mile.

The table was computed using the conversion factors:

1 nautical mile = 1.15077945 statute miles,

1 statute mile = 0.86897624 nautical mile.

b o (0.8279) accounts for ter-

restrial refraction.

This formula simplifies to: D (nm)

1.169

h f

(

statute miles

)

1.345

h f

Table 13. Geographic Range – This table gives the

geographic range or the maximum distance at which the

curvature of the earth permits a light to be seen from a par-

ticular height of eye without regard to the luminous

intensity of the light. The geographic range depends upon

the height of both the light and the eye of the observer.

The table was computed using the formula:

Table 10. Speed Table for Measured Mile – To find

the speed of a vessel on a measured nautical mile in a given

number of minutes and seconds of time, enter this table at

the top or bottom with the number of minutes, and at either

side with the number of seconds. The number taken from

the table is speed in knots. Accurate results can be obtained

by interpolating to the nearest 0.1 second.

1.17

in which D is the geographic range in nautical miles, H is

the height in feet of the light above sea level, and h is the

height in feet of the eye of the observer above sea level.

This table was computed by means of the formula:

3600

------------

, in which S is speed in knots, and T is

Table 14. Dip of the Sea Short of the Horizon – If

land, another vessel, or other obstruction is between the ob-

server and the sea horizon, use the waterline of the obstruction

as the horizontal reference for altitude measurements, and

substitute dip from this table for the dip of the horizon (height

of eye correction) given in the Nautical Almanac . The values

below the bold rules are for normal dip, the visible horizon be-

ing between the observer and the obstruction.

The table was computed with the formula:

elapsed time in seconds.

Table 11. Speed, Time, and Distance Table – To

find the distance steamed at any given speed between 0.5

and 40 knots in any given number of minutes from 1 to 60,

enter this table at the top with the speed, and at the left with

the number of minutes. The number taken from the table is

the distance in nautical miles. If hours are substituted for

minutes, the tabulated distance should be multiplied by 60;

if seconds are substituted for minutes, the tabulated dis-

tance should be divided by 60.

h f

6076.1d s

b o d s

2r o

–

D s

60 tan

----------------------

------------

The table was computed by means of the formula:

in which D s is the dip short of the sea horizon, in minutes of

arc; h f is the height of eye of the observer above sea level in

feet;

------- , in which D is distance in nautical miles,

b o (0.8321) accounts for terrestrial refraction; r o is the

mean radius of the earth, 3440.1 nautical miles; and d s is the

S is speed in knots, and T is elapsed time in minutes.

560

line at the object and the visible (sea) horizon beyond is

measured and corrected for index error. The table is entered

with the corrected vertical angle and the height of eye of the

observer in nautical miles ; the distance in yards is taken di-

rectly from the table

The table was computed from the formula:

distance to the waterline of the obstruction in nautical miles.

Table 15. Distance by Vertical Angle Measured Be-

tween Sea Horizon and Top of Object Beyond Sea Hori-

zon – This table tabulates the distance to an object of

known height above sea level when the object lies beyond

the horizon. The vertical angle between the top of the object

and the visible horizon is measured with a sextant and cor-

rected for index error and dip only. The table is entered with

the difference in the height of the object and the height of

eye of the observer and the corrected vertical angle; and the

distance in nautical miles is taken directly from the table.

An error may be introduced if refraction differs from the

standard value used in the computation of the table.

tan h s

(

–

)

(

1AB

)

where

b o d s

2r o

d s

-----

------------ and

b o h/r o

in which

b o (0.8279) accounts for terrestrial refraction, r o is

the mean radius of the earth, 3440.1 nautical miles; h is the

height of eye of the observer in feet; h s is the observed ver-

tical angle corrected for index error; and d s is the distance

to the waterline of the object in nautical miles.

The table was computed using the formula:

0.0002419

tan

–

0.7349

0.0002419

tan

-------------------------

----------------

–

-------------------------

Table 18. Distance of an Object by Two Bearings –

To determine the distance of an object as a vessel on a

steady course passes it, observe the difference between the

course and two bearings of the object, and note the time in-

terval between bearings. Enter this table with the two

differences. Multiply the distance run between bearings by

the number in the first column to find the distance of the ob-

ject at the time of the second bearing, and by the number in

the second column to find the distance when abeam.

The table was computed by solving plane oblique and

right triangles.

in which D is the distance in nautical miles,

is the correct-

ed vertical angle, H is the height of the top of the object

above sea level in feet, and h is the height of eye of the ob-

server above sea level in feet. The constants 0.0002419 and

0.7349 account for terrestrial refraction.

Table 16. Distance by Vertical Angle Measured Be-

tween Waterline at Object and Top of Object – This ta-

ble tabulates the angle subtended by an object of known

height lying at a particular distance within the observer’s

visible horizon or vice versa.

The table provides the solution of a plane right triangle

having its right angle at the base of the observed object and

its altitude coincident with the vertical dimension of the ob-

served object. The solutions are based upon the following

simplifying assumptions: (1) the eye of the observer is at

sea level, (2) the sea surface between the observer and the

object is flat, (3) atmospheric refraction is negligible, and

(4) the waterline at the object is vertically below the peak of

the object. The error due to the height of eye of the observer

does not exceed 3 percent of the distance-off for sextant an-

gles less than 20

Celestial Navigation Tables

Table 19. Table of Offsets – This table gives the cor-

rections to the straight line of position (LOP) as drawn on a

chart or plotting sheet to provide a closer approximation to

the arc of the circle of equal altitude, a small circle of radius

equal to the zenith distance.

In adjusting the straight LOP to obtain a closer approx-

imation of the arc of the circle of equal altitude, points on

the LOP are offset at right angles to the LOP in the direction

of the celestial body. The arguments for entering the table

are the distance from the DR to the foot of the perpendicular

and the altitude of the body.

The table was computed using the formulas:

and heights of eye less than one-third of

the object height. The error due to the waterline not being

below the peak of the object does not exceed 3 percent of

the distance-off when the height of eye is less than one-third

of the object height and the offset of the waterline from the

base of the object is less than one-tenth of the distance-off.

Errors due to earth’s curvature and atmospheric refraction

are negligible for cases of practical interest.

sin

3438' cot h

Table 17. Distance by Vertical Angle Measured Be-

tween Waterline at Object and Sea Horizon Beyond Ob-

ject – This table tabulates the distance to an object lying

within or short of the horizon when the height of eye of the

observer is known. The vertical angle between the water-

D/R

R 1

(

–

cos

)

in which X is the offset, R is the radius of a circle of equal

561

altitude for altitude h, and D is the distance from the intercept

to the point on the LOP to be offset.

longitude factor, F, which is the error in minutes of longi-

tude for a one-minute error of latitude. In each case, the

total error is the factor multiplied by the number of minutes

error in the assumed value. Although the factors were orig-

inally intended for use in correcting ex-meridian altitudes

and time-sight longitudes, they have other uses as well.

The azimuth angle used for entering the table can be

measured from either the north or south, through 90

Table 20. Meridian Angle and Altitude of a Body on

the Prime Vertical Circle – A celestial body having a

declination of contrary name to the latitude does not cross

the prime vertical above the celestial horizon, its nearest ap-

proach being at rising or setting.

If the declination and latitude are of the same name,

and the declination is numerically greater, the body does

not cross the prime vertical, but makes its nearest approach

(in azimuth) when its meridian angle, east or west, and alti-

tude are as shown in this table, these values being given in

italics above the heavy line. At this time the body is station-

ary in azimuth.

If the declination and latitude are of the same name and

numerically equal, the body passes through the zenith as it

crosses both the celestial meridian and the prime vertical, as

shown in the table.

If the declination and latitude are of the same name,

and the declination is numerically less, the body crosses the

prime vertical when its meridian angle, east or west, and al-

titude are as tabulated in vertical type below the heavy line.

The table is entered with declination of the celestial

body and the latitude of the observer. Computed altitudes

are given, with no allowance made for refraction, dip, par-

allax, etc. The tabulated values apply to any celestial body,

but values are not given for declination greater than 23

;orit

may be measured from the elevated pole, through 180

.If

the celestial body is in the southeast (090

– 180

) or north-

west (270

) quadrant, the f correction is applied to

the northward if the correct longitude is east of that used in

the solution, and to the southward if the correct longitude is

west of that used; while the F correction is applied to the

eastward if the correct latitude is north of that used in the

solution, and to the westward if the correct latitude is south

of that used. If the body is in the northeast (000

– 360

– 090

)or

southwest (180

) quadrant, the correction is applied

in the opposite direction. These rules apply in both north

and south latitude.

The table was computed using the formulas:

– 270

sec L cot Z

---

cos L tan Z

--------------------------

cos L tan Z

---

sec L cot Z

---------------------------

in which f is the tabulated latitude factor, L is the latitude,

Z is the azimuth angle, and F is the tabulated longitude

factor.

be-

cause the tabulated information is generally desired for the

sun only.

The table was computed using the following formulas,

derived by Napier’s rules:

Nearest approach (in azimuth) to the prime vertical:

Table 22. Amplitudes – This table lists amplitudes of

celestial bodies at rising and setting. Enter with the declina-

tion of the body and the latitude of the observer. The value

taken from the table is the amplitude when the center of the

body is on the celestial horizon. For the sun, this occurs

when the lower limb is a little more than half a diameter

above the visible horizon. For the moon it occurs when the

upper limb is about on the horizon. Use the prefix E if the

body is rising, and W if it is setting; use the suffix N or S to

agree with the declination of the body. Table 23 can be used

with reversed sign to correct the tabulations to the values

for the visible horizon.

The table was computed using the following formula,

derived by Napier’s rules:

csc

sin

csc

sec

tan

cot

On the prime vertical:

sin

csc

cos

tan

cot

In these formulas, h is the altitude, d is the declination,

L is the latitude, t is the meridian angle.

sin

sec

sin

Table 21. Latitude and Longitude Factors – The

latitude obtained by an ex-meridian sight is inaccurate if the

longitude used in determining the meridian angle is incor-

rect. Similarly, the longitude obtained by solution of a time

sight is inaccurate if the latitude used in the solution is in-

correct, unless the celestial body is on the prime vertical.

This table gives the errors resulting from unit errors in the

assumed values used in the computations. There are two

columns for each tabulated value of latitude. The first gives

the latitude factor, f, which is the error in minutes of latitude

for a one-minute error of longitude. The second gives the

in which A is the amplitude, L is the latitude of the observ-

er, and d is the declination of the celestial body.

Table 23. Correction of Amplitude Observed on the

Visible Horizon – This table contains a correction to be

applied to the amplitude observed when the center of a ce-

lestial body is on the visible horizon, to obtain the

corresponding amplitude when the center of the body is on

the celestial horizon. For the sun, a planet, or a star, apply

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