what at any rate is not based on _à priori_ considerations, yet
manifestly we may expect to find evidence as we proceed which shall
either strengthen our opinion on this point, or show it to be unsound.
We are going to make this astronomical purpose the starting-point for a
series of _à priori_ considerations, each to be tested by whatever
direct evidence may be available; and it is practically certain that if
we have thus started in an entirely wrong direction, we shall before
long find out our mistake. At least we shall do so, if we start with the
desire to find out as much of the truth as we can, and not with the
determination to see only those facts which point in the direction along
which we have set out, overlooking any which seem to point in a
different direction. We need not necessarily be in the wrong track
because of such seeming indications. If we are on the right track, we
shall see things more clearly as we proceed; and it may be that evidence
which at first seems to accord ill with the idea that we are progressing
towards the truth, may be found among the most satisfactory evidence
obtainable. But we must in any case note such evidence, even at the time
when it seems to suggest that we are on the wrong track. We may push on,
nevertheless, to see how such evidence appears a little later. But we
must by no means forget its existence. So only can we hope to reach the
truth or a portion of the truth, instead of merely making out a good
case for some particular theory.
We start, then, with the assumption that the great pyramid, called the
Pyramid of Cheops, was built for this purpose, _inter alia_, to enable
men to make certain astronomical observations with great accuracy; and
what we propose to do is to inquire what would be done by men having
this purpose in view, having, as the pyramid builders had, (1) a fine
astronomical site, (2) the command of enormous wealth, (3) practically
exhaustless stores of material, and (4) the means of compelling many
thousands of men to labour for them.
Watching the celestial bodies hour by hour, day by day, and year by
year, the observer recognizes certain regions of the heavens which
require special attention, and certain noteworthy directions both with
respect to the horizon and to elevation above the horizon.
For instance, the observer perceives that the stars, which are in many
respects the most conveniently observable bodies, are carried round, as
if they were rigidly attached to a hollow sphere, carried around an axis
passing through the station of the observer (as through a centre) and
directed towards a certain point in the dome of the heavens. That point,
then, is one whose direction must not only be ascertained, but must be
in some way or other indicated. Whatever the nature of an astronomer's
instruments or observatory, whether he have but a few simple
contrivances in a structure of insignificant proportions, or the most
perfect instruments in a noble edifice of most exquisite construction
and of the utmost attainable stability, he must in every case have the
position of the pole of the heavens clearly indicated in some way or
other. Now, the pole of the heavens is a point lying due north, at a
certain definite elevation above the horizon. Thus the first
consideration to be attended to by the builder of any sort of
astronomical observatory, is the determination of the direction of the
true north (or the laying down of a true north-and-south line), while
the second is the determination, and in some way or other the indication
of the angle of elevation above the north point, at which the true pole
of the heavens may lie.
To get the true north-and-south line, however, the astronomer would be
apt at first, perhaps, rather to make mid-day observations than to
observe the stars at night. It would have been the observation of these
which first called his attention to the existence of a definite point
round which all the stars seem to be carried in parallel circles; but he
would very quickly notice that the sun and the moon, and also the five
planets, are carried round the same polar axis, only differing from the
stars in this: that, besides being thus carried round with the celestial
sphere, they also move upon that sphere, though with a motion which is
very slow compared with that which they derive from the seeming motion
of the sphere itself. Now, among these bodies the sun and moon possess a
distinct advantage over the stars. A body illuminated by either the sun
or the moon throws a shadow, and thus if we place an upright pointed rod
in sunlight or moonlight, and note where the shadow of the point lies,
we know that a straight line from the point to the shadow of the point
is directed exactly towards the sun or the moon, as the case may be.
Leaving the moon aside as in other respects unsuitable, for she only
shines with suitable lustre in one part of each month, we have in the
sun's motions a means of getting the north-and-south line by thus noting
the position of the shadow of a pointed upright. For being carried
around an inclined axis directed northwards, the sun is, of course,
brought to his greatest elevation on any given day when due south. So
that if we note when the shadow of an upright is shortest on any day, we
know that at that moment the sun is at his highest or due south; and the
line joining the centre of the upright's base with the end of the shadow
at that instant lies due north-and-south.
But though theoretically this method is sufficient, it is open, in
practice, to a serious objection. The sun's elevation, when he is nearly
at his highest, changes very slowly; so that it is difficult to
determine the precise moment when the shadow is shortest. But the
direction of the shadow is steadily changing all the time that we thus
remain in doubt whether the sun's elevation has reached its maximum or
not. We are apt, then, to make an error as to time, which will result in
a noteworthy error as to the direction of the north-and-south line.
For this reason, it would be better for any one employing this shadow
method to take two epochs on either side of solar noon, when the sun was
at exactly the same elevation, or the shadow of exactly the same
length,--determining this by striking out a circle around the foot of
the upright, and observing where the shadow's point crossed this circle
before noon in drawing nearer to the base, and after noon in passing
away from the base. These two intersections with the circle necessarily
lie at equal distances from the north-and-south line, which can thus be
more exactly determined than by the other method, simply because the end
of the shadow crosses the circle traced on the ground at moments which
can be more exactly determined than the moment when the shadow is
shortest.
Now, we notice in this description of methods which unquestionably were
followed by the very earliest astronomers, one circumstance which
clearly points to a feature as absolutely essential in every
astronomical observing station. (I do not say "observatory," for I am
speaking just now of observations so elementary that the word would be
out of place.) The observer must have a perfectly flat floor on which to
receive the shadow of the upright pointer. And not only must the floor
be flat, but it must also be perfectly horizontal. At any rate, it must
not slope down either towards the east or towards the west, for then the
shadows on either side of the north-and-south line would be unequal. And
though a slope towards north or south would not affect the equality of
such shadows, and would therefore be admissible, yet it would clearly be
altogether undesirable; since the avoidance of a slope towards east or
west would be made much more difficult if the surface were tilted,
however slightly, towards north or south. Apart from this, several other
circumstances make it extremely desirable that the surface from which
the astronomers make their observations should be perfectly horizontal.
In particular, we shall see presently that the exact determination of
elevations above the eastern and western horizons would be very
necessary even in the earliest and simplest methods of observation, and
for this purpose it would be essential that the observing surface should
be as carefully levelled in a north-and-south as in an east-and-west
direction.
We should expect to find, then, that when the particular stage of
astronomical progress had been reached, at which men not only perceived
the necessity of well-devised buildings for astronomical observation,
but were able to devote time, labour, and expense to the construction of
such buildings, the first point to which they would direct their
attention would be the formation of a perfectly level surface, on which
eventually they might lay down a north-and-south or true meridional
line.
Now, of the extreme care with which this preliminary question of level
was considered by the builders of the great pyramid, we have singularly
clear and decisive evidence. For all around the base of the pyramid
there was a pavement, and we find the builders not only so well
acquainted with the position of the true horizontal plane at the level
of this pavement, but so careful to follow it (even as respects this
pavement, which, be it noticed, was only, in all probability, a
subsidiary and quasi-ornamental feature of the building), that the
pavement "was varied in thickness at the rate of about an inch in 100
feet to make it absolutely level, which the rock was not."[43]
But now with regard to the true north-and-south direction, although the
shadow method, carried out on a truly level surface, would be
satisfactory enough for a first rough approximation, or even for what
any but astronomers would regard as extreme accuracy, it would be open
to serious objections for really exact work. These objections would
have become known to observers long before the construction of the
pyramid was commenced, and would have been associated with the
difficulties which suggested, I think, the idea itself of constructing
such an edifice.
Supposing an upright pointed post is set up, and the position of the end
of the shadow upon a perfectly level surface is noted; then whatever use
we intend to make of this observation, it is essential that we should
know the precise position of the centre of the upright's base, and also
that the upright should be truly vertical. Otherwise we have only
exactly obtained the position of one end of the line we want, and to
draw the line properly we ought as exactly to know the position of the
other end. If we want _also_ to know the true position of a line joining
the point of the upright and the shadow of this point, we require to
know the true height of the upright. And even if we have these points
determined, we still have not a _material_ line from the point of the
upright to the place of its shadow. A cord or chain from one point to
the other would be curved, even if tightly stretched, and it would not
be tightly stretched, if long, without either breaking or pulling over
the upright. A straight bar of the required length could not be readily
made or used: if stout enough to lie straight from point to point it
would be unwieldy, if not stout enough so that it bent under its own
weight it would be useless.
Thus the shadow method, while difficult of application to give a true
north-and-south horizontal line, would fail utterly to give material
indications of the sun's elevation on particular days, without which it
would be impossible to obtain in this manner any material indications of
the position of the celestial pole.
A natural resource, under these circumstances--at least a natural
resource for astronomers who could afford to adopt the plan--would be to
build up masses of masonry, in which there should be tubular holes or
tunnellings pointing in certain required directions. In one sense the
contrivance would be clumsy, for a tunnelling once constructed, would
not admit of any change of position, nor even allow of any save very
limited changes in the direction of the line of view through them. In
fact, the more effective a tunnelling would be in determining any
particular direction, the less scope, of course, would it afford for any
change in the direction of a line of sight along it. So that the
astronomical architect would have to limit the use of this particular
method to those cases in which great accuracy in obtaining a direction
line and great rigidity in the material indication of that line's
position were essential or at least exceedingly desirable. Again, in
some cases presently to be noticed, he would require, not a tubing
directed to some special fixed point in the sky, but an opening
commanding some special range of view. Yet again it would be manifestly
well for him to retain, whenever possible, the power of using the shadow
method in observing the sun and moon; for this method in the case of
bodies varying their position on the celestial sphere, not merely with
respect to the cardinal points, would be of great value. Its value would
be enhanced if the shadows could be formed by objects and received on
surfaces holding a permanent position.
We begin to see some of the requirements of an astronomical building
such as we have supposed the earlier observers to plan.
First, such a building must be large, to give suitable length to the
direction lines, whether along edges of the building or along tubular
passages or tunnellings within it. Secondly, it must be massive in order
that these edges and passages might have the necessary stability and
permanence. Thirdly, it must be of a form contributing to such
stability, and as height above surrounding objects (even hills lying at
considerable distances) would be a desirable feature, it would be proper
to have the mass of masonry growing smaller from the base upwards.
Fourthly, it must have its sides carefully oriented, so that it must
have either a square or oblong base with two sides lying exactly north
and south, and the other two lying exactly east and west. Fifthly, it
must have the direction of the pole of the heavens either actually
indicated by a tunnelling of some sort pointed directly polewards, or
else inferable from a tunnelling pointing upon a suitable star close to
the true pole of the heavens.
The lower part of a pyramid would fulfil the conditions required for the
stability of such a structure, and a square or oblong form would be
suitable for the base of such a pyramid. We must not overlook the fact
that a complete pyramid would be utterly unsuitable for an astronomical
edifice. Even a pyramid built up of layers of stone and continued so far
upwards that the uppermost layer consisted of a single massive stone,
would be quite useless as an observatory. The notion which has been
entertained by some fanciful persons, that one purpose which the great
pyramid was intended to subserve, was to provide a raised small platform
high above the general level of the soil, in order that astronomers
might climb night after night to that platform, and thence make their
observations on the stars, is altogether untenable. Probably no fancy
respecting the pyramids has done more to discredit the astronomical
theory of these structures than has this ridiculous notion; because even
those who are not astronomers and therefore little familiar with the
requirements of a building intended for astronomical observation,
perceive at once the futility of any such arrangement, and the enormous,
one may almost say the infinite disproportion between the cost at which
the raised small platform would have been obtained, and the small
advantage which astronomers would derive from climbing up to it instead
of observing from the ground level. Yet we have seen this notion not
only gravely advanced by persons who are to some degree acquainted with
astronomical requirements, but elaborately illustrated. Thus, in
Flammariou's "History of the Heavens," there is a picture representing
six astronomers in eastern garb, perched in uncomfortable attitudes on
the uppermost steps of a pyramid, whence they are staring hard at a
comet, naturally without the slightest opportunity of determining its
true position in the sky, since they have no direction lines of any sort
for their guidance. Apart from this, their attention is very properly
directed in great part to the necessity of preserving their equilibrium.
In only one point in fact does this picture accord with à priori
probabilities--namely, in the great muscular development of these
ancient observers. They are perfectly herculean, and well they might be,
if night after night they had to observe the celestial bodies from a
place so hard to reach, and where attitudes so awkward must be
maintained during the long hours of the night.
It is perfectly clear, and is in fact one of the chief difficulties of
the astronomical theory of the pyramids, that it would only be when
these buildings were as yet incomplete that they could subserve any
useful astronomical purposes; nevertheless we must not on this account
suffer ourselves at this early stage of our inquiry to be diverted from
the astronomical theory by what must be admitted to be a very strong
argument against it. We have seen that there is such decisive and even
demonstrative evidence in favour of the theory that the pyramids were
not oriented in a general, still less in a merely casual, manner, and
this is, in reality, such clear evidence of their astronomical
significance, that we must pass further on upon the line of reasoning
which we have adopted--prepared to turn back indeed if absolutely
convincing evidence should be found against the theory of the
astronomical _purpose_ of the pyramids, but anticipating rather that, on
a close inquiry, a means of obviating this particular objection may
before long be found.
Let us suppose, then, that astronomers have determined to erect a
massive edifice, on a square or oblong base properly oriented,
constructing within this edifice such tubular openings as would be most
useful for the purpose of indicating the true directions of certain
celestial objects at particular times and seasons.
Before commencing so costly a structure they would be careful to select
the best possible position for it, not only as respects the nature of
the ground, but also as respects latitude. For it must be remembered
that, from certain parts of the earth, the various points and circles
which the astronomer recognizes in the heavens occupy special positions
and fulfil special relations.
So far as conditions of the soil, surrounding country, and so forth are
concerned, few positions could surpass that selected for the great
pyramid and its companions. The pyramids of Ghizeh are situated on a
platform of rock, about 150 feet above the level of the desert. The
largest of them, the Pyramid of Cheops, stands on an elevation free all
around, insomuch that less sand has gathered round it than would
otherwise have been the case. How admirably suited these pyramids are
for observing stations is shown by the way in which they are themselves
seen from a distance. It has been remarked by every one who has seen
the pyramids that the sense of sight is deceived in the attempt to
appreciate their distance and magnitude. "Though removed several leagues
from the spectator, they appear to be close at hand; and it is not until
he has travelled some miles in a direct line towards them, that he
becomes sensible of their vast bulk and also of the pure atmosphere
through which they are viewed."
With regard to their astronomical position, it seems clear that the
builders intended to place the great pyramid precisely in latitude 30°,
or, in other words, in that latitude where the true pole of the heavens
is one-third of the way from the horizon to the point overhead (the
zenith), and where the noon sun at true spring or autumn (when the sun
rises almost exactly in the east, and sets almost exactly in the west)
is two-thirds of the way from the horizon to the point overhead. In an
observatory set exactly in this position, some of the calculations or
geometrical constructions, as the case may be, involved in astronomical
problems, are considerably simplified. The first problem in Euclid, for
example, by which a triangle of three equal sides is made, affords the
means of drawing the proper angle at which the mid-day sun in spring or
autumn is raised above the horizon, and at which the pole of the heavens
is removed from the point overhead. Relations depending on this angle
are also more readily calculated, for the very same reason, in fact,
that the angle itself is more readily drawn. And though the builders of
the great pyramid must have been advanced far beyond the stage at which
any difficulty in dealing directly with other angles would be involved,
yet they would perceive the great advantage of having one among the
angles entering into their problems thus conveniently chosen. In our
time, when by the use of logarithmic and other tables, all calculations
are greatly simplified, and when also astronomers have learned to
recognize that no possible choice of latitude would simplify their
labours (unless an observatory could be set up at the North Pole itself,
which would be in other respects inconvenient), matters of this sort are
no longer worth considering, but to the mathematicians who planned the
great pyramid they would have possessed extreme importance.
[Illustration: Fig. 1.]
To set the centre of the pyramid's future base in latitude 30°, two
methods could be used, both already to some degree considered--the
shadow method, and the Pole-star method. If at noon, at the season when
the sun rose due east and set due west, an upright A C were found to
throw a shadow C D, so proportioned to A C that A C D would be one-half
of an equal-sided triangle, then, theoretically, the point where this
upright was placed would be in latitude 30°. As a matter of fact it
would not be, because the air, by bending the sun's rays, throws the sun
apparently somewhat above his true position. Apart from this, at the
time of true spring or autumn, the sun does not seem to rise due east,
or set due west, for he is raised above the horizon by atmospheric
refraction, before he has really reached it in the morning, and he
remains raised above it after he has really passed below--understanding
the word "really" to relate to his actual geometrical direction. Thus,
at true spring and autumn, the sun rises slightly to the north of east,
and sets slightly to the north of west. The atmospheric refraction is
indeed so marked, as respects these parts of the sun's apparent course,
that it must have been quickly recognized. Probably, however, it would
be regarded as a peculiarity only affecting the sun when close to the
horizon, and would be (correctly) associated with his apparent change of
shape when so situated. Astronomers would be prevented in this way from
using the sun's horizontal position at any season to guide them with
respect to the cardinal points, but they would still consider the sun,
when raised high above the horizon, as a suitable astronomical index (so
to speak), and would have no idea that even at a height of sixty degrees
above the horizon, or seen as in direction D A, Fig. 1, he is seen
appreciably above his true position.
Adopting this method--the shadow method--to fix the latitude of the
pyramid's base, they would conceive the sun was sixty degrees above the
horizon at noon, at true spring or autumn, when in reality he was
somewhat below that elevation. Or, in other words, they would conceive
they were in latitude 30° north, when in reality they were farther north
(the mid-day sun at any season sinking lower and lower as we travel
farther and farther north). The actual amount by which, supposing their
observations exact, they would thus set this station north of its proper
position, would depend on the refractive qualities of the air in Egypt.
But although there is some slight difference in this respect between
Egypt and Greenwich, it is but small; and we can determine from the
Greenwich refraction tables, within a very slight limit of error, the
amount by which the architects of the great pyramid would have set the
centre or the base north of latitude 30°, if they had trusted solely to
the shadow method. The distance would have been as nearly as possible
1125 yards, or say three furlongs.