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 ill.u.s.trates the preceding description. Figure 31 shows the 31 triangle, and the 21 triangle built up on the sine and co-sine of the 3, 4, 5 triangle.
The 31 triangle contains 18 26" 582? and the 21 triangle 26 33" 5419?; 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 circ.u.mference of a circle.
Footnote 6: 60 = 3 4 5
The 20, 21, 29 triangle contains 43 36" 1015? and the complement, 46 23" 4985?.
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 = 2? radius Co-tan + Co-sec = 2 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., 43, 30, and 31, 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 ill.u.s.trates the 20, 21, 29 triangle. Figure 34 shows the 52 and 73 triangles built up on the sine and co-sine of the 20, 21, 29 triangle.
The 52 triangle contains 21 48" 508? and the 73 triangle 23 11" 5498?.
Figure 35 shows how these two triangles are combined with the 20, 21, 29 triangle on the circ.u.mference, 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 Gzeh there are traces of other measures than R. B. cubits, but that the original cubit of the plan was 1685 British feet I feel no doubt.
It is a perfect and beautiful measure, fit for such a n.o.ble design, and, representing as it does the sixtieth part of a second of the Earth"s polar circ.u.mference, 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 circ.u.mference; 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 circ.u.mference 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 circ.u.mference 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 circ.u.mference 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 O5131 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 ident.i.ty of the builders of the Pyramids of Gzeh. 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 17126 British feet; they seem to vary from 170 to 172, 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 1685 British feet works as truly into the plan of the Pyramids _without fractions_ as it does into the circ.u.mference 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 alt.i.tude 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 alt.i.tude, 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 circ.u.mference 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 Gzeh.
I watched them carefully as the train pa.s.sed 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 gla.s.s 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 Gzeh appear to have been main marks; and those of Abousir, Sakkarah, 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.
I declare that the pyramids of Gzeh in all their polished glory, before the destroyer stripped them of their beautiful garments, were in every respect adapted to flash around clearly defined lines of sight, upon which the lands of the nation could be accurately threaded. The very thought of these mighty theodolites of the old Egyptians fills me with wonder and reverence. What perfect and beautiful instruments they were! never out of adjustment, always correct, always ready; no magnetic deviation to allow for. No wonder they took the trouble they did to build them so correctly in their so marvellously suitable positions.
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.
-- 10. HOW THE PYRAMIDS WERE MADE USE OF.
It appears from what I have already set forth that the plan of the Pyramids under consideration is geometrically exact, a perfect set of measures.
I shall now show how these edifices were applied to a thoroughly geometrical purpose in the true meaning of the word--to measure the Earth.
I shall show how true straight lines could be extended from the Pyramids in given directions useful in right-angled trigonometry, by direct observation of the buildings, and without the aid of other instruments.