pole, in order to determine the elevation of the true pole of the
heavens, they would be in a similar way exposed to error arising from
the effects of atmospheric refraction. They would proceed probably
somewhat in this wise:--Using any kind of direction lines, they would
take the altitude of their Polar star (1) when passing immediately under
the pole, and (2) when passing immediately above the pole. The mean of
the altitudes thus obtained would be the altitude of the true pole of
the heavens. Now, atmospheric refraction affects the stars in the same
way that it affects the sun, and the nearer a star is to the horizon,
the more it is raised by atmospheric refraction. The Pole-star in both
its positions--that is when passing below the pole, and when passing
above that point--is raised by refraction, rather more when below than
when above; but the estimated position of the pole itself, raised by
about the mean of these two effects, is in effect raised almost exactly
as much as it would be if it were itself directly observed (that is, if
a star occupied the pole itself, instead of merely circling close round
the pole). We may then simplify matters by leaving out of consideration
at present all questions of the actual Pole-star in the time of the
pyramid builders, and simply considering how far they would have set the
pyramid's base in error, if they had determined their latitude by
observing a star occupying the position of the true pole of the heavens.
They would have endeavoured to determine where the pole appears to be
raised exactly thirty degrees above the horizon. But the effect of
refraction being to raise every celestial object above its true
position, they would have supposed the pole to be raised thirty degrees,
when in reality it was less raised than this. In other words, they would
have supposed they were in latitude 30°, when, in reality, they were in
some lower latitude, for the pole of the heavens rises higher and higher
above the horizon as we pass to higher and higher latitudes. Thus they
would set their station somewhat to the south of latitude 30°, instead
of to the north, as when they were supposed to have used the shadow
method. Here again we can find how far they would set it south of that
latitude. Using the Greenwich refraction table (which is the same as
Bessel's), we find that they would have made a much greater error than
when using the other method, simply because they would be observing a
body at an elevation of about thirty degrees only, whereas in taking the
sun's mid-day altitude in spring or autumn, they would be observing a
body at twice as great an elevation. The error would be, in fact, in
this case, about 1 mile 1512 yards.
It seems not at all unlikely that astronomers, so skilful and ingenious
as the builders of the pyramid manifestly were, would have employed both
methods. In that case they would certainly have obtained widely
discrepant results, rough as their means and methods must unquestionably
have been, compared with modern instruments and methods. The exact
determination from the shadow plan would have set them 1125 yards to the
north of the true latitude; while the exact determination from the
Pole-star method would have set them 1 mile 1512 yards south of the true
latitude. Whether they would thus have been led to detect the effect of
atmospheric refraction on celestial bodies high above the horizon may be
open to question. But certainly they would have recognized the action of
some cause or other, rendering one or other method, or both methods,
unsatisfactory If so, and we can scarcely doubt that this would actually
happen (for certainly they would recognize the theoretical justice of
both methods, and we can hardly imagine that having two available
methods, they would limit their operations to one method only), they
would scarcely see any better way of proceeding than to take a position
intermediate between the two which they had thus obtained. Such a
position would lie almost exactly 1072 yards south of true latitude 30°
north.
Whether the architects of the pyramid of Cheops really proceeded in this
way or not, it is certain that they obtained a result corresponding so
well with this that if we assume they really did intend to set the base
of the pyramid in latitude 30°, we find it difficult to persuade
ourselves that they did not follow some such course as I have just
indicated--the coincidence is so close considering the nature of the
observations involved. According to Professor Piazzi Smyth, whose
observational labours in relation to the great pyramid are worthy of all
praise, the centre of the base of this pyramid lies about 1 mile 568
yards south of the thirtieth parallel of latitude. This is 944 yards
north of the position they would have deduced from the Pole-star method;
1 mile 1693 yards south of the position they would have deduced from the
shadow method; and 1256 yards south of the mean position between the two
last-named. The position of the base seems to prove beyond all
possibility of question that the shadow method was not the method on
which sole or chief reliance was placed, though this method must have
been known to the builders of the pyramid. It does not, however, prove
that the star method was the only method followed. A distance of 944
yards is so small in a matter of this sort that we might fairly enough
assume that the position of the base was determined by the Pole-star
method. If, however, we supposed the builders of the pyramid to have
been exceedingly skilful in applying the methods available to them, we
might not unreasonably conclude from the position of the pyramid's base
that they used both the shadow method and the Pole-star method, but
that, recognizing the superiority of the latter, they gave greater
weight to the result of employing this method. Supposing, for instance,
they applied the Pole-star method three times as often as the shadow
method, and took the mean of all the results thus obtained, then the
deduced position would lie three times as far from the northern position
obtained by the shadow method as from the southern position obtained by
the Pole-star method. In this case their result, if correctly deduced,
would have been only about 156 yards north of the actual present
position of the centre of the base.
It is impossible, however, to place the least reliance on any
calculation like that made in the last few lines. By _à posteriori_
reasoning such as this one can prove almost anything about the pyramids.
For observe, though presented as _à priori_ reasoning, it is in reality
not so, being based on the observed fact, that the true position lies
more than three times as far from the northerly limit as from the
southern one. Now, if in any other way, not open to exception, we knew
that the builders of the pyramid used both the sun method and the star
method, with perfect observational accuracy, but without knowledge of
the laws of atmospheric refraction, we could infer from the observed
position the precise relative weights they attached to the two methods.
But it is altogether unsafe, or, to speak plainly, it is in the logical
sense a perfectly vicious manner of reasoning, to ascertain first such
relative weights on an assumption of this kind, and having so found
them, to assert that the relation thus detected is a probable one in
itself, and that since, when assumed, it accounts precisely for the
observed position of the pyramid, therefore the pyramid was posited in
that way and no other. It has been by unsound reasoning of this kind
that nine-tenths of the absurdities have been established on which
Taylor and Professor Smyth and their followers have established what may
be called the pyramid religion.
All we can fairly assume as probable from the evidence, in so far as
that evidence bears on the results of _à priori_ considerations, is that
the builders of the great pyramid preferred the Pole-star method to the
shadow method, as a means of determining the true position of latitude
30° north. They seem to have applied this method with great skill
considering the means at their disposal, if we suppose that they took no
account whatever of the influence of refraction. If they took refraction
into account at all they considerably underrated its influence.
Piazzi Smyth's idea that they knew the _precise_ position of the
thirtieth parallel of latitude, and also the _precise_ position of the
parallel, where, owing to refraction, the Pole-star would appear to be
thirty degrees above the horizon, and deliberately set the base of the
pyramid between these limits (not exactly or nearly exactly half-way,
but somewhere between them), cannot be entertained for a moment by any
one not prepared to regard the whole history of the construction of the
pyramid as supernatural. My argument, let me note in passing, is not
intended for persons who take this particular view of the pyramid, a
view on which reasoning could not very well be brought to bear.
If the star method had been used to determine the position of the
parallel of 30° north latitude, we may be certain it would be used also
to orient the building. Probably indeed the very structures (temporary,
of course) by which the final observations for the latitude had been
made, would remain available also for the orientation. These structures
would consist of uprights so placed that the line of sight along their
extremities (or along a tube perhaps borne aloft by them in a slanting
position) the Pole-star could be seen when immediately below or
immediately above the pole. Altogether the more convenient direction of
the two would be that towards the Pole-star when below the pole. The
extremities of these uprights, or the axis of the upraised tube, would
lie in a north-and-south line considerably inclined to the horizon,
because the pole itself being thirty degrees above the horizon, the
Pole-star, whatever star this might be, would be high above the horizon
even when exactly under the pole. No star so far from the pole as to
pass close to the horizon would be of use even for the work of
orientation, while for the work of obtaining the latitude it would be
absolutely essential that a star close to the pole should be used.
A line along the feet of the uprights would run north-and-south. But the
very object for which the great astronomical edifice was being raised,
was that the north-and-south line amongst others should be indicated by
more perfect methods.
Now at this stage of proceedings, what could be more perfect as a method
of obtaining the true bearing of the pole than to dig a tubular hole
into the solid rock, along which tube the Pole-star at its lower
culmination should be visible? Perfect stability would be thus insured
for this fundamental direction line. It would be easy to obtain the
direction with great accuracy, even though at first starting the borings
were not quite correctly made. And the further the boring was continued
downwards towards the south the greater the accuracy of the direction
line thus obtained. Of course there could be no question whatever in
such underground boring, of the advantage of taking the lower passage of
the Pole-star, not the upper. For a line directly from the star at its
upper passage would slant downwards at an angle of more than thirty
degrees from the horizon, while a line directly from the star at its
lower passage would slant downwards at an angle of less than thirty
degrees; and the smaller this angle the less would be the length, and
the less the depth of the boring required for any given horizontal
range.
Besides perfect stability, a boring through the solid rock would present
another most important advantage over any other method of orienting the
base of the pyramid. In the case of an inclined direction line above the
level of the horizontal base, there would be the difficulty of
determining the precise position of points under the raised line; for
manifest difficulties would arise in letting fall plumb-lines from
various points along the optical axis of a raised tubing. But nothing
could be simpler than the plan by which the horizontal line
corresponding to the underground tube could be determined. All that
would be necessary would be to allow the tube to terminate in a
tolerably large open space; and from a point in the base vertically
above this, to let fall a plumb-line through a fine vertical boring into
this open space. It would thus be found how far the point from which the
plumb-line was let fall lay, either to the east or to the west of the
optical axis of the underground tunnel, and therefore how far to the
east or to the west of the centre of the open mouth of this tunnel. Thus
the true direction of a north-and-south line from the end of the tube to
the middle of the base would be ascertained. This would be the meridian
line of the pyramid's base, or rather the meridian line corresponding to
the position of the underground passage directed towards the Pole-star
when immediately under the pole.
A line at right angles to the meridian line thus obtained would lie due
east and west, and the true position of the east-and-west line would
probably be better indicated in this way than by direct observation of
the sun or stars. If direct observation were made at all, it would be
made not on the sun in the horizon near the time of spring and autumn,
for the sun's position is then largely affected by refraction. The sun
might be observed for this purpose during the summer months, at moments
when calculation showed that he should be due east or west, or crossing
what is technically the _prime vertical_. Possibly the so-called azimuth
trenches on the east side of the great pyramid may have been in some way
associated with observations of this sort, as the middle trench is
directed considerably to the north of the east point, and not far from
the direction in which the sun would rise when about thirty degrees (a
favourite angle with the pyramid architects) past the vernal equinox.
But I lay no stress on this point. The meridian line obtained from the
underground passage would have given the builders so ready a means of
determining accurately the east and west lines for the north and south
edges of the pyramid's base, that any other observations for this
purpose can hardly have been more than subsidiary.
It is, of course, well known that there is precisely such an underground
tunnelling as the considerations I have indicated seem to suggest as a
desirable feature in a proposed astronomical edifice on a very noble
scale. In all the pyramids of Ghizeh, indeed, there is such a tunnelling
as we might expect on almost any theory of the relation of the smaller
pyramids to the great one. But the slant tunnel under the great pyramid
is constructed with far greater skill and care than have been bestowed
on the tunnels under the other pyramids. Its length underground amounts
to more than 350 feet, so that, viewed from the bottom, the mouth, about
four feet across from top to bottom on the square, would give a sky
range of rather less than one-third of a degree, or about one-fourth
more than the moon's apparent diameter. But, of course, there was
nothing to prevent the observers who used this tube from greatly
narrowing these limits by using diaphragms, one covering up all the
mouth of the tube, except a small opening near the centre, and another
correspondingly occupying the lower part of the tube from which the
observation was made.
It seems satisfactorily made out that the object of the slant tunnel,
which runs 350 feet through the rock on which the pyramid is built, was
to observe the Pole-star of the period at its lower culmination, to
obtain thence the true direction of the north point. The slow motion of
a star very near the pole would cause any error in time, as when this
observation was made, to be of very little importance, though we can
understand that even such observations as these would remind the
builders of the pyramid of the absolute necessity of good
time-measurements and time-observations in astronomical research.
Finding this point clearly made out, we can fairly use the observed
direction of the inclined passage to determine what was the position of
the Pole-star at the time when the foundations of the great pyramid were
laid, and even what that Pole-star may have been. On this point there
has never been much doubt, though considerable doubt exists as to the
exact epoch when the star occupied the position in question. According
to the observations made by Professor Smyth, the entrance passage has a
slope of about 26° 27', which would have corresponded, when refraction
is taken into account, to the elevation of the star observed through the
passage, at an angle of about 26° 29' above the horizon. The true
latitude of the pyramid being 29° 58' 51", corresponding to an elevation
of the true pole of the heavens, by about 30° 1/2' above the horizon, it
follows that if Professor Smyth obtained the true angle for the entrance
passage, the Pole-star must have been about 3° 31-1/2' from the pole.
Smyth himself considers that we ought to infer the angle for the
entrance passage from that of other internal passages, presently to be
mentioned, which he thinks were manifestly intended to be at the same
angle of inclination, though directed southwards instead of northwards.
Assuming this to be the case, though for my own part I cannot see why we
should do so (most certainly we have no _à priori_ reason for so doing),
we should have 26° 18' as about the required angle of inclination,
whence we should get about 3° 42' for the distance of the Pole-star of
the pyramid's time from the true pole of the heavens. The difference may
seem of very slight importance, and I note that Professor Smyth passes
it over as if it really were unimportant; but in reality it corresponds
to somewhat large time-differences. He quotes Sir J. Herschel's correct
statement, that about the year 2170 B.C. the star Alpha Draconis, when
passing below the pole, was elevated at an angle of about 26° 18' above
the horizon, or was about 3° 42' from the pole of the heavens (I have
before me, as I write, Sir J. Herschel's original statement, which is
not put precisely in this way); and he mentions also that somewhere
about 3440 B.C. the same star was situated at about the same distance
from the pole. But he omits to notice that since, during the long
interval of 1270 years, Alpha Draconis had been first gradually
approaching the pole until it was at its nearest, when it was only about
3-1/2' from that point, and then as gradually receding from the pole
until again 3° 42' from it, it follows that the difference of nine or
ten minutes in the estimated inclination of the entrance passage
corresponds to a very considerable interval in time, certainly to not
less than fifty years. (Exact calculation would be easy, but it would be
time wasted where the data are inexact.)
Having their base properly oriented, and being about to erect the
building itself, the architects would certainly not have closed the
mouth of the slant tunnel pointing northwards, but would have carried
the passage onwards through the basement layers of the edifice, until
these had reached the height corresponding to the place where the
prolongation of the passage would meet the slanting north face of the
building. I incline to think that at this place they would not be
content to allow the north face to remain in steps, but would fit in
casing stones (not necessarily those which would eventually form the
slant surface of the pyramid, but more probably slanted so as to be
perpendicular to the axis of the ascending passage.) They would probably
cut a square aperture through such slant stones corresponding to the
size of the passage elsewhere, so as to make the four surfaces of the
passage perfectly plane from its greatest depth below the base of the
pyramid to its aperture, close to the surface to be formed eventually by
the casing stones of the pyramid itself.
Now, in this part of his work, the astronomical architect could scarcely
fail to take into account the circumstance that the inclined passage,
however convenient as bearing upon a bright star near the pole when that
star was due north, was, nevertheless, not coincident in direction with
the true polar axis of the celestial sphere. I cannot but think he would
in some way mark the position of their true polar axis. And the natural
way of marking it would be to indicate where the passage of his
Pole-star _above_ the pole ceased to be visible through the slant tube.
In other words he would mark where a line from the middle of the lowest
face of the inclined passage to the middle of the upper edge of the
mouth was inclined by twice the angle 3° 42' to the axis of the passage.
To an eye placed on the optical axis of the passage, at this distance
from the mouth the middle of the upper edge of the mouth would (_quam
proximé_) show the place of the true pole of the heavens. It certainly
is a singular coincidence that at the part of the tube where this
condition would be fulfilled, there is a peculiarity in the construction
of the entrance passage, which has been indeed otherwise explained, but
I shall leave the reader to determine whether the other explanation is
altogether a likely one. The feature is described by Smyth as "a most
singular portion of the passage--viz., a place where two adjacent
wall-joints, similar, too, on either side of the passage, were vertical
or nearly so; while every other wall-joint, both above and below, was
_rectangular_ to the length of the passage, and, therefore, largely
_inclined_ to the vertical." Now I take the mean of Smyth's
determinations of the transverse height of the entrance passage as 47.23
inches (the extreme values are 47.14 and 47.32), and I find that, from a
point on the floor of the entrance passage, this transverse height would
subtend an angle of 7° 24' (the range of Alpha Draconis in altitude when
on the meridian) at a distance 363.65 inches from the transverse mouth
of the passage. Taking this distance from Smyth's scale in Plate xvii.
of his work on the pyramid ("Our Inheritance in the Great Pyramid"), I
find that, if measured along the base of the entrance passage from the
lowest edge of the vertical stone, it falls exactly upon the spot where
he has marked in the probable outline of the uncased pyramid, while, if
measured from the upper edge of the same stone, it falls just about as
far within the outline of the cased pyramid as we should expect the
outer edge of a sloped end stone to the tunnel to have lain.
It may be said that from the floor of the entrance passage no star could
have been seen, because no eye could be placed there. But the builders
of the pyramid cannot reasonably be supposed to have been ignorant of
the simple properties of plane mirrors, and by simply placing a thin
piece of polished metal upon the floor at this spot, and noting where
they could see the star and the upper edge of the tunnel's mouth in
contact by reflection in this mirror, they could determine precisely
where the star could be seen touching that edge, by an eye placed (were
that possible) precisely in the plane of the floor.
I have said there is another explanation of this peculiarity in the
entrance passage, but I should rather have said there is another
explanation of a line marked on the stone next below the vertical one. I
should imagine this line, which is nothing more than a mark such "as
might be ruled with a blunt steel instrument, but by a master hand for
power, evenness, straightness, and still more for rectangularity to the
passage axis," was a mere sign to show where the upright stone was to
come. But Professor Smyth, who gives no explanation of the upright stone
itself, except that it seems, from its upright position, to have had
"something representative of setting up, or preparation for the erecting
of a building," believes that the mark is as many inches from the mouth
of the tunnel as there were years between the dispersal of man and the
building of the pyramid; that thence downwards to the place where an
ascending passage begins, marks in like manner the number of years which
were to follow before the Exodus; thence along the ascending passage to
the beginning of the great gallery the number of years from the Exodus
to the coming of Christ; and thence along the floor of the grand gallery
to its end, the interval between the first coming of Christ and the
second coming or the end of the world, which it appears is to take place
in the year 1881. It is true not one of these intervals accords with the
dates given by those who are considered the best authorities in Biblical
matters,--but so much the worse for the dates.
To return to the pyramid.