International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 2
Number 3
September 2008
271
Some New Dimensions in Sextant-Based
Celestial Navigation Aspects of Position
Solution Reliability with Multiple Sights
K.H. Zevering
Brisbane, Queensland, Australia
ABSTRACT: The traditional approach relying on sight reduction tables, a non-programmatic location of the
position fix and an inadequate allowance for observation errors is still widely pursued and advocated. In the
late 1970s the programmatic Least Squares method (LSQ) was introduced which determines a random error
fix (Fix
Q
) for any multiple sights combination. B.D Yallop & C.Y Hohenkerk (1985) expanded LSQ to
incorporate the computation of the random error margin of a fix. Several marketed PDA-based programs
apply LSQ, but none have fully incorporated the random error margin as a guide for the navigator. All existing
LSQ applications have two drawbacks. One is, all observation error is attributed to random sources, whereas
the possibility of systematic error has in fact a long theoretical and practical background in celestial
navigation. Systematic error represents a bias in statistical random error theory and can and should be allowed
for. A major drawback is that existing LSQ program applications incorporate the running fix technique (RFT)
traditionally applied in coastal navigation. It has no general validity in celestial navigation. The position circle
of an earlier celestial sight can only be mathematically correctly transferred when its Geometric Position (GP)
is transferred for the run data. A final aspect of reliability is the strategy adopted at the sight planning stage. At
least during twilight observations, navigators should aim at getting three or four sights with a total azimuth
angle >180
o
, with three successive subsights on each body. In such configurations Fix
Q
and Fix
S
will be
relatively close together, generally obviating the need to process the sights for possible systematic error.
1 THE TRADITIONAL VERSUS
PROGRAMMATIC APPROACH
The calculator with trig functions makes the use of
sight reduction (SR) tables redundant and avoids
precession and nutation problems with AP 3270
Vol 1. It does not overcome the limitations of the
non-programmatic approach. Dedicated navigation
calculators or PDAs would be a solution provided
they run theoretically correct programs. Better
known programs such as Starpilot, Astronavigation,
Celestnav incorporate the traditional running fix
technique (RFT) which is not generally applicable
and cannot or cannot meaningfully compute the
confidence ellipse (error margin) of the
programmatic LSQ fix.
Complete power failure is often cited to support
the use of SR tables and the non-programmatic
approach. On today’s yachts such a contingency is
for many reasons unrealistic and it would jeopardize
nautical navigation in many ways. A better
contingency plan is a Pocket PC running, say,
Windows Mobile/Excel which can support the full
programmatic approach in one or several
spreadsheets for any number of sights.
The traditional approach relies on a number of
precepts which are not borne out by modern methods
and analysis:
1 The fix is presumed to lie somewhere in the
middle or centre of a 'cocked hat' n-polygon
(n ≥ 3).
272
2 The position of the vessel is put at the “greatest
disadvantage”, but it means on a vertex or
position line of the cocked hat n-polygon
closest to a known danger. The error margin
implied by the cocked hat area cannot be
quantified.
3 Traditional RFT is used to account for the transfer
of an earlier sight’s position circle for the run
between sights.
None of the above precepts have general validity.
With i. we mean specifically that the fix is generally
inside the cocked hat but in irregular n-polygons in
an eccentric position (e.g see Fig. 6). A fix with
more than three sights may even lie outside the
cocked hat n-polygon. One notion propounded by
G. Huxtable
1
and apparently taught in nav classes in
England is that the chance of the celestial fix lying
outside the 3-polygon is 75%, outside the 4-polygon
87½% etc. The notion derives from random
errors in compass bearings in coastal navigation
(see Fig. 1)
2.
With random + and - deviations, the
terrestrial fix with three sights has a 75% chance of
physically lying in peripheral cocked hat areas
(Fig. 1, areas 1 to 6). There is only a 25% of the fix
lying in cocked hat areas abc and def. The terrestrial
construction is not possible unless (unknown) values
are assigned to the deviations.
Fig. 1. Errors in terrestrial compass bearings versus celestial
observation errors
The + and - deviations in terrestrial bearings are
equated with the chance of getting toward (+) and
away (-) intercept (p) combinations in celestial
navigation. The idea pursued by Huxtable is that the
celestial fix will only lie inside the triangle when the
p-values are + + + or - - -, which is simply incorrect.
The p-values are only corrections for inaccurate DR
position. With three sights the random error fix will
always lie inside the triangle and the observation
errors e1, e2 and e3 are measurable. There is a
theoretical 25% chance of getting a consistent
3-polygon (e
i
= +++, or ---). Based on the azimuth
distributions of stars listed in AP 3270 Vol. 1 this
chance is more like 40%.
2 LSQ PECULIARITIES
The programmatic approach is based on LSQ
(least squares method). The version demonstrated in
B.D. Yallop and C.Y. Hohenkerk (1986) is applied
to non-simultaneous sights and uses a RFT-based
subroutine. We refer to this version as LSQ*. As
RFT is not generally valid as a transfer technique,
also LSQ* is not generally valid. For simultaneous
sights the method is simply LSQ. Both programs
compute a random error fix and its random error
margin.
Yallop-Hohenkerk (1986, xx) assert that “it is
possible to start from any position on the Earth, and,
provided that
λ
f
(i.e Long
DR
) is kept in the range -
180
o
to +180
o
and
ϕ
f
(i.e Lat
DR
) in the range -90
o
to
+90
o
, the position solution in most cases will begin
to converge after a few iterations”. A position
solution is indeed in principle independent from the
initial assumption but different DR assumptions are
necessary to force LSQ into finding the two
alternative position fixes resulting from two sights
with intersecting position circles. This may be
demonstrated with a numerical example
3
. For two
simultaneous sights the two alternative position fixes
can be found by applying either a double sight
method like K-Z
4
or LSQ. Results with K-Z are:
Q = 0.387623; R = 0.277141; S = 1.196021;
Lat
1
= 49.8408; Lat
2
= -6.6652
MD1,1a = 79.7248; MD1,1b = 15.3635; MD2,1a =
65.7485; MD2,1b = 29.3398
Long
1
= GHA
S
+ MD1,1a = GHA
M
- MD1,1b =3.9715 W;
Long
2
= GHA
S
+ MD2,1a = GHA
M
- MD2,1b = 10.0048 E;
Lat
1
~Long
1
= 49.8408 N/3.9715 W (LHA
S
280.2752 ~ Zn
S
85.4518;
LHA
M
15.3635 ~ 205.3902); Lat
2
~Long
2
= 6.6652
S/10.0048 E (LHA
S
294.2515 ~ Zn
S
67.4722; LHA
M
29.3398 ~
Zn
M
307.5339).
Note: SinLat
1,2
= (Q ± R
½
)/S; MD = meridian difference.
The same coordinates are found with LSQ using
an appropriate initial position in each instance:
273
Northern Hemisph.:
Southern Hemisph.:
DR
49
o
50'.0 N/4
o
20'.0 W
6
o
42'.0 S/10
o
30'.0 E
1
st
iteration
1
st
iteration
Sun
Moon
Sun
Zn
85.1796
204.8322
67.3309
p
13'.99
-6'.35
-26'.44
3
rd
iteration
3
rd
iteration
Zn
85.4518
205.3902
67.4722
e
-0'.00
0'.00
-0'.00
Fix
Q
49.8408 N/3.9715 W
6.6652 S/10.0048 E
3 THE GENERAL EQUATION OF A POSITION
CIRCLE
A position circle is represented on a Mercator chart
by the general equation. To plot a relevant segment
of it on the chart it is necessary to know the
coordinates of its GP (GHA and Dec) and its radius
(90
o
– H
o
). To transfer the position circle of an
earlier sight for a run it is necessary to determine the
transferred GP in terms of its new coordinates.
To quote the ANM on this point: “If the observer
is in a ship and there is
a run between sights, the
first position
circle
must
be transferred for the run.
Fig. 2. Plotting with the general equation of a sight-run-sight
case
The general equation
6
is: 2Cos(GHA+x) = e
y
[SinH
o
(1/CosDec)
- TanDec] + (1/e
y
)[SinH
o
(1/CosDec) + TanDec, in which x and
y are measures of longitude and latitude. In terms of its
constants the equation may be rewritten as: ae
2y
- be
y
+ c = 0;
a = SinH
o
/CosDec-TanDec; b = -2cos(MD); c = SinH
o
/CosDec
+ TanDec, so that e
y+
= [-b + (b
2
- 4ac)
½
]/2a = Tan (45 + ½ϕ)
and e
y-
= [-b - (b
2
- 4ac)
½
]/2a = Tan (45 + ½ϕ).
This can be done by transferring the
geographical position and then drawing the circle.”
5
The relevant segments of the position circles and
their point(s) of intersection can only be drawn on a
(small-scale) chart with a pair of compasses if their
altitudes are large and radii (zenith distances)
consequently small. In the general case when zenith
distances are very large such a construction must use
the general equation, which was circumvented in the
pre-electronic era by applying traditional RFT
known from coastal navigation.
4 THE TRANSFER ISSUE
With traditional RFT the transferred position line is
supposed to represent a position circle segment. An
article of faith for traditional RFT supporters is that
the transferred position 'backward-projected' for the
run data is postulated to lie on the original position
circle. This concept derives directly from coastal
navigation. Thus in Fig. 4a, the position J
1
backward-projected from Fix'
Q
is supposed to lie on
the original position circle of the Moon.
This is nonetheless an untenable proposition for
two main arguments. One is that coastal or terrestrial
RFT on which celestial RFT is patterned is itself in
principle an application of GD-UT transfer (Fig. 3).
By nominating longitude intervals for x the
corresponding latitude values for ϕ can be computed
and plotted as in
Fig. 2 for two sights where the
GHA, Dec and H
o
are for convenience simplified
as 0
o
, 0
o
, 30
o
and 45
o
, 1
o
, 45
o
respectively. There is a
run due north of 1
o
between these sights, so that the
transferred GP becomes GHA* = 0
o
, Dec* = 1
o
. In
Fig. 2, the longitude scale is fixed but the latitude
scale is derived from the meridional parts formula.
The programmatic fix in this case may be obtained
without plotting by applying K-Z.
The other argument is that celestial RFT cannot
specify the transferred position circle in terms of its
radius 90
o
- H
o
and relocated GP.
4.1 The terrestrial transfer analogy
Points A and B in Fig-3 represent landmarks with
known height and also the GPs of two celestial
bodies with large altitudes. Assume in the terrestrial
case the distance to A and to B is determined by
vertical sextant angle; no bearings are used.
The relevant position fix on the chart is at F
where the transferred position circle with radius r
A
from A' (PC*
1
) and the last sight’s position circle
with radius r
B
from B (PC
2
) intersect. Had only the
274
bearings been used the fix would also have been in
F, assuming the celestial azimuths could be observed
accurately. In the celestial case when r
A
and r
B
are
short zenith distances, DF virtually equals AA' and
the angle at D the course (α). In the terrestrial case a
running fix as at F can therefore be found in all
instances by applying the GD-UT transfer principle
if r
A
and r
B
are known. In the celestial case, F is also
found in all instances with GD-UT by transferring
the 1
st
sight’s GP for the run data
7
and plotting the
relevant sections of PC*
1
and PC
2
(with the general
equation).
Fig. 3. The GD-UT principle of terrestrial RFT
The celestial case generally involves large zenith
distances (huge radii r
A
and r
B
). In this general case
the transferred position circle’s locus can no longer
be found reliably by assuming that it will pass
through points on the original position circle
transferred for the run, like J to J*. If J is the initial
(assumed) position on PC
1
it can no longer be
presumed that J* will lie on PC*
1
and vice versa.
But this is exactly what happens with celestial RFT:
the parallel ruler construction shifts JJ* so that J*
comes to lie on PC
2
at F. The nice warm feeling RFT
supporters get with this construction is that D will lie
on PC
1
. But this self-evidence is entirely caused by
the RFT construction gimmick.
4.2 The specification of the position circle
transferred with RFT/LSQ*
When zenith distances are large the plane-geometric
properties of the terrestrial analogy no longer apply
and it is impossible to correctly specify a transferred
position circle like PC*
1
in Fig. 3 as the locus of
points like J* and F whose backward-projected
positions are postulated to lie on PC
1
at J and D.
This can be demonstrated in different ways
8
but here
we choose the Moon-Run-Sun example found in the
ANM
9
(see Fig. 4).
The solution for the SH is forced by assuming a
DR at say 59.4000 S/85.1000 E. The results with
LSQ* are:
Northern Hemisphere:
Southern Hemisphere:
Moon
Sun
Moon
Sun
LHA
X
Y
TanA
A
Z
Zn
e
353.3483
-0.9481
0.1074
-0.1132
-6.4609
173.5391
173.5391
0.0000
295.3366
-0.2711
0.9001
-3.3202
-73.2386
106.7614
106.7614
-0.0000
92.8268
-0.2310
-0.9258
4.0077
75.9896
104.0104
255.9896
-0.0000
395.0077
0.7465
-0.5713
-0.7654
-37.4288
37.4288
322.5712
0.0000
DR
(Sun)
50.3496 N/14.0474 W
59.4000 S/85.1000 E
Fix'
Q
(4th it)
50.49178 N/13.8418 W
59.0949 S/85.8293 E
When GD-UT+K-Z is applied, the results are:
Northern Hemisphere:
Southern Hemisphere
Moon
Sun
Moon
Sun
Fix
Q
50.5117 N/13.8323 W
59.3735 W/85.1076 E
LHA
353.6209
295.3460
92.5608
34.2859
Zn
173.7959
106.7762
256.3396
323.3593
The plot for the NH is shown in Fig 4a. The
distance between Fix'
Q
and Fix
Q
for the NH is 1'.25,
but for the SH it is 27'.75. With RFT/LSQ*, the
backward-projected positions respectively from Fix
1
(NH) and from Fix
2
(SH) are postulated to lie on the
original position circle at J
1
(NH) and J
2
(SH) (see
sketch Fig. 4b). If this were a correct proposition, the
position circle passing through J
1
and J
2
should ave a
zenith distance equal to 90° - H
o
= 72
o
.5883.
But the zenith distance of the position circle
passing through these two points can only be
72
o
.8551
10
.
This apparent contraction in zenith distance is not
the only problem because none of the great-circle
segments in Fig. 4b indicated with broken lines
or the angles they make can be evaluated. Also, the
Zn-values of the Moon (earlier sight) shown above
for the final LSQ* iteration are the azimuth bearings
from the backward-projected positions J
1
and J
2
and
not from Fix
1
and Fix
2
.
LSQ* cannot compute the actual azimuth bearing
from the fix on the transferred GP. Even if this were
possible, the properties (coordinates of the GP and
zenith distance) of the earlier sight’s position circle
transferred for the run data cannot be computed.
We call this the 'Achilles heel' of LSQ*. The only
correct way to transfer the GP of an earlier sight is
GD-UT.
275
Fig. 4a. Plot of LSQ* transfer of an earlier sight (The Moon-
-Run-Sun case in the ANM)
Fig. 4b. Sketch of the implications of the LSQ* transfer as
in Fig. 4a
5 RANDOM AND SYSTEMATIC ERRORS
Error intercepts (e) indicate a possible combination
of random and systematic error. Both types of error
will displace a position line parallel to itself in a
certain direction along its Zn axis. Systematic error
(e
S
) is caused by instrument
error (e.g I.E, Dip) and
is equal in magnitude and sign for each sight in the
collection. LSQ computes a random error fix (Fix
Q
)
and systematic error statistically constitutes bias.
Maximum permissible systematic error (e
S
max) can
be removed as a correction to the altitudes. LSQ will
then determine a fix (Fix
S
) corrected for e
S
max.
This approach in fact reconciles conventional LSQ
concerned with random error and traditional bisector
constructions which allow for maximum systematic
error in consistent 3-polygons.
For practical navigational reasons and also lack of
space we will confine the discussion to 3- and 4-
polygons. The main factor affecting the difference in
location (distance d) between Fix
Q
and Fix
S
is 'total
azimuth angle' (TAZ), which is the smallest angle or
arc enclosing all azimuth (Zn) bearings. A statistic š
expressing the difference in location is the ratio of d
to average error intercept distance: š = d ÷ Σ|e
i
|/n.
The consistency of a collection of sights is seen from
the signs of the e’s found with LSQ: it is consistent
if they all have the same sign; if one or more signs
are different it is inconsistent (also see Fig. 1).
Navigators are to take at least three subsights
on the same body for screening aberrant sightings
in the usual manner. The programmatic approach
makes it possible to process all included subsights
independently, rather than the averages of the
subsights’ GMT and H
s
data. The various ramifications
of such an approach cannot be discussed here.
During the short twilight periods navigators
should concentrate on the best star triads indicated in
AP 3270 Vol 1 which fall in group II. Four sights
should be planned to fall in groups IV and V. As
configurations in group V are preponderant, the
effect of possible systematic error on the location of
the LSQ fix is in this way minimized. Collections I,
II and III can be predicted from the approximate
azimuths of the sights known beforehand, but this is
not possible for IV and V. When three sights fall in
group I, a fourth sight may be taken so that the
overall combination will have the favourable
azimuth distribution (i.e TAZ > 180
o
). Best results
are generally obtained from two intersecting pairs in
which the sights in each pair are widely separated in
azimuth.
3-polygon:
4-polygon:
TAZ > 180
o
Properties
TAZ < 180
o
Inconsistent
I
TAZ > 180
o
Consistent
II
TAZ < 180
o
Inconsistent
III
Consistent
IV
Inconsistent
V
1
2
3
4
5
% chanc
e
Fix
Q
Fix
S
š
š range
60
inside
outside
large
1.5 to 3.0
40
inside
inside
small
around 0.2
33
in- or outside
in- or outside
large
0.6 to 2.7
12? (P)
inside
inside
small
around 0.6
87? (P)
in- or outside
in- or outside
smallest to negligible
0.0 to 0.16
Notes: Internal vertex angles (IVA) of an n-polygon are < 180
o
. Item 1 indicates the chance of getting certain combinations in
twilight observations; if 4-polygons with TAZ > 180
o
are planned, the binomial distribution is as shown.
276
5.1 Procedures for obtaining e
S
max
Sketched in Fig. 5 are vertices 1 and 2 of an
n-polygon and adjacent IVAs A and B. The
perpendicular h on side AB is computed with the
known values of the two IVAs and the known
distance (d) between the two vertices. The distance
d equals |d’Lat|/Cos(Zn ± 90), where d’Lat derives
from the vertex latitudes computed with K-Z. The
perpendicular h indicates |e
S
max| if it is the smallest
among similar perpendiculars dropped on the other
sides: H'
o
= H
o
± h
min
/60, where H'
o
is a sight’s
altitude corrected for e
S
max; the sign of h
min
is
opposite to the sign of the corresponding error
intercept (e). An IVA is consistent (C) when the
error intercepts of its sides have the same sign;
otherwise it is inconsistent (IC). Two adjacent IVAs
may form one of the following sequences: CC,
IC-IC, C-IC (a and b) and IC-C (a and b). The IVA
angles are computed as:
C : |Zn
A
-Zn
B
| < 180 IVA = 180 - |Zn
A
-Zn
B
|;
|Zn
A
-Zn
B
| > 180 IVA = |Zn
A
-Zn
B
| - 180, and
IC: |Zn
A
-Zn
B
| < 180 IVA = |Zn
A
-Zn
B
|;
|Zn
A
-Zn
B
| > 180 IVA = 360 - |Zn
A
-Zn
B
|.
Needed in the computation of h are the ½IVAs; if
IVA = IC, ½IVA = ½(180 – IVA). If the angles of
two adjacent IVAs are indicated as respectively A
and B, then A' and B' indicate their supplements.
The formulas for computing the plane-geometric
perpendicular h for different IVA sequences are
shown in Fig. 5.
Fig. 5. Determining the h-values for consistent cocked hat
3-polygons and for n-polygons (n ≥ 3)
With group I sights, the perpendicular h from the
intersection of the bisector of the consistent (apex)
IVA and this IVA’s opposite side determines e
S
max:
The approximate method is sufficient and has the
advantage that the distance d need not be computed.
From the results for the Sun1-Sun2-Sun3 case
shown in Fig 6 |Zn
1
-Zn
3
| > 180, thus IVA
apex
=
= 251.7313 – 180 = 71.7313; e
1
= 0'.75 and e
3
=
= 0'.86. These values substituted in the approximate
method give approximate |e
S
max| = 1'.36.
5.2 Application of e
S
max to three and four sights
The navigator is in practice faced with two
situations. One is the need for updating the DR
position for which Fix
Q
is generally sufficient. The
other is the need for considering the vessel’s position
in the presence of a known danger. Depending on
the consistency of the sights, possible systematic
error may significantly affect the location of the fix
but not necessarily the error margin. A third
situation is simply that the navigator is prepared to
speculate on the preponderance of either type of
observation error and also to consider error margins
at less than 95% probability.
Removing e
S
max from a consistent 3-polygon
(group II) will virtually eliminate all residual random
error and Fix
S
is at the point of intersection of the
bisectors. As Fix
Q
and Fix
S
are relatively close, in
both practical situations mentioned Fix
Q
and its error
margin will suffice. Space does not permit to
demonstrate this, but with four inconsistent sights
(group V) the two fixes will be very close and their
respective error margins remain practically the same.
Again, Fix
Q
will suffice in all circumstances.
In Fig 6. are worked examples of an inconsistent
3-polygon (group I)
11
and consistent 4-polygon
(group IV)
12
. The inconsistent 3-polygon is
avoidable in twilight observations but running fixes
on the Sun remain most important and will often
form an inconsistent collection. The chance of
getting four consistent star sights, let alone larger
collections of stars as dished up in contrived
examples in the literature
13
is remote. In the three
Sun sights case, the DR position is wildly out and
indicates that the danger has been cleared. If Fix
Q
and its error margin is accepted, the vessel might not
change course.
277
Fig. 6. Cases falling respectively in Group I and group IV
278
Fix
S
with three inconsistent sights always lies
outside the 3-polygon. It is prudent in this case to
accept Fix
S
and change course.
Assuming long runs between sights as in this case
is common in the literature but a habit which should
not be emulated in practice. It compounds the
(unknown) inaccuracies of the DR record. As the run
data are used to transfer the position circles of earlier
sights, a large discrepancy between the fix and
the run record as in this case also tends to invalidate
the fix. In other respects the DR position should be
completely ignored as irrelevant. LSQ will
determine the same fix regardless of a wide range of
assumed (DR) positions.
In the consistent 4-polygon case, noted is first the
eccentric location of the fix. The distance between
the two fixes is significant. Allowing for e
S
max
substantially reduces the error margin, but in the
presence of a known danger, the prudent navigator
would adopt Fix
Q
.
REFERENCES
Admiralty Navigation Manual, 1938, Volumes II and III.
Bennett G.G., Celestial Navigator, 2003-2007;
Blewitt M., Celestial Navigation for Yachtsmen, 1975, 1996;
Cunliffe T., Celestial Navigation, 2001;
Yallop B.D. & Hohenkerk C.Y., Compact Data for Navigations
and Astronomy, 1986-1990.
1
See G. Huxtable, quoted in K.H. Zevering – The Navigator’s Newsletter (Foundation for the Promotion of the Art of Navigation), no. 88, p. 11-12.
2
See ANM, Vol. III, p. 165-166.
3
Data from M. Blewitt (1975, p. 30 and p. 33). GHA, Dec and H
o
of Sun 284.2467, 18.4050 and 20.5150; Moon 19.3350, 15.4900 and 53.4550.
4
For the K-Z algebraic double sight position solution method see K. Herman Zevering – “The K-
Z Position Solution For The Double Sight”, European Journal of
Navigation, Vol. 1, No. 3 (and 4), 2003, p. 43-46.
5
ANM, Vol. II, p. 43.
6
See ANM, Vol. III, p. 36, 39-40.
7
With GD-UT, the coordinates of the GP may be transferred with the rhumbline equations and are indicated as GHA* and Dec*. With the ve
ssel’s movement,
the GP of the sight taken at A transfers to A' (see Fig. 3). AA'
is strictly not a rhumbline but part of a great circle. The angle of cut of this great circle with
the meridians through respectively A and A' is not constant as the mercatorial bearing α at A suggests. Applying the convergency (c), for α could be substit
uted
α' = α ± ½|c|, where c = (d/60)SinαTan(MeanDec). In Δ PAA', Dec* of the transferred GP at A' may be determined with Lat
A
, α
' and d/60. Angle (arc) APA'
may be determined with Lat
A
, Dec* and d/60, from which follows GHA*. It can be shown that even wi
th very large displacements these adjustments have
a negligible effect compared to the rhumbline definitions of Dec* and GHA*.
8
See the discussion in Forum, The Journal of Navigation (2006), 59, p. 521-529.
9
ANM, Vol II, p 191-195; only the fix for the NH is worked out, with cosine-
haversine and the traverse table. The GMT data for Moon and Sun are: 05 59 45 and
08 40 10. The data for GHA, Dec and H
o
are: Moon 7.9783, -22.0400, 17.4117; Sun 309.1783, 5.2050, 19.9450.
The assumed DR position in the NH (Moon
sight) is 50.1667 N/50.8333 W; course 70
o
, speed 12 kn/hr.
10
Cosβ
1
= (SinH
o
– SinLat
J1
SinDec)/CosLat
J1
CosDec and Cosβ
2
= (SinH
o
- SinLat
J2
SinDec)/CosLat
J2
CosDec. The angles β
1
and β
2
are known.
SinHo =
= [c1c2(a1+a2)+b1c2+b2c1]/(c1+c2), where a1 = Cosβ
1
; a2 = Cosβ
2
; b1 = Sin Lat
J1
SinDec; c1 = CosLat
J1
CosDec; b2 = Sin Lat
J2
SinDec; c2 = CosLat
J2
CosDec. Sin Ho = [c1c2(a1+a2)+b1c2+b2c1]/(c1+c2).
11
Data from G.G. Bennett (2003, p. 164).
12
Data from M. Blewitt (1975, p 39)
13
E.g see M. Blewitt (1975, 1997); T. Cunliffe (2001).