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From: George Huxtable
Date: 2002 Feb 13, 20:46 +0000

```3.1  CHANGE OF PLAN.

In part 2, there was a prediction of what part 3 would contain. However, I
have recently had some useful feedback from listmembers, (especially Steven
Wepster) which shows that rather more remains to be said about measuring
and correcting the lunar distance. So part 3 will fill in some of those
gaps.

Part 4, which should appear in the next few weeks, will discuss the
possibility of calculating the altitudes required with a lunar distance,
instead of measuring them: also the question of observing Moon altitudes in
place of lunar distances. It will show how to obtain local apparent time
and use it with GMT to obtain longitude, as early navigators had to do.

3.2  UNDUE EMPHASIS ON ACCURACY IN MOON ALTITUDES.

In part 1, under "How is a lunar distance measured?", I stated "The
corrections are such that the Sun altitude is not needed to great accuracy,
but the Moon altitude should be measured with precision." This overstated
things.

An error of a degree in Moon altitude will, at the most, give rise to an
error of 1 minute in the parallax correction of the lunar distance,
therefore the accuracy required in Moon altitude is no better than to 20
minutes or so. This is useful in night measurements of Moon-star or
Moon-planet lunars, when the horizon has to picked out from the ripply
reflection of moonlight below the Moon.

For the refraction of Sun, stars, or planets, when altitudes are as low as
10 deg., altitude measurements to 20 min. of arc are needed, but from 20
deg up, 1 degree will suffice. Again, for stars or planets, this is helpful
when the night-horizon is hard to make out.

The conclusion is that the requirements for accuracy of the altitude
measurements are usually rather easy to meet. This leaves the observer to
concentrate all his precision on measuring the lunar distance.

3.3  UPSIDE-DOWN SEXTANT?

In part 1, under "How is a lunar distance measured?", I said-

"The horizon plays no part in the lunar distance observation itself, and
the view, straight through the sextant, is instead used to observe the
fainter of the bodies observed (usually the Moon). The brighter one
(usually the Sun) is viewed in the index mirror. To see them both, the
sextant may have to be tilted at what seems like a most unnatural angle,
pointing up in the air, and over on its side, even upside-down."

I should have added here that if you find the "sextant upside-down" posture
particularly awkward (as many do), it will do no great harm to swap the two
images over, in the two light-paths, and use the sextant in a more natural
attitude. This presumes that your sextant offers a suitable choice of

3.4 REFRACTION CORRECTIONS. (see parts 1 and 2)

In clearing a lunar distance, it's necessary to separate the refraction
corrections from those for dip and for semidiameter. In many tables these
will have been combined, sometimes with Moon parallax also, to minimise the
arithmetic for a normal sextant altitude sight. Such a combined refraction
table is not useful for lunars. You can find the refraction correction, on
its own, for any body, from the "mean refraction" table in Norie's, or in
the star correction table in the pull-out card in the Nautical Almanac.

3.5 INVERSE LINEAR INTERPOLATION FOR TIME (see part 2).

This unwieldy phrase describes the process of finding the exact GMT
corresponding to a lunar distance, given predicted values of LD at two
times. Although I stated that those times, T1 and T2, could be chosen at
intervals 1, 2, or 3 hours apart, the example quoted presumed an interval
of 3 hours. This is the maximum interval that's compatible with the
assumption of a linear change, and is the interval on which the old
lunar-distance tables were based. That 3-hour interval was chosen to
minimise the size of the tables and to minimize the amount of calculation
in their making (in the days when that mattered).

If you are calculating lunar distances for yourself, it will do no harm to
choose a shorter interval than 3 hours, if you can be sure that the GMT
will end up within that interval.

3.6  ERROR IN EXAMPLE

Please note that in part 2, an error occurred in the following passage, as
uncovered by Greg Gilbert-

"NOW TO DEAL WITH THE LUNAR DISTANCE, AT LAST!

After correction for sextant errors, the Observed Lunar Distance between
the near limbs of the Sun and Moon was 105 deg 50.5 min. There is no dip to
consider because the horizon is not involved in the measurement."

Where this reads 105deg 50.5 min, it should have been 106deg 50.5 min.
Sorry about that. It was a transcription error, which doesn't affect the
following parts of the calculation.

3.7  ABOUT SOME PROBLEMS IN CLEARING THE LUNAR DISTANCE

In "About Lunars. part 1." it was shown how to obtain a corrected Lunar
Distance, D, using Young's method.

It included the following comment, about Young's method-

"CLEARING THE LUNAR DISTANCE, LONGHAND.

The formula above is fine for electronic computation, but quite unsuitable
for longhand calculation using logs. The trouble is that some of the
quantities may go negative, and the log of a negative number is
meaningless.

In the era of lunar distances, the navigator relied on 5-figure logs and
trig tables. The many different ways devised for clearing the Lunar
distance were mostly devoted to expressing it in such a way that logs could
be used for the solution, sometimes using auxiliary tables designed for the
purpose.

If readers find the need to do this clearance longhand using logs, my
suggestion is to use Borda's method, to be found in Cotter. On request, I
will spell it out for the list."

================ (end of quote)

Bill Noyce responded by saying-

>And I would like to take him up on his offer to spell out Borda's
>method for clearing the distance longhand.  Young's formula seems
>to require converting out of logs to do additions.

==================

For modern navigators, the availability of calculators and on-board
computers has made lunars more accessible to the rest of us, and that is
the approach I would recommend. But readers are entitled to choose a
technique using tables and logs, the traditional method that was employed
right through the era of lunars.

There are several problems that crop up when trying to use Young's method
using tables, rather than a calculator. Borda offers an alternative, but
still involves popping in and out of logs, and more than once!

However, I promised Borda's method, on request, and Bill Noyce has indeed
requested it, so here goes-

3.8  BORDA'S METHOD

This is more-or-less as quoted by Cotter. The quantities m, M, s, S, d, are
given, as follows.

d   (observed lunar distance between centres)
m   (observed moon-centre altitude above true horizontal)
M   (true Moon-centre altitude above true horizontal: it has been corrected
for parallax and refraction)
s   (obs Sun/body-centre altitude above true horizontal)
S   (true Sun/body-centre altitude above true horizontal: it has been
corrected for parallax and refraction)

The result D, will be the true lunar distance, which has been corrected for
parallax and refraction.

Firstly, an angle A has to be calculated. A is just an angle that's used as
an intermediate step in the calculation. Whether it has any physical
reality, I rather doubt. Obtain A as follows-

First work out and write down  (m+s+d)/2 and (m+s-d)/2

log cos A = {log cos((m+s+d)/2) + log cos((m+s-d)/2) +log cos M + log cos S
+ log sec m + log sec s}/2

This involves working out all those 6 log cos and log sec terms, adding
them up, and dividing by two.

Having calculated a value for the right-hand side of this equation, search
for that value in the log cos table and find out what angle it corresponds
to. This is angle A. You are now "out of logs" and back in the world of
ordinary numbers, for a time.

Now work out and write down A + (M+S)/2, and A ~ (M+S)/2.

where "~" means "subtract the smaller from the larger"

then go back into logs again, obtaining D from

log sin (D/2) = {log sin (A + (M+S)/2) + log sin (A ~ (M+S)/2)}/2.,

So having calculated a value for the right-hand side of the equation,
search for that value in the log sin table and find out what angle it
corresponds to. Double the result. That gives you D, the corrected lunar

That second term of that last equation differs from what is given in
Cotter, who puts a minus sign, which I have changed to "~". Following his
notation would lead you into trying to obtain the log of a negative
quantity. I think he has got that wrong.

The way I have rewritten that expression now conforms with the following
explanation in words of how to tackle the problem, as follows (from Cotter,
but somewhat modified)-

1. Find M, S, m, s, and d, as above.

D is the true lunar distance (in which the effects of parallax and
refraction have been allowed for) that we wish to find.

2. Place under one another the apparent distance d and the apparent
altitudes m and s: and take half their sum, L. From the half-sum L,
subtract the apparent distance d. Under this place the true altudes M and
S.

3. Take from tables log cosines of L, L-d, M, and S, log secant m, and log
secant s. Add these six quantities and divide by 2. The result is the log
cosine of A. So look up this quantity in the log cosine table, and find
what angle corresponds to it. This is A.

4. Take half the sum of the true altitudes M and S. Call this B. Find the
sum of, and the difference between, A and B. Add the log sines of the sum
and the difference. Divide by 2. The result is the log sine of half the
true lunar difference, that is D/2. So look up that result in the log sine
table, find the angle that corresponds to it, and double it to obtain the
corrected lunar distance D.

(Cotter also missed out an important word in his paragraph 4, which I have
reinstated.)

3.9 BORDA'S METHOD: AN EXAMPLE

Let's try Borda's method for real using the numbers below, for Steven
Wepster's Sun-lunar in the Atlantic on 2001 April 02, just as were used in
part 2 with Young's method.

I will stick to the degrees-and-minutes notation rather than decimal
degrees, as that's what is needed for looking up the tables.

d=107 deg 22.9 min    (observed lunar distance between centres)
m= 49 deg 52.5 min    (observed moon-centre altitude above true horizontal)
M= 50 deg 29.9 min    (true Moon-centre altitude above true horizontal)
s= 21 deg 10.4 min    (obs Sun/body-centre altitude above true horizontal)
S= 21 deg 08.1 min    (true Sun/body-centre altitude above true horizontal)

Following the written-out instructions in 3.8, we have now completed step 1.

now for steps 2 and 3

d           107deg  22.9
m           049deg  52.5    log sec = 10.19080
s           021deg  10.4    log sec = 10.03036
sum (d+m+s) 178deg  25.8
half-sum L  089deg  12.9    log cos =  8.13673
L-d        -018deg  10.0    log cos =  9.97779 (same as log cos +18deg 10.0)
M           050deg  29.9    log cos =  9.80353
S           021deg  08.1    log cos =  9.96976
sum     = 58.10897
Halve   = 29.05448
cast off 20=  9.05448
from log cos table,  Angle A = 083deg 29.4min
=====================

next, step 4.

M  (copied)       050deg 29.9
S  (copied)       021deg 08.1
sum               071deg 38.0
half-sum, =B      035deg 49.0
A (copied)        083deg 29.4
sum, A and B.     119deg 18.4      log sin =  9.94052
diff., A and B.   047deg 40.4      log sin =  9.86884
sum     = 19.80936
halved  =  9.90468
from log sin table, Angle D/2 = 053deg 24.6
so Angle D   = 106deg 49.2
=======================

This result, of 106deg 49.2min, for the corrected lunar distance, can be
compared with the value obtained (in part 2) for D using Young's method
with a calculator, which was 106deg 49.3. So rather good agreement: nothing

However, few navigators would find clearing the lunar distance by Borda's
method to be a lot of fun, I admit.

3.10  AN EASIER APPROACH TO CLEARING THE LUNAR DISTANCE.

The two methods considered so far, Young's and Borda's, are "mathematically
exact". Both these methods end up by multiplying the observed lunar
distance d by a calculated factor, which is always very near 1. The
observed distance d  can be of the order of 100 deg or so. To obtain the
answer D to within 0.1 min, that multiplying factor of about 1 has to be
known to 1 part in 100,000 or thereabouts. In other words the error in that
multiplying factor should be within .00001. That is why 5-figure logs are
required, and every step in the calculation made with scrupulous attention
to accuracy.

But what if we can correct d, to obtain D, by ADDING a correction to it,
rather than using a multiplier? The correction is never more than about 60
minutes, so to make that correction to provide D to 0.1 minutes, it's only
necessary to evaluate that added correction to 0.1 in 60 minutes. An error
of 1 part in 1,000 of that added amount now becomes acceptable. That is an
accuracy that might even be approached by the careful wielder of a
slide-rule that shows trig functions.

Throughout the 19th century, much effort went into devising a solution to
clearing the lunar distance which involves such an added correction, rather
than a multiplying factor. The trouble has been that such solutions have
involved an approximation to the geometry, not an exact solution. They rely
on the fact that both angles are small, the parallax and the refraction. In
practice, those angles always are acceptably small. If observations are
made only when altitudes exceed 10 degrees, refractions are always less
than 5.5 min. Parallax of the Moon never much exceeds 60 minutes. If we
make an additional rule that lunar distances are measured only when they
exceed 10 degrees the requirements are met for good accuracy of the
"approximate solution".

3.11 LETCHER'S METHOD.

In "Self-contained celestial navigation using H.O.208" (1977), John S
Letcher describes such an approximate method for clearing the lunar
distance. The way he proposes for its use would be suitable only for lunar
distances up to 90 degrees, whereas any lunar observer would certainly
require its application to angles up to 120 degrees. I will therefore give
a modified calculation method that will cover the full range of useful
sextant angles.

I am not sure where this method comes from, as Letcher does not quote any
reference, nor does he show how it is derived from first-principles. It
might be his own invention, perhaps. To me, it seems very clever. I will
christen it "Letcher's method" until we find out more about its provenance.
If anyone recognises it from another publication, I would be interested to
learn.

Let's investigate Letcher's Method.

We will assume that all sextant observations have been corrected for index
error.

As before, it starts with the observed lunar distance d, after correction
for semidiameters. This is to be corrected by adding, separately,-

a) The correction P, for the combined parallaxes of the Moon and the Sun,
which may be a positive or negative amount, never greater than about 60
min.

b) The correction R, for the refractions of Moon and Sun, always positive,
never more than 11 min.

Both P and R need to be calculated to within 0.1 min.

These two corrections require knowledge of-

d, the observed lunar distance between centres (i.e. corrected for
semidiameters).
m, the observed altitude of the Moon's centre above the true horizon (i.e.
corrected for dip and semidiameter)
s, the observed altitude of the centre of the Sun (or other body), above
the true horizon (i.e. corrected for dip and, if necessary, semidiameter).
HP, the Moon's horizontal parallax, in minutes of arc, at the hour of
observation.
The HP of the Sun (or other body) is ignored, thus losing a bit of precision.

It is no longer necessary (as it was in the other methods) to make parallax
and refraction corrections to m and s individually, as M and S are no
longer needed. These corrections are taken into account automatically, as
part of the clearance procedure. That is partly why the method is so much
simpler.

Let's go through it.

First, obtain B = (cos d sin m - sin s) / sin d  (B is just an intermediate
angle, used in the computation).

Then the parallax correction in minutes is-

P = HP*B + (HP)^2*[(cos m)^2 - B^2)]/ (6900*tan d)

P may be a positive or negative number, in minutes, to be added or
subtracted from d.

The second term of P is usually a very small quantity, but still needs to
be evaluated.

Now for the refraction term R. This, very cleverly, includes its own
built-in calculation of the way refraction changes with altitude.

R = .95* (sin s/sin m + sin m/sin s - 2*cos d) / sin d ,in minutes of arc.

So the end result is D = d + P + R , giving the Corrected Lunar Distance.

3.7. LETCHER: WORKED EXAMPLE

Let's compare this method with the others, using Steven Wepster's Atlantic
observations of 2001 Apr 02 once again.

The inputs we need are, as before-

d = 107.382 deg    (observed lunar distance between centres)
m = 049.875 deg   (observed moon-centre altitude above true horizontal)
s = 021.173 deg   (obs Sun or body-centre altitude above true horizontal)
HP = 59.4 min.

By pocket calculator we get-

B= -0.6178

so P = -36.7 - 0.1 = -36.8 min,   and R = 3.2 min

As d (in minutes) is 107deg 22.9, then for the Corrected Lunar Distance,

D= d + P + R, or 107deg 22.9 - 36.8 + 3.2

Therefore D = 106deg 49.3 by Letcher's method.
===============

This should be compared with 106 deg 49.2, by Borda's method using tables,
and 106deg 49.3 using Young's method with a calculator. Of these, I would
choose the last, 106deg 49.3, as being the more precise. Really, what
better agreement could anyone wish for, between three different methods?

However, don't expect to always obtain that same precision using Letcher's
method. He concedes that by ignoring some of the contributions such as the
Sun's parallax and the ellipsoidal figure-of-the-Earth, errors may combine
up to a total of 0.3 minutes. Even so, when compared with the likely errors
inherent in measuring the lunar distance, due to the motion of a small
craft, a possible error of 0.3 minutes in the clearing of it may be very
acceptable. My vote would go to Letcher. But it's your choice...

My thanks to Bill Murdoch for alerting me to this useful Letcher
publication. If you can find a secondhand copy, I think it would be a
worthwhile buy. It was written in the days when scientific calculators
existed but were uncommon; hence the emphasis on using tables.

=================

Part 4, which should appear in the next few weeks, will discuss the
possibility of calculating the altitudes required with a lunar distance,
instead of measuring them: also the question of observing Moon altitudes in
place of lunar distances. It will show how to obtain local apparent time
and use it with GMT to obtain longitude, as early navigators had to do.

George Huxtable.

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

george---.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
------------------------------

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