Part 2

His cubit was, I believe, the Memphis, or Nilometric cubit--but at any rate, he made the base of Cephren 412 of them.

I therefore divided the recognized base of Cephren--viz., 707·75 British feet--by 412, and got a result of 1·7178 British feet for his cubit. Therefore, his measures multiplied by 1·7178 and divided by 1·685 will turn his cubits into R.B. cubits.

I thus make Cheops, Cephren, and Mycerinus, respectively, 81·56, 101·93, and 106·93 R.B. cubits above the datum that J. J. Wild calls Nile Level. According to Bonwick's "Facts and Fancies," p. 31, high water Nile would be 138½ ft. below base of Cheops (or 82·19 R.B. cubits).

Piazzi Smyth makes the pavement of Cheops 1752 British inches (or 86·64 R.B. cubits) above _average Nile Level_, but, by scaling his map, his _high Nile Level_ appears to agree nearly with Wild.

It is the _relative levels_ of the Pyramids, however, that I require, no matter how much above Nile Level.

Cephren's base of 420 cubits being 101·93 cubits, and Cheops' base of 452 cubits being 81·56 cubits above Wild's datum, the difference in level of their bases is, 20·37 cubits.

The ratio of base to altitude of Cheops being 330 to 210, therefore 20·37 cubits divided by 210 and multiplied by 330 equals 32 cubits; and 452 cubits minus 32 cubits, equals 420.

Similarly, the base of Mycerinus is 5 cubits _above_ the base of Cephren, and the ratio of base to altitude 32 to 20; therefore, 5 cubits divided by 20 and multiplied by 32 equals 8 cubits to be _added_ to the 210 cubit base of Mycerinus, making it 218 cubits in breadth at the level of Cephren's base.

Thus, a horizontal section or plan at the level of Cephren's base would meet the slopes of the Pyramids so that they would on plan appear as squares with sides equal to 218, 420, and 420 R.B. cubits, for Mycerinus, Cephren, and Cheops, respectively.

Fig. 21. R.B. Cub. Apex of Cephren above Base Cheops 295·98 Apex of Cheops above Base Cheops 287·77 Apex of Mycerinus above Base Cheops 156·51 Base Cephren above base of Cheops 20·37 Base Mycerinus above base of Cheops 25·37

Piazzi Smyth makes the top of the tenth course of Cheops 414 pyramid inches above the pavement; and 414 divided by 20·2006 equals 20·49 R.B. cubits.

But I have already proved that Cheops' 420 cubit base measure occurs at a level of 20·37 cubits above pavement; therefore is this level the level of the top of the tenth course, for the difference is only 0·12 R.B. cubits, or 2½ inches.

* * * * *

I wish here to note as a matter of interest, but not as affecting my theory, the following measures of Piazzi Smyth, turned into R.B. cubits, viz.:--

PYR. INCHES. R.B. CUBITS. King's Chamber floor, above pavement 1702· = 84·25 Cheops' Base, as before stated 9131·05 = 452·01 King's Chamber, "True Length," 412·132 = 20·40 " " "True First Height," 230·389 = 11·40 " " "True Breadth," 206·066 = 10·20

He makes the present summit platform of Cheops 5445 pyramid inches above pavement. My calculation of 269·80 R.B. cub. (See Fig. 21) is equal to 5450 pyramid inches--this is about 18 cubits below the theoretical apex.

* * * * *

Figure 21 represents the comparative levels and dimensions of Mycerinus, Cephren, and Cheops.

The following peculiarities are noticeable:--That Cheops and Cephren are of equal bases at the level of Cephren's base;--that, at the level of Cheops' base, the latter is only half a cubit larger;--that, from the level of Mycerinus' base, Cheops is just double the height of Mycerinus;--and that from the level of Cephren's base, Cephren is just double the height of Mycerinus; measuring in the latter case, however, only up to the level platform at the summit of Cephren, which is said to be about eight feet wide.

The present summit of Cephren is 23·07 cubits above the present summit of Cheops, and the completed apex of Cephren would be 8·21 cubits above the completed apex of Cheops.

In the summit platforms I have been guided by P. Smyth's estimate of _height deficient_, 363 pyr. inches, for Cheops, and I have taken 8 feet base for Cephren's summit platform.

§ 6. GEOMETRICAL PECULIARITIES OF THE PYRAMIDS.

_In any pyramid, the apothem is to half the base as the area of the four sides is to the area of the base._

Thus--Ratio apothem to half base Mycerinus 32 to 20 " " " " Cephren 33 to 20 " " " " Cheops 34 to 21

AREA OF THE FOUR SIDES. AREA OF THE BASE.

Mycerinus 70560· 44100 Cephren 291060· 176400 Cheops 330777·90 204304

All in R.B. cubits.

Therefore--32 : 20 : : 70560· : 44100 33 : 20 : : 291060· : 176400 34 : 21 : : 330777·90 : 204304

[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 4¼ British inches. I have therefore the authority of Herodotus to support the theory which I shall subsequently set forth, that this pyramid was the exponent of lines divided in mean and extreme ratio.

By taking the dimensions of the Pyramid from what I may call its _working level_, that is, the level of the base of Cephren, this peculiarity shows more clearly, as also others to which I shall refer. Thus--base of Cheops at working level, 420 cubits, and apothem 340 cubits; base area is, therefore, 176400 cubits, and area of one face is (420 cubits, multiplied by half apothem, or 170 cubits) 71400 cubits. Now the square root of 71400 would give altitude, or side of square equal to altitude, 267·207784 cubits: but the real altitude is √(340²-210²) = √71500 = 267·394839. So that the error of Herodotus's proposition is the difference between √714 and √715.

Footnote 2: Proctor is responsible for this statement, as I am quoting from an essay of his in the _Gentleman's Magazine_. R. B.

This leads to a consideration of the properties of the angle formed by the ratio _apothem_ 34 to _half base_ 21, peculiar to the pyramid Cheops. (_See Figure 22._)

Fig. 22. Diagram illustrating relations of ratios of the pyramid Cheops.

Calling apothem 34, _radius_; and half base 21, _sine_--I find that--

Radius is the square root of 1156 Sine 441 Co-sine 715 Tangent 713 Secant 1869 and Co-versed-sine 169

So it follows that the area of one of the faces, 714, is a mean between the square of the altitude or co-sine, 715, and the square of the tangent, 713.

Thus the reader will notice that the peculiarities of the Pyramid Cheops lie in the regular relations of the _squares_ of its various lines; while the peculiarities of the other two pyramids lie in the relations of the lines themselves.

Mycerinus and Cephren, born, as one may say, of those two noble triangles 3, 4, 5, and 20, 21, 29, exhibit in their lineal developments ratios so nearly perfect that, for all practical purposes, they may be called correct.

Thus--Mycerinus, [3]20² + 25² = 1025, and 32² = 1024. and Cephren, [4]80² + 105² = 17425, and 132² = 17424. or [5]400² + 431² = 345761, and 588² = 345744.

See diagrams, Figures 11 to 14 inclusive.

In the Pyramid Cheops, altitude is _very nearly_ a mean proportional between apothem and half base. Apothem being 34, and half base 21, then altitude would be √(34²-21²) = √715 = 26·7394839, and--

21 : 26·7394839 :: 26·7394839 : 34, nearly.

Here, of course, the same difference comes in as occurred in considering the assumption of Herodotus, viz., the difference between √715 and √714; because if the altitude were √714, then would it be _exactly_ a mean proportional between the half base and the apothem; (thus, 21 : 26·72077 :: 26·72077 :: 34.)

Footnote 3: Half base to altitude.

Footnote 4: Half base to altitude.

Footnote 5: Half diagonal of base to altitude.

In Cheops, the ratios of apothem, half base and edge are, 34, 21, and 40, very nearly, thus, 34² + 21² = 1597, and 40² = 1600.

The dimensions of Cheops (from the level of the base of Cephren) to be what Piazzi Smyth calls a [Pi] pyramid, would be--

Half base 210 R.B. cubits. Altitude 267·380304, &c. " Apothem 339·988573, &c. "

Altitude being to perimeter of base, as radius of a circle to circumference.

My dimensions of the pyramid therefore in which--

Half base = 210 R.B. cubits. Altitude = 267·394839 &c. " Apothem = 340 "

come about as near to the ratio of [Pi] as it is possible to come, and provide simple lines and templates to the workmen in constructing the building; and I entertain no doubt that on the simple lines and templates that my ratios provide, were these three pyramids built.

§ 6A. THE CASING STONES OF THE PYRAMIDS.

Figures 23, 24, and 25, represent ordinary casing stones of the three pyramids, and Figures 26, 27, and 28, represent angle or quoin casing stones of the same.

The casing stone of Cheops, found by Colonel Vyse, is represented in Bonwick's "Pyramid Facts and Fancies," page 16, as measuring four feet three inches at the top, eight feet three inches at the bottom, four feet eleven inches at the back, and six feet three inches at the front. Taking four feet eleven inches as _Radius_, and six feet three inches as _Secant_, then the _Tangent_ is three feet ten inches and three tenths.

Fig. 23. Cheops Casing Stone.

Fig. 24. Cephren Casing Stone.

Fig. 25. Mycerinus Casing Stone.

Fig. 26. Cheops Angle or Quoin Stone.

Fig. 27. Cephren Angle or Quoin Stone.

Fig. 28. Mycerinus Angle or Quoin Stone.

Thus, in inches (√(75²-59²)) = 46·30 inches; therefore the inclination of the stone must have been--slant height 75 inches to 46·30 inches horizontal. Now, 46·30 is to 75, as 21 is to 34. Therefore, Col. Vyse's casing stone agrees exactly with my ratio for the Pyramid Cheops, viz., 21 to 34. (_See Figure 29._)

Fig. 29. =Col. Vyse's Casing Stone.= 75 : 46·3 :: 34 : 21

This stone must have been out of plumb at the back an inch and seven tenths; perhaps to give room for grouting the back joint of the marble casing stone to the limestone body of the work: or, because, as it is not a necessity in good masonry that the back of a stone should be exactly plumb, so long as the error is on the right side, the builders might not have been particular in that respect.

Fig. 59. (Temple of Cheops, standing at angle of wall.)

Figure 59 represents such a template as the masons would have used in building Cheops, both for dressing and setting the stones. (The courses are drawn out of proportion to the template.) The other pyramids must have been built by the aid of similar templates.

Such large blocks of stone as were used in the casing of these pyramids could not have been completely dressed before setting; the back and ends, and the top and bottom beds were probably dressed off truly, and the face roughly scabbled off; but the true slope angle could not have been dressed off until the stone had been truly set and bedded, otherwise there would have been great danger to the sharp arises.

* * * * *

I shall now record the peculiarities of the 3, 4, 5 or Pythagorean triangle, and the right-angled triangle 20, 21, 29.

§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.

Fig. 30 to 35. PECULIARITIES OF THE TRIANGLES

The 3, 4, 5 triangle contains 36° 52′ 11·65″ and the complement or greater angle 53° 7′ 48·35″

Radius 5 = 60 whole numbers.[6] Co-sine 4 = 48" Sine 3 = 36" Versed sine 1 = 12" Co-versed sine 2 = 24" Tangent 3¾ = 45" Secant 6¼ = 75" Co-tangent 6⅔ = 80" Co-secant 8⅓ = 100"

Tangent + Secant = Diameter or 2 Radius Co-tan + Co-sec = 3 Radius Sine : Versed-sine :: 3 : 1 Co-sine : Co-versed sine :: 2 : 1

Figure 30 illustrates the preceding description. Figure 31 shows the 3·1 triangle, and the 2·1 triangle built up on the sine and co-sine of the 3, 4, 5 triangle.

The 3·1 triangle contains 18° 26′ 5·82″ and the 2·1 triangle 26° 33′ 54·19″; the latter has been frequently noticed as a pyramid angle in the gallery inclinations.

Figure 32 shows these two triangles combined with the 3, 4, 5 triangle, on the circumference of a circle.

Footnote 6: 60 = 3 × 4 × 5

The 20, 21, 29 triangle contains 43° 36′ 10·15″ and the complement, 46° 23′ 49·85″.

Expressed in whole numbers--

Radius 29 = 12180[7] Sine 20 = 8400 Co-sine 21 = 8820 Versed sine 8 = 3360 Co-versed sine 9 = 3780 Tangent = 11600 Co-tangent = 12789 Secant = 16820 Co-sec = 17661

Tangent + Secant = 2⅓ radius Co-tan + Co-sec = 2½ radius Sine : Versed sine :: 5 : 2 Co-sine : Co-versed sine :: 7 : 3

Footnote 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.[8]

Footnote 8: 12 = 3 × 4, and 420 = 20 × 21

I am led to believe from study of the plan, and consideration of the whole numbers in this 20, 21, 29 triangle, that the R.B. cubit, like the Memphis cubit, was divided into 280 parts.

The whole numbers of radius, sine, and co-sine divided by 280, give a very pretty measure and series in R.B. cubits, viz., 43½, 30, and 31½, or 87, 60, and 63, or 174, 120 and 126;--all exceedingly useful in right-angled measurements. Notice that the right-angled triangle 174, 120, 126, in the sum of its sides _amounts to_ 420.

Figure 33 illustrates the 20, 21, 29 triangle. Figure 34 shows the 5·2 and 7·3 triangles built up on the sine and co-sine of the 20, 21, 29 triangle.

The 5·2 triangle contains 21° 48′ 5·08″ and the 7·3 triangle 23° 11′ 54·98″.

Figure 35 shows how these two triangles are combined with the 20, 21, 29 triangle on the circumference, and Figure 36 gives a general view and identification of these six triangles which occupied an important position in the trigonometry of a people who did all their work by right angles and proportional lines.

Fig. 36. Ratios of Leading Triangles.

§ 8. GENERAL OBSERVATIONS.

It must be admitted that in the details of the building of the Pyramids of Gïzeh there are traces of other measures than R. B. cubits, but that the original cubit of the plan was 1·685 British feet I feel no doubt. It is a perfect and beautiful measure, fit for such a noble design, and, representing as it does the sixtieth part of a second of the Earth's polar circumference, it is and was a measure for all time.

It may be objected that these ancient geometricians could not have been aware of the measure of the Earth's circumference; and wisely so, were it not for two distinct answers that arise. The first being, that since I think I have shown that Pythagoras never discovered the Pythagorean triangle, but that it must have been known and practically employed thousands of years before his era, in the Egyptian Colleges where he obtained his M.A. degree, so in the same way it is probable that Eratosthenes, when he went to work to prove that the earth's circumference was fifty times the distance from Syene to Alexandria, may have obtained the idea from his ready access to the ill-fated Alexandrian Library, in which perhaps some record of the learning of the builders of the Pyramids was stored. 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 those magnificent and, as I shall prove, useful edifices to be stripped of their beautiful garments of polished marble.

My second answer is that the correct cubit measure may have been got by its inventors in a variety of other ways; for instance, by observations of shadows of heavenly bodies, without any knowledge even that the earth was round; or it may have been evolved like the British inch, which Sir John Herschel tells us is within a thousandth part of being one five hundred millionth of the earth's polar axis. I doubt if the circumference of the earth was considered by the inventor of the British inch.

It was a peculiarity of the Hindoo mathematicians that they tried to make out that all they knew was _very old_. 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.

While I was groping about in the dark searching for this bright needle, I tried on the plan many likely ancient measures.

For a long time I worked in Memphis or Nilometric cubits, which I made 1·7126 British feet; they seem to vary from 1·70 to 1·72, and although I made good use of them in identifying other people's measures, still they were evidently not in accordance with the design; but the R.B. cubit of 1·685 British feet works as truly into the plan of the Pyramids _without fractions_ as it does into the circumference of the earth.

Here I might, to prevent others from falling into one of my errors, point out a rock on which I was aground for a long time. I took the base of the Pyramid Cheops, determined by Piazzi Smyth, from Bonwick's "Pyramid Facts and Fancies" (a valuable little reference book), as 763.81 British feet, and the altitude as 486.2567; and then from Piazzi Smyth's "Inheritance," page 27, I confirmed these figures, and so worked on them for a long time, but found always a great flaw in my work, and at last adopted a fresh base for Cheops, feeling sure that Mr. Smyth's base was wrong: for I was absolutely grounded in my conviction that at a certain level, Cheops' and Cephren's measures bore certain relations to each other. I subsequently found in another part of Mr. Smyth's book, that the correct measures were 761.65 and 484.91 British feet for base and altitude, which were exactly what I wanted, and enabled me to be in accordance with him in that pyramid which he appears to have made his particular study.

For the information of those who may wish to compare my measures, which are the results of an even or regular circumference without fractions, with Mr. Smyth's measures, which are the results of an even or regular diameter without fractions, it may be well to state that there are just about 99 R.B. cubits in 80 of Piazzi Smyth's cubits of 25 pyramid inches each.

§ 9. THE PYRAMIDS OF EGYPT, THE THEODOLITES OF THE EGYPTIAN LAND SURVEYORS.

About twenty-three years ago, on my road to Australia, I was crossing from Alexandria to Cairo, and saw the pyramids of Gïzeh.

I watched them carefully as the train passed along, noticed their clear cut lines against the sky, and their constantly changing relative position.

I then felt a strong conviction that they were built for at least one useful purpose, and that purpose was the survey of the country. I said, "Here be the Theodolites of the Egyptians."

Built by scientific men, well versed in geometry, but unacquainted with the use of glass lenses, these great stone monuments are so suited in shape for the purposes of land surveying, that the practical engineer or surveyor must, after consideration, admit that they may have been built mainly for that purpose.

Not only might the country have been surveyed by these great instruments, and the land allotted at periodical times to the people; but they, remaining always in one position, were there to correct and readjust boundaries destroyed or confused by the annual inundations of the Nile.

The Pyramids of Egypt may be considered as a great system of landmarks for the establishment and easy readjustment at any time of the boundaries of the holdings of the people.

The Pyramids of Gïzeh appear to have been main marks; and those of Abousir, Sakkârah, Dashow, Lisht, Meydoun, &c., with the great pyramids in Lake Moeris, subordinate marks, in this system, which was probably extended from Chaldea through Egypt into Ethiopia.

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 land of Egypt was valuable and maintained a dense population; every year it was mostly submerged, and the boundaries destroyed or confused. Every soldier had six to twelve acres of land; the priests had their slice of the land too; after every war a reallotment of the lands must have taken place, perhaps every year.

While the water was lying on the land, it so softened the ground that the stone boundary marks must have required frequent readjustment, as they would have been likely to fall on one side.

By the aid of their great stone theodolites, the surveyors, 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. It is possible that at the time of Diodorus and Herodotus, (and even when Pythagoras visited Egypt,) theology may have so smothered science, that the uses of the pyramids may have been forgotten by the very priests to whom in former times the knowledge belonged; but "a respectful reticence" which has been noticed in some of these old writers on pyramid and other priestly matters would rather lead us to believe that an initiation into the mysteries may have sealed their lips on subjects about which they might otherwise have been more explicit.

The "_closing_" of one pyramid over another in bringing any of their many lines into true order, must even now be very perfect;--but now we can only imagine the beauties of these great instrumental wonders of the world when the casing stones were on them. We can picture the rosy lights of one, and the bright white lights of others; their clear cut lines against the sky, true as the hairs of a theodolite; and the sombre darkness of the contrasting shades, bringing out their angles with startling distinctness. Under the influence of the Eastern sun, the faces must have been a very blaze of light, and could have been seen at enormous distances like great mirrors.