VMs: Identifying VMS stars, and the longitude problem

[Jorge Stolfi <stolfi@xxxxxxxxxxxxx>]

  > [Rene:] Yes, I tried this for longitude, oblique ascension etc,
  > but a match could not be achieved, even just looking at the nr. of
  > 9-pointed stars per VMS zodiac sign, making no assumption about
  > the order in which they appear in each sign. (It is not guaranteed
  > that there should be an order) Clearly, one should allow also for
  > incidental mistakes in the figures, plus the uncertainy which
  > tables, if any, are behind it.
  
Perhaps Fourier analysis can see through all that noise? Say that you
make two lists of pairs (longitude,magnitude): (VL[i],VM[i]) taken
from the VMS Zodiac diagrams, and (CL[i],CM[i]) taken from the star
catalogs, excluding perhaps stars above a certain latitude. Then look
at the two lists as periodic impulse sequences, and compute their
Fourier coefficients VF, CF of frequency 1. If the V pairs are indeed
related to the C pairs, even with large random shifts in longitude and
many errors in the magnitudes, comparison of the two coeffs may
confirm the relationship and reveal the general nature of the
measurements (risings, settings, or culminations).

Assuming that brighter stars are more likely to be used in the VMS
than fainter ones, I would truncate both lists at some minimum
magnitude, and encode the magnitudes in linear rather than logarithmic
scale (i.e. 16,8,4,2,1 rather than 1,2,3,4,5). But these are just
steps in the dark...

  > [Dan Gibson:] The Arabs used several ways to determine latitude...
  
Yes, but latitude ("what parallel") is the easy one. Just measure the
height of something celestial over the horizon, on a north-south line;
put in a correction for the season of year; and you will have a very
accurate and absolute value for the longitude. The culmination of any
celestial body will do; it doesn't matter whether the event happens at
different times in different places. This trick surely was known 
from remote antiquity.

Elmar's question, and my guessed aswer, are about longitude ("what
meridian"), which has been a mostly unsolved navigation problem until
the 17th-18th century. For that you need to relate local measurements
to those made at some reference longitude, *and you need to know the
time interval between those measurements*, within a fraction of one hour.

My guess is that Ptolemy and other ancient astronomers could have
determined the relative longitude of cities by comparing the local
times (or the moon's position over the horizon) when a lunar eclipse
was reported to have reached its maximum. Obviously this method is
useless for sea navigation (unless one has very accurate predictions
of lunar eclipses, which presumably were not available until
recently).

Note however that one can get very rough longitudes by estimating city
distances from travel time. From distances in the North-South
direction you get the conversion factor between miles and degrees,
which you then use to convert East-West distances into degrees.

(Solving the "longitude problem" for ships would have meant fame and
fortune to the inventor. Galileo was among the many frustrated
inventors-to-be. After discovering the four satellites of Jupiter, he
noticed that their regular movements around the planet could provide
the needed "clock". He tried to sell his method to several kings of
Europe, but none was willing to pay for his secret. The problem was
solved for good only with the development of accurate ship clocks, a
century or so later.)

  > [Pam:] As for your questions about Ptolemy, he knew about
  > "transits". . . that is, I think you mean aspects
  > between planets.
  
Sorry, I meant in the modern sense of planets crossing the sun's disk.
Could pre-telescope astronomers observe a transit of Venus, say by
looking at a pinhole-camera projection of the sun? (Presumably they
would have been capable of computing the approximate time of the
transit, and thus would know when to look.)

  > Say, a lunar eclipse is taking place at night. One person in an
  > Eastern location sees the eclipse occur on the Western horizon.
  > ... calculations can be taken to show the time of the eclipse in
  > different locations, and the differences can be noted.

Yes, that is precisely what I meant.  Was this method actually used?
How accurate could it be? 

All the best,

--stolfi
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