§13. The Size and Shape of the Pyramids indicated by the Plan 77 §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._ Illustration: General Plan of Gïzeh Group 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._) Fig. 1. Cheops, Mycerinus 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._) Fig. 2. Cheops, Cephren