The Solution of the Pyramid Problem Part 6

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 compa.s.s, 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 circ.u.mferenter, 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 s.p.a.ce 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 Gzeh group our surveyor could also tie in and perfect his work by sights to the Sakkarah group with Sakkarah 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 Gzeh 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 Gzeh 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 Gzeh 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 compa.s.s 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 compa.s.ses, 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 Ca.s.sini 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 Gzeh 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 Gzeh, 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 ill.u.s.trate this beautiful law from which it will be seen that the Gzeh 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 23.7.5024 W W 23.7.5024 N N 23.7.5024 E E 23.7.5024 N

Fig. 53. Cephren from points bearing

S 23.7.5024 E W 23.7.5024 S N 23.7.5024 W E 23.7.5024 N

Fig. 54. Mycerinus from points bearing

S 17.1.404 W W 17.1.404 N N 17.1.404 E E 17.1.404 S

Fig. 55. Mycerinus from points bearing

S 17.1.404 E W 17.1.404 S N 17.1.404 W E 17.1.404 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 ill.u.s.trated 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.