In Cheops, the ratios of apothem, half base and edge are, 34, 21, and 40, very nearly, thus, 34² + 21² = 1597, and 40² = 1600. The dimensions of Cheops (from the level of the base of Cephren) to be what Piazzi Smyth calls a [Pi] pyramid, would be-- Half base 210 R.B. cubits. Altitude 267·380304, &c. " Apothem 339·988573, &c. " Altitude being to perimeter of base, as radius of a circle to circumference. My dimensions of the pyramid therefore in which-- Half base = 210 R.B. cubits. Altitude = 267·394839 &c. " Apothem = 340 " come about as near to the ratio of [Pi] as it is possible to come, and provide simple lines and templates to the workmen in constructing the building; and I entertain no doubt that on the simple lines and templates that my ratios provide, were these three pyramids built. § 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 three inches as _Secant_, then the _Tangent_ is three feet ten inches and three tenths. Fig. 23. Cheops Casing Stone. Fig. 24. Cephren Casing Stone. Fig. 25. Mycerinus Casing Stone. Fig. 26. Cheops Angle or Quoin Stone. Fig. 27. Cephren Angle or Quoin Stone. Fig. 28. Mycerinus Angle or Quoin Stone. Thus, in inches (√(75²-59²)) = 46·30 inches; therefore the inclination of the stone must have been--slant height 75 inches to 46·30 inches horizontal. Now, 46·30 is to 75, as 21 is to 34. Therefore, Col. Vyse's casing stone agrees exactly with my ratio for the Pyramid Cheops, viz., 21 to 34. (_See Figure 29._) Fig. 29. =Col. Vyse's Casing Stone.= 75 : 46·3 :: 34 : 21 This stone must have been out of plumb at the back an inch and seven tenths; perhaps to give room for grouting the back joint of the marble casing stone to the limestone body of the work: or, because, as it is not a necessity in good masonry that the back of a stone should be exactly plumb, so long as the error is on the right side, the builders might not have been particular in that respect. Fig. 59. (Temple of Cheops, standing at angle of wall.) Figure 59 represents such a template as the masons would have used in building Cheops, both for dressing and setting the stones. (The courses are drawn out of proportion to the template.) The other pyramids must have been built by the aid of similar templates.