Now there are various entities involved, say L for the local
north-south, east-west, up-down directions as seen by the voyagers (depending
on the orientation of the earth and on the latitude and longitude of the
observer on it), S for the sun, M for the moon, P for the planets, and A for
the distant stars (astros). The voyagers apparently only used their records of
the LS angles (how many solar days they had seen, what I might call local solar
time) in comparing with the islanders and discovering the missing day. If they
had accurately measured enough angles between distant objects like S, P, and A,
they could have gotten a measure of what I might call sidereal time (time
determined by the motions of things out in the heavens as seen from the
position of the earth but fairly independent of where one is on the earth),
which would not have been missing a day. Measuring angles between M and any of
S, P, or A would also have given a reasonably good indication of this "sidereal
time," though there would be some error from the fact that the direction to the
moon does depend to one degree or so on where on earth one views it from.
Measuring angles between M, P, or A and L would have had the effect of both
sidereal time and of where on earth the crew was, so that could give yet other
indications of the time.
So I think Glenn is right that the situation is complicated, and there
are many sorts of time measures the crew could have gotten with various
measurements they might have made. If they had compared enough of them, they
could presumably have found a discrepancy with their apparent implicit
assumption that they were at fixed longitude (i.e., in using their local solar
time as a measure of the total passage of time).
However, I would emphasize two points on this:
(1) I don't think there would be any significant discrepancy between
Ptolemy's system and Kepler's for the predictions (since the orbital
eccentricities are low enough that the quadratic corrections from them are very
small), provided that one explicitly used in both systems the fact that the
observations were being made from different positions on a round earth. Of
course, if one makes the implicit assumption that one stays at fixed longitude
(as the crew apparently did) in using Ptolemy's system and then corrects that
mistake in Kepler's system, then there will be a discrepancy, but it is between
the fixed-longitude and the observer-moving-over-the-earth assumptions and not
between Ptolemy's and Kepler's system.
(2) From what Glenn and others have told me so far, there doesn't seem
to be evidence that the voyagers tried to measure time by the angles between
objects purely in the heavens, as they did not remark on noticing any
discrepancy with their local solar time (by the implicit fixed-longitude
assumption) until they compared notes with the islanders.
One minor thing I have realized since writing my previous message on
this is that if the voyagers had carried a table of the times of the full moon,
and if one could ignore or correct for such factors as the fact that the moon's
orbit is not in the same plane as that of the earth around the sun, and the
refraction of light in the atmosphere, they could presumably have observed
whether an almost full moon rose before the sun set (in which case it would be
before being full) or after (in which case it would have been later). By this
means one might suppose that they could have determined to within one day when
the moon was full and then noticed a discrepancy with their count of local
solar days.
It also occurred to me that if one had sufficient calculations
available (which is very unlikely for Magellan's crew to have had), then by
measuring the direction to the moon relative to the local cardinal directions
(the ML angles) when the moon occulted various stars (determined by the MA
angles), one should have been able to get a measure of the longitude of the
observation point to roughly the same accuracy as the measurement of the moon
direction. These calculations would have been complicated by the fact that the
time at which a moon occults a star depends on where on earth one observes the
event (which if uncorrected would presumably lead to an error of order the
angular size of the earth as seen from the moon, somewhat more than one
degree), but it makes me wonder whether anyone tried to use that method for
determining longitude before accurate chronometers were developed for that
purpose.
Another point I am curious about is Glenn's claim that Ptolemy's system
would have predicted that the moon, if of fixed physical size, would have an
angular size that varied by 100% during the month. Why didn't Ptolemy's system
have the moon going round the earth on a circle that was eccentric by an amount
proportional to the eccentricity of its Keplerian orbit? I would have guessed
that the latter were used, and that then the error would be only quadratic in
the eccentricity and hence probably unmeasurable. In fact, I am also curious
as to how early one could measure the actual angular size variations in the sun
and moon between apogee and perigee.
Glenn, if you know of some observation that the voyagers made that was
clearly in conflict with Ptolemy's system even when one puts in the fact that
the voyagers were observing from different positions on a round earth, I would
be curious as to what it is and how you calculated the discrepancy. But unless
someone can show this, I don't see how their observations (once their implicit
fixed-longitude mistake was corrected) could possibly have given any vidence as
to whether or not it was the earth that was rotating and revolving, rather than
the heavenly bodies about the earth.
Don Page