§14. A Simple Instrument for laying off "Primary Triangles" 79 §14a. General Observations 80 §15. Primary Triangulation 82 §16. The Pentangle or Five-pointed Star the Geometric Symbol of the Great Pyramid 91 Table showing the comparative Measures of Lines 96 §17. The manner in which the Slope Ratios of the Pyramids were arrived at 104

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.

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THE SOLUTION OF THE PYRAMID PROBLEM.

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 convenience, I find it necessary before I can establish the theory I advance, to endeavor to determine the proportions and measures of one of the principal groups. I take that of Gïzeh as being the one affording most data, and as being probably one of the most important groups.

I shall first try to set forth the results of my investigations into the peculiarities of construction of the Gïzeh Group, and afterwards show how the Pyramids were applied to the national work for which I believe they were designed.

§ 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 situate at the acute angle of a right-angled triangle more beautiful than the so-called triangle of Pythagoras, because more practically useful. I have named it the 20, 21, 29 triangle. Base, perpendicular, and hypotenuse are to each other as twenty, twenty-one, and twenty-nine.

The Pyramid Cephren is situate on the greater angle of this triangle, and base and perpendicular are as before described in the Pythagorean triangle upon which Mycerinus is built. (See Fig. 2.)

Figure 3 represents the combination,—A being Cheops, F Cephren, and D Mycerinus.

Lines DC, CA, and AD are to each other as 3, 4, and 5; and lines FB, BA, and AF are to each other as 20, 21, and 29.

The line CB is to BA, as 8 to 7; the line FH is to DH, as 96 to 55; and the line FB is to BC, as 5 to 6.

The Ratios of the first triangle multiplied by forty-five, of the second multiplied by four, and the other three sets by twelve, one, and sixteen respectively, produce the following connected lengths in natural numbers for all the lines.

Figure 4 connects another pyramid of the group—it is the one to the southward and eastward of Cheops.

In this connection, A Y Z A is a 3, 4, 5 triangle, and B Y Z O B is a square.

I may also point out on the same plan that calling the line FA radius, and the lines BA