Chapter II: Part 2

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.

In the existing system whole angles, such as 40, 45, or 50 degrees, are surrounded by lines, most of which can only be described in numbers by interminable fractions.

In the ancient system, lines are only dealt with, and every angle in the table is surrounded by lines measuring whole units, and described by the use of a couple of simple numbers.

Connecting this with our present system of trigonometry would effect a saving in calculation, and general use of certain peculiar angles by means of which all the simplicity and beauty of the work of the ancients would be combined with the excellences of our modern instrumental appliances. Surveyors should appreciate the advantages to be derived from laying out traverses on the hypotenuses of "primary" triangles, by the saving of calculation and facility of plotting to be obtained from the practice.

The key to these old tables is the fact, that in "primary" triangles the right-angled triangle formed by the sine and versed sine, also by the co-sine and co-versed-sine, is one in which base and perpendicular are measured by numbers without fractions. These I will call "satellite" triangles.

Thus, to the "primary" triangle 20, 21, 29, the ratios of the co-sinal and sinal satellites are respectively 7 to 3, and 2 to 5. (_See Figure 35._) To the 48, 55, 73 triangle the satellites are 11, 5 and 8, 3 (_Fig. 62_); to the 3, 4, 5 triangle they are 2, 1 and 3, 1 (_Fig. 30_); and to the 28, 45, 53 triangle, they are 9, 5 and 7, 2 (_Fig. 61_). The primary triangle, 7, 24, 25, possesses as satellites the "primary" triangle, 3, 4, 5, and the ordinary triangle, 4, 1; and the primary triangle 41, 840, 841, is attended by the 20, 21, 29 triangle, as a satellite with the ordinary triangle 41, 1, and so on.

Fig. 61. The 28-45-53 Triangle.

Fig. 62. The 48-55-73 Triangle.

Since any ratio, however, whose terms, one or both, are represented by fractions, can be transformed into whole numbers, it evidently follows that every conceivable relative measure of two lines which we may decide to call co-sine and co-versed-sine, becomes a satellite to a corresponding "primary" triangle.

Now, since the angle of the satellite on the circumference must be _half_ the angle of the adjacent primary triangle at the centre, it follows that in constructing a list of satellites and their angles, the angles of the corresponding primary triangles can be found. For instance--

Satellite 8, 3, contains 20° 33′ 21·76″
Satellite 2, 7, contains 15° 56′ 43·425″

Each of these angles doubled, gives the angle of a "primary" triangle as follows, viz.:--

The 48, 55, 73 triangle = 41° 6′ 43·52″
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.

+-------+-------+-------+------+-------+------+------+------+
| 1, 2 | 2, 3 | 3, 4 | 4, 5 | 5, 6 | 6, 7 | 7, 8 | 8, 9 |
| | | | | | | | |
| 1, 3 | 2, 5 | 3, 5 | 4, 7 | 5, 7 | | 7, 9 | |
| | | | | | | | |
| 1, 4 | 2, 7 | 3, 7 | 4, 9 | 5, 8 | | | |
| | | | | | | | |
| 1, 5 | 2, 9 | 3, 8 | | 5, 9 | | 7, 1 | |
| | | | | | | | |
| 1, 6 | | | | 5, 11 | | | |
| | | | | | | | |
| 1, 7 | | 3, 11 | | 5, 13 | | | |
| | | | | | | | |
| 1, 8 | | 3, 13 | | | | | |
| | | | | | | | |
| 1, 9 | | | | | | | |
| | | | | | | | |
| 1, 11 | | | | | | | |
| | | | | | | | |
| 1, 13 | | | | | | | |
| | | | | | | | |
| 1, 15 | | | | | | | |
| | | | | | | | |
| 1, 17 | | | | | | | |
+-------+-------+-------+------+-------+------+------+------+

Primary triangles may be found from the _angle of the satellite_, but it is an exceedingly round-about way. I will, however, give an example.

Let us construct a primary triangle from the satellite 4, 9.

Rad. × 4
-------- = ·4444444 = Tangt. < 23° 57′ 45·041″
9

∠ 23° 57′ 45·041″ × 2 = 47° 55′ 30·083″.

therefore the angles of the "primary" are 47° 55′ 30·083″.

and 42° 4′ 29·917″.

The natural sine of 42° 4′ 29·917″ = ·6701025.

The natural co-sine 42° 4′ 29·917″ = ·7422684.

The greatest common measure of these numbers is about 102717, therefore--

Radius 10000000 ÷ 102717 = 97
Co-sine 7422684 ÷ 102717 = 72
Sine 6701025 ÷ 102717 = 65

and 65, 72, 97 is the primary triangle to which the satellites are 4, 9, and 5, 13. (_See Fig_. 63.) The figures in the calculation do not balance exactly, in consequence of the insufficient delicacy of the tables or calculations.

Fig. 63.

The connection between primaries and satellites is shown by figure 64.

Fig. 64.

Let the triangle ADB be a satellite, 5, 2, which we will call BD 20, and AD 8. Let C be centre of semi-circle ABE.

AD : DB :: DB : DE = 50 (_Euc. VI_. 8)

AD + DE = AE = 58 = diameter

AE ÷ 2 = AC = BC = 29 = radius

AC-AD = DC = 21 = co-sine

and DB = 20 = sine

From the preceding it is manifest that--

sine²
----- + ver-s = dia.
ver-s

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--

(1) a = sine.

a² + b²
(2) ------- = radius.
2b

a² - b²
(3) ------- = co-sine.
2b

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--

} 5 = sine = 5
}
} 5² + 1²
From } ------- = radius = 13
Satellite } 2 × 1
5, 1. }
} 5² - 1²
} ------- = co-s. = 12
} 2 × 1

* * * * *

} 5 = sine = 5 } {20
} } {
} 5² + 2² } {
From } ------- = radius = 7¼ } × 4 {29
Satellite} 2 × 2 } {
5, 2. } } {
} 5² + 2² } {
} ------- = co-s. = 5¼ } {21
} 2 × 2 } {

* * * * *

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.
a² - b = perpendicular.
a × 2b = base.

"PRIMARY" LOWEST RATIO.
Thus-- | 5² + 1² = 26 = 13
Satellite 5,1 | 5² - 1² = 24 = 12
| 5 × 2 × 1 = 10 = 5
--------------- |----------------------------
and-- | 5² + 2² = 29 = 29
Satellite 5,2 | 5² - 2² = 21 = 21
| 5 × 2 × 2 = 20 = 20

Having found the lowest ratio of the three sides of a "primary" triangle, the lowest whole numbers for tangent, secant, co-secant, and co-tangent, if required, are obtained in the following manner.

Take for example the 20, 21, 29 triangle, now 20 × 21 = 420, and 29 × 420 = 12180, a new radius instead of 29 from which with the sine 20, and co-sine 21, increased in the same ratio, the whole canon of the 20, 21, 29 triangle will come out in whole numbers.

Similarly in the triangle 48, 55, 73, radius 73 × 13200 (the product of 48 × 55) makes radius in whole numbers 963600, for an even canon without fractions. This is because sine and co-sine are the two denominators in the fractional parts of the other lines when worked out at the lowest ratio of sine, co-sine, and radius.

After I found that the plan of the Gïzeh group was a system of "primary" triangles, I had to work out the rule for constructing them, for I had never met with it in any book, but I came across it afterwards in the "Penny Encyclopedia," and in Rankine's "Civil Engineering."

The practical utility of these triangles, however, does not appear to have received sufficient consideration. I certainly never met with any except the 3, 4, 5, in the practice of any surveyor of my acquaintance.

(For squaring off a line nothing could be more convenient than the 20, 21, 29 triangle; for instance, taking a base of 40 links, then using the whole chain for the two remaining sides of 42 and 58 links.)

Table of Some Primary Triangles and their Satellites.

ANGLE OF PRIMARY PRIMARY SATELLITE. ANGLE OF SATELLITE
DEG. MIN. SEC. RAD. CO.-S. SINE. DEG. MIN. SEC.
2 47 39·70 841 840 41 41 1 1 23 49·85

6 43 58·62 145 144 17 17 1 3 21 59·31

8 47 50·69 85 84 13 13 1 4 23 55·34

10 23 19·89 61 60 11 11 1 5 11 39·94

12 40 49·37 41 40 9 9 1 6 20 24·68

14 14 59·10 65 63 16 8 1 7 7 29·55

16 15 36·73 25 24 7 7 1 8 7 48·36

18 55 28·71 37 35 12 6 1 9 27 44·35

22 37 11·51 13 12 5 5 1 11 18 35·75

25 3 27·27 85 77 36 9 2 12 31 43·63

25 59 21·22 89 80 39 13 3 12 59 40·61

28 4 20·94 17 15 8 4 1 14 2 10·47

30 30 36·49 65 56 33 11 3 15 15 18·24

31 53 26·85 53 45 28 7 2 15 56 43·42

36 52 11·65 5 4 3 3 1 18 26 5·82

41 6 43·52 73 55 48 8 3 20 33 21·76

42 4 30·08 97 72 65 13 5 21 2 15·04

43 36 10·15 29 21 20 5 2 21 48 5·07

46 23 49·85 29 20 21 7 3 23 11 54·92

47 55 29·92 97 65 72 9 4 23 57 44·96

48 53 16·48 73 48 55 11 5 24 26 38·24

53 7 48·35 5 3 4 2 1 26 33 54·17

58 6 33·15 53 28 45 9 5 29 3 16·57

59 29 23·51 65 33 56 7 4 29 44 41·75

61 55 39·06 17 8 15 5 3 30 57 49·53

64 0 38·78 89 39 80 8 5 32 0 19·39

64 56 32·73 85 36 77 11 7 32 28 16·36

67 22 48·49 13 5 12 3 2 33 41 24·24

71 4 31·29 37 12 35 7 5 35 32 15·64

73 44 23·27 25 7 24 4 3 36 52 11·63

75 45 0·90 65 16 63 9 7 37 52 30·45

77 19 10·63 41 9 40 5 4 38 39 35·31

79 36 40·11 61 11 60 6 5 39 48 20·05

81 12 9·31 85 13 84 7 6 40 36 4·65

83 16 1·38 145 17 144 9 8 41 38 0·69

87 12 20·30 841 41 840 21 20 43 36 10·15

Reference to the plan ratio table at the commencement, and to the tables here introduced, will shew that most of the primary triangles mentioned are indicated on the plan ratio table principally by the lines corresponding to the ratios of the satellites. Thus--

PRIMARY TRIANGLE INDICATED BY

17, 144, 145. Triangle FP, PA, AF on plan.
13, 84, 85. Plan ratio of SJ to SU, 7 to 6.
11, 60, 61. Plan ratio BC to FB, 6 to 5, and DN to NR, 61
to 60.
12, 35, 37. Plan ratio EO to AY, 37 to 12, and EA to AY,
35 to 12.
5, 12, 13. Plan ratio CY to BC, 3 to 2; JE to EX, 3 to 2;
CA to YA, 5 to 1; and NZ to ZA, 12 to 5.
8, 15, 17. Plan ratio FB to BY, 5 to 3, and AC to BC, 15
to 8.
33, 56, 55. Plan ratio YX to AY, 7 to 4; AB to BO, 7 to 4;
and EA to AZ, 7 to 4.
28, 45, 53. Exists on plan, AB, BJ, JA.
3, 4, 5. Pervades the plan, and is also indicated by plan
ratio GX to DG, 2 to 1; SU to SV, 2 to 1;
and CY to YZ, 3 to 1.
48, 55, 73. Exists on plan, FW, WV, VF--and is also indicated
by plan ratio FO to OZ, 8 to 3.
65, 72, 97. Plan ratio AC to CH, 9 to 4; MY to YZ, 9 to 4.
20, 21, 29. Exists on plan FB, BA, AF; and plan ratio, GU
to DG, 5 to 2.

It seems probable that could I add to my pyramid plan the lines and triangles that the missing eleven pyramids would supply, it would comprise a complete table on which would appear indications of all the ratios and triangles made use of in right-angled trigonometry, a "_ratiometer_" in fact.

I firmly believe that so far as I have gone it is correct--and it is possible, therefore, with the start that I have made, for others to continue the work, and add the eleven pyramids to the plan in their correct geometrical position. 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.

§ 13. THE SIZE AND SHAPE OF THE PYRAMIDS INDICATED BY THE PLAN.

I pursued my investigations into the slopes and altitudes of the pyramids without reference to the plan, after once deciding their exact bases.

Now it will be interesting to note some of the ways in which the plan hints at the shape and size of these pyramids, and corroborates my work.

The dimensions of _Cheops_ are indicated on the plan by the lines EA to YA, measuring 840 and 288 R.B. cubits respectively, being the half periphery of its horizontal section at the level of Cephren's base, and its own altitude from its own base. (_See Fig_. 5.)

The line EA, in fact, represents in R.B. cubits the half periphery of the bases of either Cheops or Cephren measured at the level which I have set forth as the _plan level_, viz., base of Cephren.

The ratio of Cephren's base to Cephren's altitude is indicated on the plan by the ratios of the lines BC to EB, or FO to OR, viz., 32 to 21. (_See Fig._ 4.)

The altitude of Mycerinus above Cephren's base appears on plan in the line EF, measuring 136 R.B. cubits.

The line EO on plan measures 888 cubits, which would be the length of a line stretched from the apex of Cheops to the point E, at the level of Cheops' base.

This merits consideration:--the lines EA and AY are connected on plan at the centre of Cheops, and the lines EO and EA are connected on plan at the point E.

Now the lines EO, EA and AY are sides of a "primary triangle," whose ratio is 37, 35, 12, and whose measure in cubits is 888, 840, and 288; and if we suppose the line EA to be stretched horizontally beneath the pyramids at the level of the base of Cheops from E to A on plan, and the line AY to be a plumb line hanging from the apex of Cheops to the level of his base, then will the line EO just stretch from the point E to the apex of Cheops, and the three lines will connect the two main pyramids by a vertical triangle of which EA, AY and EO form the base, perpendicular, and hypotenuse. Or, to explain it in another manner: let the line EA be a _cord_ stretching horizontally from A at the centre of the base of Cheops to the point E, both ends being at the same level; let the line AY be a _rod_, lift it on the end A till it stands erect, then is the end Y the apex of Cheops. Now, the line EO would just stretch from the top of the rod AY to the point E first described.

It is a singular coincidence, and one that may be interesting to students of the _interior_ of the Pyramids, that the side EP, of the small 3, 4, 5 triangle, EP, PF, FE, in the centre of the plan, measures 81·60 R.B. cubits, which is very nearly eight times the "_true breadth_" of the King's chamber in Cheops, according to Piazzi Smyth; for 81·60/8 = 10·20 R.B. cubits, or 206·046 pyramid inches (one R.B. cubit being 20·2006 pyramid inches). The sides of this little triangle measure 81·60, 108·80, and 136, R.B. cubits respectively, as can be easily proved from the plan ratio table.

§ 14. A SIMPLE INSTRUMENT FOR LAYING OFF "PRIMARY TRIANGLES."

A simple instrument for laying off "primary triangles" upon the ground, might have been made with three rods divided into a number of small equal divisions, with holes through each division, which rods could be pinned together triangularly, the rods working as arms on a flat table, and the pins acting as pointers or sights.

One of the pins would be permanently fixed in the table through the first hole of two of the rods or arms, and the two other pins would be movable so as to fix the arms into the shape of the various "primary triangles."

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.

§ 14_a_. 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.

A re-examination of these pyramids on the ground with the ideas suggested by the preceding pages in view, may lead to interesting discoveries.

For instance, it is just possible that the very accurately and beautifully worked stones in the walls of the King's chamber of Cheops, may be found to indicate the ratios of the rectangles formed by the bases and perpendiculars of the triangulations used by the old surveyors--that on these walls may be found, in fact, corroboration of the theory that I have set forth. I am led to believe also from the fact that Gïzeh was a central and commanding locality, and that it was the custom of those who preceded those Egyptians that history tells of, to excavate mighty caverns in the earth--that, therefore, in the limestone upon which the pyramids are built, and underneath the pyramids, may be found vast excavations, chambers and galleries, that had entrance on the face of the ridge at the level of High Nile. From this subterraneous city, occupied by the priests and the surveyors of Memphis, access may be found to every pyramid; and while to the outside world the pyramids might have appeared sealed up as mausoleums to the Kings that it may have seen publicly interred therein, this very sealing and closing of the outer galleries may have only rendered their mysterious recesses more private to the priests who entered from below, and who were, perhaps, enabled to ascend by private passages to their very summits. The recent discovery of a number of regal mummies stowed away in an out of the way cave on the banks of the Nile, points to the unceremonious manner in which the real rulers of Kings and people may have dealt with their sovereigns, the pomp and circumstance of a public burial once over. It is just possible that the chambers in the pyramids may have been used in connection with their mysteries: and the small passages called by some "ventilators" or "air passages," sealed as they were from the chamber by a thin stone (and therefore no ventilators) may have been _auditory passages_ along which sound might have been projected from other chambers not yet opened by the moderns; sounds which were perhaps a part of the "hanky panky" of the ancient ceremonial connected with the "mysteries" or the "religion" of that period.

Down that "well" which exists in the interior of Cheops, and in the limestone foundations of the pyramid, should I be disposed to look for openings into the vast subterraneous chambers which I am convinced _do_ exist below the Pyramids of Gïzeh.

The priests of the Pyramids of Lake Moeris had their vast subterranean residences. It appears to me more than probable that those of Gïzeh were similarly provided. And I go further:--Out of these very caverns may have been excavated the limestone of which the pyramids were built, thus killing two birds with one stone--building the instruments and finding cool quarters below for those who were to make use of them. In the bowels of that limestone ridge on which the pyramids are built will yet be found, I feel convinced, ample information as to their uses. A good diamond drill with two or three hundred feet of rods is what is what is wanted to test this, and the solidity of the pyramids at the same time.

§ 15. PRIMARY TRIANGULATION.

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.

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--

SATELLITE. PRIMARY. ANGLE.

5 2 20 21 29 43° 36′ 10·15″

7 3 21 20 29 46° 23′ 49·85″

and so on. Such a set of tables would be a boon to sailors, architects, surveyors, engineers, and all handi-craftsmen: and I make bold to say, would assist in the intricate investigations of the astronomer:--and the rule for building the tables is so simple, that they could easily be achieved. The architect from these tables might arrange the shape of his chambers, passages or galleries, so that all measures, not only at right angles on the walls, but from any corner of floor to ceiling should be even feet. The pitch of his roofs might be more varied, and the monotony of the buildings relieved, with rafters and tie-beams always in even measures. The one solitary 3, 4, 5 of Vitruvius would cease to be his standard for a staircase; and even in doors and sashes, and panels of glass, would he be alive to the perfection of rectitude gained by evenly-measured diagonals. By a slight modification of the compass card, the navigator of blue water might steer his courses on the hypotenuses of great primary triangles--such tables would be useful to all sailors and surveyors who have to deal with latitude and departure. For instance, familiarity with such tables would make ever present in the mind of the surveyor or sailor his proportionate northing and easting, no matter what course he was steering between north and east, "the _primary_" embraces the _three ideas in one view_.

In designing trussed roofs or bridges, the "primaries" would be invaluable to the engineer, strain-calculations on diagonal and upright members would be simplified, and the builder would find the benefit of a measure in even feet or inches from centre of one pin or connection to another.

For earthwork slopes 3, 4, 5; 20, 21, 29; 21, 20, 29; and 4, 3, 5 would be found more convenient ratios than 1 _to_ 1, and 1½ _to_ 1, etc. Templates and battering rules would be more perfect and correct, and the engineer could prove his slopes and measure his work at one and the same time without the aid of a staff or level; the slope measures would reveal the depth, and the slope measures and bottom width would be all the measures required, while the top width would prove the correctness of the slopes and the measurements.

To the land surveyor, however, the primary triangle would be the most useful, and more especially to those laying out new holdings, whether small or large, in new countries.

Whether it be for a "squatter's run," or for a town allotment, the advantages of a diagonal measure to every parallelogram in even _miles_, _chains_, or _feet_, should be keenly felt and appreciated.

This was, I believe, _one_ of the secrets of the speedy and correct replacement of boundary marks by the Egyptian land surveyors.

I have heard of a review in the "Contemporary," September, 1881, referring to the translation of a papyrus in the British Museum, by Dr. Eisenlohr--"_A handbook of practical arithmetic and geometry," etc., "such as we might suppose would be used by a scribe acting as clerk of the works, or by an architect to shew the working out of the problems he had to solve in his operations_." I should like to see a translation of the book, from which it appears that "_the clumsiness of the Egyptian method is very remarkable_." Perhaps this Egyptian "_Handbook_" may yet shew that their operations were not so "_clumsy_," as they appear at first sight to those accustomed to the practice of modern trigonometry. 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.