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PYRAMID PROBLEM
PYRAMID PROBLEM
OR, PYRAMID DISCOVERIES. WITH A NEW THEORY AS TO THEIR ANCIENT USE. BY ROBERT BALLARD, M. INST. C.E., ENGLAND; M. AMER, SOC. C.E. CHIEF ENGINEER OF THE CENTRAL AND NORTHERN RAILWAY DIVISION OF THE COLONY OF QUEENSLAND, AUSTRALIA. NEW YORK: JOHN WILEY & SONS. 1882 Copyright , 1882, By JOHN WILEY & SONS . PRESS OF J. J. LITTLE & CO., NOS. 10 TO 20 ASTOR PLACE, NEW YORK....
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NOTE.
NOTE.
In preparing this work for publication I have received valuable help from the following friends in Queensland:— E. A. Delisser , L.S. and C.E., Bogantungan, who assisted me in my calculations, and furnished many useful suggestions. J. Brunton Stephens , Brisbane, who persuaded me to publish my theory, and who also undertook the work of correction for the press. J. A. Clarke , Artist, Brisbane, who contributed to the Illustrations. Lyne Brown , Emerald,—(photographs). F. Rothery , Emerald,—(model
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LIST OF WORKS CONSULTED.
LIST OF WORKS CONSULTED.
Penny Cyclopædia. ( Knight, London. 1833. ) Sharpe's Egypt. "Our Inheritance in the Great Pyramid." Piazzi Smyth. "The Pyramids of Egypt." R. A. Proctor. ( Article in Gentleman's Magazine. Feb. 1880. ) "Traite de la Grandeur et de la Figure de la Terre." Cassini. ( Amsterdam. 1723. ) "Pyramid Facts and Fancies." J. Bonwick. With the firm conviction that the Pyramids of Egypt were built and employed, among other purposes, for one special, main, and important purpose of the greatest utility and co
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§ 1. THE GROUND PLAN OF THE GIZEH GROUP.
§ 1. THE GROUND PLAN OF THE GIZEH GROUP.
I find that the Pyramid Cheops is situated on the acute angle of a right-angled triangle—sometimes called the Pythagorean, or Egyptian triangle—of which base, perpendicular, and hypotenuse are to each other as 3, 4, and 5. The Pyramid called Mycerinus, is situate on the greater angle of this triangle, and the base of the triangle, measuring three , is a line due east from Mycerinus, and joining perpendicular at a point due south of Cheops. ( See Figure 1. ) I find that the Pyramid Cheops is also
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Plan Ratios connected into Natural Numbers.
Plan Ratios connected into Natural Numbers.
The above connected natural numbers multiplied by eight become...
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§ 2. THE ORIGINAL CUBIT MEASURE OF THE GIZEH GROUP.
§ 2. THE ORIGINAL CUBIT MEASURE OF THE GIZEH GROUP.
Mr. J. J. Wild, in his letter to Lord Brougham written in 1850, called the base of Cephren seven seconds. I estimate the base of Cephren to be just seven thirtieths of the line DA. The line DA is therefore thirty seconds of the Earth's Polar circumference. The line DA is therefore 3033·118625 British feet, and the base of Cephren 707·727 British feet. I applied a variety of Cubits but found none to work in without fractions on the beautiful set of natural dimensions which I had worked out for my
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§ 3. THE EXACT MEASURE OF THE BASES OF THE PYRAMIDS.
§ 3. THE EXACT MEASURE OF THE BASES OF THE PYRAMIDS.
A stadium being 360 R.B. cubits, or six seconds—and a plethron 60 R.B. cubits, or one second, the base of the Pyramid Cephren is seven plethra, or a stadium and a plethron, equal to seven seconds, or four hundred and twenty R.B. cubits. Mycerinus' base is acknowledged to be half the base of Cephren. Piazzi Smyth makes the base of the Pyramid Cheops 9131·05 pyramid (or geometric) inches, which divided by 20·2006 gives 452·01 R.B. cubits. I call it 452 cubits, and accept it as the measure which ex
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§ 4. THE SLOPES, RATIOS, AND ANGLES OF THE THREE PRINCIPAL PYRAMIDS OF THE GIZEH GROUP.
§ 4. THE SLOPES, RATIOS, AND ANGLES OF THE THREE PRINCIPAL PYRAMIDS OF THE GIZEH GROUP.
Before entering on the description of the exact slopes and angles of the three principal pyramids, I must premise that I was guided to my conclusions by making full use of the combined evolutions of the two wonderful right-angled triangles, 3, 4, 5, and 20, 21, 29, which seem to run through the whole design as a sort of dominant. From the first I was firmly convinced that in such skilful workmanship some very simple and easily applied templates must have been employed, and so it turned out. Buil
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§ 5. THE EXACT DIMENSIONS OF THE PYRAMIDS.
§ 5. THE EXACT DIMENSIONS OF THE PYRAMIDS.
Figures 15 to 20 inclusive, show the linear dimensions of the three pyramids, also their angles. The base angles are, Cheops, 51° 51′ 20"; Cephren, 52° 41′ 41″; and Mycerinus, 51° 19′ 4″. In Cheops, my dimensions agree with Piazzi Smyth—in the base of Cephren, with Vyse and Perring—in the height of Cephren, with Sir Gardner Wilkinson, nearly—in the base of Mycerinus, they agree with the usually accepted measures, and in the height of Mycerinus, they exceed Jas. J. Wild's measure, by not quite on
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§ 6. GEOMETRICAL PECULIARITIES OF THE PYRAMIDS.
§ 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. All in R.B. cubits. [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
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§ 6A. THE CASING STONES OF THE PYRAMIDS.
§ 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 thr
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§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.
§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.
The 3, 4, 5 triangle contains 36° 52′ 11·65″ and the complement or greater angle 53° 7′ 48·35″ 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 notice
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§ 8. GENERAL OBSERVATIONS.
§ 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
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§ 9. THE PYRAMIDS OF EGYPT, THE THEODOLITES OF THE EGYPTIAN LAND SURVEYORS.
§ 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
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§ 10. HOW THE PYRAMIDS WERE MADE USE OF.
§ 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. An
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§ 11. DESCRIPTION OF THE ANCIENT PORTABLE SURVEY INSTRUMENT.
§ 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
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§ 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.
§ 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.
TABLE TO EXPLAIN FIGURE 60. Main Triangular Dimensions of Plan are Represented by the Following Eight Right-angled Triangles. 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:— 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 suitabl
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Table of Some Primary Triangles and their Satellites.
Table of Some Primary Triangles and their Satellites.
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— 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 i
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§ 13. THE SIZE AND SHAPE OF THE PYRAMIDS INDICATED BY THE PLAN.
§ 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
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§ 14. A SIMPLE INSTRUMENT FOR LAYING OFF "PRIMARY TRIANGLES."
§ 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
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§ 14a. GENERAL OBSERVATIONS.
§ 14a. GENERAL OBSERVATIONS.
I must be excused by geometricians for going so much in detail into the simple truths connected with right-angled trigonometry. My object has been to make it very clear to that portion of the public not versed in geometry, that the Pyramids of Egypt must have been used for land surveying by right-angled triangles with sides having whole numbers. 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 insta
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§ 15. PRIMARY TRIANGULATION.
§ 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, m
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§ 16. THE PENTANGLE OR FIVE POINTED STAR THE GEOMETRIC SYMBOL OF THE GREAT PYRAMID.
§ 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 noble truths, and a solemn emblem of important duties. Its geometric significance, however, has long been lost sight of. It is said to have constituted 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 S
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Table Showing the Comparative Measures of Lines.
Table Showing the Comparative Measures of Lines.
( Fig. 67. ) The triangle DXH represents a vertical section of the pentagonal pyramid; the edge HX is equal to HN, and the apothem DX is equal to DE. Let DH be a hinge attaching the plane DXH to the base, now lift the plane DXH until the point X is vertical above the centre C. Then the points A, E, B, O, N of the five slant slides, when closed up, will all meet at the point X over the centre C. We have now built a pyramid out of the pentangle, whose slope is 2 to 1, altitude CX being to CD as 2
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§ 17. THE MANNER IN WHICH THE SLOPE RATIOS OF THE PYRAMIDS WERE ARRIVED AT.
§ 17. THE MANNER IN WHICH THE SLOPE RATIOS OF THE PYRAMIDS WERE ARRIVED AT.
The manner in which I arrived at the Slope Ratios of the Pyramids, viz., 32 to 20, 33 to 20, and 34 to 21, for Mycerinus , Cephren , and Cheops , respectively ( see Figures 8, 7 and 6), was as follows:— First, believing in the connection between the relative positions of the Pyramids on plan ( see Fig . 3, 4 or 5), and their slopes, I viewed their positions thus:— Mycerinus, situate at the angle of the 3, 4, 5 triangle ADC, is likely to be connected with that "primary" in his slopes. Cephren, si
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