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.