Fig. 22. Diagram ill.u.s.trating relations of ratios of the pyramid Cheops.
Calling apothem 34, _radius_; and half base 21, _sine_--I find that--
Radius is the square root of 1156 Sine 441 Co-sine 715 Tangent 713 Secant 1869 and Co-versed-sine 169
So it follows that the area of one of the faces, 714, is a mean between the square of the alt.i.tude or co-sine, 715, and the square of the tangent, 713.
Thus the reader will notice that the peculiarities of the Pyramid Cheops lie in the regular relations of the _squares_ of its various lines; while the peculiarities of the other two pyramids lie in the relations of the lines themselves.
Mycerinus and Cephren, born, as one may say, of those two n.o.ble triangles 3, 4, 5, and 20, 21, 29, exhibit in their lineal developments ratios so nearly perfect that, for all practical purposes, they may be called correct.
Thus--Mycerinus, [3]20 + 25 = 1025, and 32 = 1024.
and Cephren, [4]80 + 105 = 17425, and 132 = 17424.
or [5]400 + 431 = 345761, and 588 = 345744.
See diagrams, Figures 11 to 14 inclusive.
In the Pyramid Cheops, alt.i.tude is _very nearly_ a mean proportional between apothem and half base. Apothem being 34, and half base 21, then alt.i.tude would be v(34-21) = v715 = 267394839, and--
21 : 267394839 :: 267394839 : 34, nearly.
Here, of course, the same difference comes in as occurred in considering the a.s.sumption of Herodotus, viz., the difference between v715 and v714; because if the alt.i.tude were v714, then would it be _exactly_ a mean proportional between the half base and the apothem; (thus, 21 : 2672077 :: 2672077 :: 34.)
Footnote 3: Half base to alt.i.tude.
Footnote 4: Half base to alt.i.tude.
Footnote 5: Half diagonal of base to alt.i.tude.
In Cheops, the ratios of apothem, half base and edge are, 34, 21, and 40, very nearly, thus, 34 + 21 = 1597, and 40 = 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.
Alt.i.tude 267380304, &c. "
Apothem 339988573, &c. "
Alt.i.tude being to perimeter of base, as radius of a circle to circ.u.mference.
My dimensions of the pyramid therefore in which--
Half base = 210 R.B. cubits.
Alt.i.tude = 267394839 &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 (v(75-59)) = 4630 inches; therefore the inclination of the stone must have been--slant height 75 inches to 4630 inches horizontal. Now, 4630 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 : 463 :: 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.
Such large blocks of stone as were used in the casing of these pyramids could not have been completely dressed before setting; the back and ends, and the top and bottom beds were probably dressed off truly, and the face roughly scabbled off; but the true slope angle could not have been dressed off until the stone had been truly set and bedded, otherwise there would have been great danger to the sharp arises.
I shall now record the peculiarities of the 3, 4, 5 or Pythagorean triangle, and the right-angled triangle 20, 21, 29.
-- 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.
Fig. 30 to 35. PECULIARITIES OF THE TRIANGLES
The 3, 4, 5 triangle contains 36 52" 1165? and the complement or greater angle 53 7" 4835?
Radius 5 = 60 whole numbers.[6]
Co-sine 4 = 48"
Sine 3 = 36"
Versed sine 1 = 12"
Co-versed sine 2 = 24"
Tangent 3 = 45"
Secant 6 = 75"
Co-tangent 6? = 80"
Co-secant 8? = 100"