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# Egyptian Cubit Rods and Cubits Part II

## Turin Cubit Rods C Data

The following are from Senigalliesi's data. I use his scale designations. I list the rods in sequence according to my estimate of their practical usefulness and mechanical simplicity. The designation 'w' is Senigalliesi's difference between the longest and shortest respective inscribed line intervals. 's' is his calculated standard deviation of the intervals. He did not attempt to calculate the standard deviation if the number of data points were less than a half-dozen. Even so, calculation of the standard deviation for the small number of data points available from the rods, 30 or less, involves mathematical assumptions that may not be true.

Specimen #1: Turin Museum Supplement #8391.

A wood rod, hinged in the center. Marked off in palms with the right-most palm divided into one-half palm first and then 2 digits from the end. The left-most section has palms 6 and 7 combined into one length.

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Common Scale:

Length = 20.81

(7) Palm mean length = 2.973

w = 0.155 (3.93 mm)

(2) Digit mean length = 0.737

w = 0.045 (1.14 mm)

Hinge spacing = 0.014

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This rod length exceeds the royal cubit of 20.625 by 0.185 (4.7 mm).

The discrepancy between an assumed 4 X 0.737 digit length = 2.948 and the mean palm of 2.973 is that only one palm (with only two digits) was so inscribed.

The interval differences between the palm inscribed lines were more variable than those for the digits, by a magnitude of 3:1. If we take the mean palm interval length and add and subtract one-half of 'w' we would obtain differences from 2.89 to 3.05 . Of course, this is not legitimate to determine the actual shortest and longest intervals nor the individual variability. Thus we can recognize the limits of the methods of Senigalliesi.

This is probably the best example of a working rod, but the lack of inscribed digit subdivisions would make it impossible to use in fine measurements. Difference in length from the Royal Egyptian Cubit may reflect loss of standards and reveal the reduced accuracy by which later Egyptian monuments and works were constructed, not to the same degree of resolution found in structures of the Old Kingdom. Or it might have been a rod used for roughly checking construction by Kha, not intended for actual installation, but a self-respecting architect would not resort to a rod that was not accurate.

Note that some of the palm intervals would be in excess of 3.0 English inches while the mean palm length is a mere 0.69 mm from 3.0 inches. This difference is much less than the variability between palms intervals. One could easily assume that the intended palm length was an integral 3.0 English inches.

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, "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 'w' 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.

Although he implies that the fine divisions continue for the length of the rod, his photograph does not show thus, but merely for the length of six subdivisions from one end. Lepsius, Plate 4b, confirms this fact.

The last is based on 30 fine inscribed lines only, covering 6 subdivisions, extrapolated the total length of the rod. Hence the discrepancy may be due to the extrapolation. Senigalliesi made no attempt to reconcile the numbers.

I shall later compare the 'w' values reported by Senigalliesi with those reported by Petrie.

Use of the designators "palms" and "digits" strains normal understanding. The large scale divisions are 12, each of those is divided into 6 parts, and then 6 of those are each divided into 5. The division of the total scale length into 12 is similar to the division of the English foot. This does not follow the usual division of the Egyptian cubit into 28 digits, nor does this scale display any commonly accepted Egyptian cubit divisions. This may have been one of causes for the conclusion as a fake by Lepsius. However, this strong departure from the usual cubit divisions should not cause us to conclude that it was a fake, but rather that a need existed. If we take such view we might then seek such length in actual Egyptian (or other) constructions from that period.

The mean major interval length, multiplied by three would make 5.03 inches, a mere 0.76 mm from the round number of 5.0. Compare this difference with the 0.69 mm difference for integral

English inches on the Kha working rod. The curiosity of cubit rod divisions that multiply into integral English inches is found repeatedly on extant rods. See discussion in Part IV.

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Scale D:

Useful Length = 20.448

Major interval mean length = 3.408, six palms, each divided into 4 digits.

The scale is oriented starting at each end with a small gap in the center.

w = 0.052 (1.32 mm)

Subdivision mean length = 0.852

w = 0.045 (1.14 mm)

s = 0.011

Calculated standard deviation spread in digit interval length at 0.852 = +/- 0.023

Length of Surface D = 20.509

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This scale is significantly different from a standard palm and digit scale of 2.95 and 0.737 inches but matches that of non-Egyptian palms and cubits. Refer to Petrie information in Part III and later discussion.

Here Lepsius might have found satisfaction for his desire of a six-palm subdivision.

In summary, this Bronze rod with three different length scales of 20.247, 20.124, and 20.448 suggests three different measurement systems. The rod seems to have been held to a seemingly sacred length of nearly one royal cubit, while sacrificing integrity of scale illustrated by the odd number of 27 digits, and gaps in each of the three scales.

According to Lepsius, Plate 4b, Scales B and C start at opposite ends of the rod, with their respective short pieces so arranged. Scale D starts at both ends with a short gap in the center. The difference between the total rod length of 20.508 and D useful length of 20.448 would make the gap length equal to 0.061. The arrangement of the three scales, with their orientation to the ends, shows a deliberate design. Perhaps the arrangement was intended to easily distinguish the different scales, but their respective unique scaling would preclude such need.

I conclude that the rod was not a fake, but an attempt to match a need where the three different measurement systems were under active use. Refer to comparisons with other rods. Since this includes non-Egyptian cubit systems it may date from a period with considerable cross-cultural exchange, and construction methods. The rod may represent design to match measuring systems from different geographical regions.

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.

'w' 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.

Specimen # 2: Turin Museum Supplement #8647

Hieroglyphic inscriptions cover the two ends and four sides. The front contains the measurement scale. The division of the digits into subdivisions, starting with 2 and progressing to 16, is from the left end, not the right as usually found. None of these are grouped into palms. (Was Kha left handed?) Apparently the craftsmen knew this rod was ornamental. They were careless in their subdivisions. Digit 12 is divided into 14 subdivisions instead of 13. The following digits pick up correctly. The digits were inscribed to 15. Digit 16 is divided into two parts, top to bottom. Thereafter 3 palms were not subdivided; the total length would make 28 digits. The total length exceeded the Royal Cubit by 0.007, hardly discernible in a practical sense.

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Scale C:

Length = 20.632

(16) Digits mean length = 0.737

w = 0.090 (2.29 mm)

s = 0.023

Calculated standard deviation spread in digit interval length at 0.737 = +/- 0.045

(3) Palm divisions mean length = 2.947

w = 0.117 (2.97 mm)

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This rod meets the accepted ideal Egyptian digit length of 0.737, and the accepted ideal palm length of 2.946.

Senigalliesi chose digit 15 to measure the intervals of digit subdivisions. He did not report measurements on other digit subdivisions.

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The length of digit 15 divided into 16 subdivisions = 0.751

The mean length of the 16 subdivisions = 0.047

w = 0.019 (0.48 mm)

s = 0.005

Calculated standard deviation spread in subdivision interval length at 0.047 = +/- 0.009

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Note that the length of digit 15 is virtually 3/4 inches.

Specimen #3: Turin Museum Catalog #6347: The Amenemope Rod

The (in)famous Lepsius rod, decorated with hieroglyphs on the back, top, and bevel face. Digit divisions on the bevel face, with subdivisions of the 15 right-most digits on the front. There are no palm divisions. The front is blank for the length of 8 digits from the left, and then another five digits. Lepsius did not properly illustrate these blank spaces. No attempt was made to show palms. Compare with Specimen #2, the Kha ornamental rod.

Most of the subdivisions of the digits were carelessly inscribed according to the following list. (The number of inscribed lines should be one more than the number of the digit.)

Digit

# 4 = 6

# 5 = 7

# 6 = 8

# 8 = 10

#10 = 12

#11 = 13

#13 = 12

#14 = 15

Lepsius incorrectly counted the number of subdivisions in digits #7 and #11. (This assumes that his graphical drawing correctly replicates the squeeze he obtained from the Turin Museum.) He assigned 4 subdivisions to digit #2, but it appears to me that it is correctly scribed with 3 evenly-spaced lines and another line splitting the first subdivision into 2 parts.

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Scale A (bevel face):

Length = 20.614

(28) Digit mean length = 0.736

w = 0.204 (5.18 mm)

s = 0.043

Calculated standard deviation spread in subdivision interval length at 0.736 = +/- 0.102

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Scale E (front):

(Since the digit marks follow those of Scale A Senigalliesi did not measure them independently.)

Length of the digit with 16 subdivisions = 0.780

Mean length of subdivisions of this digit = 0.049

w = 0.013 (0.33 mm)

s = 0.004

Calculated standard deviation spread in subdivision interval length at 0.049 = +/- 0.007

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Michael St. John kindly sent me a full-size proof copy of the drawing published by Lepsius. From that I was able to make some estimates of the variation in digit lengths and palm lengths. For example, the first digit is in excess of 0.90 inches. The digit with the 16 subdivisions is 0.780 in length. Because of these extra-wide digits others had to be less than the mean. Some were less than 0.700. Senigalliesi shows a spread in digit lengths of 0.204, far outside any useful rod application. Again this illustrates the carelessness used in manufacture of the rod. This rod could only be classified as ornamental, yet, unfortunately, it was taken by many Egyptologists as indicative of a real measuring device.

St. John performed a rigorous comparative symbolic analysis of this rod with the Maya rod on display in the Louvre (Paris 1), and another from the Louvre not on display (Paris 2). He provides no dimensions for the rods nor their inscribed interval lengths. All three rods have inscribed lines that extend from the top surface to the bevel face to the front. The Paris 1 (Maya) and Paris 2 rod have digits grouped in the sequence 1, 2, 3-6 (4 for one palm), 7-8 (2), 9-11 (3), and 12-13 (2). When these are compared with the Amenemope rod we readily see that there was no sacredness to palm divisions, and that many rods were not so inscribed.

Next - Part III - Petrie Rods