the earth's orbit around the sun is of the same figure; and we shall
hereafter see this to be true of all the planetary orbits. The path of
the earth, however, departs very little from a circle; that of the moon
differs materially from a circle, being considerably longer one way than
the other. Were the orbit a circle having the earth in the centre, then
the radius vector, or line drawn from the centre of the moon to the
centre of the earth, would always be of the same length; but it is found
that the length of the radius vector is only fifty-six times the radius
of the earth when the moon is nearest to us, while it is sixty-four
times that radius when the moon is furthest from us. The point in the
moon's orbit nearest the earth is called her _perigee_; the point
furthest from the earth, her _apogee_. We always know when the moon is
at one of these points, by her apparent diameter or apparent velocity;
for, when at the perigee, her diameter is greater than at any time, and
her motion most rapid; and, on the other hand, her diameter is least,
and her motion slowest, when she is at her apogee.
The moon's nodes constantly shift their positions in the ecliptic, from
east to west, at the rate of about nineteen and a half degrees every
year, returning to the same points once in eighteen and a half years. In
order to understand what is meant by this backward motion of the nodes,
you must have very distinctly in mind the meaning of the terms
themselves; and if, at any time, you should be at a loss about the
signification of any word that is used in expressing an astronomical
proposition, I would advise you to turn back to the previous definition
of that term, and revive its meaning clearly in the mind, before you
proceed any further. In the present case, you will recollect that the
moon's nodes are the two points where her orbit cuts the plane of the
ecliptic. Suppose the great circle of the ecliptic marked out on the
face of the sky in a distinct line, and let us observe, at any given
time, the exact moment when the moon crosses this line, which we will
suppose to be close to a certain star; then, on its next return to that
part of the heavens, we shall find that it crosses the ecliptic sensibly
to the westward of that star, and so on, further and further to the
westward, every time it crosses the ecliptic at either node. This fact
is expressed by saying that _the nodes retrograde on the ecliptic_;
since any motion from east to west, being contrary to the order of the
signs, is called retrograde. The line which joins these two points, or
the line of the nodes, is also said to have a retrograde motion, or to
revolve from east to west once in eighteen and a half years.
The _line of the apsides_ of the moon's orbit revolves from west to
east, through her whole course, in about nine years. You will recollect
that the apsides of an elliptical orbit are the two extremities of the
longer axis of the ellipse; corresponding to the perihelion and aphelion
of bodies revolving about the sun, or to the perigee and apogee of a
body revolving about the earth. If, in any revolution of the moon, we
should accurately mark the place in the heavens where the moon is
nearest the earth, (which may be known by the moon's apparent diameter
being then greatest,) we should find that, at the next revolution, it
would come to its perigee a little further eastward than before, and so
on, at every revolution, until, after nine years, it would come to its
perigee nearly at the same point as at first. This fact is expressed by
saying, that the perigee, and of course the apogee, revolves, and that
the line which joins these two points, or the line of the apsides, also
revolves.
These are only a few of the irregularities that attend the motions of
the moon. These and a few others were first discovered by actual
observation and have been long known; but a far greater number of lunar
irregularities have been made known by following out all the
consequences of the law of universal gravitation.
The moon may be regarded as a body endeavoring to make its way around
the earth, but as subject to be continually impeded, or diverted from
its main course, by the action of the sun and of the earth; sometimes
acting in concert and sometimes in opposition to each other. Now, by
exactly estimating the amount of these respective forces, and
ascertaining their resultant or combined effect, in any given case, the
direction and velocity of the moon's motion may be accurately
determined. But to do this has required the highest powers of the human
mind, aided by all the wonderful resources of mathematics. Yet, so
consistent is truth with itself, that, where some minute inequality in
the moon's motions is developed at the end of a long and intricate
mathematical process, it invariably happens, that, on pointing the
telescope to the moon, and watching its progress through the skies, we
may actually see her commit the same irregularities, unless (as is the
case with many of them) they are too minute to be matters of
observation, being beyond the powers of our vision, even when aided by
the best telescopes. But the truth of the law of gravitation, and of the
results it gives, when followed out by a chain of mathematical
reasoning, is fully confirmed, even in these minutest matters, by the
fact that the moon's place in the heavens, when thus determined, always
corresponds, with wonderful exactness, to the place which she is
actually observed to occupy at that time.
The mind, that was first able to elicit from the operations of Nature
the law of universal gravitation, and afterwards to apply it to the
complete explanation of all the irregular wanderings of the moon, must
have given evidence of intellectual powers far elevated above those of
the majority of the human race. We need not wonder, therefore, that such
homage is now paid to the genius of Newton,--an admiration which has
been continually increasing, as new discoveries have been made by
tracing out new consequences of the law of universal gravitation.
The chief object of astronomical _tables_ is to give the amount of all
the irregularities that attend the motions of the heavenly bodies, by
estimating the separate value of each, under all the different
circumstances in which a body can be placed. Thus, with respect to the
moon, before we can determine accurately the distance of the moon from
the vernal equinox, that is, her longitude at any given moment, we must
be able to make exact allowances for all her irregularities which would
affect her longitude. These are in all no less than sixty, though most
of them are so exceedingly minute, that it is not common to take into
the account more than twenty-eight or thirty. The values of these are
all given in the lunar tables; and in finding the moon's place, at any
given time, we proceed as follows: We first find what her place would be
on the supposition that she moves uniformly in a circle. This gives her
_mean_ place. We next apply the various corrections for her irregular
motions; that is, we apply the _equations_, subtracting some and adding
others, and thus we find her _true_ place.
The astronomical tables have been carried to such an astonishing degree
of accuracy, that it is said, by the highest authority, that an
astronomer could now predict, for a thousand years to come, the precise
moment of the passage of any one of the stars over the meridian wire of
the telescope of his transit-instrument, with such a degree of accuracy,
that the error would not be so great as to remove the object through an
angular space corresponding to the semidiameter of the finest wire that
could be made; and a body which, by the tables, ought to appear in the
transit-instrument in the middle of that wire, would in no case be
removed to its outer edge. The astronomer, the mathematician, and the
artist, have united their powers to produce this great result. The
astronomer has collected the data, by long-continued and most accurate
observations on the actual motions of the heavenly bodies, from night to
night, and from year to year; the mathematician has taken these data,
and applied to them the boundless resources of geometry and the
calculus; and, finally, the instrument-maker has furnished the means,
not only of verifying these conclusions, but of discovering new truths,
as the foundation of future reasonings.
Since the points where the moon crosses the ecliptic, or the moon's
nodes, constantly shift their positions about nineteen and a half
degrees to the westward, every year, the sun, in his annual progress in
the ecliptic, will go from the node round to the same node again in less
time than a year, since the node goes to meet him nineteen and a half
degrees to the west of the point where they met before. It would have
taken the sun about nineteen days to have passed over this arc; and
consequently, the interval between two successive conjunctions between
the sun and the moon's node is about nineteen days shorter than the
solar year of three hundred and sixty-five days; that is, it is about
three hundred and forty-six days; or, more exactly, it is 346.619851
days. The time from one new moon to another is 29.5305887 days. Now,
nineteen of the former periods are almost exactly equal to two hundred
and twenty-three of the latter:
For 346.619851 × 19=6585.78 days=18 y. 10 d.
And 29.5305887 × 223=6585.32 " = " " " "
Hence, if the sun and moon were to leave the moon's node together, after
the sun had been round to the same node nineteen times, the moon would
have made very nearly two hundred and twenty-three conjunctions with the
sun. If, therefore, she was in conjunction with the sun at the beginning
of this period, she would be in conjunction again at the end of it; and
all things relating to the sun, the moon, and the node, would be
restored to the same relative situation as before, and the sun and moon
would start again, to repeat the same phenomena, arising out of these
relations, as occurred in the preceding period, and in the same order.
Now, when the sun and moon meet at the moon's node, an eclipse of the
sun happens; and during the entire period of eighteen and a half years
eclipses will happen, nearly in the same manner as they did at
corresponding times in the preceding period. Thus, if there was a great
eclipse of the sun on the fifth year of one of these periods, a similar
eclipse (usually differing somewhat in magnitude) might be expected on
the fifth year of the next period. Hence this period, consisting of
about eighteen years and ten days, under the name of the _Saros_, was
used by the Chaldeans, and other ancient nations, in predicting
eclipses. It was probably by this means that Thales, a Grecian
astronomer who flourished six hundred years before the Christian era,
predicted an eclipse of the sun. Herodotus, the old historian of Greece,
relates that the day was suddenly changed into night, and that Thales of
Miletus had foretold that a great eclipse was to happen _this year_. It
was therefore, at that age, considered as a distinguished feat to
predict even the year in which an eclipse was to happen. This eclipse is
memorable in ancient history, from its having terminated the war between
the Lydians and the Medes, both parties being smitten with such
indications of the wrath of the gods.
The _Metonic Cycle_ has sometimes been confounded with the Saros, but it
is not the same with it, nor was the period used, like the Saros, for
foretelling eclipses, but for ascertaining the _age_ of the moon at any
given period. It consisted of nineteen tropical years, during which time
there are exactly two hundred and thirty-five new moons; so that, at the
end of this period, the new moons will recur at seasons of the year
corresponding exactly to those of the preceding cycle. If, for example,
a new moon fell at the time of the vernal equinox, in one cycle,
nineteen years afterwards it would occur again at the same equinox; or,
if it had happened ten days after the equinox, in one cycle, it would
also happen ten days after the equinox, nineteen years afterwards. By
registering, therefore, the exact days of any cycle at which the new or
full moons occurred, such a calendar would show on what days these
events would occur in any other cycle; and, since the regulation of
games, feasts, and fasts, has been made very extensively, both in
ancient and modern times, according to new or full moons, such a
calendar becomes very convenient for finding the day on which the new or
full moon required takes place. Suppose, for example, it were decreed
that a festival should be held on the day of the first full moon after
the Vernal equinox. Then, to find on what day that would happen, in any
given year, we have only to see what year it is of the lunar cycle; for
the day will be the same as it was in the corresponding year of the
calendar which records all the full moons of the cycle for each year,
and the respective days on which they happen.
The Athenians adopted the metonic cycle four hundred and thirty-three
years before the Christian era, for the regulation of their calendars,
and had it inscribed in letters of gold on the walls of the temple of
Minerva. Hence the term _golden number_, still found in our almanacs,
which denotes the year of the lunar cycle. Thus, fourteen was the golden
number for 1837, being the fourteenth year of the lunar cycle.
The inequalities of the moon's motions are divided into periodical and
secular. _Periodical_ inequalities are those which are completed in
comparatively short periods. _Secular_ inequalities are those which are
completed only in very long periods, such as centuries or ages. Hence
the corresponding terms _periodical equations_ and _secular equations_.
As an example of a secular inequality, we may mention the acceleration
of the _moon's mean motion_. It is discovered that the moon actually
revolves around the earth in a less period now than she did in ancient
times. The difference, however, is exceedingly small, being only about
ten seconds in a century. In a lunar eclipse, the moon's longitude
differs from that of the sun, at the middle of the eclipse, by exactly
one hundred and eighty degrees; and since the sun's longitude at any
given time of the year is known, if we can learn the day and hour when
an eclipse occurred at any period of the world, we of course know the
longitude of the sun and moon at that period. Now, in the year 721,
before the Christian era, Ptolemy records a lunar eclipse to have
happened, and to have been observed by the Chaldeans. The moon's
longitude, therefore, for that time, is known; and as we know the mean
motions of the moon, at present, starting from that epoch, and
computing, as may easily be done, the place which the moon ought to
occupy at present, at any given time, she is found to be actually nearly
a degree and a half in advance of that place. Moreover, the same
conclusion is derived from a comparison of the Chaldean observations
with those made by an Arabian astronomer of the tenth century.
This phenomenon at first led astronomers to apprehend that the moon
encountered a resisting medium, which, by destroying at every revolution
a small portion of her projectile force, would have the effect to bring
her nearer and nearer to the earth, and thus to augment her velocity.
But, in 1786, La Place demonstrated that this acceleration is one of the
legitimate effects of the sun's disturbing force, and is so connected
with changes in the eccentricity of the earth's orbit, that the moon
will continue to be accelerated while that eccentricity diminishes; but
when the eccentricity has reached its minimum, or lowest point, (as it
will do, after many ages,) and begins to increase, then the moon's
motions will begin to be retarded, and thus her mean motions will
oscillate for ever about a mean value.
LETTER XVIII.
ECLIPSES.
----"As when the sun, new risen,
Looks through the horizontal misty air,
Shorn of his beams, or from behind the moon,
In dim eclipse, disastrous twilight sheds
On half the nations, and with fear of change
Perplexes monarchs: darkened so, yet shone,
Above them all, the Archangel."--_Milton._
HAVING now learned various particulars respecting the earth, the sun,
and the moon, you are prepared to understand the explanation of solar
and lunar eclipses, which have in all ages excited a high degree of
interest. Indeed, what is more admirable, than that astronomers should
be able to tell us, years beforehand, the exact instant of the
commencement and termination of an eclipse, and describe all the
attendant circumstances with the greatest fidelity. You have doubtless,
my dear friend, participated in this admiration, and felt a strong
desire to learn how it is that astronomers are able to look so far into
futurity. I will endeavor, in this Letter, to explain to you the leading
principles of the calculation of eclipses, with as much plainness as
possible.
An _eclipse of the moon_ happens when the moon, in its revolution around
the earth, falls into the earth's shadow. An _eclipse of the sun_
happens when the moon, coming between the earth and the sun, covers
either a part or the whole of the solar disk.
The earth and the moon being both opaque, globular bodies, exposed to
the sun's light, they cast shadows opposite to the sun, like any other
bodies on which the sun shines. Were the sun of the same size with the
earth and the moon, then the lines drawn touching the surface of the sun
and the surface of the earth or moon (which lines form the boundaries of
the shadow) would be parallel to each other, and the shadow would be a
cylinder infinite in length; and were the sun less than the earth or
the moon, the shadow would be an increasing cone, its narrower end
resting on the earth; but as the sun is vastly greater than either of
these bodies, the shadow of each is a cone whose base rests on the body
itself, and which comes to a point, or vertex, at a certain distance
behind the body. These several cases are represented in the following
diagrams, Figs. 39, 40, 41.
[Illustration Figs. 39, 40, 41.]
It is found, by calculation, that the length of the moon's shadow, on an
average, is just about sufficient to reach to the earth; but the moon is
sometimes further from the earth than at others, and when she is nearer
than usual, the shadow reaches considerably beyond the surface of the
earth. Also, the moon, as well as the earth, is at different distances
from the sun at different times, and its shadow is longest when it is
furthest from the sun. Now, when both these circumstances conspire, that
is, when the moon is in her perigee and along with the earth in her
aphelion, her shadow extends nearly fifteen thousand miles beyond the
centre of the earth, and covers a space on the surface one hundred and
seventy miles broad. The earth's shadow is nearly a million of miles in
length, and consequently more than three and a half times as long as the
distance of the earth from the moon; and it is also, at the distance of
the moon, three times as broad as the moon itself.
An eclipse of the sun can take place only at new moon, when the sun and
moon meet in the same part of the heavens, for then only can the moon
come between us and the sun; and an eclipse of the moon can occur only
when the sun and moon are in opposite parts of the heavens, or at full
moon; for then only can the moon fall into the shadow of the earth.
[Illustration Fig. 42.]
The nature of eclipses will be clearly understood from the following
representation. The diagram, Fig. 42, exhibits the relative position of
the sun, the earth, and the moon, both in a solar and in a lunar
eclipse. Here, the moon is first represented, while revolving round the
earth, as passing between the earth and the sun, and casting its shadow
on the earth. As the moon is here supposed to be at her average distance
from the earth, the shadow but just reaches the earth's surface. Were
the moon (as is sometimes the case) nearer the earth her shadow would
not terminate in a point, as is represented in the figure, but at a
greater or less distance nearer the base of the cone, so as to cover a
considerable space, which, as I have already mentioned, sometimes
extends to one hundred and seventy miles in breadth, but is commonly
much less than this. On the other side of the earth, the moon is
represented as traversing the earth's shadow, as is the case in a lunar
eclipse. As the moon is sometimes nearer the earth and sometimes further
off, it is evident that it will traverse the shadow at a broader or a
narrower part, accordingly. The figure, however, represents the moon as
passing the shadow further from the earth than is ever actually the
case, since the distance from the earth is never so much as one third of
the whole length of the shadow.
It is evident from the figure, that if a spectator were situated where
the moon's shadow strikes the earth, the moon would cut off from him the
view of the sun, or the sun would be totally eclipsed. Or, if he were
within a certain distance of the shadow on either side, the moon would
be partly between him and the sun, and would intercept from him more or
less of the sun's light, according as he was nearer to the shadow or
further from it. If he were at _c_ or _d_, he would just see the moon
entering upon the sun's disk; if he were nearer the shadow than either
of these points, he would have a portion of this light cut off from his
view, and more, in proportion as he drew nearer the shadow; and the
moment he entered the shadow, he would lose sight of the sun. To all
places between _a_ or _b_ and the shadow, the sun would cast a partial
shadow of the moon, growing deeper and deeper, as it approached the true
shadow. This partial shadow is called the moon's _penumbra_. In like
manner, as the moon approaches the earth's shadow, in a lunar eclipse,
as soon as she arrives at _a_, the earth begins to intercept from her a
portion of the sun's light, or she falls in the earth's penumbra. She
continues to lose more and more of the sun's light, as she draws near to
the shadow, and hence her disk becomes gradually obscured, until it
enters the shadow, when the sun's light is entirely lost.