The Solution of the Pyramid Problem Part 10

of the King's chamber in Cheops, according to Piazzi Smyth; for 8160/8 = 1020 R.B. cubits, or 206046 pyramid inches (one R.B. cubit being 202006 pyramid inches). The sides of this little triangle measure 8160, 10880, 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 Gzeh 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 pa.s.sages 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 circ.u.mstance 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 pa.s.sages called by some "ventilators" or "air pa.s.sages," sealed as they were from the chamber by a thin stone (and therefore no ventilators) may have been _auditory pa.s.sages_ 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 Gzeh.

The priests of the Pyramids of Lake Moeris had their vast subterranean residences. It appears to me more than probable that those of Gzeh 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 circ.u.mference 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' 1015?

7 3 21 20 29 46 23' 4985?

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 a.s.sist 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, pa.s.sages 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 gla.s.s, would he be alive to the perfection of rect.i.tude gained by evenly-measured diagonals. By a slight modification of the compa.s.s 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 lat.i.tude 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 1 _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.

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; and that this system of geometric triangulation or right-angled trigonometry was the method practised, seems in the preceding pages to be fairly established. I claim, therefore, that I have discovered and described the main secret of the pyramids, that I have found for them at last a practical use, and that it is no longer "_a marvel how after the annual inundation, each property could have been accurately described by the aid of geometry._" I have advanced nothing in the shape of a theory that will not stand a practical test; but to do it, the pyramids should be _re-cased_. Iron sheeting, on iron or wooden framework, would answer. I may be wrong in some of my conclusions, but in the main I am satisfied that I am right. It must be admitted that I have worked under difficulties; a glimpse at the pyramids three and twenty years ago, and the meagre library of a nomad in the Australian wilderness having been all my advantages, and time at my disposal only that s.n.a.t.c.hed from the rare intervals of leisure afforded by a arduous professional life.

After fruitless waiting for a chance of visiting Egypt and Europe, to sift the matter to the bottom, I have at last resolved to give my ideas to the world as they stand; crude necessarily, so I must be excused if in some details I may be found erroneous; there is truth I know in the general conclusions. I am presumptuous enough to believe that the R.B.

cubit of 1685 British feet was the measure of the pyramids of Gzeh, although there may have been an astronomical 25 inch cubit also. It appears to me that no cubit measure to be depended on is either to be got from a stray measuring stick found in the joints of a ruined building, or from any line or dimensions of one of the pyramids. I submit that a most reasonable way to get a cubit measure out of the Pyramids of Gzeh, was to do as I did:--take them as a whole, comprehend and establish the general ground plan, find it geometric and harmonic, obtain the ratios of all the lines, establish a complete set of natural and even numbers to represent the measures of the lines, and finally bring these numbers to cubits by a common multiplier (which in this case was the number eight). After the whole proportions had been thus expressed in a cubit evolved _from_ the whole proportions, I established its length in British feet by dividing the base of Cephren, as known, by the number of my cubits representing its base. It is pretty sound evidence of the theory being correct that this test, with 420 cubits neat for Cephren, gave me also a neat measure for Cheops, from Piazzi Smyth's base, of 452 cubits, and that at the same level, these two pyramids become equal based.

I have paid little attention to the inside measurements. I take it we should first obtain our exoteric knowledge before venturing on esotoric research. Thus the intricate internal measurements of Cheops, made by various enquirers have been little service to me, while the accurate measures of the base of Cheops by Piazzi Smyth, and John James Wild's letter to Lord Brougham, helped me amazingly, as from the two I established the plan level and even bases of Cheops and Cephren at plan level--as I have shown in the preceding pages. My theory demanded that both for the building of the pyramids and for the construction of the models or subsidiary instruments of the surveyors, simple slope ratios should govern each building; before I conclude, I shall show how I got at my slope ratios, by evolving them from the general ground plan.

I am firmly convinced that a careful investigation into the ground plans of the various other groups of pyramids will amply confirm my survey theory--the relative positions of the groups should also be established--much additional light will be then thrown on the subject.

Let me conjure the investigator to view these piles _from a distance_ with his mind's eye, as the old surveyors viewed them with their bodily eye. Approach them too nearly, and, like Henry Kinglake, you will be lost in the "_one idea of solid immensity._" Common sense tells us they were built to be viewed from a distance.

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.

The modern astronomer from the top of an observatory points his instruments direct at the stars; the Egyptian astronomer from the summit of his particular pyramid directed his observations to the rising and setting of the stars, or the positions of the heavenly bodies in respect to the far away groups of pyramids scattered around him in the distance; and by comparing notes, and with the knowledge of the relative position of the groups, did these observers map out the sky. Solar and lunar shadows of their own pyramids on the flat trenches prepared for the purpose, enabled the astronomer at each observatory to record the annual and monthly flight of time, while its hours were marked by the shadows of their obelisks, capped by copper pyramids or b.a.l.l.s, on the more delicate pavements of the court-yards of their public buildings.

We must grasp that their celestial and terrestrial surveys were almost a reverse process to our own, before we can venture to enquire into its details. It then becomes a much easier tangle to unravel. That a particular pyramid among so many, should have been chosen as a favoured interpreter of Divine truths, seems an unfair conclusion to the other pyramids;--that the other pyramids were rough and imperfect imitations, appears to my poor capacity "a base and impotent conclusion;"--(as far as I can learn, _Mycerinus_, in its perfection, was a marvel of the mason's art;) but that one particular pyramid should have anything to do with the past or the future of the lost ten tribes of Israel (whoever that fraction of our present earthly community may be), seems to me the wildest conclusion of all, except perhaps the theory that this one pyramid points to the future of the British race. Yet in one way do I admit that the pyramids point to our future.

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.

Thirty-six centuries hence, they may look down on a civilization equally strange, in which our history, language, and religion, Hebrew race, and British race, may have no place, no part.

If the thoughts of n.o.ble poets live, as they seem to do, old Cheops, that mountain of ma.s.sive masonry, may (like the brook of our Laureate), in that dim future, still be singing, as he seems to sing now, this idea, though not perhaps these words:

"For men may come, and men may go, But I go on for ever."

"Ars longa, vita brevis." Man's work remains, when the workman is forgotten; fair work and square, can never perish entirely from men's minds, so long as the world stands. These pyramids were grand and n.o.ble works, and they will not perish till their reputation has been re-established in the world, when they will live in men's memories to all generations as symbols of the mighty past. To the minds of many now, as to Josephus in his day, they are "_vast and vain monuments,_" records of folly. To me they are as monuments of peace, civilization and order--relics of a people living under wise and beneficent rulers--evidences of cultivation, science, and art.

-- 16. THE PENTANGLE OR FIVE POINTED STAR THE GEOMETRIC SYMBOL OF THE GREAT PYRAMID.

From time immemorial this symbol has been a blazing pointer to grand and n.o.ble truths, and a solemn emblem of important duties.

Its geometric significance, however, has long been lost sight of.

It is said to have const.i.tuted the seal or signet of King Solomon (1000 B.C.), and in early times it was in use among the Jews, as a symbol of safety.

It was the Pentalpha of Pythagoras, and the Pythagorean emblem of health (530 B.C.).

It was carried as the banner of Antiochus, King of Syria (surnamed Soter, or the Preserver), in his wars against the Gauls (260 B.C.).

Among the Cabalists, the star with the sacred name written on each of its points, and in the centre, was considered talismanic; and in ancient times it was employed all over Asia as a charm against witchcraft. Even now, European troops at war with Arab tribes, sometimes find, under the clothing, on the b.r.e.a.s.t.s of their slain enemies, this ancient emblem, in the form of a metal talisman, or charm.

The European Goethe puts these words into the mouth of Mephistopheles:

"I am hindered egress by a quaint device upon the threshold,--that five-toed d.a.m.ned spell."