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Bill wrote-

>... Frankly, the magnitude of the Moon's v surprised me when I perused it
>in the almanac.

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

Compared with everything else we see in the sky, the Moon's motion is
rather wild, tugged about as it is by the combined gravities of Earth and
Sun, and to a lesser extent, planets also.. The speed variation, with
respect to the star background, is mainly due to the Moon's orbit being so
eccentric. The Moon's apogee distance can be more than 14% greater than at
perigee. which implies that the angular speed of the Moon at perigee (with
respect to the star background) can be 30% greater than it is at apogee.
Combine this with the fact that the Moon moves MUCH faster, across the
stars, than anything else does, and it will be clear why such large
variations are called for in the Moon's "v" term in the almanac
predictions.

And the Moon's changes in declination are similarly wild. Where the Sun's
declination swings through 57 degrees and back in the course of a year, the
Moon makes such a swing every month. What's more. the Moon's swing in some
years is all of 67 degrees, and in other years is as low as 46 degrees.

Predicting the detailed motion of the Moon (for use by navigators) was an
immense challenge to astronomers and mathematicians of the 18th (and 19th)
century, and stimulated many important advances in mathematics.

===============
Responding to my comment-

"Most planets move across the sky at a speed close to (and greater than)
that of the Sun, in which case a small positive value of v, applied to the
nominal Sun motion of 15 deg per hour, gives rise to an added correction.
Venus frequently shows a retrograde motion in the sky, which gives rise to
a negative v correction, so in the Venus case navigators need to apply any
subtractional ability they may possess."

Bill wrote-

>I recollect there are other planets that exhibit retrograde motion.  If my
>recollection is correct, may I safely assume they are not navigational
>planets?

A good question, to which the answer is "No, you may not"!

I over-simplified the matter, somewhat. A planet that shows retrograde
motion does so with respect to the star background, which we know moves in
GHA, with Aries, at a rate of
15deg 02.46' per hour. If its motion in GHA is at any time less than that
figure, then it is, at that moment, moving "backwards", showing retrograde
motion, with respect to the stars.

Most of the time, planets are moving round the sky just slightly faster
than the stars do, but now and again a planet's path makes a "loop" against
the star background, and the planet moves backwards, for a time, among the
stars. This does NOT, of course, relate to the planet actually moving
backward in its orbit! It's a perspective effect, due to the changing
viewpoint of the observer, from the moving Earth.

A planet's "v" correction is made, not with respect to that motion of the
stars, but with respect to the nominal motion of the Mean Sun, which is
exactly 15 deg per hour. The Sun itself shows "retrograde motion" with
respect to the stars.

So it's quite possible for a planet to show retrograde motion, with respect
to the stars, even if its v value is zero or slightly positive. In the case
of Venus (and only Venus, I think), the retrograde motion can be
sufficiently fast that it can be moving across the sky even more slowly
than the Sun does, and in that extreme situation a negative value of v
applies.

So, I confused matters rather, by tying together retrograde motion with a
negative value of v; it's a bit more complicated than that. Bill's
perceptive question has provided an opportunity to undo that confusion.

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

In response to my comment-

>> It fits in with the notion
>> that navigators do not understand how to subtract.

Bill wrote
>I am puzzled about the notion George has suggested that navigators do not
>understand how to subtract. Where did it come from? 

Well, our trade of celestial navigation was founded, mostly, in the 18th
century. Although navigators were indeed taught how to subtract one
positive number from a larger positive number (even in sexagesimal),
everyone in that age seems to have avoided manipulating  actual negative
quantities if it was at all possible. Or so it seems to me.

Whereas a modern high-school kid would be expected to subtract -9 from -16
and (sometimes) get the right answer, I get the picture, from reading
19th-century navigation manuals, that such concepts were alien to the
mindset of those days.

Instead, it was common for quantities to be given NAMES, rather than signs.
So latitudes and declinations were labelled North and South, not + and -.
And elaborate rules were devised which said things like "If the names
differ add, but if they are the same, take the smaller from the larger, and
label the result appropriately". But really such rules are only a
complicated implementation of sign-depended subtraction.

Similarly, the cleverness of logs was brought into trig computations, and
this added its own problems, because the log of a negative number has no
meaning. There were tricks to overcome this in a truly mathematical way, as
used by Chauvenet, but instead, wordy rules to get round the problem and
avoid such negatives were devised.

Indeed, the practice, in navigational logs, of adding ten, or tens, to the
characteristic of a log, was another trick to avoid negative values, at the
expense of understanding and consistency.

Such matters have only started to change over the last few years, it seems
to me, with the introduction of calculator and computer formulae to
navigation, these having no difficulties with signed quantities.

George.

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
================================================================

   

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