Ranger Hope © 2016
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Traditional coastal navigation begins from a known point or a fix from land based features plotted on a suitable chart. During the passage, course steered from the ships compass and speed from the ships log provides dead reckoning plots of the ships progress, confirmed by additional fixes en route. Whenever the opportunity arises a confirmation of continuing compass accuracy is checked by land based transits.
When on passage out of sight of land the navigator relies on the continuing accuracy of the dead reckoning or GPS for notifying of change in position. If she experiences compass error she is none the wiser and if she experiences power failure navigation position is lost. However, two basic computational processes using Nautical (navigational tables) are available. The first is to compare the predicted bearing of the sun at rising or setting with the observed compass reading (using azimuths or amplitudes tables) and the second to determine the changes in position without plotting on a chart (by traverse tables).
In order to use the tabulated data an underpinning knowledge of terrestrial and celestial navigational definitions is required.
Latitude and Longitude
Latitude and longitude are used as reference points for any geographical point (GP) on the earth’s surface. They are the angles at the centre of the earth between a GP that is North or South a base line of the equator (latitude) and a GP that is East or West of a base of the prime meridian of Greenwich (longitude).
Small circles and Great circles
The planes of circles drawn around the earth’s greatest girth cut the
earth’s centre and are called great
circles, being the shortest distance between two points on the spherical
earth’s surface. The planes of circles drawn around the earth’s lesser girths
do not cut the earth’s centre and are called small circles. All longitude lines follow great circles and are called meridians.
All latitude lines apart from the Equator follow small circles and are called parallels.
Charts and Mercator projection
In order to represent the spherical earth on a flat surface the Mercator projection is commonly used for coastal charts in low latitude (less than 60ş N/S). The projection allows rhumb lines with constant true bearing to be plotted over short distance with fair accuracy. Distance can also be measured using the parallels of latitude as a scale (1ş = 60 nm). Measures cannot be taken using the Longitude scale as distortion occurs as the segments of longitude narrow to meet at the Poles.
As the earth spins once every twenty four hours its longitude can be equated to time:
180şE + 180şW = 360ş
360ş ÷ 24 = 15ş
A reference point
is taken as the Greenwich Prime Meridian (0ş). The time at that meridian was
called Greenwich Mean Time (GMT) but is now called Universal Time Constant
(UTC) or just Universal Time (UT). So each 15ş of longitude is equivalent to one hour, or each 1ş of longitude is equivalent to 4 minutes.
For convenience the world is divided into 12 hourly time zones East of UT (+
before) of Greenwich Prime Meridian and 12 hourly time zones West of UT (after)
of Greenwich Prime Meridian.
Charts and Gnomic projection
Straight lines drawn on gnomic charts are great circles of shortest
distance from point to point but with constantly changing compass direction, particularly so between
GP’s with similar latitudes. However, these charts
are required in the high latitude zones for distance measuring where the Mercator
projection is too distorted.
Gnomic & Mercator
 Great Circle Sailing
A rhumb
line drawn on a Mercator chart below represents a small circle of constant direction
that is not the shorter great circle route.
The straight
line drawn on a Gnomic chart below represents a great
circle with constantly changing direction. This line can be
transposed from the GP below as waypoint legs onto the Mercator chart above.
Drawings above are Courtesy of BowditchAmerican
Practical Navigator
The earth’s spin and its revolution
The earth
spins (approx) once every 24 hours as it revolves around the sun in an
elliptical orbit of (approx) 365Ľ days. Its axis remains tilted with the consequence that the sun is
overhead (in zenith) varying from 23.5şN to 23.5şS over the yearly orbit. When the sun’s rays are directly overhead the heat
to earth’s surface is concentrated causing summer. This results in the opposing
seasons experienced by Northern and Southern Hemispheres.
Geographic position (GP), Time
and Longitude
The earth’s elliptical orbit causes it to accelerate on
approach to the sun and decelerate in moving away. Our mechanical or atomic
clocks maintain a constant beat referred to as mean time. As 15ş of
longitude is equivalent to 1 mean hour of time passing, or 1ş is equivalent to, mean 4 minutes mean time
and longitude can be equated. Local mean time is therefore unique at every GP’s
latitude. However, for convenience the navigator divides the world into 12
hourly time zones East (+UT) and 12 hourly time West (UT).
Greenwich Hour Angle GHA & Local Hour Angle
(LHA)
The earth is considered as at the centre of a celestial
sphere on which stars and planets are projected. The position on the celestial
dome directly overhead of any GP is called its Zenith. GP’s are described by
the coordinates Lat. & Long – Celestial positions are described by Dec. and GHA. GHA increases from 0ş to 360ş measured clockwise (westward)
following the apparent motion of the heavenly bodies (HB) overhead.
The angle
between the observer’s longitude and a heavenly body GHA is called the Local
Hour Angle (LHA) and is always measured clockwise (westward) from the observer.
Hence if the observer is in:
East
longitude the LHA is the GP longitude East + the HB’s
GHA.
West longitude the LHA is the GP longitude West  the HB’s GHA.
Coastal passage requiring charted land based features are unavailable beyond sight of land. Other mathematical solutions of determining passage progress without a chart are available, collectively called the Sailings.
Plane sailing  These solutions provide sufficiently accurate single course/distance, difference in latitude and departure for passages of less than a few hundred miles. They assume the earth traversed (crossed over) is a flat surface (a plane) consequently difference in longitude cannot be calculated, that requiring spherical trigonometry.
Traverse sailing  Extends plane sailing to summing multiple rhumb lines of changes in course/distance over a passage.
Parallel sailing – The historic practice of maintaining an East or West heading along a parallel of latitude (by altitude of a heavenly body) until dead reckoning (departure) indicated that the final position sought was directly North or South.
Mid latitude sailing – A course that is not East/West will start and finish in different latitudes. In converting departure into difference in longitude the mean (average) of the two latitudes is applied.
Mercator sailing – A mathematical alternative to plotting on a Mercator chart that uses computational tables of meridional difference and difference in longitude rather than difference of latitude and departure.
Great circle sailing – Courses and distances that follow great circles, those being the shortest distance between two points of a sphere.
Composite sailing –Great circle sailing adapted for the purpose of routing convenience.
Checking for Compass error
Without terrestrial features a practical way for a navigator to check the accuracy of the ships steering compass is to compare its bearing of a heavenly body with that predicted. The predicted position of GHA and Declination of heavenly bodies are tabulated in Nautical Almanacs. The most accurate bearings can be observed from sun on rising and on setting. Due to refraction in the earth’s atmosphere the moment of sunrise and sunset is actually when the sun is half a diameter above the horizon, this normally being the tabulated time.
Bearings used by coastal navigators are measured from North in a clockwise fashion through 360ş back to North – not so the tabulated quadrant notation of azimuths or amplitudes.
Azimuths are tabulated from North or South towards East or West, i.e.:
N 20 şE = 020ş N 20 şW = 340ş S 20 şE = 160ş S 20 şW = 200ş
Amplitudes are tabulated from East or West towards North or South, i.e.:
E 20 şN = 070ş E 20 şS = 110ş W 20 şN = 290ş W 20 şS = 250ş
The symbols AZI and AMP may help to memorise the different reference points of azimuths or amplitudes, being oriented as from the heavy lines of Z or M.
Three methods commonly used to find compass error from the sun:
1. a scientific calculator, or
2. azimuth or amplitude tables and form, or
3. the ABC tables and form.
In all three methods the time, the ship’s approximate position, the local variation and ship’s compass bearing of a rising or setting sun must be known. The time of the risings or settings in local time (at the ship’s longitude) is tabulated in the current Nautical Almanac. The ship’s Mean Time of the observation must be converted into Universal Time (UT) so the sun’s declination can be extracted from the tables for method 1 & 2, and the sun’s declination and GHA can be extracted from the tables for method 3.
Finding time of Sunrise or Sunset, Declination and GHA
Nautical almanacs and online sources publish annual details of celestial positions. Follow the steps below to use the Nautical Almanac:
Tabulated range is N58ş 08:45 & N60ş  09:02.
The difference in time is 08:45  09:02 = +00:17.
The
difference in latitude is N58ş  N60ş = N2ş = +120’.0.
The correction is time range ÷ latitude range x difference in
latitude, eg: (17 ÷ 120) x 90 = +12’.7. Time
of local sunrise = 08:45 + 12’.7 = 08:57’.7
08:58’
local sunrise – 10:03hr position east = 22:55 UT on 1/1/11
(0.2 ÷ 60’) x 55 = 0.18 so
declination = S22ş 58’.2 – 0’.18 = S22ş 58’.02
164ş 05’.3  149ş 05’.6 = 14ş 59’.7 (say 15ş ÷ 60) x 55 = 13.75ş = 13ş 45’
149ş 05’.6 + 13ş 45’ = GHA 162ş 50’.6
1. Scientific calculator method:
Example  find the amplitude for sunrise with a tabulated
Declination of 22ş 30’S
in Latitude of 33ş 45’S. Convert the degree/minutes to degrees/decimals
and enter the formula Sin Amp=Sin Dec ÷
Cos Lat. Enter
the quantities to calculate Sin Amp; covert this to Amp using the invert sine key pads. Calculator key pads differ, but a common system is shown below.
Sin 
22.50 
÷ 
Cos 
33.75 
= 
Inv 
Sin 
= 
27.4 
Next covert Amp to 360ş notation, East if rising, West if setting and named as the declination, in the case above, E 27.4ş S or 117.4ş T then compare with ships compass for error.
Follow the worked example A1 below, then try the questions A2 and A3.
2. Azimuths or amplitudes tables and form
After finding time of sunrise/sunset and declination from a current Nautical Almanac use nautical tables (Nories or Burtons) as per their instructions in the Explanation of the Tables. (click on link to see table extracts):
Follow the worked example A1 below, then try the questions A2 and A3.
3. The ABC tables and form.
After finding time of sunrise/sunset, declination and GHA from the current Nautical Almanac calculate the LHA. Then use nautical tables (Nories or Burtons) as their instructions in the Explanation of the Tables.
(click on link to see tables):
Follow the worked example B1 below, then try the questions B2 and B3.
Worked Examples
Required:
Time, approx lat & long, variation and ship’s bearing of sun on rising or setting
Nautical Almanac of the sun’s GHA and Declination
Nautical (Navigational) Tables – Nories, Burtons, etc
Amplitude tables (example only)
Traverse tables (example only)
At sunrise on 11^{th} January 2011 in latitude 45° 00’S
and longitude 75° 30’E
the sun was observed (half a diameter over the horizon) to bear 110° C
Find the time of sunrise, true amplitude and deviation if variation = 9° E
(Determine declination to nearest 1°)
From Almanac
Date…11/1/11.......…Heavenly body …Sun……Compass bearing…110° (C)
Latitude….45° 00’............ (N) (S) ....... Longitude………75° 30’..…(E) (W)
Average local time of rising (E) or setting (W)…..04h 27m …...
Time correction (if required
to < 1 degree)
Time difference from UT (E) or (+W) 5h 00m hours
Time difference from UT (E) or (+W) 0h 02m mins
05h
02m hours &
mins
UT (GMT) (04h 27m 05h 02m) Date 10/1/12 23h 25m hours & mins
UT tabulated Declination (hours) S21° 53.70m hours
d correction (minutes) + or  0.4 x 25’/60 = 0.17m mins
Heavenly body declination at chosen long. S21° 53.53m (N) or (S)
From amplitude or azimuth tables
Enter with declination and latitude (apply same name as declination)
Tabulated/calc. Amplitude (E) (W) E 32°S (N) (S)
or Azimuth (N) (S) S 58°E (E) (W)
(T) True Azimuth 90 +
E 32°S = 122°T (Amplitude tables)
180  S 58°E = 122°T (Azimuth tables)
(T) True bearing 122°T
W+ (V) Variation 9°E
(M) Magnetic 113°M
E (D) Deviation 3°E
(C) Compass 110°C
From calculator
Sin Amp=Sin Dec ÷
Cos Lat= 0.5272 Amp =
31.82 = E 31° 49’ S = 121.8°T
Question A2 using the Azimuth/Amplitude form or calculator:
At sunset on 12^{th} January 2011 in latitude 60° 00’N
and longitude 20° 25’W
the sun was observed (half a diameter over the horizon) to bear 327.5° C
Find the time of sunset, true amplitude and deviation if variation=10° W
(Determine declination to nearest 1°)
From Almanac
Date……………Heavenly body ……………Compass bearing…….…….(C)
Latitude………………..…(N) (S) Longitude……………………...…(E) (W)
Average local time of rising (E) or setting (W)……..……………...
Time correction (if required
to < 1 degree)
Time difference from UT (E) or (+W) ……..……………... hours
Time difference from UT (E) or (+W) _______________ mins
_______________ hours & mins
UT (GMT) (Local +/ Long) Date ________ _______________ hours & mins
UT tabulated declination (hours) ……..……………... hours
d correction (minutes) + or  _______________ mins
Heavenly body declination at chosen long. _______________ (N) or (S)
From amplitude/azimuth tables or calculator
Enter with declination and latitude (apply same name as declination)
Tabulated/calc. Amplitude (E) (W..................................(N) (S)
or
Azimuth (N) (S)..................................(E) (W)
(T) True Azimuth …………………
(T) True bearing………………..
W+ (V) Variation ..………………
(M) Magnetic ..………………
E (D) Deviation ..………………
(C) Compass ..………………
From calculator
Sin Amp=Sin Dec ÷
Cos Lat=
Question A3 using the Azimuth/Amplitude form or calculator:
At sunrise on 16^{th} November 2011 in latitude 35° 00’S
and longitude 155° 30’E
the sun was observed (half a diameter over the horizon) to bear 116° C
Find the time of sunrise, true amplitude and deviation if variation = 10° E
(Determine
declination to nearest 1°)
From Almanac
Date……………Heavenly body ……………Compass bearing…….…….(C)
Latitude………………..…(N) (S) Longitude……………………...…(E) (W)
Average local time of rising (E) or setting (W)……..……………...
Time correction (if required
to < 1 degree)
Time difference from UT (E) or (+W) ……..……………... hours
Time difference from UT (E) or (+W) _______________ mins
_______________ hours & mins
UT (GMT) (Local +/ Long) Date ________ _______________ hours & mins
UT tabulated declination (hours) ……..……………... hours
d correction (minutes) + or  _______________ mins
Heavenly body declination at chosen long. _______________ (N) or (S)
From amplitude/azimuth tables or calculator
Enter with declination and latitude (apply same name as declination)
Tabulated/calc. Amplitude (E) (W..................................(N) (S)
or
Azimuth (N) (S)..................................(E) (W)
(T) True Azimuth …………………
(T) True bearing………………..
W+ (V) Variation ..………………
(M) Magnetic ..………………
E (D) Deviation ..………………
(C) Compass ..………………
From calculator
Sin Amp=Sin Dec ÷
Cos Lat=
At sunrise on 11^{th} January 2011 in latitude 45° 00’S
and longitude 75° 30’E
the sun was observed (half a diameter over the horizon) to bear 110° C
Find the time of sunrise, true amplitude and deviation if variation = 9° E
(Determine declination to nearest 1°)
1 
Date Ship 
11/1/11 
Smt Ship mean time 
04h 27m 
HB Heavenly body 
Sun 



2 
DR Lat. CP 
45° 00’ 
N / S 
DR Long. CP 
75° 30’ 
E / W 


Variation Local 
9° E 
E / W 
Bearing of HB 
110°C 
şC 



3 
Convert SMT to UT (GMT) 
4 
Extract GHA (almanac)
& Calculate LHA 

Smt 
04h 
27m 
UT/GHA hour 
163° 
06’.0 

Zone
Time 
east +
west 
5h 
2m 
UT/GHA min
& sec increment 
6° 
15’.0 

(75 ÷ 15) 
(0.5 ÷ 15)x60 

UT 
23h 
25m 
UT/GHA 
169° 
21’.0 

UT
Date 
10/1/11 

DR
long +
E 
W 
75° 
30’.0 

Tab Inc & corrections 6°21’.3 Calc. (178°05.7163°06’.0) = (14° 59’.7 ÷ 60) =14’.99 x 25= 6°15’ 
LHA 
244° 
51’.3 

^{ } 

5 
Calculate Declination 



Dec. at
hour 
21° 53’.7 
N
/ S 

‘d’ minute correction Int. 0’.4 x 25/60 
0’.17 
+
or – as
table trends 

Dec. 
21° 53’.53 
N / S
‘d’ 



6 
Enter ABC tables 
10 
Compass error check 

LHA/Latitude 
A= 0.47 
N / S name opp. to Lat except LHAs 90270 
T 
123°T 


7 
LHA/Dec 
B= 0.45 
N / S name
same as Dec 
V 
09°E 


8 
A+orB 
0.47 + 0.45 
N / S name
same as greatest 
M 
114°M 


A+orB /lat 
C= 0.92 

D 
4°E 


Quadrant
notation = 
S 57° E 
W if LHA 000180 E if LHA 180360 
C 
110°C 




9 
Convert to 360şT notation 




123°T 

Question B2 using the ABC tables:
At sunset on 12^{th} January 2011 in latitude 60° 00’N
and longitude 20° 25’W
the sun was observed (half a diameter over the horizon) to bear 327.5° C
Find the time of sunset, true amplitude and deviation if variation=10° W
(Determine declination to nearest 1°)
1 
Date Ship 

Smt Ship mean time 

HB Heavenly body 




2 
DR Lat. CP 

N / S 
DR Long. CP 

E / W 


Variation Local 

E / W 
Bearing of HB 

şC 



3 
Convert SMT to UTC (GMT) 
4 
Extract GHA (almanac)
& Calculate LHA 

Smt 


UT/GHA hour 



Zone
Time 
east +
west 


UT/GHA min
& sec 



UT 


UT/GHA 



UT
Date 


DR
long +
E 
W 




LHA 



^{ } 

5 
Calculate Declination 



Dec. at
hour 

N
/ S 

‘d’ minute correction 

+
or – as
table trends 

Dec. 

N / S ‘d’ 



6 
Enter ABC tables 
10 
Compass error check 

LHA/Latitude 
A= 
N / S name
opp. to Lat except LHAs 90270 
T 



7 
LHA/Dec 
B= 
N / S name
same as Dec 
V 



8 
A+orB 

N / S name
same as greatest 
M 



A+orB /lat 
C= 

D 



Quadrant
notation = 
N / S E / W 
W if
LHA 000180 E if
LHA 180360 
C 





9 
Convert to 360şT notation 






Question B3 using the ABC tables:
At sunrise on 16^{th} November 2011 in latitude 35° 00’S
and longitude 155° 30’E
the sun was observed (half a diameter over the horizon) to bear 116° C
Find the time of sunrise, true amplitude and deviation if variation = 10° E
(Determine
declination to nearest 1°)
1 
Date Ship 

Smt Ship mean time 

HB Heavenly body 




2 
DR Lat. CP 

N / S 
DR Long. CP 

E / W 


Variation Local 

E / W 
Bearing of HB 

şC 



3 
Convert SMT to UT (GMT) 
4 
Extract GHA (almanac)
& Calculate LHA 

Smt 


UT/GHA hour 



Zone
Time 
east +
west 


UT/GHA min
& sec 



UT 


UT/GHA 



UT
Date 


DR
long +
E 
W 




LHA 



^{ } 

5 
Calculate Declination 



Dec. at
hour 

N
/ S 

‘d’ minute correction 

+
or – as
table trends 

Dec. 

N / S ‘d’ 



6 
Enter ABC tables 
10 
Compass error check 

LHA/Latitude 
A= 
N / S name
opp. to Lat except LHAs 90270 
T 



7 
LHA/Dec 
B= 
N / S name
same as Dec 
V 



8 
A+orB 

N / S name
same as greatest 
M 



A+orB /lat 
C= 

D 



Quadrant
notation = 
N / S E / W 
W if
LHA 000180 E if
LHA 180360 
C 





9 
Convert to 360şT notation 






Charts for ocean areas are relatively blank. To remove the need to buy many blank charts to record a position, mathematical calculation rather than geometrical plotting is adopted using trigonometry and/or traverse tables.
Trigonometry
Trigonometry defines and provides formulas regarding
relationships between sides and an angle within a right angle triangle. These
definitions include:
The opposite side is opposite angle θ.
The adjacent side is adjacent (next to) to angle θ.
The hypotenuse is the longest of the sides.
These trigonometric formulas include:
Sine (sin) of θ 
= 
Opposite 
= 
b 
Hypotenuse 
a 






Cosine (cos) of θ 
= 
Adjacent 
= 
c 
Hypotenuse 
a 






Tangent (tan) of θ 
= 
Opposite 
= 
b 
Adjacent 
c 
These formulas can be remembered by the mnemonic of:
SOHCAHTOA
The sine, tangent or cosine of the angle θ can be found
tabulated in published books of tables, or can be found using the function on a
scientific calculator.
Also the reciprocal trigonometrically functions and formulas could be used. When using a calculator, the shift key is used to select reciprocal functions.
1 
θ 
= 
Secant (sec) θ 
= 
Hypotenuse 
= 
a 
Sin 
Adjacent 
c 









1 
θ 
= 
Cosecant (cosec) θ 
= 
Hypotenuse 
= 
a 
Cos 
Opposite 
b 









1 
θ 
= 
Cotangent (cot) θ 
= 
Adjacent 
= 
c 
Tan 
Opposite 
b 
If any two of the components of the formula is known then the last can be calculated using substitution if required (rearranging the order of the formula).
Finding Difference in Latitude and Departure
using trigonometry
This solution assumes that the earth surface is a flat plane, in other words it is a plane sailing solution that is accurate limited to a few hundred miles. A right angle triangle and (θ angle) ABC is used for solutions:
We start from position B steering 301şT and log 348 nm travelled along
line a
(the hypotenuse) to reach a more northerly position of C. This course is
the angle between ABC (301ş  270ş) = 31ş subtended north of the line c
(the adjacent). The line b (the opposite) is equivalent to
our change in latitude.
Sine (sin) of θ 
= 
Opposite 
= 
b 
Hypotenuse 
a 
We find the length of the opposite using the formula and sine function of a scientific calculator. Sine 31ş = 0.515.
Sine = Opposite ÷ Hypotenuse or Hypotenuse x Sine = Opposite Therefore 348 nm x 0.515 = 179.23 nm, entered on a calculator as:
348 
x 
Sin 
31 
= 
179.23 
As our initial latitude at B was 33ş 00’.0 S then our difference in latitude having steamed north to C will be 179.23 nm ÷ 60 = North 2.9872ş = 2ş 59’
Our new latitude C is 33ş 00’.0 S  2ş 59’ = 30ş 01’.0 S.
As our initial longitude at B is 160ş 00’.0 E the departure (our East/West shift in miles) will be the length of the adjacent (BA).
Cosine (cos) of θ 
= 
Adjacent 
= 
c 
Hypotenuse 
a 
We find the length of the adjacent using the formula and cosine function of a scientific calculator. Cos 31ş = 0.978
Cos = Adjacent ÷ Hypotenuse or Adjacent = Hypotenuse x Cosine
Therefore 348 nm x 0.978 = 298 nm, entered on a calculator as:
348 
x 
cos 
= 
298 nm 
We have travelled 298 nm in a westerly direction (our departure). How this relates to our change in degrees of longitude depends on the latitudes we were travelling in. Remember, longitudes are meridians all meeting at a point at the poles while having a separation of 60 nm per degree at the equator. The varying equivalence of departure and difference in longitude dependant on latitude is best found by using the traverse tables, as explained below.
Distance and courses using Traverse tables
Trigonometric
solutions are tabulated in the traverse tables using the definitions below:
Rhumb line  A course line of constant true bearing. It crosses all meridians at the same angle and appears on a Mercator chart as a straight course line. It is adequate for short distances of less than a few hundred miles.
Distance A separation of between one point and another expressed in units of length, for navigational purposes these being divisions of a nautical mile (equal to 1852 metres).
Departure The easting or westing of a vessel’s progress measured in nautical miles. Note this is only equivalent to Difference Longitude on the equator.
Difference in latitude The angular distance from one parallel of latitude to another, being approximately 60 nm per degree or 1 nm per minute.
Difference in
longitude  The angular distance from one meridian of longitude to another,
being approximately between 60 nm per degree at the equator to 0 nm at the
poles.
Traverse course is tabulated from North or South towards East or West, i.e.:
N 68 şE = 068ş
N 68 şW = 292ş S 68 şE = 112ş
S 68 şW = 248ş
Enter or extract your required course on the page with the
quadrant notation of your course. Read the table from the top down or the
bottom up dependant on the 360ş
notation of your course. See example below:
Example C1.
At 09:00 in latitude 40° 00’S and longitude 150° 00’E a ship changes course to 112° T and steams at 8.4 knots until 19:00. Assuming no drift and steady course and speed, calculate the 19:00 position.
Finding Distance and Quadrant notation to enter tables.
19:00
09:00
10 hours x 8.4 = Distance 84
nm steamed. Course 112° = S 68° E
Finding Difference in Latitude and Departure
From Page 68ş reading
from bottom up D. Lat = 31’.5 Dep = 77.9 miles
Old Latitude 40° 00’.0 S
Change in Latitude +31’.5 S
New Latitude 40° 31’.5
S
Finding Difference in Longitude at Mid Latitude
Mid Lat = (40° 31’.5  40° 00’.0) ÷ 2 + old lat = 40° 15’S
From Page top 40ş use D’Long
and Dep
columns for nearest to Dep
77.9
Find
D’Long
101’ Dep 77.4
D’Long 102’
Dep
78.1
Interpolate for D’Long
correction for 77.9.
(D’Long increases 101’ to 102’ = 1’ between 77’.4 and
78’.1)
78’.1
77’.9
Interp =
77’.4 77’.4
0’.7 0’.5
0’.7 Dep increase is equivalent to 1’.0 D’Long increase.
0’.5 Dep
increase is equivalent to
0’.5 ÷ 0’.7 = +
0’.71
D long
101’.0
Corr. + 0’.71
101’.71 ÷ 60 = 1° 41.71’E
Old Longitude 150° 00’.00E
Change in longitude +1° 41’.71E
New
longitude 151° 41’.71E
Question C2.
On 1/2/11 at 12:00 in latitude 25° 00’S and longitude 100° 00’E a ship changes course to 220° T and steams at 14 knots until noon on 2/2/11.
Assuming no drift and steady course and speed, calculate the last position.
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Question C3.
Work up the change in position of the yacht that beats into the wind from position 31° 30’S and longitude 153° 30’E.
Course 
Distance run 
Diff of Lat 
Departure 
Notes 



North 
South 
East 
West 

035°T 
20 nm 





260°T 
35 nm 





125°T 
8 nm 





290°T 
15 nm 

















Answer A2 using the Azimuth/Amplitude form or
calculator:
At sunset on 12^{th} January 2011 in latitude 60° 00’N
and longitude 20° 25’W
the sun was observed (half a diameter over the horizon) to bear 230° C
Find the time of sunset, true amplitude and deviation if variation= 10° W
(Determine declination to nearest 1°)
Date…12/1/11.......…Heavenly body …Sun……Compass bearing…230° (C)
Latitude….60° 00’............ (N) (S) ....... Longitude………20° 25’..…(E) (W)
Average local time of rising (E) or setting (W)…..15h 22m …...
Time correction (if required
to < 1 degree)
Time difference from UT (E) or (+W) +1h 20m hours
Time difference from UT (E) or (+W) +0h 1.7m mins
UT (GMT) (15h 22m + 1h 22m) 12/1/12 16h 44m hrs & mins
UT tabulated Declination (hours) S21° 37.5m hours
d correction (minutes) + or  (0.4 x 44/60) = 0.3m mins
Heavenly body declination at chosen long. S21° 37.2m (N) or (S)
From amplitude or azimuth tables
Enter with declination and latitude (apply same name as declination)
Tabulated/calc. Amplitude (E) (W) 47.1° (N) (S)
or Azimuth (N) (S) 42.9° (E) (W)
(T) True
Azimuth 270  W 47.1° S = 222.9° (Amplitude tables)
180 + S 42.9° W = 222.9° (Azimuth tables)