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8/15/2019 Norie's Nautical Tables 1991 (Partial)
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8/15/2019 Norie's Nautical Tables 1991 (Partial)
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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
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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
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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
8/15/2019 Norie's Nautical Tables 1991 (Partial)
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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
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EXPL N nO ND
USE OF THE T BLES
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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 :-
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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.
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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
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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.
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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.
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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
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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
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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
8/15/2019 Norie's Nautical Tables 1991 (Partial)
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
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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.
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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.
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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 .
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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.
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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
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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.
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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.
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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
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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
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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.
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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
-
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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
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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.
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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
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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