Norie's Nautical Tables 1991 (Partial)

8/15/2019 Norie's Nautical Tables 1991 (Partial)

1/82


2/82

NORIE

S

N UTIC L

TABLES

WITH

EXPL

N

TIO

N S

OF

THEIR

USE

E ITE Y

C

PT

A I N A G BLA NCE S c

IMR Y L U R IE N

OR

IE A N D W

IL

SON

LTD

SA I NT IVES CA MBRIDGESH I R E ENGL ND

SCANNED By MKP - 2005


3/82

CONTENTS

]

re

face.

7

Explanation a nd use

of

the T

ab

l

es.

9

I. COMPUTA

TI

ON T A

BLES

Trave rse

Ta

bles. 2

Meridional Pa

ns

.

14

Loga rithms. 1 3

Logarithms o fTri

g

Function

s

8

Log. a nd Na tural Havcrsines. 242

al ural Functions of Angles. 34Y

Squ

are

s of Numbe rs. 364

Cubes of Numbers, 366

Squa re R

oots

of Numb

ers,

36R

Cube Roots of Numbers. 372

. TA

BL

ES FOR USE IN CELESTI A L N A VIGATION

A B C Azimut h

T lb

les. 380

Amplitudes and

Cor

rect ion

s.

429

Ex -Me ridian Tables. 432

Change of -lour A ngle with A ltitude, 449

Chnnge

of Altitude in

Onc

Minute

of

Tim

e, 4

51

Di

p of S

ea

Horizon . 453

Monthly Mea n of the Sun s Se mi-diamete r and

Para llax in A ltit

ude

. 453

Augmentation of Moon s Semi-

diame

ter. 453

Red uc tio n o f the Moon s Para llax. 453

M

ea

n Refrac

ti

on. 454

Add itional Refra ctio n Corrections, 454

Correction of Moon s Meridional Passage,

45

5

Su n's To tal Correc tio n. 456

SlU r STow l

Co

rrec tio n. 462

Moon s TOIa l Co

rr

ection L

owcr

Limb . 466

Moon s Tolal

Co rr

cc tion Upper Lim b. 479

I l l T A BLES FOR USE I N COA STA L NAV IGATION

Day s

Run -

Average Speed 494

R

adar

Ran ge.

5 1

Rada r -Plo th r SSpeed and DiSI,

lR

ce. 502

Mea sured Mile Speed , 503

Di stance by Ve rt ical Angle. 510

Extreme Range. 5 16

Dis

tan

ce o f the Sea Ho rizo n . 51 8

Dip of

the Shore Horizon. 5 19

Correction required

1

convert a

Radio

Great Cirde

Bea ring to Merca tori al Bearing, 520

I

V

CONVE

RSI

ON

AN

D I)H YS I

CAL

TA

B

LES

Arc into Time. 522

Time into A rc, 523

Hours a nd Minutes to a Dec imal of a Da y, 524

Atmosphe ric Press ure Conversion, 5

25



4/82

vi

Te mpe rature Conversion. Degrees Elhn:nhe it

Degrees Celsius - Deg ree :- Fah renhe it. 526

SI - British U ni ts, 527

Brit ish Ga

ll

ons - Litres - Briti sh G,llIo n5, 52 )

British Ga lil) 1s - US

Gallo

ns - BritIsh Gallll Ils. 53 )

US

Gallolls -

Li

tres -

US Ga

llons .

531

interna tior l


5/82

PREF CE

Onc hundred nd eighty y

ears of

publi

ca

tion involving ma ny new ed it i

ons and

reprints atlord a

searching test

of the

usefulness

and

value

of

,my publication desi

gned

0

m

eet

the exacting r

e·

quirclllcnts o f navigators

and the

shipping industry. Since J W. Norie published the first edition

of his C

OMPLETE

SET OF

N UTI

C L

T BLES ND

E

PITOME

OF

PR

CT IC L

NAVIGA Tl ON in 18 3. the tabl es have und erg

one

a continuing proc

es

s of change to maintain

their usefulness to the modern practical Ililv iga tor.

During Ihe se years many changes 10 the tab les have been necessary in both CO ri l Cl1I a nd

presclll llio n to co

nf

o rm wi th changing tec hniques of naviga lion. bUI the aim of the ed ito rs has

al

ways bee n to have us

er

friendly nav igllliona i t;lbles which

co

uld be used q

ui

c

kl

y a nd cas ily un-

der shipboard condi tions. This has resulted in many

chan

ges 10 the tab l

es

aimed at removing

much of the tedium of interpol atio n. so enab ling the nav i

gato

r to ob ta

in

the answers to na viga-

ti

ona

l

prob

le ms quickly with the minimum

of probab

il ity

of

crror. Ce rtain tables and

dat

a arc

also incl

uded

which

arc

not readil y avai l

ab

le on

board

ship

or

ar

c on ly used in

the

examinatioll

room, but th e physica l dimensions o f the book impose stri ct limits on what can be included with

the result th at it is impossible to include a ll the tables the editor would wish.

[n

the

pr

ese nt e

dition,

the

Star's To

tal Correction

Ta

b[e inside the back cover.

the

M

oon's

Total

Co

rr

ec

tion Table

and

the

exte

nde d Slar s To la[

Co

rrec tion

Tab

le have

been

redes igned to

r

educ

e ime rpolation

to

a minimum.

The ed ito r wishes to thank a ll those who have suggested improvements to the tab les. and will

we

lcome any f

urth

er h

elpf

ul c riticism which users

of

the Tables

ma

y c

ar

e

to

makc .

A G BL NCE

London [

99

1



6/82

EXPL N nO ND

USE OF THE T BLES



7/82

I COMPUTATION TABLES

TR VERSE

T BLE

Pages 2 -

93

Th

e

se

Tables afford an easy and expeditious means of solving a

problems that resolve thems

elves

inlo the sohllion of ri

gh t-

angled p la ne tria ngles.

The

y can thus

be

applied to a ll the fo rm s

of

Sai lings cltcept Great Circle Sa ilin

g;

but they a re specia lly useful in reso lving a Traverse.

On

this

account they are

ca

lled Trave rse Tabks,

and

the terms

Course, Distance,

Djff

erence

of

Latitude

and

Departure a re used as names of the different parts involved.

Th

e Traverse Table has now been brough t into line with the requirements of the modern com pass

notation

by

the inclusion. at the top a nd foot of each page, of the number of degrees of the new

(0- 360) circular system of reckoning. correspondi ng to the value printed at centre of tit le in con

formity with th e older quadrantal notation. the latter form being retained f

or

its application to the

so lution

of

ce rtain

pr

oblems in th e Sailings and for

it

s utili ty in the co

nv

e

rsi

on of Departure to

Difference of Long itude and vice versa, as ex plained later in thi s

ar

ticle.

The figures denoting th e number

of

degrees under the new a rrangement are placed in the

appr

o

priate quadrants o f a small diagrammatic symbol, represe nting the

ca

rdinal points of the compass,

a nd in these posi tions they introduce the equivalents in the new notation co rresponding to the

number of degrees

of

the old system, show n at centre o f titles, when pertain ing to the respect ive

quad rants. T he arrangement will be better understood by reference 10 an

ex

ample ; thus, on page

58 - 28 DEG REES

For old

Read new

11

5r

208

0

I

or v ice versa, and . as examples of the

w

it

h ca ption 72

DEGR

EES -

reverse process, o n page 38, but this time from

th

e foot

Fo r new

Read old

I

N7rE

_ 1 _

I l

ogo

S7rE

1

5r I

S7rw

28

80

1

I

N7rw

It

will be obse rved that , in the new notation of the Traverse Tab le, the th ree-figure degrees

co rresponding to Easte

rl

y co urses are placed in the symbol diagram towa rd s the right-ha nd side

of the pa ge, in contrad isti nction to those for Wes te

rl

y equivalents wh ich are printed on the left .

The courses, in both the old and new notation, are di splayed at the top and bottom of the pages,

while the

Di

stances are arranged

in

order in the columns marked Dist.

The

Difference

of

Lati tude

and Dep

lITw

re co rresponding to any given Run on any given Course will e found in the columns

ma rked D. La t. and Dep ., re specti ve l

y

of the page for the

gi

ve n Course

and

opposite the given

Di stance. But it must mos t carefully be observed th a t when the required Course is found a t the

lOp

of the page, the Difference of Lat itude and Depa rture also are to be taken from the columns

as named at t

he l p

o f the page; and when the Course appears at t

he

foot

of the page, the relevant

quantities too must be taken from the columns as named at the

foot

of the page.

When a ny of the given qua ntit ies (

ex

cept the Cou rse which is never to e changed) exceeds the

limi ts of the tables, any a liquot pa rt, as a half or a third, is to be taken , and the qua ntities found

are

to be d

ou

bled or t rebled ; tha t is, they a re to be multiplied by the same

fig

ure as the given quant ity

was di

vi

ded

by.

And since the Di

ff

erence of Latitude

and

Departure correspo nding to any

gi

ve

n

Co

urse and Dista

nc

e are to

e

found oppos

it

e the

Di

stance on that page which contains the

Cour

se

.

it follows tha t if any two of

th

e four parts be g iven, and these two be found in

th

eir proper places

in

th

e tab les {he othe r two will e found in thei r respective places o n the same page.

Th

e following exampl

es

wi

ll

i

ll

ustrate the appl

ic

at ion of the tables 10 Plane

Sa

iling :-



8/82

EXPLANAT ION

OF TH

E TAIJLES

1 Example :

F

in

d t

he

difference of latitude and deparlure made good

by

a ship in sailing 84 miles

on

a course 1

12

°. (S68°E., Old Style).

Co ur

se

I1r is found in the Tab

le

at foo t Oppos ite 84

in

the Distance column on that

page

we

gel:- O. Lat. 3

1'

5,

Dep. 77·9.

The D. Lat. is named S

and

Departure

E,

bec

;

use

it

is noted

that

I r s

sh

o

wn

in the South a

nd

Eas t quadrant of t

he

compass

sy

mbo l.

Example:

Find the

cou

rse and distance made good by a ship whose difference of lati tude was found

to

e

431

mi

les

S,

and

d e o a r t u r ~ 1

32

W.

431 and 132 are not to e found alongside each other, but in the

Table on

age 37 e find 431·3

and

13

1

9,

and these a rc

sulTic

ient ly near to the desired value for all practical pur These give

197

°,

or

17° o

ld

s

tyl e,

as a course,

and

451 as a dista nce. Hcnce-

Cou

rs

e 517° W, or 197°, a nd Distance 451 miles.

These tables may also, as has already been stated, be

a p p l i ~ d

to solving

problems

in Parallel and

Midd le Latitude 5a ilings. In solving these

prob

lems the

Cou

rse old nOlal ion) a l lhe

top

o r

bottom

of

page beco mes the l

at

itude or Midd le l atitude, the Distance column becomes a Di

ff.

l on

gi lUd

e

co lumn, and the O. Lat. column

be

comes a Oep. co lumn. To facilitate t

he

taking

out

of these

quantit

ies

the

D.

Lon

g.

and Ocp. arc brac ketcd together. and the words

D. Long

. and

Dep.

are also

printed in ita

li

cs

at

the t

op

of the ir respective co lumns when the Latitude

or

Midd

le

Latitude, as

co urse, is at the

fop;

but at the bottom of their respect ive co lumns when Latit ude or

Mi

dd le Latitude ,

as cou rse is at the b0110m .

7 Exampl

e In Latitude o r Middle Lat itude 4

7

the depa rture made good was 260 ,5; requi red Ihe

di

ffe re

nce of Lo ngitude.

With 47° as course at the

bo

t om of the page, look

in

the co lumn with Dep

.

pr inted

in

italics at t

he

bot/om, ju st over the end of the bracket; and opposite to 260 -5-wiILl2:t found 382 in the

D.

Long.

column,

wh

ich is the Difference

of

Lon

gi

tude requ ired .

? o3L

6 ~ )

1/

Example:

A ship.

af

ter sailing

Ea

st 260 , 5, had

h C T L u d e

6° 22 ,

R

~ q u i

the

parallel

of

Latitude on which s

he

sailed.

6°

22

eq uals 382 .

Oppo

s

it

e 382 in

D. Lon

g_

co lumn is 260 · 5 in

Dep,

co lumn en

te

red from t

he

b ,om,

aod th, pamlle] wh;ch ,he sa;],d;s LaL T

MERIDIONAL

PARTS

For

the

Spheroid)

P ~ g e s 94 .

t02

Th

is

table

is

used in resolving o b ~ m s by Mercator

s Sailing

and in co nstruct ing charts on

Merca tor s projection . T

he

mer

id

i

ona

l

parts are

to

e

taken

out

fo r the

~ ~ s

answering to the

g

iv

en latitude

at th

e f p o r

bottom,

and for Ihe

m

a t eith

er side

co lumn.

Th

us, the meridional

parts corresponding

1

the lat itude 49° 57 are 345]-88.



9/82

2

EXPL N

TION

OF

TH E

T BLES

LOG RITHMS

Pl>ges

103

-

11 7

)

This ta ble gives co rrect to

ve

signi

cant gures t

he

mantissae

(o

r frac

ti

onal parts) o f the co mmon

logarit hms of numbers. The operator must decide for himseJfthe integral or whole number

part

of

the loga rithm (called the characteri stic) acco rding to the position of the decimal point in th e natural

numb

e

r.

The rules for determining the characteristic can be dem onstrated by the following:

10 {)()()

10 ' 10g.1O

10

{)()()

4

I {)()()

-

10

log HI

I ()()()

3

100 10'

1 g10

100

-

2

10 10 '

1

0g.1O 10

I

I

10

log l/)

I

0

0 -1

10

- -

1 g10

0-1

-

I

0-01

\og·)O

0-01

- 2

0·001

10

-

1

10g 10

0-001

- 3

0·0001

10

g.

10

o-{)()()

I

- 4

T he above, which may be extended infinitely in both directions, shows that the log. of, say. 342

must lie be tween 2 and

3.

Similarly, the log.

of

29·64 must be between I and 2.

From

the tab le it

will be f

ou

nd that log.

,53

40

3

and Jog.

29·64 = 1·47 188.

The

se

statements could be ex-

pressed

liS

follows:- 10 :t (?3 ~

- IO

U

:J.I 1 3 =

3

42

I

)1

.m M

= 29·64

or numbers greater than I the rule fo r finding the charac teri stic is- T he characteristic is the

nu mber which is I less than the number

of

figures before the decimal point. If there are five f i g u r e ~

before the decimal point the characteristic

is

4;

if there is onc figure before the decimal

point

the

characteristic is

0, and

so on. Thu s:-

log. 5378

log . 537·8

log. 53·78

log. 5·378

3-73062

2-73062

1·73062

0-73062

~

For numbers less Ihan I the ru le for nding the

ch

a racter istic i5- henega ti ve characte

ri

stic of the

log. o f a

number

less tha n I is the number which is I mo re than the

numb

er of no ughts between the

decim al point and the first significant gure. Thus:-

log.

0·5378

1-7306

2

log . 0 ·05378

-

2· 73062

log. 0{l05378

-

3·73062

log. O{)005378 4· 73062

Tabular logarithms To avoid the nega tive ch

ar

ac ter istics, logari th ms in tabu lar form are obta ined

by adding

10

to th e characteri st ic.

Example log 0·5378 = 1·73062 or in tabu l

ar

f

orm

9·73062

rc)g

0·005378

= ) ·7

3062 o r in tabula r fo rm 7·73062

In the ta bles of logarithms of

tr

igonometrical function s the characteris ti c is given in both form s at the

t

op

of eac h co lumn

of

logarithm

s.

Example log. sin . 5° 30 = '2 98 157 or 8·98 157 / r ~ ~ O

log. cot.

5°

30 =

1·0 1642 or 11·01642

PAGE 106

PAGE 110

PAGE 140



10/82

I .XP LAI\.ATION 0 1- TH E r ABL ES

13

1 erpo J/ iOIl

When the

numbe

r

whose

lo

gari

thm is req u ired

co

nsists

of

ro ur

si

gni fic

am

figu res o r less t

he

mant issa is ta ken

rrom

the ma in pa rt of

th

e table. Wh

ere

there a re fi ve sig nific

an

t figures

th

e

di

ffe

r

en

ce

ror

t

he

fifth figure is o

bta

ined f

rom

t

he

relevan t

sec

tio n o f the D

col

um n.

E.rampl

:

log . 140·27 =

2· 4675

f

21

= 2

·1

4696

(i'0-5

e

-10 ;

)

Ir the

num ber

consists

or

mo

re t

ha

n six sign i

fi

ca n t

fi

gures the

app

ro

x

ima

te

logar

i

thm ca

n

be

round by

sim p

le proportion.

E

xa

mp

le: l

og

. 140·277

=

2·1 4675

(r

rom ma in

tab

le )

+ 2

(rrom

D

col

umn

)

+ 2 (by

simp le propo rti

on)

2· 1469S

Ta

find ,l te

nllmht

r,

N,

IrllOse

log.

is k ,lOlI"l/. Ir the

number

is

requ

ir

ed

to

four

significant fi

gures

o r

less al l that is nec

essary

is to fi nd

th

e se ries o f d igi ts co

rr

esponding

to

t

he

tabu la t

ed

ma m issa wh ich is

lIt"an's

to Ihe

one

given. T he c

har

acteristic

of

the log . wi

ll

deter m ine

the

pos i

tion

of the

de

c

ima

l

po int. Th us: -

G;, , log. N ~ 1·8 7109. (

11?\

Nea rest tabulated

man

tissa

87 111

gives digi ts 7432. P.3

:»

T he charac tcri Slic being I , there a re two H

gure

s

be

rore the deci ma l po in t.

The

requ

ired number,

N

is

the

refo re 74·32.

Th

e fo

ll

o wing

examp

les wi

ll

s

er

ve to illu strate the

pro

c

edur

e whe n mo re th

an

rou r

sig

nifi

ca nt

figu res

a re req uired . Suppose

the number, N.

co

rrect

to five significan t figures is r

equi

red whe n log.

N

is

kn

own

to

be

2 ·27 104.

Example :

log. N

=

2

·27 1

04

T he nex t less

tabu

la ted ma nl issa

·27091

gives

the

d igit s

1866.

D ~

e

10 y)

But ,27 104 - ·2709 1 =

13

I ..J

:. En

tering the 18Q ..189 section

of

the D co lumn the fift h figure is 6 ror a D va lue

of

12 a nd bv

sim ple p

ro

po

rti

on

the

sixth

fi

g

ur

e is

th

e ref

or

e 5. -

tlle di gi ts o f Ihe number are IS6665

The ch a racter

is

t ic

of

the m is 2.

:.

the n

umbe

r N is 186·6

65

LOGS . of TR IG . FUNCTIONS

P

.

gesI1

8 · 241 )

Whilst

pr

ese rvi ng t he

basic

layo ut which has been a o f '

No

rh s , and

Node s

alo ne, since

J. W. Node produced the o r iglllal ed ition , c hanges

have been

introd uced whi

ch

ma ke

th

e table a

muc

h

more effic ien t ins trument in conform ing with the mode

rn tech

n ique o r

as

tro no

mic

a l na viga tion.

For a

ll

a nglcs rrorn 0 to

9{)

the ta b le is n

ow

co mp ete

ly

downward

r

ead

ing a nd

for

tha t

reaso

n

alone

should be p ractica

ll }

blunder-p

ro

of. In

th

e ma in pari

of

the t

able

f ro m 4

0

to 86° the lo

g. ru

nct ions

of

angles

are

tabulated

ro

r o nc

minu

te

inter

va ls

of the an

gles a nd p ro p

or

t

io

na

l parts r

or

frac tio ns o f

one

minu

te (from 0 ' ·[ to 0 ' ·9) a re given. In the

rema

inder

of

the ta ble, where that

sys

t

em

ceases to

be pr

;lctic

ab

le, l

og

. func

tio

ns

ar

e

tab

ula t

ed

fo r in te rva ls o r

0 · 1

o r

0 ·2 as

nec

es

sa ry a nd d ifferences

between success ive tab lLla

tion

s lITe given. This me

an

s t

ha

t, except in special and

ra

re cases, in ter-

polation is re

duced

wh

en ta

king o ut a ny l

og

.

fun

ct ion

oran

an

gle

and

th

ere is

no

need to re

so

rt

10

the

qu

estio

na ble p ractice or ro

und

ing o

ff

a ngles to the neares t m in

ut

e in o rde r to 'save troub le'. Wi

th

this t:able it is no

mo

re o f a n effo

rt to work

acc

ura

tely t

han

it is to

work

roughly. How fa r a

nav

i

gator is ju st

il

i

ed

in wo rking to tent

hs

o r a minute is a m;.t ler wh ich C;1Il

be

argued

ab out

indefinilcly.

PAGE 113



11/82

14

EXP

LANATION

OF THE TABLES

but since the

Nau

tica l Almanac gives hour a ngles and dec li nat ions 10 tent hs ora minute and a mod

ern

sex tant with a decima l vernier enables rea

ding

s to be la ken 10 tent hs

of

a min ute as wel

l, it

would

see m o

nl y l

og

ical to use navigat ion tables which. with the minimum of effort .

pro

v

id

e fo r the sa me

o rder of precision.

Th e ch

ara

cter ist ic of the loga rithm is given at the top of eac h function s column

in

ind icial

fo

rm

with the tabular form in brackets.

Example :

log. sin. 5° 09 '

= 2.

953 10 or 8·953

10

T hus the na viga

tor

can use whichever fo rm

of

cha racteristic is preferr ed

though it

must

e

apprec iated

tha t t he two

fo

rms cannot

be

interchanged withi n a calcu la tion.

Occasionall

y,

it may be necessa ry to find the logs. of trigonomet rica l funct ions of grea ter

than 90°. No difficu lty

h o u

be ex perie nced in such

s e ~

as the ~ e c o n d

third

and fo urt h quad

r

an

t equi vale nt s o f the

quadra

nt g l e ~

afe

pla in

ly

indi

ca

ted .

It

be noted . however. that

the ta ble is upward reading fo r angles be tween 90° a nd 180

0

and also

fo

r those between 270° a nd

360°. but dolt nn·

rd

re adi ng

fo

r angles be tween 180° and 270°. In all cases,

how

eve r. the name of

the rat io bei ng used

appear

s at the

top

of the page. When

applying

proport ional parts c

ar

e s hou ld

be taken to notice in whic h direc ti on the log . functi on is in c reas ing, i.e.

upwards

or

downwa

rd s.

Examples:

f: 12' Io

g ,

;0

-

1

0

38'·7

=

· 45754

l

¥ )

=

2· 45798

0

8-45798

( ~ 12.3 , -l

og-

tao_

177 57'·5

2 (55240 _ 7 ~ O

2-55205

0

8·55205

~ log . cosec. 26 04 ' ·4

0-(35712 - 10)

- 0-35702

0< 10·35702

log . sec. 333

0

25'-3

0-(04852 -

2)

- 0-04850

0< 10·04850

log. cos. 138 17 '· 6 1'(87300 +

7)

-

1.87307

0 '

9·87307

log. sin .

62°

19 ·8

1·(94720 + 5) 1.94725 0< 9·94725

log. cot.

11 7° 53'·0 1·72354

0<

9·72354

log. cos.

8]0 15'·3

-

1- 0 70 18 - 32) 1.06986

0<

9{16986

To find the ang

le

whose log. functi on is g iven is equally simple. Fo r instance, to fin d 8

wh

en log .

sin . 0

=

1-66305

or

9·66305, notice that the next less tabulated log. sin . is 1-66295

or

9·66295 which

g ives the angle 27 24'·0. T he excess 10 give s a n additional 0' ·4. Hence. 8 = 27° 24' ·4.

In prac t

ic

e, the above processes will , o f course, e perfo rmed mentally.

HAVERSINES

P ~ g e 3

242

.

348)

To make the tables clearer and 10 make interpolat ion almost com pletely unnecessary the

tab

les are

presented as fo llows: -

1 The Log.

Ha

ve rsines

ar

e printed in bo ld type and the Natura l Hav

er

sines in light type.

2. n the range 0° to

90

° and 270° la 360° (the ra nge mos t frequently used) havers

in

es

are

tabu

la ted a t 0

·2

' in tervals a nd the proportiona l parts fo r 0-1' a re given a t the foot o f each page.

3. n the remainder o r the tab le haversines are tabulated a t 1·0' inte rvals and the prop

or

tional

pa rts fo r 0·2' a re gi

ve

n a t the top of each column_

4. The ch

ar

ac teris tic

or

the loga

rithms

is given a t the top of eac h co lum n in the nega t ive index

fo rm

together

with the

tabu

l

ar

form

in

bracket

s.

PAGE 127

PAGE 129



12/82

EXPLAN AT ION OF

TH

E TABLES

ANG

LE

LOG . HAV ER SINE

15'33

'-

0

2·26249 o r 8·26249

W

33

'-6

2-26304 or 8·26304

IS 33 ',7

2' 26

313 or 8· 26313

344° 10' ·0

2

'27807 or 8·27807

3440

10

' ·4 2'2777 1 or 8·27

77

1

344

0

10

',5

2·27762 or 8· 27762

95

°

25

',0

1,738 15 or 9·73815

95

0

25' -6

1·73822 or 9·73 822

26

3 37',0

1 ·744

75 or

9·74475

263

0

37',8 1·74466 o r 9·74466

DeriM iO

l

of Haversine Formulae

cos. a - cos. b cos. c

Cos.

A

= - - _

b ' ,

(fundamental fo rmula).

Slfl . 5

10

. C

cos. a - cos.

b

co

s.

c

.. I - cos. A = I -

-

510. SIfl . c

sin.

b.

sin. c - cos. a

+

co

s. b

cos. c

i.e. v

er

s A

= .

b - ,

Sil l . SIR . C

:. cos. (b

.......

c) - cos. a

=

sin. b

si

n. c

ve

r

so

A,

or - cos. a

= - -

cos. (b ...... c) + sin. b sin. c

ve

r

so

A.

By add ing uni ty to each side this becomes-

1 - cos. a

= -

cos. (b

......

c)

+

s.in. b s in. c versoA,

: . ver

so

a

=

verso(b

.......

c)

-I-

si

n.

b sin. c versoA.

NAT . HAV.

0-018

30

-

0-01832

0·0 1832

0-01

897

0-01895

0-

01895

_

0·54720

0-5

4729

0·55559

0

·5

5547

whence ha

v.

a = hay . (b ...... c)

+

s

in

. b sin. c hav. A

. . . . . . . .

1

)

By t

ra

nsposi ng we obtai

n

h

aY

. A

=

{hay. a - hav. (b c

))

cosec. b

costt C•

•• •• (2)

and hay. (b __ c)

=

hav. a - hay. A

si

n. b sin. c.

. . . . .

.

. . . .

.

. . .

(3)

I

D 2.S8

1/

1

1 /

11

P B ( I

I

f

These three

ve

rsions o r the spherical ha

ve

rsine rormula are rrequ ently adapted for nav igat ional

purposes as fo llows .

I)

H

av

. z

=

hav. (I

' ;t

d)

' -I-

hay . h cos . I cos . d .

(2) Hav. h

=

[hav. z - hav.

(I

.t

cl ; ']

se

c.

I sec. d.

(3)

Hav. mer.

ze

n. dist.

=

haY. z - hay. h cos. I cos . d.

{

:

where d =

• (I ...... d) when I and d have the same name,

(1+ d) when I

and

d have different names .

zenith distance,

latitude,

declina tion,

ho ur angle.

PAGE 258

PAGE 334



13/82

16

EXPLANATION

OF THE

TABLES

xamples

(I )

Find

zenith distance when h = 66

Q

49 3, 1 = 31°

10',2

N., d = 19 24'·7 N.

h

d

z

66° 49'

-)

31° 10 2

)9 24', 7

° 45 5

60°

40

'-

9

Hav.7.

=

hay. h cos. I cos . d

+ haY. (I t

d) .

L hay. ]·48 1

73

o r 9·48 173

L.

cos. \·93228 or 9·93228

L.

cos.

],97458

or

9·97458

L.

haY.

1·38859 or 9·38859 N. hay. 0·24468

N. hay. 0·01049

N. hay

. 0·25517

Ca

lc

ul

ated

u nith

distance

=

60 40'·9

and

is

used

for

co

mparing with the true zenith distance to

find the intercept when establishing the position line by the Marc St. HiJ aire

or

Intercept method.

(2) Find the

hour

an

gle when I

=

41 ? 21'·6 N., d

=

9

34'·\ S.

, z

=

63° 45'-8.

z

(I + d)

1

d

h

63 45' ·8

50 55 '

-7

H

av

. h

=

{hay. z - hay . (I d)] sec. I sec. d.

N. hay. 0-27896

N. h

ay

. 0,18485

N h

aY

0·09411

L. hay. 2·97364 or 8·97364

L.

sec. 0·12461 or 10 '

12461

L.

sec. 0·00608 o r JO.OO608

L.

ha

y. 1·104

33

or 9'10433

H

ou

r angle =

41

46 9 if b

ody

is w. of the meridian,

or

hour angle = 3 18 13 '· \ if body is E.

of

the meridian,

an

d is used for finding the

computed

longi

tude

when establi shing the position line by

the chro n

ometer method .

(3) Find the mer. zen. disl. when h

=

355 57 2, I

=

48 12 ' ·5 N., d = 12 1)' ·7 S ., z

=

60 2 1 ·6.

h

I

d

z

355

57

' ,2

48 \2'·5

12 13 7

Hav. mer. zen. disc

=

hay. z - hay. h cos.

1

cos.

d.

L. ha

v.

H)957

1

or

7·0957 1

L.

cos. \ ·82375 or 9·82375

L.

cos. \·99003 or 9·99003

L.

hav. 4·90949 or 6·90949

N.

ha y. (HJOO

81

N . hay

. 0·25273

mer. zen. di st. . .

60

15

' ,2 . . , . • .

. .•

. .

. .

N hay. 0·25192

The mer. zen. disc , 60 15 2, when combined wi th the dec lina

tion

g

iv

es the lat itude

of

the point

where the

po

si t ion line

(at

right angles

to

the di re


14/82

EXPLANAT ION

OF

THE TABL

ES

17

The haw rs

in

e formulae a

nd

grelll circle sailing calculations

Form ula ( I) is used to find the gre

at

circle distance rrom one poi

nt

to

anothe

r and rormula (2) is

used to find the initial and final courses.

The

vertex or the tr

ack and

the lat i

tude

or

th

e point where

t

he

track

cu

ts any speci

fi

ed meridian

can

t

he

n

be

ro und by

ri

ght

an

gled sp

he ri

ca

l trigono me

try.

Example :

Find the great ci rcl e d is tance and the initial

co

urse on the track fro m

A

pr

22

'

N .,

25

° 2S' W

.)

to B (40°

OS'

N., 73° 17' W

.)

.

To find fht

great

ci

rcle distance

Hay. AB

=

ha y.

( PA

-

PO

) ha

y.

P sin. PA sin. PB.

P 4r

49'·0

L.

hav.

1·21 55

0

or

9·2 15

50

PA 72° 38' ,0

L.

si

n.

1·9 79

74 or 9·979 74

PB

49 52 0

L.

s

in

.

' -88340 o r 9·88340

L

haY

. ,·07864 o r 9·07864 N hay. 0 ' 11985

(PA -

PB

)

r

46',0

.

.

.

N

h

aY

. 0·038 96

AB

46°

58

' ,2

. N.

hav.

O·

1

5881

:.

G reat circle d ista nce

=

18 18·2 miles.

To

find

(he init i

al

cOl/rse

HaY.

A

=

(ha

y.

PB -

haY.

( P

A......,

AB

)J c o ~

PA

cOse


15/82

2. To fi

nd

cos. soe

7':

O·16677

-0 -00086

cos. 80 27' = 0 ' 165

91

3.

To

find s

ec. 76° 4

4'

:

4·34689

+ 0·01078

Sec.

76 44 ' = 4·35767

EXP

L N TIO

N

OF

THE

T BLES

(

P

Cq ,

; S2. )

(cos. SO

24 ') J

(3

' from differe nce t

ab

le agai nst

.

s

ubtr

acted)

(sec. 76°

42 ' )

(lh

lsec

.

76° 42' -sec

.

76

> 48

' 1;

diffe

rence mu st be

obtai

ned by thi s

method

because the

mean

diffe re nc

es

are n

ot

s u

ffi

cie ntly accu r

ate)

4 . To

c

onv

ert

31 46'

to radians;

3 1° = 0 ·54105

ra

dian

s

O O ~ L

46 '

=

0·01338

ra

dian

s

I' ..J

3 1 46 ' =0 ·55443 radian

s

5. To co

n

ve

rt

1·648

r

adia

ns to

deg

rees:

1 radian

=

51 17·

7'

0·648

rad

ia ns

=

37 07·

7'

1·648

radian

s

= 94°

25 -4

'

SQU RES ND CUBES OF NUMBERS

P

i g es 364 · 367)

T hese

tab

les

wi

ll give squa res

and

cubes of numbers to fou r significant figures.

To

o

tain

rhe

squun or

cube

o

a numb..,.'

(a) If th e

number is

between I and [0

and

cons ists

of

three significa nt figures (o r less) the sq uare or

cube is t

aken

fr

om

the ma in

pa rt

of the tables, but if there are fo ur

si

gnificant flgu res the Mean

Difference section is a lso used.

Example:

2.824

2

7·63

J

= 7·952

(fro m main ta ble)

1 23

(from Mean Di fference

7·975

58·22

(from

main

tab

le) -

2

(from M

ean

Difference)

58·24

(b) All ot

her numbers are converted into

sci

en t

ific

n

otat

ion a nd the s

quare

or

cub

e

ob

tained for t

he

si

gnificant

fi

gures

as

before.

Example:

46J ·8

t

0 ·000 0725

J3

(4 ·638 x 1O:)t = 2 1·51 x I(P

2'

15

1

x

I }>

or

215 100

(7'251 x = 381 ·3 X 10 -

1

;

3·813 X

10

-

13

o r 0·000 000 000 000 3813

PAGE 352

PAGE 363

PAGE 362



16/82

EXPLANA

TI

ON OF

THE TABLES

19

SQUARE

AND CUBE ROOTS OF

NUMBERS

~ g f U

368

.

37 7

)

These tables, which give the square and cube rools o f numbers to f

ou

r significa nt figures, are in the

following fo rm :

(a) Square roots of numbers between

I

and 10

and

between 10

and 100 .

(b) Cube

r

oots of

numbers between

I and 10.

between 10 and 1

00

a nd belween 100 and 1000.

The following examples illustrate the method

of

obtaining square

or

cube

roots

using the tables:

,1839-2

1

4523 ,1

78620000

,10-000

72

47

I. Change Ihe nllmber inlO scielllijic nOlation.

v 4·523 x jCP

v 7·

247 X 10-

1

2 Adjust Ihe position of rhl decimal point to make the index of 10 exactl), divil ible by

Ihe

r

oo

l being

found.

v 45·23 )I P

= v 72 4·7x 10-=4

3.

En

ter lhe

l a b /

s

hoK

n below and x l r a l 1 Ihe required r

I of Ihe signijil OnI figures.

TABl.E

Of

SQUARE

ROOTS 1- 10

2-897

TABl.E Of SQUARE

ROOTS 10-100

6·725

TABLE

OF CU8E

ROOTS 10-100

4·284

4.

Del

t rmint the square

or l u ~

roo t

of

the poK't'r of 10.

,1 ,0 _ 10

,1839-2

2·897 x

10

or 28 -

97

V I

Ol

10

:_

145-23

_ 6·725 x 10

or 67-25

3v 1()G

: . 3

Y78

620 000

4·284 x

I

Q

or

4

28·

4

TABLE OF CUBE

ROOTS 100-1000

- 8-982

v

10-- 10-

1

:.

O-OOO 7247

8·982

x

10

-

or 0·08982

Mean Difference

co

lumns

are

not required in the table of square roo ts of num bers between 5·5 and

9·9 nor in the tables of cube roots of numbers between 1-0 and

10

·0 and between 55-0 and

100

·0.

When any of the above tables

are

being used 10 find the roolS of numbers wi th four significa nt

figures

in

terpolat ion ca n be

ca

rr ied out mentally.



17/82

11. TABLES FOR CELESTIAL NA VIGAnON

A B & C AZIMUTH TABLES

Pages 38

- 428

)

To conform with the method

of

present ing data in the Nautical Almanac the

hour

angles in

Tables A and B are given in degrees and minutes

of

arc

from OC 15' to 359"

45 .

If

the H.A. is between 0 and 180

0

the

body

is west

of

the meridian and its hour ang le will appear

in the

upper

row

of H.A.s

at

either

the top

or

bottom

of

the page.

If

the H.A. is between

180C and

360

0

the body is east of the meridian and ils

hour

angle will

appear

in the lower row.

The A Band

C

values and the azimuth are derived by employing the we ll known formula which

connects four adjacent parts of

a spherical triangle.

It

can be shown, for instance,

that

in spherical

triangle A B C:-

cot. a sin. b = cot. A sin. C

+

cos. b cos. C.

cot. a sin : b - cos .

C

cos . b = sin.

C

cot. A.

x.

p

,

o

•

j

f :V

: 11 Z

The figure shows the astronomica l triangle PZX with

the four

adjacent parts

PX, P,

PZ

and

Z representing, in

that order, polar distance, hour ang le, co-

latitude

and

azimuth.

Applying the

above formula

to this

particular

case,

we have:- -

cot. PX

sin.

PZ

-

cos

. P. cos. PZ = P. cot. Z.

Dividing by sin. P. sin.

PZ

, this becomes--

cot. PX . sin. PZ cos. P cos. PZ

sin. PZ

sin. P

sin. P

i.c.

N

sin . P

cot. PX

tan. dee .

In the tables;-

sin.

PZ

sin. P

cosec. P - cot. P cot.

PZ

= cot. Z cosee. PZ,

cosec. H.A. - cot. H.A. tan. lat.

= cot. azi. sec. lat.

cot. Z

sin. PZ

cot . H.A. tan. lat. is tabulated as A, and tan. decl. cosec. H. A. is

tabulated

as B.

Hence (A

; ; B)

cos. lal. = , cot.

azimuth.

(A

; ;

B), referred

to

for convenience as

.C

, forms the primary argument in Table C with l

at.

as

the secondary argument. With these two arguments the azimuth

is

found.

As an example,

consider

the case where hour angle = 48°, lat.

= sr

N., and deel. = 15° N.



18/82

lat.

H

.A

.

decl.

EX

PL

ANATION

OF

THE TABLES

ZO 00' N

48 00'

15

°

00 N

L. tan. 0' 107 19

L cot. 9.9 5444 L coscc.

+ L. tan.

0·12893

9·42805

Log. A 0·06 163 Log.

B 9·

55698

(?oy 3%.

~ 5

)

_ ~

_

(A is named oppos

it

e to lat.; B has the same name as decl.) (A

.:t

name

as

A which

is

numerically greater

than

B.).

C 0·792 S.

fOJ M ~ Azi. 64 00'

Log.

L. cos.

L

co l.

(

)

lal.

52 00 ' N.

. . Azimuth = S. 64

o

£

or

244 .

8)

'C' -

0·792 S.

21

(Same

(The az i

mu

th ta kes the names of the

C

factor

an

d

hour

anglc.)

( p ; _

394 - 345 )

Reference to the tables wi

ll

show that fo r the above d

ata

A

= \ S a

nd 8

= 036

N. T he

combination of these is 0·79 S., which in Table C with la l. sr gives az im uth S. 64°·2 W.

The ru les for

naming

and combi ning A and B a nd for nam ing t

he

azi mut h

are

given

on

e

ac

h

page of the ap

propr

ia te wb

le

. It

is

imponant that they shou ld be

ap

p

li

ed corre(;tly.

umgilllde Correction

The quanti ty (A t B) or 'C', besides being one of the arguments for finding the azimuth from

t

ab

le C, is a lso the ' longit u

de co

rr

ect ion fac to r' o r the e

rr

o r in longitude d ue to

an

error

of

I' of

lat itude. Th is ca n often be

ve

ry usefu l to those accUSlOmed to working sights by the longitude

me

thod.

A simple sketch showing the di rect ion

of

the position line will at once make it clear which way

the longitude

corr

ection should be

app

lied.

It

will easil y he

appa

rent that wh

en

work ing a sight

by

t

he

longitude method :-

(a) when the position

li

ne lies

N.E.

jS.W. (body in N.W. or S.E.

quadrant),

if the assumed lat itude

is too far north the com puted lo ngitude will

be

too far east, and if the latitude is too far south t

he

longitude will be too fa r west ;

(b) when t

he

positi

on

line lies N.W. jS.E. (body in N. E. or S.W. quad rant) the reverse holds good.

Example Suppose a sight worked with lal. 49° 06' N. gi ves longit ude 179

0

46 ,0 W. a nd azimuth

S. 70

°

·5

E., the value of

C

being 0·54. If the correct

la

t itude turned

ou

l 10 be 49° 46' N., i.e.

40 ' error, the e rr

or

in longitude would be 40 x 0 ' ,54

or

21'·6. We should Iherefore have

Com

puted long.

179

° 46 0 W.

Correct ion 2

1

6 E.

Cor rect long.

179

0

24 4 W.

This

is

a case where the la titude being

too

far sou t

h,

the

com

puted longi tude

is

1

00

f

ar

west.

Exampl

s on

he use of the tables

In each of the fo

ll

owing cases

fin

d the longitude correct i

on

factor and the t r

ue

azimuth .

PAGE 394

PAGE 395

PAGE 414



19/82

22

EX I)

LA

NA

TIO

N

OF TH

E TABLES

Example I: H.A. 3 10°, la . 48° N., ded. 20" N.

Fr

om

Tab

le

A with H.A. 310". lal. 48° N., A = 0-93

S. poqe. ? J4

-

y::y

5

From Table B with H A 310· , decl. 20" N., B

=

0-48 N ...)

Lo ng.

corr'n.

factor = A - B

=

C =

0-

45 S.

Fro m Table C with C 0-45 S • la t. 48° N. , T. Azi. =

S 73

°

2

g

~ A ;. e.J

w

( ..( /\

1 1//- 360

0

A

is

named

S.

opposite to la

t.

because H.A. is

11 1

between 90° a

nd

270 °,

B is named N. beca use the dec

l.

;s N.

C

=

A - 8

as

A a nd B ha

ve

d

iffe

r

ent

names,

and

is named

S. as th

e grea ter

qu

ant ity is

S.

The az imuth is na med S. beca use C is S • and E. beca use H.A. is between 180" and 360",

Example

2: H.A. 244", la l.

41

" S

.

decl. 5" S.

Fr

om Table A wi th H.A. 244", lat. 4 1" 5.,

From

T

ab

le B with H.A.

24

4", decl. 5" 5.,

Long.

corr' n.

factor

=

A

+

B

=

A = 0-42

S.

B = 0·10 .

c

0·52 S.

Fr

om Table C with C 0·52 S., lat . 41 ° S.,

T.

Azi.

= S. 68

°·6 E.

A

is

named

S.

same

as

lat. because H.A. is

be

t

wee

n 90° a nd 270°.

B is named S. because the dec l. is S.

C = A + B

as

A and B have the same name (both 5.).

The azimu th is named S. because C is 5., and E. because H.A. is between 180" a nd 360°.

Example 3: H.A.

108

°, 1at.

61

" N

.

decl. 20° N.

From

Supp

lementary Table A with H.A.

10

8", lat.

61

° N.,

From T

ab le

B with H.A. 108", dec

l. 20

" N.,

Long. corr'n. factor = A + B =

A

0·59

N.

B

0·38

N.

c

0·97

N.

From Tab le C witn C 0·97 N., lal. 61" N., T. Azi. = N. 64"'8 w.

A is named N. same

as

lat. because H.A. is between 90"

an

d 270".

B is na med N. because the decl. is

N.

C = A + B as A and B have the same n

am

e (both N.).

The azimuth

is

named N. because C is N., and

W.

because H.A. is between 0" and 180".

Use q( A BC Tables or Great Circle Sailing

These tables provide a ready means of finding tne initia l gr

ea

t c

ir

cle course

fr

om onc point

0

another. Suppose, for example the in itial course from P (49° 30' N., 5° 00' W.) to Q (46° 00' N.,

53° 00' W.) is req uired. The procedure is simply to treat d. long. as hou r angle, lat. of P. as

lat itude ,

and

lat. ofQ a s declinnlion . Th us:-

HA:::::

5 ~ _ 5 ; : ; . U D

Fr

om

Ta ble A w

it

h H.A. 48°, lat. 49° 30'

N.

, A

=

1·06

S l.f Il 10 \ _ I QO

Fr

om

Tnble B wi

thH .A.48

", decl. 46° ()(),N. , 8

=

1

· )9N

.

\j 1. 5} 01'_

'-JI

I I

- ( eke ;

4b

O

A - B C

~ - = -

N \P3

4fI

From Table C with C 0·33 N" lat. 49" 30' N., T. Azi. = N. 77°·9

W.

i.e . Initi al a.c.

Cour

se = N. 77 °·9 W. or 282°·1.

Th e final co urse, if requ ired, may be obta ined in a similar way

by

fi nding the initial course from

Q

to P and reversing it.

PAGE 394

PAGE 395

PAGE 412



20/82

EXPLA NAT

IO

N OF THE TA BLES

AMPLITUDES and

CORRECTIONS

Explan tion wilh rabla)

(pages 4

29

.

431

)

EX-MERIDIAN

TABLE

I

(P g

es

43

2 · 44 3)

3

•A' is the change in the altitude of a b


21/82

EX

PLA

NA

TI ON

0 1- E TA

BLE

S

ample I: In O. R. Lat. 48

c

13

' N ., D.R. Long. 7" 20' W., the True Altitudeof the sun was 19° 52'.

Sun' s LHA

356

" 00 ' . Declination 2 1

0

39' S. Detennine th e Position Line.

Table

(Different Name)

La L 48° I ' N.

Dec .

21° 39'

S.

A = [ ·3

(yy

4

3/':,

)

Tabl 1

LHA 3

56

0

00 '

A =

Red for.

=

4' ·)

.) = [ ' ·3

Reduct ion = 5'·6

T.

Alt.

Reduct ion

T . Mer. AIL

T . M er. Z. Disl.

D

ecL

Lac

True A

zi

mut h

19

° 52',0 S.

5'·6

19° 57' ,6 S.

70°

02

' -4 N.

21 ° 39'·05.

48° 23'·4 N.

(

' fey

41[1-1

from Az. Tables 176°

Posit ion Line passes 086° and 266°

th r

ough L

1.

t. 48° 23',4 N., Long. 7° 20' W.

ample 2: D .R. La

t.

4 r 12' N., D.R. Long. 24° 32' W. ,

th

e True Altitude

of

Antares was 21 0 28'.

St:l r's LHA 357

0

00' . Declination 26

0

18' '0 5. Determine the Position Lin

e.

Table I TaM- 11 T. All.

(Different Name) LHA 357" 00' Reduction

Lat. 42° 12' N .

Oecl. 26° 18' S.

A = 1 '4

A

=

1· ·4

Red . for 1 ·0

=

2' -4

-4 = 0',96

Red uc

ti

on

=

3',36

T. Mer. Alt.

T. Mer. Z. Dist.

Dec .

La .

True AzinHuh

2 1° 31' -4 S.

68

0

28

' ,6 N.

26

0

[8 '·0 S.

42°

10

' ·6 N.

from A

z. Tab

les 177 °

Position Line passes OSY an d 2 6 7 through La c 42°

10

" 6 N. Long. 24° 32' W.

Although the latitude and declination of a circumpolar body are always of the same name, 'A ' for

Lower Transit observa

ti

ons is tabulated in the lower part of th e " Latitude and Declination Different

Name " sec

ti

o n of Table I ,

When near

it

s Lower

Tran

sit the L

oc

al Hour Angle is less th an 180

0

when west oh he meridian and

mo re than 180° when

eas

t

of

it.

The

Hour Angle

to

use whe n entering

Ta

ble in this

ca

se is (1 80°....,

LHA

.

ample 3:

D.R

La

t.

42

0

10' N. , Long. 21 0 30' W., the True Altitude ofD ubhe was 14° 20'.

Star

's

LHA 176

0

30' . D eclination

62

" 0 1' N . Determine the Pos ition Line.

Table I

Table 11 T. All. 14° 20' ·0

(Same Nam e)

LHA

176° 30'

Reduct ion

- 2'· )

= 3" 30'

~ ~

Lat. 42 ' 10' N.

A

=

0"·7

T. Mer. All.

17'· 7

Oecl. 62 ' 01' N .

Reductio n

2'· 29

Po la r Dist.

27" 59',0

For Lower T ra nsi t- 2'· )

A=

0 '7

Lat. 42" 16',7 N.

"True Azimuth

fro m Az. Ta bles 358

0

Positi

on

Li

ne

passes 088" a nd 268° th rough La . 4

2"

16' ,7 N. , Long. 21° 30' W.

PAGE 439

PAGE 444



22/82

EXPLAN;\TI ON OF THE

TAB

L

ES

xamp le 4:

D.R Lat. 50"

02' S.,

D.R Lon

g. 67" 20' W

., the

True

Altitude o f Achemar was

17°

20'. Star' s LHA

184" 20'.

Dec linat ion 57

° 29'

S.

Determine

the Position Line.

Table I Tahle II

T. Alt . Ir 20 ', 0

(Same Name) LHA 184"

20

'

Reduction -

)',5

=

4"

20

'

A

= 0

-'

7

T.

Mer. Alt. 17

° 16

',5

= 3' ,5 Polar Dis t. 32"

31

' ,0

Lat

50' 02' S,

Red uc tio n

Dec .

57" 29

' S.

For Lower T r L llsit- 3 5

~

0 ' 7

Lat. 49 ° 47 ',5 S.

True Azimuth

from Az. Ta

bl

es 177 ·5"

Posi

tion

Line passes 08

7 ·5 and

26

7 ·5

through 49° 47'

·55.,

Long. 6

7

20'

w.

EX-MERIDIAN

TABLE

III

( P ~ g e 48

This Table

contai

ns a Sec

ond Corre

ction. which, when the amount of the Main Cor rect ion is

considerable, enables the process of Redu c

ti

on to Meridian to e applied wit h advan tage on much

larger hour angles than could otherwise e the

Cdse.

x mpl

e

D.R Lat.

3I 00' N

., D.R long. 1

24

"

00'

W.

,

the

True

Altitude of the

Sun

was 5

01' .

Sun's LHA 347

0

30' . Declination 2°

00

' S. D etermine the Posi

ti

on Line.

Table I

Tab

le /I

T. AIt .

(Different Name) LHA 347° 30' 1st Co rrection

Lat. 31° 00' N .

Dec .

r

00' S.

A

~

3 , 1

A = Y' I 2nd Corrttt i

on

Red. fo r Y·O =

125

' ,0

·1 = 4', 2

1st Correct ion = 129 ' ,2

T.

Mer. Alt.

T. Mer. Z. Di sl.

Dec .

55" 01'·0 S.

2' 09

2

+

3"6 -

57" 06 6 S.

3r 53 4 N.

2' (I() .(I

S.

Entering Table I11 with 129 ' as First Correct ion

and 56

0

as

Altitude

we

have 3·6' Subtracti\'c

for Second Correc tion.

Lat. 30" 53' -4 N.

Tru

e Azimuth

from Az. Tables

158

0

Position Line

pa

sses 068°

and

248"

through

La L 30" 53',4 N., Lon

g.

1

24

° 00'

W.

EX-MERIDIAN TABLE IV

Pig 4 4$)

This Tab le gives the limits

of

H

our

Angle or T ime before or

f

ter the time

of

the Meridia n

Pussage when an

Ex

-Meridian observati on can e taken . When the observation is taken within

the lime limit prescribed by thi s Table the Second Correction from Tab le I1I is ne

gl

igible. The

Table is entered with

'A'

ta ke n from

Ta

ble L

Given Lat. 3] N., Declination ISON., find the lim its of Hour Angle for taking an Ex-Meridian

observation.

For

LOlL

37"

and

Declination 18°, 'S me Name', Tab le I gives for 'A'. Entering Table IV

with

4'

·6 as 'A the lime limit

ab

reast is found to

be

24 minutes.



23/82

26

EXPL N TION

OF

THE T BLES

CH NGE

of HOUR NGLE with LTITUDE

Pages 449

-

450)

The formula used in calculating the values tabulated

i s :

Change of H.A. (in mins.) due to I change of Alt cosec. Az. sec. Lat.

The table gives in minutes

of arc

the

error

in

hour

angle resulting from

an

altitude I' in error.

This is of particular value to those navigators who work the ir sights by the 'Longitude by Chrono

meter' method.

I t

will be seen

that

the

error is

least in the case of a body on the prime vertical

and that

it increases

as

the azimuth

decreases-very

rapidly as the azimuth becomes very small.

From the table the observer can readily find the least azimuth on which the altitude of a body

should

e

observed in order

that

the resulting longitude may not exceed a chosen

lim

it of error.

Another

use to which th

is

table can be

put is

to find the correct longitude when a sight has been

worked using an altitude in error

by a known amount.

, xample

I:

In latitude

18

° what should e the lowest value

of

azimuth in order

that

ail

error

.ill

of I in the altitude may not produce more than 2' of error in the computed longitude'

Under lat. 18° and against azi. 32°, the error for I of alt. is found to be 1',98. Accordingly,

the observation should be taken on a bearing greater than

3r .

(In lat.

36°,

it will be seen,

an

azimuth of

about 39

0

would constitute the limit. In lat.

63°

the

error

would exceed 2' even when the body was on the P.V.)

xample

2: A sight worked in lat.

54° by

the 'Longitude Method ' resulted in a deducted longi

tude of 64° 14',5 W. and azimuth

N

65°

E

Afterwards it was discovered

that

the sextant index

error

of 2'

w

off the

arc

had been

app

li

ed

the wrong way. Find the correct longitude.

Since the longitude is found by comparing the L.H.A. of the body with its G.H.A., it is evident

that

the

error

in the L.H.A. will

e

the

error

in the computed longitude. The index

error

of 2"5,

which should have been added, was subtracted , so

that

the altitude used was 5' too small.

The table shows that in lat. 54°, when the az i. is 65 ", the error in H.A. is 1',88 per I ' of alt. For

5' , therefore, the

error

will be 5 x 1'·88

=

9',40.

As the real altitude

wa3

greater

than

the value used, the observer must

be nearer

to the body

than

his computed longitude would lead him to suppose. With an

easterly

azimuth this means

that

the

Westerly

L.H.A. should be greater,

and

therefore the observer's west longitude should

be

smaller. Hence:-

Computed long

. . . .

. . . . . . . . .

. . 14'·5W.

Error 9',4 to subtract

Correct long.

I t

will

be

appreciated

that

this

is

much quicker

than

re-working the sight.

CH NGE OF

LTITUDE

IN

ONE MINUTE

OF TIME

Pages 451 _

452)

This Table contains the change in the alt itude of a celestial body in minutes and tenths of arc

in one minute of time. It is useful for finding the correction to be applied to the computed altitude

of a heavenly body when the time

of

observation differs from

that u ~ e

in the

computation

of the

altitude. When the star

is

East

of

the Meridiar.. the correction from the Table

is

subtractive from

the computed altitude if the time of observation is earlier than

that

used in the

computation

of the

PAGE 448



24/82

EX PLA NATION OF TH E TABLES

altitude; it is additive if

the

time

of

observation is later. When

the star

is West

of

the meridian

th

e correct ion is

addit

ive if t he time

of

observation is before that used when

com

puting the altitude ,

it is subtractive if the lime o f observation is after.

ormula

Cha nge

of

altitude in one minute

of

tim e

=

15' Sin. Az. Cos . Lat.

The change in 6 seconds of time is found by shifting the decimal poi nt one place to the left.

Th

e change in

I

second

of

time is found

by

ca

ll

ing the qu antities in the Table seconds instead

of

minute

s.

Example:

In La t.

51

30 ' N.

on

the Meri

dian of

Greenwich

on October

26

1h

,

192

5 a t 8 h. 0 m.

p.m. the

computed altitude

of the

sta

r

Altair

was

31

09 '·2.

Find

the true altitude at 8 h.

10

m.

p.m., the Az. being S.49°

37

' W. Opposite 52

0

in the Lat . Col. and under in the Az. Col. is

l 7 IJof arc which is the change of a l

titude

in I min. of time, and 7'· 1

x

10 minutes gives 71' or

I 11', which is the correction to app ly to the com puted altitude.

/ \

Com

pu ted All. .. . . . . . 37 09 2

P

rs

- y52 )

CO H.t

oSubt.

. . . . . . .

I'

WO

True All. required

35 ' 58',2

DIP

of the

SEA

HORIZON

The

tabu

lated

va

lues

are

derived from

th

e formu la

D ip

(in minutes) = 1'7

6-vh

where h =

height of eye in metres.

Thu s,

for example, when

h

=

3 m

(98

ft),

dip . = 9

6

.

Heights of eye

are

gi

ve

n in metres, ranging from'O· 5

m

to 50·0 In and al

so

in the equiva lent feet

(1 5 ft to 164 ft ).

MONTHLY

MEAN OF

THE SUN

' S SEMIDIAMETER

AND SUN

' S PARALLAX

IN

ALTITUDE

Pag/il 453

Correction fo r pa ra llax is to be t

aken ou

l opposite the Su n's Alt itude and is always

addithe.

Example:

The sun 's parallax

co

rresponding to 51 of altitude is 0'·

1.

AUGMENTATION OF THE MOON S SEMIDIAMETER

Pagtl453

)

REDUCTION

OF

THE

MOON S

PARALLAX

P ~ } e 453

PAGE 452



25/82

EXPLANATIO N OF TH E TABLES

MEAN

REFRACTION

Page 454)

This table con tai ns the Refrac

ti

on of the heavenly bodies, in m

in

utes and d

e.::imals

at a mea n

state of the atmos phere, a nd corres pond ing to their appa rent a ltitudes. Th

is

correc

ti

on is always

to be

s

lIbrro

cl

ed from th e

appar

ent altitude of the object.

Example

The mean refraction for the apparent

al

titude 10

°

50', is

4·

'9.

Caution For low altitudes all refraction tables are more or less inaccurate.

ADDITIONAL

REFRACTION

CORRECTIONS

PJge 454)

The mean re

fr

act

io

n

va

lues g i

ve

n in the Mean Refra c

ti

on table a re for a n atmospher ic pressure

of

t,000 mb (29·5 in) and anairternpe rature

of IO

C (SO°F . If the atmospher

ic

pressure or tempera

ture

di

ffer from th ese values additional co rrect io ns mu st be a pp

li

ed to the a pparent altitude. These

corrections a re conta ined in the

ta

bles Additional Refraction Co rrections for Atmospheric Pres sure'

and

Add

itiona l Refract i

on

Corrections for Air Temperature'

Example Find the true altitude

of

the sun when t

he

observed altitude

of

t

he

sun's lower lim b

was 6

) ,

height of eye 16 m (85 ft), a tm ospher

ic

pressure

1020

mb

(30 1

in), a ir tempera

tu re

0°C (32° F .

Observed alt itude sun's

low

er limb

-

6°

00

Total correction 0 - 01'S'

-

-

True altit ude

5'

8S

Correction for temperature

-OA

Co rrection f

or

pressure -0 2

Correc

te

d altitude 5° 57-9'

If the altitude is greater than 5

0

00' th e error due to appl ying these corr

ec

tions to the true a

lt

itude

can be ignored

in

prac

ti

c

e.

N.B To convert baro meter readings from me rcury inches to millibars, o r vi

ce

-versa, see page

499 .

To

con

ve

rt temperatures

rr

om Fah re

nh

e

it

to Celsius, or vice-versa,

se

e page 494 .

The adjustment

or

mean refraction as shown above is important only when the alt itude is small.

It should be borne

in

mind

that

on account

or

uncertai n refraction positio n lines obtained from

sights taken when t

he

altitude

of

the body is less than lO° or so sho uld not be re

ed up on imp lici

tl

y.

Moreover, due to th e effect

of

atmospheric re fract ion on dip it is unw

ise

to place too much re

li

an

ce

on sights taken , whatever the a

lt

itud

e.

when there is cause fo r abnormal refraction to be suspected.



26/82

EX PLANATI ON OF TH E TA BLES

"

CORRECTION

o f

MOON ' S

MERIDIAN

PASSAGE

PiJge 4

)

The correction obtained

fr

om this ta ble is to be app lied to the time of me r idian passage gi ven in

t

he

Nau tical Almanac (i.e. the time of trans

it

at Gree

nwich) in o rder to find the time of the local

tr.lns

it

according to the obse r

ve

r's longitude.

D X longitude

Corr

ec

tion =

wh

ere D is the d i

ffe

rence between the times o f successi

ve

transits.

360

When th

e obser

ve

r is in

£ S

longi tude, D is the difference between the time of Inlnsil on the day

of

observation arxf the time of transit on the preceding da

y.

When in W

st

lo ngitooe

it

is the differ

ence be

tw

ee

n the times o n the day of observa tion a nd the oJlQwing da

y.

£xamplt·: From Nau . Aim. L M .T .

of

moon's upper trans

it

at Greenwich ls: -

h.

m.

I Si July

18

44

diff. 48m.

2nd Ju

ly

19 32

diff. 5Jm .

3rd July

20 25

Find

G.M.T.

of moon

s

uppe r tra

ns it

o n 2nd July (a) in long itude 1

56

0

E., (b)

in longitude

63

W.

Jul y

a)

LM

.T. of transi t at Greenwich

. .

. .

. . .

. . . .

2

Co rr ·n. for

0

48m . long. 1

56

0

E • • . _ .

. .

. . .

L M T . of loca l t ransit

E

as

t longitude in ti me units

G.M.T.

of

local transit ( 1

56

0

E.) . . .

..

. . . . . .

b

)

L M

.T. of tra ns

it

at Greenwich

Corr'n. for D 53 ., lo n

g.

63

0

W.

L.M

.T. of loca llran si t . . _ . , .

Wes t longitude in t

im

e un its . . . .. . _ . • . .

a .M.T. of local transit (63

0

W.) . .

2

2

J uly

2

2

2

h. m.

19

32

- 20·8

19

·2

-

10

24

8

47·2

h.

m.

19

32

+9 '2

19

41

·2

+4

12

23

53·2

-



27/82

30

EXPLANATIO

N

OF

THE T

AB

L

ES

SUN S TOTAL CORRECTION

Pages 456-461 and Iniith Front

over

)

This is a combined table for the correction of both Lower Limb and Upper Limb altitudes of the

Sun. To simplify interpolat ion for intenned iate altitudes and heights ofeye, the tabulation is based on

co

lumn

ar

and line

ar

correction differences

of

0.2.

The co

rrections in

th

e main table give the

com

bin

ed

effect

of

dip , refraction, parallax in altitude and

an assumed semi-diameter

of

16.0 . Subsidiary corrections at the f

oo

t

of

the

tab le

give the monthly

va

ri

at i

ons

of the semi-diameter from the assumed va

lu

e of 16.0 .

Th

e

co

rrections and subsidiary

corrections are added to or s

ubtra

cted from

the

observed altitude as show in the table.

Example J

Obs. AIL Sun

s L.L

0(-- \

Co

rm. for obs. al t. 25

f J b I

and H.

E. 12.0m

.

A)

True

Alt

.

of

Sun s

centre

24

57.2

+ 8.0

+

0.1

2505.3

Exa

mp

le 2

Obs

.

AIL

Sun

s U.L

Corm

. for obs. a lto 34

and H

.E.

19.7m

Subs idiary corm. for June

True A lt.

of

Sun

s

centre

STAR S

TOTAL CORRECTION

( I SIL

Pages 462-465 and InSIde S,

ck

Cover)

33 45.6

- 24

+ 0.2

3320

.7

y53 )

This table corrects

th

e

combined

eff

ec

ts

of dip and

refractio n .

To

si mplify interpolation for in

t

er

medi

ate

alti tudes and heights

of

eye.

the

table is b;:lsed o n

co lumnar

and l

inear co

rrection

diffe rences of 0 .2.

This

tab

le can also be u

sed

for the

co

rrect io n o f observed altitudes

of

the planets, bu t in the

case of Ve nus and Mar s the small additional co rrection given in the Nautical Almanac for

para

ll

ax and phase may he necessary. T he size

of

these

co

rrections vary with the d

ate and

the

altitude of the planet.

MOON S

TOTAL

CORRECTION

Lower

Limb

-p8ge:s466-478;

~ Limb

- pages

4

79-49I

J

This t

ab

le

co

rr

ec

ts the

co

mbin

ed

effects

of

d ip. at m

os

phe

ri

c refrac

ti

on . augme

nted

se

mi

diamet

e r and para

ll

ax in altitude. '1l1e

dip co

mponent used in lhe main tahl e is ,. cons

tan

t 12 .3.

therefor

e the subsidiary

co rre

c

tion

given at the foot o f the pages must be added to the main

cor

rection. T he argument for this subsidiary co rrection is the observer 's height of eye .

No account has b

een taken of

the red uction with latitudc of th e

moon s

hori zo ntal parallax,

bu t in ge neral this is of no practica l significance. in cases whcre a

hi

gh degree of accu rac y is rc

quired it will be necessary to appl y th e

co

rrec tions

scpa

rately toge th e r wi th the ad

ju

stment of

the ret rac tion co rrec tio n for the

preva

iling at mosphe ric pressure and tempe ra ture .

T he main correctio ns

AL

W A VS added 10 bot h the lower limb and upper limb obse rved

altitudes of the m

oo

n , the

dip co

rr

ec ti

on is then

d

and for up

per li

mb ob

serva

tions

3(]

must be

subt

racted from

the

result.

Example

I

Moon 's

Hor .

Pax.

( from N.

Aim.)

= 57 .5

Obs. All. moo n's lower lim b

Co rr

ec ti

on from main tuble

Co rr

ec tion

fo r heigh t o f eye 13.5m

T rue altitude o f m

oo

n

=

=

+

47 . 1

= + 5 .8

=

:lif'40' ..

\

Example

2

M

oo

n 's Ho

T.

Pax. ( from N. Aim. )

=

59 ' 0

O hs. All. moon 's upper limb

Correc ti

o n from mai n tabl e

Correctio

n for heigh t of

eye

33m

T rue altitude o t moon

= 69 36'.0

=

+ 22 .0

= + 2' ,2

7

0 00

' .2

- 30

69 30 . 2

PAGE 458

PAGE 459

PAGE 471



28/82

Ill. TABLES FOR COASTAL NAVIGATION

D Y

S

RUN

-

VER GE

SPEED

T BLE

This table prov ides a rapid means of finding the average speed directly from the arguments

steaming time and

d

istance run , It wi

ll

be apprecialed that there is no necessity 10 con

ve

rt

minutes into decimals of a da y, a nd th

at

no logarithms or co- loga rithms a re required. Sim ple

ad ditio n is all that is needed .

The

scope of the t

ab

le has been made wide enough to cover cases of hi gh speed vessels (up to

40

kn

ots o r so) on easter ly or we sterly cou rses in high latitudes where cha nge o f longitude between

o

ne

local noon and the ne xt may amount to some 30°, or 2 hours of time.

Distances a re tabulated as multiples

of

100 miles. Increments of speed for multip les of

IQ

miles

and multiples of 1 mile are obtained simply

by

shifting the decimal point one or two places to the

left , rtsptttive ly.

£xanrp/( ; Gi ve n st

eaming

tim e 23 h. 29 m., distance 582 miles, find the ave rage speed .

D

is

tance in miles

500

8

0

2

582

Speed in

knots

21 ·29 1

3 4066

0·08517

24·78277

Th

at is. av

.:

rage speed

co

rrect to two places of decimals, which a re g.:nerally

co

nsidered suffici.:nt,

is

24·78 knots

• Enter with 800 miles and shift d.:cima l point I place to the left

t Ent.:r with 200 miles and shift decimal po int 2 places to the left

R D R R NGE

T BLE

{Pag9501}

RADAR PLOTTER S SPEED

ND DIST NCE

T BLE

g f l 5021

ME SURED MILE SPEED T BLE

~

503 509

J

This table is arranged in crit ical tab le fo rm and gi ves speeds correct to th

.:

nearest hundredth

of

a knot without interpolatio n. If the tim e

argument

is

an

exa

ct

tabulated value, the speed

immediately a bo ve it shou ld be taken.



29/82

32

EXPL N TION

OF

TH E T BLES

I. If the time reco rded for the measured mile i5 9 m. 16·2 s., the speed is 6·47 knots.

2. f

th

e time is 4 m.

55

·3 s., the spttd is 12·19 knots.

3. If the time

is

3 m. 52 ·3 s. , the speed is 15·49 knots.

4. Suppose a ship on tr ials makes

si

x runs over a measured mile, three against th e tide and three

with the tide, such th at the timings

by

stop- watch are as follows: -

First run against tide . . .

First run with tide . . .

Second fun agai nst tide . . .

Second fun with t ide

Third run against tide . . . . . . . . . . . . . .

. .

. • . . . .

Third run with tide

. .

. . . . . . . . .

. . .

. . .

. . . .

•

. . . .

Th

en

total time for 6 miles is ..

:.Average time for 1 mile

is

m.

s.

3 28· 8

3 18·4

3 30-0

3 1J.8

3 31

3 16·7

20

2H

3

2H

rom

th

e

fa

ble the average speed or

rh

e six runs is

17·66

knot

s

Strictly speaking, the average speed should be computed by finding the 'mean of means',

in

which case the

work

would

be arra

nged as follows.

RU N

SPEED

1ST

2ND 3RO

4TH MEAN OF

m.

s.

KNOTS

MEAN

M

EA

N MEAN

MEAN

MEANS

ISI

3

28·8 17·24

17· 690

2nd

3 18A

18'

14

IH650

17·640

17·66000

3rd

3

30·0

17· 14

17·6550

17·655625

17

·6

70

17 ·65 125 17·6528125

4th 3

17·8

18·20

17·6475

17·650000

17·625

17·64875

5th

3

31· 1

17·05

17·6500

17·675

6th

3

16·7

18·30

-

6)

106·07

4)

70·6175

IH

8

17·6544

'I'

Ordinary

Ordinary mean True mean

mean speed

o

second means· speed

At speeds greater than

about

19

1

knots it wi

ll

be not iced that in certain cases a

cha

nge of a

tenth of a seco nd in the time will make a d ifference o f mo re t

ha

n one hundr

edth

o f a knot in

the

tabulated speed.

Fo

r example, if the time for o

ne

mile is betwee n 2 m. 38·7 s. and 2 m. 38·8 s.

the speed.

co

rrect to two places of decima ls,

co

uld be either

22

·68 or 22 ·67 knot

s.

In very high speed vesse

ls

the recorded t ime fo r a measured mi

le

may

be

0 sma

ll as

to

be

beyond

the scope of the table. Even so, a reasonably accurate speed is easily

obt

a ined by entering the

table w

it

h double the recorded t ime, a nd then

doub li

ng the speed so

obta in

ed. For

in

stance, i

a mile

is

run in I

m.

55·25 . enter with 3

m.

50

·4 s. Thi s gives 15·62 knots which is half the required

speed of 3

1·

24 knots

and

this wi

ll be

correct w

it

hin 0·02 of a knot). By calculat ion th,e co rrect

speed

is

actually 31·250 knots

• Th

is

is usually regarded as being sufficien

tl

y accurate



30/82

EXPLANATIO N

OF HIE

TABLE S

JJ

Be

si

des its orthodox use for speed trial purposes, the table will be found useful to navigators

for other purposes.

For exampl

e, su

ppose it is decided to alter course after the ship has run 6 mi

les

on

a

certain

heading fr

om

a pos

iti

on line obtained at 1432 , the speed

of

the ship being

11·75

knots. The table

shows that

at

this speed the ship will run one mile in a

li

ttle over

5

m.

6

s.

, or

6

miles in about

30t minutes. Therefore, the course should

be

altered at 1502t

In

ce rtain circumstances it

mi

ght be considered convenient to pl ot the radar target

of

ano the r

vesse

l at

re

gular intervals corresponding to one mile runs of one s o

wn

vessel. Suppose the speed

to

be

9·70 kn o ts ,

whi

ch the table shows to co rrespond to a mil e in about 6

m.

s. Then, if the

stop-watch is started from zero at the t ime

of

the first observat ion , successive observations should

be taken as nea rly as

pr

ac ticable when the watch sho

ws

6 m. 11

s.

, 12 m. 22 s • 18 m. 33s. ,

24

m. 44

s.,

a

nd

so o

n.

DISTANCE

BY

VERTICAL

ANGLE

Pages ~

This table gives the distance of an obse r

ve

r fro m objects of kn own height when the angle be t

wee

n

base

a

nd

the summit is known. The ta bles are for distances up to 7 mi les so that the whole object

from b

as

e to summit will be in view when the hei ght

of

eye is more than 12 metres (39 feet)

Observers whose height

of

eye is le ss than th is must apply a cor

re

cti on for

Documents

Norie's Nautical Tables 1991 (Partial)