§ 10. HOW THE PYRAMIDS WERE MADE USE OF.
It appears from what I have already set forth that the plan of the Pyramids under consideration is geometrically exact, a perfect set of measures.
I shall now show how these edifices were applied to a thoroughly geometrical purpose in the true meaning of the word—to measure the Earth.
I shall show how true straight lines could be extended from the Pyramids in given directions useful in right-angled trigonometry, by direct observation of the buildings, and without the aid of other instruments.
And I shall show how by the aid of a simple instrument angles could be exactly observed from any point.
This Survey theory does not stand or fall on the merits of my theory of the Gïzeh plan. Let it be proved that this group is not built on the exact system of triangulation set forth by me, it is still a fact that its plan is in a similar shape, and any such shape would enable a surveyor acquainted with the plan to lay down accurate surveys by observations of the group even should it not occupy the precise lines assumed by me.
And here I must state that although the lines of the plan as laid down herein agree nearly with the lines as laid down in Piazzi Smyth's book, in the Penny Cyclopædia, and in an essay of Proctor's in the Gentleman's Magazine, still I find that they do not agree at all satisfactorily with a map of the Pyramids in Sharp's "Egypt," said to be copied from Wilkinson's map.
We will, however, for the time, and to explain my survey theory, suppose the plan theory to be correct, as I firmly believe it is.
And then, supposing it may be proved that the respective positions of the pyramids are slightly different to those that I have allotted to them on my plan, it will only make a similar slight difference to the lines and angles which I shall here show could be laid out by their aid.
Let us in the first place comprehend clearly the shape of the land of Egypt.
A sector or fan, with a long handle—the fan or sector, the delta; and the handle of the fan, the Nile Valley, running nearly due south.
The Pyramids of Gïzeh are situate at the angle of the sector, on a rocky eminence whence they can all be seen for many miles. The summits of the two high ones can be seen from the delta, and from the Nile Valley to a very great distance; how far, I am unable to say; but I should think that while the group could be made general use of for a radius of fifteen miles, the summits of Cephren and Cheops could be made use of for a distance of thirty miles; taking into consideration the general fall of the country.
It must be admitted that if meridian observations of the star Alpha of the Dragon could be made with accuracy by peeping up a small hole in one of the pyramids, then surely might the surveyors have carried true north and south lines up the Nile Valley as far as the summit of Cheops was visible, by "plumbing in" the star and the apex of the pyramid by the aid of a string and a stone.
True east and west lines could have been made to intersect such north and south lines from the various groups of pyramids along the river banks, by whose aid also such lines would be prolonged.
Next, supposing that their astronomers had been aware of the latitude of Cheops, and the annual northing and southing of the sun, straight lines could have been laid out in various sectoral directions to the north-eastward and north-westward of Cheops, across the delta, as far as the extreme apex of the pyramid was visible, by observations of the sun, rising or setting over his summit. (That the Dog-star was observed in this manner from the north-west, I have little doubt.)
For this purpose, surveyors would be stationed at suitable distances apart with their strings and their stones, ready to catch the sun simultaneously, and at the very moment he became transfixed upon the apex of the pyramid, and was, as it were, "swallowed by it." (See Figure 37.) The knowledge of the pyramid slope angle from different points of view would enable the surveyor to place himself in readiness nearly on the line.
Surely such lines as these would be as true and as perfect as we could lay out nowadays with all our modern instrumental appliances. A string and a stone here, a clean-cut point of stone twenty miles away, and a great ball of fire behind that point at a distance of ninety odd million miles. The error in such a line would be very trifling.
Such observations as last mentioned would have been probably extended from Cephren for long lines, as being the higher pyramid above the earth's surface, and may have been made from the moon or stars.
In those days was the sun the intimate friend of man. The moon and stars were his hand-maidens.
How many of us can point to the spot of the sun's rising or setting? We, 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 Mœris, 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]
[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.
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.
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Cheops model Fig. 56.
Cephren Model Fig. 57.
Mycerinus model Fig. 58.
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.