THE STARS--THEIR RISINGS AND SETTINGS.
From what has been stated it is not too much to assume that the
Egyptians observed, and taught people to observe, the sun on the
horizon.
This being so, the chances are that at first they would observe the
stars on the horizon too, both stars rising and stars setting; this
indeed is rendered more probable by the very careful way in which early
astronomers defined the various conditions under which a star can rise
or set, always, be it well remembered, in relation to the sun.
It must not be forgotten that the ancients had no telescopes, and had
to use their horizon as the only scientific instrument which they
possessed. They spoke of a star as rising or setting cosmically,
achronically, or heliacally.
The cosmic rising meant that the star rose, and the cosmic setting
meant that the star set, at the same moment as the sun--that is, that
along the eastern horizon we should see the star rising at the moment
of sunrise, or along the western horizon a star setting at the moment
of the sun setting; but unless certain very obvious precautions were
taken it is clear that neither the rising nor the setting star would be
seen, in consequence of the presence of daylight. The achronical rising
or setting is different from the cosmic in this respect--that we have
the star rising when the sun is setting, or setting when the sun is
rising. Finally we have the heliacal rising and setting; that is taken
to be that the star appeared in the morning a little in advance of the
sunrise, or set at twilight a little later than the sun.
It is quite clear that if we observe a star rising in the dawn, it will
get more and more difficult to observe the nearer the time of sunrise
is approached. Therefore, what the ancients did was to determine a
time before sunrise in the early dawn at which the star could be very
obviously and clearly seen to rise. The term "heliacal rising" was
coined to represent a star rising visibly in the dawn, therefore,
before the sun. Generally throughout Egypt the sun was supposed to be
something like 10° below the horizon when a star was stated to rise
_heliacally_.
The following table from Biot should make matters quite clear:--
┌ ┌ True or Cosmic Sun rising.
│ │ ┌ Sun not yet risen, but
│ Morning ┤ Apparent or ┤ depressed below horizon
│ │ Heliacal │ sufficiently to enable
│ └ └ the star to be seen.
│
Star at Eastern ┤ ┌ True or Achronic Sun setting.
Horizon (Rising) │ │ ┌ Sun set, and depressed
│ Evening ┤ Apparent or ┤ below horizon sufficiently
│ │ Heliacal │ to enable the
└ └ └ star to be seen.
┌ ┌ True or Cosmic Sun setting.
│ │ ┌ Sun set and depressed
│ Evening ┤ Apparent or ┤ below horizon sufficiently
Star at Western │ │ Heliacal │ to enable the
Horizon (Setting)┤ └ │ star to be seen.
│
│ ┌ True or Achronic Sun rising.
│ │ ┌ Sun not yet risen, but
│ Morning ┤ Apparent or ┤ depressed below horizon
│ │ Heliacal │ sufficiently to enable
└ └ └ the star to be seen.
It is Ideler's opinion that, in Ptolemy's time, in the case of stars
of the first magnitude, for heliacal risings and settings, if the star
and sun were on the same horizon, a depression of the sun of 11° was
taken; if on opposite horizons, a depression of 7°. For stars of the
second magnitude these values were 14° and 8½°. But if temples were
employed as I have suggested, even cosmic and achronic risings and
settings could be observed in the case of the brightest stars.
But it must not be imagined that, even in Egypt, all stars can be
observed the moment they are above the horizon. In the morning,
especially, there are mists, so that all but the brightest stars are
often invisible till they are 1° or 2° high. On this point I quote
Biot:--
"Comme le rapporte Nouet, l'astronome de l'expédition française, on
n'y aperçoit jamais à leur lever les étoiles de 2° et de 3° grandeur
même dans les plus belles nuits, à cause d'une bande constant de
vapeurs qui borde l'horizon.[38] Aussi en expliquant le calcul
des levers héliaques dans l'Almageste, Ptolémée a-t-il soin de
remarquer[39] que les annonces qu'on voudrait faire de ces phénomènes
seront toujours très-incertaines, à cause de l'état des couches d'air
dans lesquelles on les observe, et à cause de la difficulté optique
qu'on éprouve à saisir la première apparition, comme il dit lui-même
en avoir fait l'expérience."[40]
Before we begin to consider the question of stars at all, we must
be able to describe them--to speak of them in a way that shall
define exactly which star is meant. We can in these days define a
star according to its constellation, or its equatorial or ecliptic
co-ordinates, but all these means of reference were unknown to the
earliest observers. Still we may assume that the Egyptians could define
some of the stars in some fashion; and it is evident that we here
approach a matter of the very highest importance for our subject, to
which I shall have to return in a subsequent chapter.
So far as we have been dealing with the sun and the observations of the
sun at rising and setting, we have taken for granted that the amplitude
of the sun at the solstices does not change; the amplitude of 26° at
Thebes for the solstices is practically, though as we have seen not
absolutely, invariable for a thousand years; but one of the results
of astronomical work is that the _stars_ are known to behave quite
differently. In consequence of what is called _precession_ the stars
change their place with regard to the pole of the equator; and further,
in consequence of this movement, the position of the sun among the
stars at the solstices and equinoxes changes also.
In reference to the sun's path we considered what are called the
ecliptic and the equatorial co-ordinates. The ecliptic defines the
plane in which the earth moves round the sun, and 90° from that plane
we have the pole of the heavens; celestial latitude we found reckoned
from the plane of the ecliptic north and south up to the pole of the
heavens, and celestial longitude was reckoned along the plane of
the ecliptic from the first point of Aries. We had also declination
reckoned from the equator of the earth prolonged to the stars, and
right ascension reckoned along the equator from the first point of
Aries.
The pole of the heavens or of the ecliptic, then, we must regard as
practically, but not absolutely, fixed; but the pole of the earth's
equator is not fixed, it slowly moves round the pole of the heavens.
_In consequence of that movement there is a change of declination in a
star's place._
Going back to the diagram (p. 49), we find that the amplitude of a
body rising or setting at Thebes or anywhere else depends upon its
declination; so that if from any cause the declination of a star
changes, its amplitude must change.
That is the first point where we meet with difficulty, because if
the amplitude changes it is the same as saying that the place of
star-rising or star-setting changes; that is, a star which rises in the
east in a certain amplitude this year will change its amplitude at some
future time.
In the last chapter I referred to one of the difficulties of modern
inquiries into the orientation of ancient temples, which arises from
the fact that the sun has not always, at the solstices, risen or set at
exactly the same points of the horizon. We now find ourselves face to
face with the fact that the stars do not rise or set at the same points
century after century. We saw that the change in the position of the
sun on the horizon at the solstices is due to a very small change of
obliquity of the ecliptic, so that in a matter of something like 6,000
years the position of the sun at sunrise and sunset on the horizon
may be varied by, roughly speaking, 1 degree. But in the case of the
stars the matter is very much more serious, because in the course of
something like 13,000 years the rising-or setting-places of a star may
vary by something like 47° along the horizon north or south.
So that in the cases both of sun and stars there is no real fixity in
the places of rising or setting, although of course those who made the
first observations and built the first temples were not in a position
to know this.
The real cause of this precessional movement which causes the stars to
change their places lies in the fact that the earth is not a sphere,
its equatorial diameter being longer than its polar diameter, so that
there is a mass of matter round the equator in excess of what we should
get if the earth were spherical. Suppose that matter to be represented
by a ring. The ring is differently presented to the sun, one part being
nearer than the other, the nearer part being attracted more forcibly.
If we take the point in the ring nearest the sun where there is the
greatest attraction, and draw a line to the opposite point where the
attraction is least, we can show that the case stands in this way: the
sun's pull may be analysed into two forces, one of them represented by
the line joining the centre of the sun and the centre of the ring, and
another at right angles to it let fall from the point most strongly
attracted on to the first line. The question is, what will that force
at right angles do?
The figure below represents a model illustrating the rotation of the
earth on its axis, and the concurrent revolution of the sun round the
earth once a year. To represent the downward force it is perfectly fair
if I add a weight. The moment this is done the axis of the gyroscope
representing the earth's axis, instead of retaining its direction to
the same point as it did before, now describes a circle round the pole
of the heavens.
[Illustration: MODEL ILLUSTRATING THE PRECESSION OF THE EQUINOXES.]
It is now a recognised principle that there is, so to speak, a wobble
of the earth's axis round the pole of the heavens, in consequence of
the attraction of the sun on the nearer point of this equatorial ring
being greater than on the part of the ring further removed from it.
That precessional movement is not quite so simple as it is shown by the
model, because what the sun does in this way is done to a very much
larger extent by the moon, the moon being so very much nearer to us.
In consequence, then, of this luni-solar precession we have a variation
of the points of intersection of the planes of the earth's equator and
of the ecliptic; in consequence of that we have a difference in the
constellations in which the sun is at the time of the solstices and the
equinoxes; and, still more important from our present point of view,
we have another difference, viz., that the declinations, and therefore
the amplitudes, and therefore the places of setting and rising of the
stars, change from century to century.
Now that we have thus become acquainted with the physical cause of
that movement of the earth's axis which gives rise to what is called
the precession of the equinoxes, we have next to enter with somewhat
greater detail into some of the results of the movement.
The change of direction of the axis in space has a cycle of something
between 25,000 and 26,000 years. As it is a question of the change of
the position of the celestial equator, or rather of the pole of the
celestial equator, amongst the stars in relation to the pole of the
heavens, of course the declinations of stars will be changed to a very
considerable extent; indeed, we have seen that the declination of a
star can vary by twice the amount of the obliquity, or say 47°, so
that a star at one time may have zero declination--that is, it may lie
on the equator--and at another it may have a declination 47° N. or S.
Or, again, a star may be the pole star at one particular time, and at
another it will be distant from the pole no less than 47°. Although we
get this enormous change in one equatorial co-ordinate, there would
from this cause alone be practically no change with regard to the
corresponding ecliptic co-ordinate--that is to say, the position of
the star with reference to the earth's movement round the sun. This
movement takes place quite independently of the direction of the
axis, so that while we get this tremendous swirl in declination, the
latitudes of the stars or their distances from the ecliptic north or
south will scarcely change at all.
[Illustration: STAR-MAP. REPRESENTING THE PRECESSIONAL MOVEMENT OF THE
CELESTIAL POLE FROM THE YEAR 4000 B.C. TO THE YEAR 2000 A.D. (_From
Piazzi Smyth._)
_Symbols adapted to represent the magnitudes or brightnesses of the
stars_, _1^{st}._ ⊙, _2^{nd}._ ⬭, _3^{rd}._ ∆, _4^{th}._ ⊡.]
Among other important results of these movements dependent upon
precession we have the various changes in the pole-star from period
to period, due to the various positions occupied by the pole of the
earth's equator. We thus see how in this period of 25,000 years or
thereabouts the pole-stars will change, for a pole-star is merely the
star near the pole of the equator for the time being. At present, as
we all know, the pole-star is in the constellation Ursa Minor. During
the last 25,000 years the pole-stars have been those lying nearest to
a curved line struck from the pole of the heavens with a radius equal
to the obliquity of the ecliptic, which, as we have seen, is liable to
change within small limits; so that about 10,000 or 12,000 years ago
the pole-star was no longer the little star in Ursa Minor that we all
know, but the bright star Vega, in the constellation Lyra. Of course
25,000 years ago the pole-star was practically the same as it is at
present.
Associated with this change in the pole-star, the point of intersection
of the two fundamental planes (the plane of the earth's rotation and
the plane of the earth's revolution) will be liable to change, and the
period will be the same--about 25,000 years. Where these two planes
cut each other we have the equinoxes, because the intersection of the
planes defines for us the vernal and the autumnal equinoxes; when the
sun is highest and lowest half-way between these points we have the
solstices. In a period of 25,000 years the star which is nearest to an
equinox will return to it, and that which is nearest a solstice will
return to it. During the period there will be a constant change of
stars marking the equinoxes and the solstices.
The chief points in the sun's yearly path then will change among the
stars in consequence of this precession. It is perfectly clear that if
we have a means of calculating back the old positions of stars, and
if we have any very old observations, we can help matters very much,
because the old observations--if they were accurately made--would tell
us that such and such a star rose with the sun at the solstice or at
the equinox at some special point of ancient time. If it be possible to
calculate the time at which the star occupied that position with regard
to the sun, we have an astronomical means of determining the time,
within a few years, at which that particular observation was made.
Fortunately, we have such a means of calculation, and it has been
employed very extensively at different periods, chiefly by M. Biot in
France, and quite recently by German astronomers, in calculating the
positions of the stars from the present time to a period of 2000 years
B.C. We can thus determine with a very high degree of accuracy the
latitude, longitude, right ascension, declination, and the relation
of the stars to an equinox, a solstice, or a pole, as far back as we
choose. Since we have the planes of the equator and ecliptic cutting
each other at different points in consequence of the cause which I
have pointed out--the attraction of the sun and moon--we have a fixed
equator and a variable equator depending upon that. In consequence of
the attraction of the planets upon the earth, the plane of the ecliptic
itself is not fixed, so that we have not only a variable equator, but
also a variable ecliptic. What has been done in these calculations is
to determine the relations and the results of these variations.
The calculations undertaken for the special purposes of this book will
be referred to later.
A simpler, though not so accurate a method consists in the use of a
processional globe. In this we have two fixed points at the part of the
globe representing the poles of the heavens, on which the globe may
be rotated; when this is done the stars move absolutely without any
reference to the earth or to the plane of the equator, but purely with
reference to the ecliptic. We have, then, this globe quite independent
of the earth's axis, flow can we make it dependent upon the earth's
axis? We have two brass circles at a distance of 23½° from each
pole of the heavens (north and south); these represent the circle
described by the pole of the earth in the period of 25,000 years. In
these circles are forty-eight holes in which I can fix two additional
clamping screws, and rotate the globe with respect to them by throwing
out of gear the two points which produced the ecliptic revolution.
If I use that part of the brass circle which is occupied by our
present pole-star, we get the apparent revolution of the heavens with
the earth's axis pointing to the pole-star of to-day. If we wish to
investigate the position of things, say 8,000 years ago, we bring the
globe back again to its bearings, and then adjust the screws into the
holes in the brass circles which are proper for that period. When
we have the globe arranged to 6000 years B.C. (_i.e._, 8,000 years
ago), in order to determine the equator at that time all we have to
do is to paint a line on the globe-in some water-colour, by holding a
camel's-hair pencil at the east or west point of the wooden horizon.
That line represents the equator 8,000 years ago. Having that line, of
course, the intersection of the equator with the ecliptic will give us
the equinoxes, so that we may affix a wafer to represent the vernal
equinox. Or if we take that part of the ecliptic which is nearest
to the North Pole, and, therefore, the N. declination of which is
greatest, viz., 23½° N., we have there the position of the sun at
the summer solstice, and 23½° S. will give us the position of the
sun at the winter solstice. So by means of such a globe as this it is
possible to determine roughly the position of the equator among the
stars, and note those four important points in the solar year, the two
equinoxes and the two solstices. I have taken a period of 8,000 years,
but I might just as easily have taken a greater or a smaller one. By
means of this arrangement, therefore, we can determine within a very
small degree of error, without any laborious calculations, the distance
of a star north or south of the equator, _i.e._, its declination, at
any point of past or future time.
The positions thus found, say, for intervals of 500 years, may be
plotted on a curve, so that we can, with a considerable amount of
accuracy, obtain the star's place for any year. Thus the globe may be
made to tell us that in the year 1000 A.D. the declination of Fomalhaut
was 35° S., in 1000 B.C. it was 42°, in 2000 it was about 44°, in 4000
it was a little over 42° again, but in 6000 B.C. it had got up to about
33°, and in 8000 B.C. to about 22°.
The curve of Capella falls from 41° N. at 0 A.D. to 10° at 5500 B.C.,
so we have in these 5500 years in the case of this star run through a
large part of that variation to which I have drawn attention.
I have ascertained that the globe is a very good guide indeed within
something like 1° of declination. Considering the difficulty of the
determination of amplitudes in the case of buildings, it is clear that
the globe may be utilised with advantage, at all events to obtain a
first approximation.