Space Forum / Astronomy / July 2008

Hi
The problem with this method was that the Moon was not always
seen at night. Ships can't wait for the Moon.
Using a clock, one could use Moon, Sun or stars.
Dwight

[Hammond]
  Does "Longitude" by Dava Sobel mention Swedenborg or his
method of observing the moon between two stars?
  Swedenborg published his Lunar Method in Swedish in 1718,
in Latin in 1721 and after he became famous 50 years later
in either Latin or English.  He discussed the method
personally with both Flamsteed and Halley in London in 1721.
  Could someone look in the Index of "Longitude" (Sobel's
book) and see if Swedenborg is listed?
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On Jul 7, 5:42 pm, will...@cfa.harvard.edu (Steve Willner) wrote:

As discussed here Tobias Mayer or his widow received part of the
famous Longitude prize for a similar method:

Precomputed lunar distance tables.
http://www.math.uu.nl/people/wepster/ldtab.html

I became interested in the story of the longitude after seeing a Nova
episode based on Dava Sobel's book "Longitude". One extraordinary
episode recounted in the book told of an entire English fleet lost in
heavy fog while just a few miles off the coast of Britain because they
did not have an accurate knowledge of their longitude. Sobel described
that a common sailor came forward to give the admirals and captains of
the ships who assembled together to decide their location   his
opinion of their position and for this he was promptly hanged on the
spot. The naval laws were strict about sailors even attempting to do
their own calculations because this was so critical it was felt mutiny
would arise if the ships crew did not trust the captains judgment on
the matter. It turned out that that sailors calculations were correct
and had the admirals and captains listened the fleet would have been
saved.
Sobels dramatic telling led me to speculate about some methods
mariners of the time might have been able to accurately determine
their longitude. I copied two emails below that I wrote to the author
of the "Precomputed lunar distance tables" web site about a suggested
method. It also concerned the position of the Moon. But I believe it
would have been easier to implement than requiring accurate
determination of the angular distance between the Moon and stars.

 Bob Clark
==============================================
Date : Wed, Nov 29, 2006 02:50 AM EST
From : "Robert Clark" <****@****>
To : ****@math.uu.nl
Subject : The apparent size of the Moon to solve the "problem of the
longitude"?

Hello. I saw your web page discussing "lunars" after a web search.
I like many other scientists was fascinated by the story described by
Dava Sobel about the quest for an accurate determination of the
longitude, finally solved in the 18th century.
I wondered if it would have been possible to get it from the change
in the apparent size of the Sun according to distance. However, the
change in this distance is relatively small. So I thought instead
about the Moon.
The idea is that a pinhole camera projects an image whose size is
proportional to the size of the source and inversely proportional to
its
distance:

Finding the Size of the Sun and Moon.
http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_03.html

Note that this can still work to measure the apparent size of the
Moon even when it's not a full Moon as long as part of the limb is
visible.
As you described on your web page on the calculation of the "lunars",
Tobias Mayer calculated very accurate positions for the Moon. I
presume this means he also would have known very accurate distances
from the Earth to the Moon. Note that the distances are not always the
same even for the perigee or for the apogee; so these predictions
would have to be made for each particular day in
each particular year.
Here's a modern calculator for this using the most up to date data
for the Moon's orbit:

Lunar Perigee and Apogee Calculator.
http://www.fourmilab.ch/earthview/pacalc.html

And this web page shows the distance to the Moon changes by as much
as 14% during its orbit, resulting in a dramatic change in its
apparent size:

Lunar Image Gallery - Scenic Phenomenon
http://www.perseus.gr/Astro-Lunar-Scenes-Apo-Perigee.htm

So instead of tables of angular distances between the Moon and
certain stars, there would be tables of the size of the Moon at at a
certain fixed longitude, and you could deduce how far away you were
from that location by the size you measured at your own location.
There would be "measuring boxes" made of a uniform size so the image
projected would be the same size for the same apparent size Moon, with
gradations marked on the inside to easily read off the size.
As discussed on the pinhole camera page, the size of the projected
image is dependent on the distance between the pinhole and the
projection surface. So this length would have to be greatly and
accurately standardized.
You would also need the boxes to be made of material that expanded
very little with temperature variations. This is a problem John
Harrison
encountered for the production of his accurate watches. I believe this
is a much easier problem with a static box than with a complicated
moving mechanism like a watch. For instance glass is used for
telescope lenses and mirrors because of its stability against size
variations on temperature change. It's much easier to make a static
glass box than a watch made of glass.
The change in distance and therefore apparent size is 1 part in 7
over half an orbit, 14 days. So it's 1 part in 7*14 = 98 per day,
assuming the distance changing uniformly. (We could include in our
calculations the deviation from the uniformity assumption by taking
into account the elliptical shape of the orbit.) Then it's 1 part in
98*24= 2352 per hour. If we wanted to reach the accuracy required for
the Longitude Prize by *direct measurement* we would have to multiply
this number by 15 to get it to 1 part in 35,280. However, we could
instead use interpolation as used for the "lunars".
The Moon's diameter is about 3500 km and the distance at perigee is
about 350,000. So the ratio of actual size to distance is about 1 to a
100. This means the ratio of size of  projected image to length of the
box would also be 1 to 100. If the measuring box was 10 meters long,
the size of the image would be about 10 cm. So at the 1 part in 2352
accuracy level we would need to measure to within an accuracy of 42
microns across the 10 cm image. This is about the width of a human
hair, which should be within the measuring accuracy available for the
18th century. For visually observing this small distance magnifying
lenses would be sufficient.
How badly would diffraction of the atmosphere effect the accuracy of
this method?

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

Date : Tue, Dec 05, 2006 07:00 PM EST
From : "Robert Clark" <****@****>
To : ****@math.uu.nl
Subject :  Re[2]: The apparent size of the Moon to solve the "problem
of the longitude"?

Thanks for the response. I didn't think of the fact that the distance
would change little at the max and min distance.
However, I thought of a way to make the measurements easier in
general. What you could do would be to use a telescope to make the
image larger. The telescope would be used like a film projector to
make a larger image on a screen.
This page shows pretty decent scopes were made in the 18th century:

18th-century telescopes.
http://www.antiquetelescopes.org/18thc.html

The use of the telescope for astronomy dates back to Galileo of
course in the 17th century:

17th-century telescopes.
http://www.antiquetelescopes.org/17thc.html

I don't know the relationship between the size of the image and the
size of the objective but I presume it would also depend on the focal
length of the scope and the distance to the screen.
The presumption is you could make a larger image say 1 meter size at
a shorter distance to the screen, so that you wouldn't need a 10 meter
distance like I first suggested, by using a larger lens or mirror and
the appropriate focal length.
Tables would be used as before to indicate the expected size of the
image at the reference location according to the time of day at the
reference location.
For finding local time required in the calculation, my reading of
Sobel's book suggests portable clocks of the time would be accurate to
within a few minutes within a single day, which would be all that is
required for determination of longitude. (Harrison's accomplishment
was to create a clock that would be accurate to within a few seconds
per day so over a sea voyage it only be off by a few minutes.) So you
would just set your clock at the local noon say and you would only
need it to be accurate to within a few minutes at the night time
observations.
Another possibility occurs to me for finding the expected size
according to the time you were observing. Wouldn't the position of the
Moon from North, determined by the Pole star or compass, change as the
night progressed? It seems to me you could have the tables for the
reference location give the distance in degrees from North at a
particular time and also give the expected size at that position.
Then for the mariners making their observations they would find the
angular distance of the Moon from north, check the table for the
expected size at this angular distance, then compute their longitude
from the deviation of their measured size from the size given in the
table.
As for the required calculations, I was startled by this discovery of
the
capabilities of this calculating machine for determining positions of
the Moon and known planets dating from 100 to 200 B.C.:

Ancient calculator demystified at last
Greeks’ 2,100-year-old Antikythera Mechanism was used in astronomy.
http://www.msnbc.msn.com/id/15953550/

The device worked by a complicated combination of interconnected
gears. This was certainly within the capabilities of the 18th century.
Admittedly it's construction details were lost until revealed
recently. But there were human-like "automatons" made of gears made in
the 18th century and I believe calculating devices could also have
been made at this time if someone had thought of it.

 Bob Clark
==========================================

If Earth, like Mars,  had two smaller moons instead one large one
how much easier would it be to locate latitude and longitude with
an astrolabe?
Or better yet, enough moons so that three or four could be seen
at all times instead of only when one is above the horizon?
Suppose we put a radio transmitter on every moon so that
we were not dependent on a visual sighting, but could "see"
right through cloud whenever we wanted to?
Suppose each moon could tell us where it was on that radio
signal so that we didn't need an astrolabe?
An even better improvement would be to have each moon
carry an accurate clock and tell us the exact time it was there
when it sent the radio transmission.
This would be sci-fi, of course... unless it was called GPS,
finally solved in the 20th century.
How badly would diffraction (oops - refraction) of the atmosphere
effect the accuracy of this method?
Quite a lot really, you could easily be up to 100 feet out of
position vertically and 30 feet off horizontally. How terrible.

I beg to differ. GPS depends solely on satellite position at ONE time,
NOW, and does not change from hour to hour. It is merely triangulation
no matter how you cut it. I do not need "real time" to calculate where
I was yesterday, I only need the positional data from the satellites
yesterday at some instant, common to all.
Time has nothing whatever to do with it, except insofar as the position
of the satellite changes as a function of time.
The ratio of the US dollar to the British pound is 4:1 (in WW II).
Time has nothing whatever to do with it, except insofar as the
ratio changes as years go by.