THE TWO HORIZONS.
It is not only of the first importance for our subject, but of great
interest in itself, to study some of the astronomical problems
connected with this horizon worship, which in the previous chapter we
have found to be common to the early peoples of India and Egypt.
[Illustration: APPARENT MOVEMENT OF THE STARS TO AN OBSERVER AT THE
NORTH POLE.]
We must be perfectly clear before we go further what this horizon
really is, and for this some diagrams are necessary.
The horizon of any place is the circle which bounds our view of the
earth's surface, along which the land (or sea) and sky appear to meet.
We have to consider the relation of the horizon of any place to the
apparent movements of celestial bodies at that place.
We know, by means of the demonstration afforded by Foucault's pendulum,
that the earth rotates on its axis, but this idea was, of course,
quite foreign to these early peoples. Since the earth rotates with
stars, infinitely removed, surrounding it on all sides, the apparent
movements of the stars will depend very much upon the position we
happen to occupy on the earth: this can be made quite clear by a few
diagrams.
[Illustration: APPARENT MOVEMENT OF THE STARS TO AN OBSERVER AT THE
EQUATOR.]
An observer at the North Pole of the earth, for instance, would see
the stars moving round in circles parallel to the horizon. No star
would either rise or set--one half of the heavens would be always
visible above his horizon, and the other half invisible; whereas an
observer at the South Pole would see that half of the stars invisible
to the observer at the northern one, because it was the half below the
N. horizon. If the observer be on the equator, the movements of the
stars all appear as indicated in the above diagram--that is, all the
stars will rise and set, and each star in turn will be half its time
above the horizon, and half its time below it. But if we consider the
position of an observer in middle latitude, say in London, we find that
some stars will always be above the horizon, some always below--that
is, they will neither rise nor set. All other stars will both rise and
set, but some of them will be above the horizon for a long time and
below for a short time, whereas others will be a very short time above
the horizon and a long time below it.
[Illustration: THE CELESTIAL SPHERE VIEWED FROM A MIDDLE LATITUDE.
AN OBLIQUE SPHERE.]
At _O_ we imagine an observer to be in latitude 45° (that is, half-way
between the equator in latitude 0°, and the North Pole in latitude
90°), hence the North Celestial Pole will be half-way between the
zenith and the horizon; and close to the pole he will see the stars
describing circles, inclined, however, and not retaining the same
distance from the horizon. As the eye leaves the pole, the stars rise
and set obliquely, describe larger circles, gradually dipping more and
more under the horizon, until, when the celestial equator is reached,
half their journey is performed below it. Still going south, we find
the stars rising less and less above the horizon, until, as there were
northern stars that never dip below the horizon, so there are southern
stars which never appear above it. _D D'_ shows the apparent path of
a circumpolar star; _B B′ B″_ the path and rising and setting points
of an equatorial star; _C C′C″_ and _A A′ A″_ those of stars of
mid-declination, one north and the other south.
[Illustration: A TERRESTRIAL GLOBE WITH WAFER ATTACHED TO SHOW THE
VARYING CONDITIONS OF OBSERVATION IN A MIDDLE LATITUDE.]
Wherever we are upon the earth we always imagine that we are on the
top of it. The idea held by all the early peoples was that the earth
was an extended plain: they imagined that the land that they knew and
just the surrounding lands were really in the centre of the extended
plain. Plato, for instance, as we have seen, was content to put the
Mediterranean and Greece upon the top of his cube, and Anaximander
placed the same region at the top of his cylinder.
We can very conveniently study the conditions of observation at the
poles of the earth, the equator, and some place in middle latitude, by
using an ordinary terrestrial globe. The wooden horizon of the globe
is parallel to the horizon of a place at the top of the globe, which
horizon we can represent by a wafer. In this way we can get a very
concrete idea of the different relations of the observer's horizon in
different latitudes to the apparent paths of the stars.
We have next to deal with the astronomical relations of the horizon
of any place in connection with the worship of the sun and stars at
the times of rising or setting, when, of course, they are on or near
the horizon; and in order to bring this matter nearer to the ancient
monuments, it will be convenient to study this question for Thebes,
where they exist in greatest number and have been most accurately
described.
To adjust things properly we must rectify the globe to the latitude of
25° 40′ N., or, in other words, incline the axis of the globe at that
angle to the wooden horizon.
It will be at once seen that the inclination of the axis to the horizon
is very much less than in the case of London. Since all the stars
which pass between the North Pole and the horizon cannot set, all
their apparent movement will take place above the horizon. All the
stars between the horizon and the South Pole will never rise. Hence,
stars within the distance of 25° from the North Pole will never set
at Thebes, and those stars within 25° of the South Pole will never
be visible there. At any place the latitude and the elevation of the
pole are the same. It so happens that all these places with which
archæologists have to do in studying the history of early peoples,
Egypt, Babylonia, Assyria, China, Greece, &c., are in middle latitudes,
therefore we have to deal with bodies in the skies, which do set, and
bodies which do not; and the elevation of the pole is neither very
great nor very small. In each different latitude the inclination of the
equator to the horizon, as well as the elevation of the pole, will
vary, but there will be a strict relationship between the inclination
of the equator at each place and the elevation of the pole. Except at
the poles themselves the equator will cut the horizon due east and due
west. Therefore every celestial body which rises or sets to the north
of the equator will cut the horizon between the east or west point and
the north point; those bodies which do not set will, of course, not cut
the horizon at all.
The sun, and stars near the equator, in such a latitude as that of
Thebes, will appear to rise or set at no very considerable angle to the
vertical; but when we deal with stars rising or setting near to the
north or south points of the horizon they will seem to skim along the
horizon instead of rising or setting vertically.
Now it will at once be obvious that there must be a strict law
connecting the position of the sun (or a star) with its place of rising
or setting. Stars at the same distance from either of the celestial
poles will rise or set at the same point of the horizon, and if a star
does not change its place in the heavens it will always rise or set in
the same place.
Here it will be convenient to introduce one or two technical terms.
Every celestial body, whether we deal with the sun, moon, planet,
or star, occupies at any moment a certain place in the sky, partly,
though not wholly, defined by what we term its declination, _i.e._,
its distance from the celestial equator. This declination is one of
the two co-ordinates which are essential for enabling us to state
accurately the position of any body on the celestial vault; and we must
quite understand that if all these bodies rise and set, and rise and
set visibly, the place of their rising or setting must be very closely
connected with their declination. Bodies with the same declination will
rise at the same points of the horizon. When the declination changes,
of course the body will rise and set in different points of the horizon.
Next we define points on the horizon by dividing the whole
circumference into four quadrants of 90° each = 360°, so that we can
have _azimuths_ of 90° from the north or south points to the east and
west points.
Azimuths are not always reckoned in this way, navigators preferring
one method, while astronomers prefer another. Thus azimuth may also
be taken as the distance measured in degrees from the south point in
a direction passing through the west, north, and east points. On this
system, a point can have an azimuth varying from 0° to 360°.
[Illustration: SHOWING AMPLITUDES RECKONED FROM THE EAST OR WEST POINTS
TO N.P., NORTH POINT OF HORIZON, AND S.P., SOUTH POINT OF HORIZON.]
It is next important to define the term _amplitude_. The amplitude of
a body on the horizon is its distance north and south of the east and
west points; it is always measured to the nearest of these two latter
points, so that its greatest value can never exceed 90°. For instance,
the south point itself would have an amplitude of 90° south of west
(generally written W. 90° S.), or 90° south of east (E. 90° S.), while
a point 2° to the westward of south would have an amplitude of W. 88°
S., and not E. 92° S.
We can say then that a star of a certain declination will rise or set
at such an _azimuth_, if we reckon from the N. point of the horizon, or
at such an amplitude if we reckon from the _equator_. This will apply
to both north and south declinations.
The following table gives for Thebes the amplitudes of rising or
setting (north or south) of celestial bodies having declinations from
0° to 64°; bodies with higher declinations than 64° never set at Thebes
if they are north, or never rise if they are south, as the latitude
(and therefore the elevation of the pole) there is nearly 26°.
AMPLITUDES AT THEBES.
─────────┬──────────┬────────┬─────────┬────────┬──────────
Declin │ Ampli │ Declin │ Ampli │ Declin │ Ampli
-ation. │ -tude. │ -ation.│ -tude. │ -ation.│ -tude.
─────────┼──────────┼────────┼─────────┼────────┼──────────
° │ ° ′ │ ° │ ° ′ │ ° │ ° ′
0 │ 0 0 │ 22 │ 24 33 │ 44 │ 50 25
1 │ 1 7 │ 23 │ 25 41 │ 45 │ 51 41
2 │ 2 13 │ 24 │ 26 49 │ 46 │ 52 57
3 │ 3 20 │ 25 │ 27 58 │ 47 │ 54 14
4 │ 4 26 │ 26 │ 29 6 │ 48 │ 55 32
5 │ 5 33 │ 27 │ 30 15 │ 49 │ 56 51
6 │ 6 40 │ 28 │ 31 23 │ 50 │ 58 12
7 │ 7 47 │ 29 │ 32 32 │ 51 │ 59 34
8 │ 8 53 │ 30 │ 33 41 │ 52 │ 60 58
9 │ 9 59 │ 31 │ 34 51 │ 53 │ 62 23
10 │ 11 6 │ 32 │ 36 1 │ 54 │ 63 51
11 │ 12 13 │ 33 │ 37 11 │ 55 │ 65 21
12 │ 13 20 │ 34 │ 38 21 │ 56 │ 66 54
13 │ 14 27 │ 35 │ 39 31 │ 57 │ 68 31
14 │ 15 34 │ 36 │ 40 42 │ 58 │ 70 12
15 │ 16 41 │ 37 │ 41 53 │ 59 │ 71 59
16 │ 17 49 │ 38 │ 43 5 │ 60 │ 73 55
17 │ 18 56 │ 39 │ 44 17 │ 61 │ 76 1
18 │ 20 3 │ 40 │ 45 30 │ 62 │ 78 25
19 │ 21 10 │ 41 │ 46 43 │ 63 │ 81 19
20 │ 22 17 │ 42 │ 47 56 │ 64 │ 85 42
21 │ 23 25 │ 43 │ 49 10 │ │
─────────┴──────────┴────────┴─────────┴────────┴──────────
The absolute connection, then, between the declination of a heavenly
body and the amplitude at which it rises and sets is obvious from the
above table: given the declination we know the amplitude; given the
amplitude we know the declination.
Suppose we were dealing with a sea horizon: all the bodies rising or
setting at the same instant of time would be in a great circle round
the heavens, for the plane of the sensible horizon is parallel to the
geocentric one.
But there are some additional points to be borne in mind. Ordinarily we
should determine that the amplitude being so and so, the declination
of the body which rose or set with that amplitude would be so and so,
taking the horizon to be an all-round horizon like a sea one. But that
would not be quite true, because we generally see the sun, to take an
instance, some little time before it really rises and after it has set,
owing to refraction. So that if we see the sun setting, say, north of
west, we know that when we see it setting it appears really a little
further to the north than it actually was at the moment of true sunset,
because refraction gives us the position of the sun just below the
true horizon. That is one point that we have to consider. Another is
that, of course, we as a rule do not deal with sea horizons. Here we
find a hill, there some other obstacle; so that it is necessary to make
a correction depending on the height of the hill or other obstacle
above the sea-or true-horizon at the place. Only when we take these
things completely into consideration, can we determine absolutely the
declination, or distance from the celestial equator, of the body at the
moment of rising or setting. Still, it is worth while noting that when
only approximations are required, the refraction-and hill-corrections
have a tendency to neutralise each other in the northern hemisphere.
Refraction will tend to carry the sunrise or sunset place more to the
north, hills will cause the body to appear to rise or set more to the
south.
[Illustration: DIAGRAM SHOWING THE VARIOUS AMPLITUDES AT WHICH STARS OF
DIFFERENT DECLINATIONS RISE AND SET IN DIFFERENT LATITUDES.]
It is important to point out that these corrections vary very
considerably in importance according to the declination of the star
with which we have to deal. With a high north or south declination the
amplitude increases very rapidly, and the more it increases the more
the corrections for refraction and elevation above the true horizon to
which I have referred become of importance. In all cases the correction
has to be made so that the amplitude will be increased or decreased
from the true amplitude by this effect of refraction, according as the
body--whether sun or star--is seen to the north or south of the equator.
In the diagram given on page 49, the various amplitudes are shown at
which bodies of different declinations appear to rise and set in places
with latitudes ranging from 19° to 51° N. It is a diagram to which
frequent reference will be made in the sequel.