with our clocks, and our watches, and our compasses, rarely observe the
sun or stars. But in a land and an age when the sun was the only clock,
and the pyramid the only compass, the movements and positions of the
heavenly bodies were known to all. These people were _familiar_ with the
stars, and kept a watch upon their movements.
How many of our vaunted educated population could point out the Dog-star
in the heavens?--but the whole Egyptian nation hailed his rising as the
beginning of their year, and as the harbinger of their annual blessing,
the rising of the waters of the Nile.
Fig. 38. From the North West Bearing 315° Sun in the West.
Fig. 39. From the South East Bearing 135° Sun in the West.
Fig. 40. From the North East Bearing 45° Sun in the East.
Fig. 41. From the South West Bearing 225° Sun in the East.
It is possible therefore that the land surveyors of Egypt made full use
of the heavenly bodies in their surveys of the land; and while we are
pitifully laying out our new countries by the circumferenter and the
compass, we presume to speak slightingly of the supposed dark heathen
days, when the land of Egypt was surveyed by means of the sun and the
stars, and the theodolites were built of stone, with vertical limbs five
hundred feet in height, and horizontal limbs three thousand feet in
diameter.
Imagine half a dozen such instruments as this in a distance of about
sixty miles (for each group of pyramids was effectually such an
instrument), and we can form some conception of the perfection of the
surveys of an almost prehistoric nation.
The centre of Lake Moeris, in which Herodotus tells us two pyramids
stood 300 feet above the level of the lake, appears from the maps to be
about S. 28° W., or S. 29° W. from Gïzeh, distant about 57 miles, and
the Meidân group of pyramids appears to be about 33 miles due south of
Gïzeh.
Figures 38, 39, 40 and 41, show that north-west, south-east, north-east,
and south-west lines from the pyramids could be extended by simply
plumbing the angles. These lines would be run in sets of two's and
three's, according to the number of pyramids in the group; and their
known distances apart at that angle would check the correctness of the
work.
A splendid line was the line bearing 43° 36′ 10·15″, or 223° 36′
10·15″ from Cheops and Cephren, the pyramids covering each other, the
line of hypotenuse of the great 20, 21, 29 triangle of the plan. This I
call the 20, 21 line. _(See Figure_ 42.)
Figure 43 represents the 3, 4, 5 triangle line from the summits of
Mycerinus and Cheops in true line bearing 216° 52' 11·65". This I call
the south 4, west 3 line.
The next line is what I call the 2, 1 line, and is illustrated by figure
44. It is one of the most perfect of the series, and bears S. 26° 33'
54·9" W. from the apex of Cephren. This line demonstrates clearly why
Mycerinus was cased with red granite.
Not in memory of the beautiful and rosy-cheeked Nitocris, as some of the
tomb theory people say, but for a less romantic but more useful object;
simply because, from this quarter, and round about, the lines of the
pyramids would have been confused if Mycerinus had not been of a
different color. The 2, 1 line is a line in which Mycerinus would have
been absolutely lost in the slopes of Cephren but for his red color.
There is not a fact that more clearly establishes my theory, and the
wisdom and forethought of those who planned the Gïzeh pyramids, than
this red pyramid Mycerinus, and the 2, 1 line.
Hekeyan Bey, speaks of this pyramid as of a "_ruddy complexion_;" John
Greaves quotes from the Arabic book, Morat Alzeman, "_and the lesser
which is coloured_;" and an Arabic writer who dates the Pyramids three
hundred years before the Flood, and cannot find among the learned men of
Egypt "_any certain relation concerning them_" nor any "_memory of them
amongst men_," also expatiates upon the beauties of the "_coloured
satin_" covering of this one particular pyramid.
Fig. 42. South 21. West 20. Bearing 223°.36'.10·15".
Fig. 43. South 4. West 3. Bearing 216°.52'.11·65".
Fig. 44. South 2. West 1. Bearing 206°.33'.54·18".
Fig. 45. South 96. West 55. Bearing 209°.48'.32·81".
Fig. 46. South 3. West 1. Bearing 198°.26'.5·82".
Fig. 47. South 5. West 2. Bearing 201°.48'.5".
Fig. 48. South 7. West 3. Bearing 203°.11'.55".
Figure 45 represents the line south 96, west 55, from Cephren, bearing
209° 48' 32·81"; the apex of Cephren is immediately above the apex of
Mycerinus.
Figure 46 is the S. 3 W. 1 line, bearing 198° 26' 5.82"; here the dark
slope angle of the pyramids with the sun to the eastward occupies half
of the apparent half base.
Figure 47 is the S. 5, W. 2 line, bearing 201° 48' 5"; here Cephren and
Mycerinus are in outside slope line.
Figure 48 is the S. 7 W. 3 line, bearing 203° 11' 55"; here the inside
slope of Cephren springs from the centre of the apparent base of
Mycerinus.
I must content myself with the preceding examples of a few pyramid
lines, but must have said enough to show that from every point of the
compass their appearance was distinctly marked and definitely to be
determined by surveyors acquainted with the plan.
§ 11. DESCRIPTION OF THE ANCIENT PORTABLE SURVEY INSTRUMENT.
I must now commence with a single pyramid, show how approximate
observations could be made from it, and then extend the theory to a
group with the observations thereby rendered more perfect and delicate.
We will suppose the surveyor to be standing looking at the pyramid
Cephren; he knows that its base is 420 cubits, and its apothem 346½
cubits. He has provided himself with a model in wood, or stone, or
metal, and one thousandth of its size--therefore his model will be O.42
cubit base, and O.3465 cubit apothem--or, in round numbers, eight and
half inches base, and seven inches apothem.
This model is fixed on the centre of a card or disc, graduated from the
centre to the circumference, like a compass card, to the various points
of the compass, or divisions of a circle.
The model pyramid is fastened due north and south on the lines of this
card or disc, so that when the north point of the card points north, the
north face of the model pyramid faces to the north.
The surveyor also has a table, which, with a pair of plumb lines or
mason's levels, he can erect quite level: this table is also graduated
from the centre with divisions of a circle, or points of the compass,
and it is larger than the card or disc attached to the model.
This table is made so that it can revolve upon its stand, and can be
clamped. We will call it the _lower limb_. There is a pin in the centre
of the lower limb, and a hole in the centre of the disc bearing the
model, which can be thus placed upon the centre of the table, and
becomes the _upper limb_. The upper limb can be clamped to the lower
limb.
The first process will be to clamp both upper and lower limbs together,
with the north and south lines of both in unison, then revolve both
limbs on the stand till the north and south line points straight for
the pyramid in the distance, which is done by the aid of sights erected
at the north and south points of the perimeter of the lower limb. When
this is adjusted, clamp the lower limb and release the upper limb; now
revolve the upper limb until the model pyramid exactly covers the
pyramid in the distance, and shows just the same shade on one side and
light on the other, when viewed from the sights of the clamped lower
limb--and the lines, angles, and shades of the model coincide with the
lines, angles, and shades of the pyramid observed;--now clamp the upper
limb. Now does the model stand really due north and south, the same as
the pyramid in the distance; it throws the same shades, and exhibits the
same angles when seen from the same point of view; just as much of it is
in shade and as much of it is in light as the pyramid under observation;
therefore it must be standing due north and south, because Cephren
himself is standing due north and south, and the upper limb reads off on
the lower limb the angle or bearing observed.
So far we possess an instrument equal to the modern circumferenter, and
yet we have only brought one pyramid into work.
If I have shown that such an operation as the above is practically
feasible, if I have shown that angles can be taken with moderate
accuracy by observing one pyramid of 420 cubits base, how much more
accurate will the observation be when the surveyor's plane table bears
a group of pyramids which occupy a representative space of about 1400
cubits when viewed from the south or north, and about 1760 cubits when
viewed from the east or west. If situated a mile or two south of the
Gïzeh group our surveyor could also tie in and perfect his work by
sights to the Sâkkarah group with Sâkkarah models; and so on, up the
Nile Valley, he would find every few miles groups of pyramids by aid of
which he would be enabled to tie his work together.
If the Gïzeh group of pyramids is placed and shaped in the manner I have
described, it must be clear that an exact model and plan, say a
thousandth of the size, could be very easily made--the plan being at the
level of the base of Cephren where the bases of the two main pyramids
are even;--and if they are not exactly so placed and shaped, it may be
admitted that their position and dimensions were known to the surveyors
or priests, so that such models could be constructed. It is probable,
therefore, that the instrument used in conjunction with these pyramids,
was a machine constructed in a similar manner to the simple machine I
have described, only instead of there being but one model pyramid on the
disc or upper limb, it bore the whole group; and the smaller pyramids
were what we may call vernier points in this great circle, enabling the
surveyor to mark off known angles with great accuracy by noticing how,
as he worked round the group of pyramids, one or other of the smaller
ones was covered by its neighbours.[9]
Footnote 9: See general plan of Gïzeh Group op. page 1.
The immensity of the main pyramids would require the smaller ones to be
used for surveys in the immediate neighbourhood, as the surveyor might
easily be too close to get accurate observations from the main pyramids.
The upper limb, then, was a disc or circular plate bearing the model of
the group.
Cheops would be situated in the centre of the circle, and observations
would be taken by bringing the whole model group into even line and even
light and shade with the Gïzeh group.
I believe that with a reasonable-sized model occupying a circle of six
or seven feet diameter, such as a couple of men could carry, very
accurate bearings could have been taken, and probably were taken.
The pyramid shape is the very shape of all others to employ for such
purposes. A cone would be useless, because the lights and shades would
be softened off and its angles from all points would be the same. Other
solids with perpendicular angles would be useless, because although they
would vary in width from different points of view they would not present
that ever changing angle that a pyramid does when viewed from different
directions.
After familiarity with the models which I have made use of in
prosecuting these investigations, I find that I can judge with great
accuracy _from their appearance only_ the bearing of the group from any
point at which I stand. I make bold to say that the pocket compass of
the Egyptian surveyor was a little model of the group of pyramids in his
district, and he had only to hold it up on his hand and turn it round in
the sun till its shades and angles corresponded with the appearance of
the group, to tell as well as we could tell by our compasses, perhaps
better, his bearing from the landmarks that governed his surveys.
The Great Circle of Gold described by Diodorus (_Diod. Sic. lib. X.,
part 2, cap. 1_) as having been employed by the Egyptians, and on which
was marked amongst other things, the position of the rising and setting
of the stars, and stated by him to have been carried off by Cambysses
when Egypt was conquered by the Persians, is supposed by Cassini to have
been also employed for finding the meridian by observation of the rising
and setting of the sun. This instrument and others described by writers
on Egypt would have been in practice very similar to the instrument
which I have described as having been probably employed for terrestrial
observations.
The table or disc comprising the lower limb of the instrument, might
have been supported upon a small stand with a circular hole in the
centre, so arranged that the instrument could be either set up alone and
supported by its own tripod, or rested fairly on the top of any of those
curious stone boundary marks which were made use of, not only to mark
the corners of the different holdings, but to show the level of the Nile
inundations. (_See Figure 49, copied from Sharpe's Egypt,_ _vol. I.,
p_. 6.) The peculiar shape of the top of these stone landmarks, or
"sacred boundary stones," appears suitable for such purposes, and it
would have been a great convenience to the surveyor, and conducive to
accuracy, that it should be so arranged that the instrument should be
fixed immediately over the mark, as appears probable from the shape of
the stone.
Fig 49. Sacred boundary stone.
A noticeable point in this theory is, that it is not in the least
essential that the apex of a pyramid should be complete. If their
summits were left permanently flat, they would work in for survey
purposes quite as well, and I think better, than if carried to a point,
and they would be more useful with a flat top for defined shadows when
used as sun dials.
In the Gïzeh group, the summit of Cheops appears to me to have been left
incomplete the better to get the range with Cephren for lines down the
delta.
In this system of surveying, there is always a beautiful connection
between the horizontal bearings and the apparent or observed angles
presented by the slopes and edges of the pyramid. Thus, in pyramids like
those of Gïzeh, which stand north and south, and whose meridional
sections contain less, and whose diagonal sections contain more than a
right angle, the vertex being the point at which the angle is
measured--this law holds:-- That the smallest interior angle at the
vertex, contained between the inside edge and the outside edge, will
exhibit the same angle as the bearing of the observer's eye from the
apex of the pyramid _when the angle at the apex contained by the outside
edges appears to be a right angle_.
Figures 50 to 55 inclusive illustrate this beautiful law from which it
will be seen that the Gïzeh surveyors possessed, in this manner alone,
eight distinctly defined bearings from each pyramid.
Fig. 50. Cheops from points bearing
° ′ ″
S 19.12.22 W
W 19.12.22 N
N 19.12.22 E
E 19.12.22 S
Fig. 51. Cheops from points bearing
° ′ ″
S 19.12.22 E
W 19.12.22 S
N 19.12.22 W
E 19.12.22 N
Fig. 52. Cephren from points bearing
° ′ ″
S 23.7.50·24 W
W 23.7.50·24 N
N 23.7.50·24 E
E 23.7.50·24 N
Fig. 53. Cephren from points bearing
° ′ ″
S 23.7.50·24 E
W 23.7.50·24 S
N 23.7.50·24 W
E 23.7.50·24 N
Fig. 54. Mycerinus from points bearing
° ′ ″
S 17.1.40·4 W
W 17.1.40·4 N
N 17.1.40·4 E
E 17.1.40·4 S
Fig. 55. Mycerinus from points bearing
°- ′ ″
S 17.1.40·4 E
W 17.1.40·4 S
N 17.1.40·4 W
E 17.1.40·4 N
* * * * *
Fig. 56. Cheops model.
Fig. 57. Cephren model
Fig. 58. Mycerinus model
* * * * *
I recommend any one desirous to thoroughly comprehend these matters, to
make a plan from my diagram, _Figure_ 5, using R.B. cubits for measures,
and to a suitable scale, on a piece of card-board. Then to cut out of
the card-board the squares of the bases of the pyramids at the level of
Cephren, viz., 420, 420 and 218 cubits respectively, for the three main
pyramids. One hundred cubits to the inch is a convenient scale and
within the limits of a sheet of Bath board.
By striking out the models on card-board in the manner shown by diagrams
(_see Figures_ 56, 57, and 58) they can be cut out with a
penknife--cutting only _half through_ where the lines are
_dotted_--bent up together, and pasted along the edges with strips of
writing paper about half an inch wide.
These models can be dropped into the squares cut out of the card-board
plan, thus correcting the error caused by the thickness of the
card-board base, and if placed in the sun, or at night by the light of
_one_ lamp or candle properly placed to represent the sun in the
eastward or westward, the clear cut lines and clear contrasting shades
will be manifest, and the lines illustrated by my figures can be
identified.
When inspecting the model, it is well to bear in mind that the eye must
be kept very nearly level with the table, or the pyramids will appear as
if viewed from a balloon.
* * * * *
I believe that the stones were got up to the building by way of the
north side of each pyramid. The casing on the south, east, and west, was
probably built up as the work proceeded, and the whole of these three
faces were probably thus finished and completed while there was not a
single casing stone set on the north side. Then the work would be closed
up until there remained nothing but a great gap or notch, wide at the
bottom, and narrowing to the apex. The work on the north side would then
be closed from the sides and top, and the bottom casing stone about the
centre of the north side, would be the last stone set on the building.
These old builders were too expert not to have thus made use of all the
shade which their own building would thus afford to a majority of the
workmen.
* * * * *
Many of the obelisks were probably marks on pyramid lines of survey.
The pyramid indeed may have been a development of the obelisk for this
purpose.
Their slanting sides might correspond with some of the nearly upright
slant angles of the pyramids, in positions opposite certain lines.
Reference to several of my figures will show how well this would come
in.
Herodotus speaks of two obelisks at Heliopolis, and Bonwick tells us
that Abd al Latif saw two there which he called Pharaoh's Needles. An
Arab traveller, in 1190, saw a pyramid of copper on the summit of the
one that remained, but it is now wanting. Pharaoh's Needles appear to
have been situated about 20 miles NE. of the Gïzeh group, and their
slope angles might have coincided with the apparent slope angles of
Cephren or Cheops on the edge nearest the obelisk.
The ancient method of describing the meridian by means of the shadow of
a ball placed on the summit of an obelisk points to a reasonable
interpretation for the peculiar construction of the two pillars, Jachin
and Boaz, which are said to have been situated in front of the Hebrew
Temple at Jerusalem, and about which so much mysterious speculation has
occurred.
They were no doubt used as sun-dials for the morning and afternoon sun
by the shadow of the balls or "chapiters" thrown upon the pavement.
Without presuming to dispute the objects assigned by others for the
galleries and passages which have been discovered in the pyramid Cheops,
I venture to opine that they were employed to carry water to the
builders. They are connected with a well, and the well with the Nile or
canal. Whether the water was slided up the smooth galleries in boxes, or
whether the cochlea, or water screw, was worked in them, their angles
being suitable, it is impossible to conjecture; either plan would have
been convenient and feasible.
These singular chambers and passages may indeed possibly have had to do
with some hydraulic machinery of great power which modern science knows
nothing about. The section of the pyramid, showing these galleries, in
the pyramid books, has a most hydraulic appearance.
The tremendous strength and regularity of the cavities called the King's
and Queen's chambers, the regularity and the _smallness_ of most of the
passages or massive stone connecting pipes, favor the idea that the
chambers might have been reservoirs, their curious roofs, air chambers,
and the galleries or passages, connecting pipes for working water under
pressure. Water raised through the passages of this one pyramid nearest
to the canal, might have been carried by troughs to the other pyramids,
which were in all probability in course of construction at the same
period of time. A profane friend of mine thinks that the sarcophagus or
"sacred coffer" in the King's chamber may have been used by the chief
architect and leading men of the works as a _bath_, and that the King's
chamber was nothing more or less than a delightful bath room.
* * * * *
The following quotation from the writing of an Arabian author (Ibn Abd
Alkokm), is extracted from Bonwick's "Pyramid Facts and Fancies," page
72:--"The Coptites mention in their books that upon them (the Pyramids)
is an inscription engraven; the exposition of it in Arabicke is
this:--'I, Saurid the King built the Pyramids (in such and such a time),
and finished them in six years; he that comes after me, and says he is
equal to me, let him destroy them in six hundred years; and yet it is
known that it is easier to pluck down than to build; _and when I had
finished them, I covered them with sattin, and let him cover them with
slats._'"
The italics are my own. The builder seems to have entertained the idea
that his work would be partially destroyed, and afterwards temporarily
repaired or rebuilt. The first part has unfortunately come true, and it
is possible that the last part of the idea of King Saurid may be carried
out, because it would not be so very expensive an undertaking for any
civilized nation in the interest of science to re-case the pyramids of
Gïzeh, so that they might be once more applied to land-surveying
purposes in the ancient manner.
It would not be absolutely necessary to case the whole of the pyramid
faces, so long as sufficient casing was put on to define the angles. The
"_slats_" used might be a light wooden framework covered with thin
metal. The metal should be painted white, except in the case of
Mycerinus, which should be of a reddish color.
§ 12. PRIMARY TRIANGLES AND THEIR SATELLITES;--OR THE ANCIENT SYSTEM OF
RIGHT-ANGLED TRIGONOMETRY UNFOLDED BY A STUDY OF THE PLAN OF THE
PYRAMIDS OF GIZEH.
=Main Triangular Dimensions of Plan are Represented by the Following
Eight Right-angled Triangles.=
TABLE TO EXPLAIN FIGURE 60.
+--------------------------++-------------------------------------+
| AB 28} × { 84} × { 672 || DG 3} × { 72} × {576 |
| BJ 45} 3 {135} 8 {1080 || GE 4} 24 { 96} 8 {768 |
| JA 53} {159} {1272 || ED 5} {120} {960 |
+--------------------------++-------------------------------------+
| DC 3} × {135} × {1080 || FW 48} × { 48} × {384 |
| CA 4} 45 {180} 8 {1440 || WV 55} 1 { 55} 8 {440 |
| AD 5} {225} {1800 || VF 73} { 73} {584 |
+--------------------------++-------------------------------------+
| EB 3} × { 63} × {504 || FB 20} × { 80} × {640 |
| BA 4} 21 { 84} 8 {672 || BA 21} 4 { 84} 8 {672 |
| AE 5} {105} {840 || AF 29} {116} {928 |
+--------------------------++=====================================+
| FH 3} × { 96} × { 768 || |
| HN 4} 32 {128} 8 {1024 || Note.--In the above table the first |
| XF 5} {160} {1280 || column _is the Ratio_, the second |
+--------------------------++ _the connected Natural Numbers_, and|
| AY 3} × { 36} × { 288 || the third column represents _the_ |
| YZ 4} 12 { 48} 8 { 384 || _length each line in R.B. cubits_. |
| ZA 5} { 60} { 480 || |
+--------------------------++-------------------------------------+
Fig. 60.
Reference to _Fig. 60_ and the preceding table, will show that the main
triangular dimensions of this plan (imperfect as it is from the lack of
eleven pyramids) are represented by four main triangles, viz:--
Ratio.
C A D C .. .. 3, 4, 5
F B A F .. .. 20, 21, 29
A B J A .. .. 28, 45, 53
F W V F .. .. 48, 55, 73
Figures 30 to 36 illustrate the two former, and _Figures_ 61 and 62
illustrate the two latter. I will call triangles of this class "primary
triangles," as the most suitable term, although it is applied to the
main triangles of geodetic surveys.
We have only to select a number of such triangles and a system of
trigonometry ensues, in which base, perpendicular, and hypotenuse of
every triangle is a whole measure without fractions, and in which the
nomenclature for every angle is clear and simple.
An angle of 43° 36′ 10·15″ will be called a 20, 21 angle, and an
angle of 36° 52′ 11·65″ will be called a 3, 4 angle, and so
on.