Robert Ballard

“ 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. ”

“ Thirty-six centuries ago, they, already venerable with antiquity, looked proudly down on living labouring Israel, in helpless slavery, in the midst of an advanced civilization, of which the history, language, and religion are now forgotten, or only at best, slightly understood. ”

“ By continuing the system of evolution by which I defined the position of Cephren, and the little pyramid to the south-east of Cheops, after I had obtained Cheops and Mycerinus, may be rebuilt, at one and the same time, a skeleton of the trigonometrical tables of a forgotten civilization, and the plan of those pyramids which are its only link with the present age. ”

“ 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. ”

“ The pyramid builders may perhaps have made the entombment of their Kings one of their exoteric objects, playing on the morbid vanity of their rulers to induce them to the work, but in the minds of the builders before ever they built must have been planted the intention to make use of the structures for the purposes of land surveying. ”

“ The triangle DXH represents a vertical section of the pentagonal pyramid; the edge HX is equal to HN, and the apothem DX is equal to DE. Let DH be a [Pg 97] hinge attaching the plane DXH to the base, now lift the plane DXH until the point X is vertical above the centre C. Then the points A, E, B, O, N of the five slant slides, when closed up, will all meet at the point X over the centre C.
We have now built a pyramid out of the pentangle, whose slope is 2 to 1, altitude CX being to CD as 2 to 1.
Note also that
Let us now consider the Pentangle as the symbol of the Great Pyramid Cheops. ”

“ Builders do not mark a dimension on a plan which they cannot measure, nor have a hidden measure of any importance without some clear outer way of establishing it.
This made me "go straight" for the slant ratios. When the Pyramids were cased from top to bottom with polished marble, there were only two feasible measures, the bases and the apothems; [1] and for that reason I conjectured that these would be the definite plan ratios.
[1] The "Apothem is a perpendicular from the vertex of a pyramid on a side of the base."—Chambers' Practical Mathematics, p. 156. ”

“ Modern surveyors stand near their instruments, and send their flagmen to a distance; the Egyptian surveyor was one of his own flagmen, and his instruments were towering to the skies on the distant horizon. These mighty tools will last out many a generation of surveyors. ”

“ 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 [Pg 58] 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. ”

“ And therefore I claim that there is no reason why the pyramid builders should not have known as much about the circumference of the earth as the modern world that has calmly stood by in its ignorance and permitted [Pg 39] those magnificent and, as I shall prove, useful edifices to be stripped of their beautiful garments of polished marble. ”

“ I may not have got the exact "hang" of the Egyptian method of land surveying—for I do not suppose that even their "clumsy" method is to be got at intuitively; but I claim that I have shewn how the Pyramids could be used for that purpose, and that the subsidiary instrument described by me was practicable.
I claim, therefore, that the theory I have set up, that the pyramids were the theodolites of the Egyptians, is sound. That the ground plan of these pyramids discloses a beautiful system of primary triangles and satellites I think I have shown beyond the shadow of a doubt ”

“ 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 [Pg 53] 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. ”

“ 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 [Pg 54] 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. ”

“ 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 ”

“ We will suppose the surveyor to be standing looking at the pyramid Cephren; he knows that its base is [Pg 52] 420 cubits, and its apothem 3461⁄2 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. ”

“ Draw two diameters of the outer circle passing through the centre square at right angles to each other, and each diameter parallel to sides of the square; then [Pg 99] will the parts of these diameters between the square and the outer circle represent the four apothems of the four slant sides of the pyramid. Connect the angles of the square with the circumference of the outer circle by lines at the four points indicated by the diameters, and the star of the pyramid is formed, which, when closed as a solid, will be a correct model of Cheops. ”

“ The answer is—from the practical impossibility of such extreme accuracy in such a mass of masonry, that it points alike to all, and may as fairly be considered the exponent of the one as of the others. Piazzi Smyth makes Cheops 761·65 feet base, and 484·91 feet altitude, which is very nearly what he calls a Π pyramid, for which I reckon the altitude would be about 484·87 feet with the same base: and for a pyramid of extreme and mean ratio the altitude would be 484·34 feet. ”

“ If Astronomers agree that observations of a pole star could have been accurately made by peering up a small gallery on the north side of one of the pyramids only a few hundred feet in length, I feel that I shall have little difficulty in satisfying them that accurate measurements to points only miles away could have been made from angular observations of the whole group. ”

“ Primary triangulation would be useful to men of almost every trade and profession in which tools or instruments are used. Any one might in a short time construct a table for himself answering to every degree or so in the circumference of a circle for which only forty or fifty triangles are required.
[Pg 83]
It would be worth while for some one to print and publish a correct set of these tables embracing a close division of the circle, in which set there should be a column showing the angle in degrees, minutes, seconds and decimals, and also a column for the satellite, thus—
and so on. ”

“ Modern savants appear to take the opposite stand for any little information they happen to possess.
The cubit which is called the Royal Babylonian cubit and stated to measure O·5131 metre, differs so slightly from my cubit, only the six-hundredth part of a foot, that it may fairly be said to be the same cubit, and it will be for antiquaries to trace the connection, as this may throw some light on the identity of the builders of the Pyramids of Gïzeh. Few good English two-foot rules agree better than these two cubits do. ”

“ The formula to find the "primary triangle" to any satellite is as follows:—
Let the long ratio line of the satellite or sine be called a, and the short ratio line or versed-sine be called b. Then—
Therefore various primary triangles can be constructed on a side DB (Fig. 64) as sine, by taking different measures for AD as versed-sine. For example—
[Pg 72]
Finally arises the following simple rule for the construction of "primaries" to contain any angle—Decide upon a satellite which shall contain half the angle—say, 5, 1. Call the first figure a, the second b, then—
a2 + b2 = hypotenuse. ”

“ In any pyramid, the apothem is to half the base as the area of the four sides is to the area of the base.
All in R.B. cubits.
[Pg 29]
[2] Herodotus states that "the area of each of the four faces of Cheops was equal to the area of a square whose base was the altitude of a Pyramid;" or, in other words, that altitude was a mean proportional to apothem and half base; thus—area of one face equals the fourth of 330777·90 or 82694·475 R.B. cubits, and the square root of 82694·475 is 287·56. But the correct altitude is 287·77, so the error is 0·21, or 41⁄4 British inches. ”

“ In Cheops, my dimensions agree with Piazzi Smyth—in the base of Cephren, with Vyse and Perring—in the height of Cephren, with Sir Gardner Wilkinson, nearly—in the base of Mycerinus, they agree with the usually accepted measures, and in the height of Mycerinus, they exceed Jas. J. Wild's measure, by not quite one of my cubits. ”

“ 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. ”

“ 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. ”

“ By the aid of their great stone theodolites, the surveyors, [Pg 43] who belonged to the priestly order, were able to readjust the boundaries with great precision. That all science was comprised in their secret mysteries may be one reason why no hieroglyphic record of the scientific uses of the pyramids remains. ”

“ 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. ”

“ Tangent + Secant = 21⁄3 radius
Co-tan + Co-sec = 21⁄2 radius
Sine : Versed sine :: 5 : 2
Co-sine : Co-versed sine :: 7 : 3
[7] 12180 = 20 × 21 × 29
It is noticeable that while the multiplier required to bring radius 5 and the rest into whole numbers, for the 3, 4, 5 triangle is twelve, in the 20, 21, 29 triangle it is 420, the key measure for the bases of the two main pyramids in R.B. cubits. ”

“ The 28, 45, 53 triangle = 31° 53′ 26·85′′
The angles of the satellites together must always be 45°, because the angle at the circumference of a quadrant must always be 135°.
From the Gïzeh plan, as far as I have developed it, the following order of satellites begins to appear, which may be a guide to the complete Gïzeh plan ratio, and to those "primary" triangles in use by the pyramid surveyors in their ordinary work. [Pg 69]
Primary triangles may be found from the angle of the satellite, but it is an exceedingly round-about way. ”

“ Thus:—Suppose NHO of figure 69 to be the NHO of a heptangle instead of a pentangle, then does NH represent apothem, and NO represent base of the pyramid Mycerinus, while the co-sine of the angle NHM (being MH minus versed sine) will be equal to the altitude of the pyramid. The angle NHM in the heptangle is, 38° 34′ 17·142′′, and according to my plan of the pyramid Mycerinus, the corresponding angle is 38° 40′ 56′′. ”

“ Thus with the two main arms pinned to the cross arm in the 21st and 29th hole from the permanently pinned end, with the cross arm stretched to twenty divisions, a 20, 21, 29 triangle would be the result, and so on.
§ 14a. GENERAL OBSERVATIONS.
I must be excused by geometricians for going so much in detail into the simple truths connected with right-angled trigonometry. My object has been to make it very clear to that portion of the public not versed in geometry, that the Pyramids of Egypt must have been used for land surveying by right-angled triangles with sides having whole numbers. ”