Hi Jammer,

I deliberately ignored your post since I was talking about problem 56 of the RMP, and the calculation of seked and the practical application of such which has nothing to do with the actual calibration of a cubit rod from the fourth dynasty or any other period. There is no dichotomy to address and I think there might possibly be a misconception of what what you believe the cubit to be.

Of course we all know the cubit is divided into 28 digits, but there are rods showing the cubit is also divided into 27 and 24 digits as shown in the cubit rods at the Turin Museum.

[www.egyptorigins.org]
Specimen #4: Turin Museum catalog #6349.
Nearly square 0.57 X 0.68 bronze rod with three different scales. This rod is short of a royal cubit of 20.625 by 0.12, hence could not be divided into standard 7 palms, or 28 digits. The design of the rod shows that the length was intentional and that a piece was not lost from one end. Two of the scales start at opposite ends. The different scales show that rod was designed as a measuring device with need for three different measuring systems. The several scales offer unique opportunity to show different working types then in use in Egypt.
This rod was earlier doubted as authentic. As stated by Lepsius, A The material, shape, subdivision and inscription seem to prove it a fake.@ The division of the scales, with careful scribing of the lines, show that the design was intentional, and not merely a crude imitation. The rod length did not differ from the royal cubit more than differences found in other cubit rods. The confusion of Lepsius was due to his lack of perception of the decline of metrological standards, and social evolution with contact from other societies.
Senigalliesi noted the sensitivity of this rod to temperature variations. The physical weight is less than three pounds, certainly not enough to make it difficult for use.
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Scale B divided into 27 parts:
Useful Length = 20.247
No Palm divisions
(27) Digit mean length = 0.750
w = 0.077 (1.96 mm)
s = 0.017
Calculated standard deviation spread in digit interval length at 0.750 = +/- 0.038
A short section at the end of the rod was left over from the divisions that started at the other end = 0.260
Total Surface B length = 20.507
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Note that Senigalliesi=s mean digit length calculates to exactly 3/4 inch, and that four digits would make exactly 3.0 English inches. The individual digits may have varied from 0.712 to 0.788. This range covers the length of 0.737 accepted for the usual Egyptian digit length.
The Aw@ digit value reported by Senigalliesi is 1.5 times greater on this rod than on the Kha working rod. However, with only two digits on the Kha rod conclusions from comparison is doubtful.

Had the rod been made 0.5 inches longer, it could have accommodated 28 divisions at 0.75 digit length. That would have made the rod exactly 21 inches in length. The manufacturer must have decided before hand that the division into 27 of 0.75-inch digits was more important than the 21-inch total length. The question then arises why the total length was held to some sacred cubit value less than 21 inches.
Obviously, rods inscribed only with digits were useful working devices.
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Scale C divided into 12 major intervals, 72 subdivisions, and 360 fine divisions (according to Senigalliesi). The scale starts at the end opposite Scale B with a short section of rod left over as in Scale B.
Useful Length = 20.124
(12) Major interval mean length = 1.677
w = 0.058 (1.47 mm)
(72) Subdivision mean length = 0.279
w = 0.053 (1.34 mm)
s = 0.011
Calculated standard deviation spread in digit interval length at 0.279 = +/- 0.027
(360) Fine division mean length = 0.056
w = 0.020 (0.51 mm)
s = 0.004
Calculated standard deviation spread in fine division interval length at 0.056 = +/- 0.010
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Senigalliesi does not indicate the total rod length from this surface. Presumably it is the same as Surface B. The difficulty with Senigalliesi=s methods is again seen in that 12 X 1.677 = 20.124, 72 X 0.279 = 20.088, and 360 X 0.056 = 20.16.


Specimen #5: Turin Museum catalog #6348
A green basalt rod, with three different scales. This rod is very near an ideal length of 20.625 royal cubit. It would not have made a practical measuring instrument. It might have served as a standard except for the considerable variability on digit intervals. Compare with Petrie report of a stone standard in Part III.
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Scale A:
Length = 20.626
(24) Digits mean length = 0.859
w = 0.039 (0.99 mm)
s = 0.017
Calculated standard deviation spread in digit interval length at 0.859 = +/- 0.036
This scale has no palm divisions; there are no subdivisions of the digits.
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The digit length corresponds to those found in non-Egyptian cubits. Refer to later discussion.
24-digit divisions shows possible cross-cultural mathematics in that 24 X 0.859 = 20.616 versus 28 X .737 = 20.636, the difference being only in the fourth place in the numbers. 20.625 royal cubit divided by 24 = 0.8594 and divided by 28 = 0.7366.
Clearly, 28-digit divisions were not sacrosanct.
Aw@ of 0.039 would give digit range of 1.0 mm from0.84 to 0.88 inches, not suitable to an administrative measurement standard.
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Scale D:
4 subdivisions of 5.156.
w = 0.091 (2.31 mm)
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Scale E:
3 subdivisions of 6.875
w = 0.041 (1.04 mm)
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The reason for these last two scales might be convenient divisions of the royal cubit into thirds and fourths.

There are 3 different systems demonstrated by the cubit being divided into 24, 27 and 28 subdivisions, with the actual length of the rod being irrelevant. Clearly, 28-digit divisions were not sacrosanct and indicative of each cubit rod being designed to be job specific.

Cubit rod of 24 divisions 20 34/55 / 24 = 189/220 or (0.859090...inches)Specimen #5 scale "A"
Cubit rod of 27 divisions 20 1/4 / 27 = 3/4 or (0.75...inches) Specimen #4 scale "B"
Cubit rod of 28 divisions 20 34/55 / 28 = 81/110 or (0.7363636...inches)Specimen #4 scale "A"

Now, it just so happens that the numbers 24 * 27 * 28 when multiplied equals 18144…18144 just happens to equal the number of inches in 1/2 perimeter base G1 in inches. Or (1/2 * 9! / 10)
As you say this may not be definitive proof, but I cannot say any more without revealing the actual source and definition of for the cubit, which I do not want to do at this time.

Regards