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Egyptian Cubit Rods and Cubits C Part IV

Assessment

Total Rod Lengths

Total rod, and rod palm and digit lengths offer insight. Where Petrie offers two measured lengths I take the mean. For total rod length I show difference from an ideal of 20.625, also shown in (mm). I evaluate the palm and digit lengths.

Lepsius provided a table of 12 rods, but several of them were broken and assumptions had to be made as to their overall length. I include only those on which reliable measurements could be made.

The many examples of broken rods offered by Von Adelheid Schwab-Schott can only be evaluated on the basis of digits, and then only according to the individual pieces. Since the rods I discuss below show such great variability on both palm and digit length it would be dangerous to make assumptions about total rod length, digit length, or what might be inferred from such a fractured group.

For want of a better method I classify them as "Egyptian," "Short Egyptian," and "Foreign."

Egyptian rods:

Rod Material Length in
inches
Difference
from ideal
Difference
from ideal
in mm

Turin Specimen #1

Kha hinged:

20.810

+0.185 (+4.699)

Turin Specimen #4

bronze:

20.508

-0.117 (+2.97)

Turin Specimen #5

basalt:

20.626

+0.001 (essentially zero)

Turin Specimen #2

Kha ornamental:

20.632

+0.007 (+0.178)

Turin Specimen #3

Amenemope ornamental:

20.614

-0.011 (-0.279)

Petrie Item #1

square wood:

20.600

-0.025 (-0.635)

Lepsius #6

slate:

20.728

+0.103 (+2.616)

Lepsius #10

yellow hard wood:

20.708

+0.083 (+2.108)

Lepsius #14

double length:

41.295
(1/2 = 20.647)

+0.022 (+0.559)

Lepsius #2

Maya (Paris 1):
20.866 front and
20.669 back mean of

20.768

+0.143 (+3.62)
Ideal is basic cubit rod length of 20.625

Short Egyptian rods:

Rod Material

Length in
inches

Difference
from ideal

Difference
from ideal
in mm

Petrie Item #6

short wood rod

17.604

-0.075

(-1.905)

Petrie Item #8

rectangular wood rod

17.780

+0.101

(+2.565)

Ideal is calculated from 6/7 of 20.625 = 17.679

Foreign rods:

Rod Material

Length in
inches

Difference
from ideal

Difference
from ideal
in mm

Petrie Item #9

rectangular wood

21.046

+0.421

(+10.693)

Petrie Item #10

rectangular wood (Ptolemaic?)

21.482

+0.857

(+21.768)

Petrie Item #11

rectangular wood

21.07

+0.445

(+11.303)

Lepsius #9

wood with Coptic inscription

21.209

+0.584

(+14.834)

Ideal is basic cubit rod length of 20.625

The first set varies from +0.185 to -0.117 around 20.625, with a mean difference of +0.027, (0.69 mm). This is +2.96 to -1.87 sixteenths of an inch (+4.70 to -2.97 mm).

The ideal of 17.679 for the short rods leads to differences similar to those found in regular length rods, from +0.101 to -0.075, with mean of +0.013 (0.33 mm).

A graphical plot of the differences around 20.625 shows an apparent random distribution. No causative assignment to the differences is evident except manufacturing carelessness. Evolutionary drift over time in any one direction is not indicated.

These are variations we would not tolerate in modern instruments. Notably, the Kha working rod has the largest deviation from the nominal. He was architect to the King. The evidence strongly supports a conclusion that State administrative control of standards no longer existed. On the other hand, some idea of a nominal value must have existed, and must have been preserved in a master standard, otherwise how did the Egyptian rods vary around that nominal? Perhaps the nominal, or master standard, was known but no attempt was made at State control. If we take that view we then naturally ask why Kha would not want his working rod to meet the nominal value? We find from the tabulation above that his ornamental rod does meet that standard. How can we explain the discrepancy?

Were all of the construction projects, buildings, and monuments under his supervision executed according to his working rod? Can we verify from monumental evidence during his period of control?

The Kha evidence presents a puzzle that has no easy answer.

The foreign rods are all greater in length than the Egyptian cubit with a mean difference of +0.576. From the limited number of four examples this is close to a difference of +0.5  or +0.6 inches.

Aristotle said, "Thus the mathematical sciences originated in the neighborhood of Egypt, because the priestly class was allowed leisure." The foreign rods show a relationship to the royal Egyptian cubit to suggest they had their origin in Egypt, or were from some source common to both the Egyptians and surrounding societies. Refer to further discussion below.

Relative Quality of the Rods

We can compare the relative quality of cubit rods by examining the resolution in placement of scribed lines for palms and digits on the respective examples. The "w" values determined by Senigalliesi and Petrie offer insight. (Because of the mathematical uncertainty, and what I refer to as smothering of metrological penetration, I shall not engage in discussion of Senigalliesi's "s" values. What superficially appears as a help in mathematical analysis actually confuses our understanding.)

Palm Comparisons, rod length, palm mean length, "w" values (mm):

Rod Material

Length in
inches

Palm Length
in inches

"w"
values

"w"
values
in mm

Turin Specimen #1 hinged wood 20.81

2.973

0.155

(3.94)

Turin Specimen #2

ornamental wood

20.632

2.947

0.117

(2.97)

Turin Specimen #4
Scale D

bronze 20.448 3.408 0.052 (1.32)

Petrie Item #1

square wood

20.460

3.410

0.028

(0.011)

Petrie Item #6

thick bar of wood
Fine and sharp cuts

17.604

2.934

0.118

(0.274)

Petrie Item #8

rectangular wood

17.775

2.963

0.251

(6.37)

Petrie Item #9

rectangular wood

21.046

3.508

0.089

(2.26)

Petrie Item #10

rectangular wood

21.482

3.580

0.21

(5.34)

Petrie Item #11

rectangular wood

21.070

3.511

0.24

(6.10)

None of the other rods had distinct palm divisions. Palm divisions of those rods are based on assumptions and division of digits into four parts.

Note that there are two main groups, one group very close to 3.0 inches, and another group close to 3.5 inches. The mean of the first is 2.954 inches; the mean of the second is 3.483 inches. The first mean multiplied by 7 makes a cubit of 20.678 inches. The second mean multiplied by 6 makes a cubit of 20.900 inches.

Curiously, Turin #4 bronze rod, Scale D, and Petrie #1 square wood were the lowest of the rod length of "3.5." They were almost in a class by themselves.

The mean length of the rods of the second group is 20.909 inches. This strongly suggests that two different rods were in use, one with a palm length of 20.625/7 = 2.945 inches, and one with a palm length of 21.0/6 = 3.500 inches.

This is illustrated better by the graph.

The palm "w" values are from 0.089 (2.26 mm) to 0.251 (6.37 mm). For a three-inch palm interval this is error from 3.0 to 8.3 percent. Thus we can see that the relative quality of the rods was poor. There was no exception to this poor quality, merely some poorer than others. The rod with the best control of length between palm divisions is Petrie Item #1. The rod with poorest control is Petrie Item #8, at 1/4 inch. The Kha architectural hinged rod is not especially good; in fact, it ranks fourth among the examples. The Kha ornamental rod ranks third while holding best to the length of the royal cubit. The lack of precision of the palm intervals is more readily grasped from this list. Today, if we had measurement rulers that held three-inch intervals with errors of 1/16 inch we would reject them. Only Petrie Item #1 is better than that. The Old Kingdom monumental evidence shows control of measurements to refinement not found in any of these rods.

We might infer that the concern of Lepsius about short rods used alongside regular rods can be understood as a general social decline in measurement standards, and borrowing from one society to another. Thus his views were heavily influenced by social developments in later periods of Egyptian history. However, the short rods might merely be a means to obtain a more manageable instrument, since they seem to follow the simple mathematical ratio of 6/7 of a regular rod. (Both of the Petrie examples use the common palm length.) Contrary to Lepsius I can visualize two different working lengths side by side, if they were both based on the same units. Compare English foot and yard sticks.

Digit Comparisons, mean length, "w" values (mm):

Rod Material

Scale on
Rod

Digit
Length

"w"
values

"w"
values
in mm

Turin Specimen #1

hinged wood rod

0.737

0.045

(1.14)

Turin Specimen #4

bronze rod

scale B

0.750

0.077

(1.96)

Turin Specimen #4

bronze rod

scale D

0.852

0.045

(1.14)

Turin Specimen #5

basalt rod

scale A

0.859

0.039

(0.99)

Turin Specimen #2

gold covered wood rod

0.737

0.090

(2.29)

Turin Specimen #3

wood (Amenemope rod)

scale A

0.736

0.204

(5.18)

Petrie Item #3

flat slip of wood broken

1.027

0.38

(9.65)

Petrie offered no other useful digit lengths.

None of the other complete rods are reported with distinct digit divisions to permit analysis.

Note that the digit length is near 0.737 on four of the rods. Two rods, Turin #4D, and #5A have a digit length near 0.850. This is more than 0.1 inch higher than the 0.737 of the "standard" digit length.

The digit "w" values are from 0.039 (0.99 mm) to 0.204 (5.18 mm). For a 0.75 digit length this is error from 5.2 to 27.0 percent. Here the poor quality of the rods is more evident. (I exclude the Petrie flat slip of wood from this calculation since it was atrocious in lack of control of intervals.)

We see that the Amenemope and Kha ornamental rods had the most deviation of the inscribed digit intervals, except for Petrie's flat slip of wood. This confirms the earlier conclusion that the ornamental examples could not have been intended as practical working rods. Clearly the ornamental rods were for commemorative purposes, possessing more than double the variability in line intervals of the other rods. Again this raises doubts about their use by Lepsius to derive typical working rods.

We would be unfair to the architects and working men of ancient Egypt if we were to use the poor quality ornamental rods as an indication of their daily lives. Modern scholarly estimates of the technical abilities of the ancient Egyptians may have been founded much on the poor evidence of those rods. This evidence from the New Kingdom and later periods was then extrapolated back to falsely assess the technical refinements of the Old Kingdom. The exceedingly fine resolution of the Old Kingdom structures was demonstrated, as Petrie said, by the fact that one could cover the error for the entire perimeter of the pyramid of Khufu of 3,000 feet with one's thumb. Such refinement is far beyond that obtainable with the ornamental rods, and, in fact, with extant rod examples from these later historic periods.

The evidence of Kha's personal cubit rod, the poor quality of the New Kingdom ornamental rods, the great variability among rods, incorporation of non-Egyptian palm and digit lengths, and comparison with the precise constructions of the Old Kingdom to exact royal cubit values suggests a deterioration of measurement standards and State administrative control over the millennia.

Only Scale D of the bronze rod had all six palms divided into four digits. None of the other extant examples of seven (or six) palms did so. Such rod is a postulated model without evidentiary support.

All rods seem to be designed according to need, not according to ideal models. This is especially evident in the Kha working rod, the Bronze rod with three different and non-corresponding scales, Petrie Item #3 designed around one-inch digits, the short rods, and the different scales among the rods. However, these remarks are not intended to imply that many identical design rods did not exist. The difficulty lies in the fact that the evidence does not support one ideal model; many different models existed. Then Lepsius pursued an ideal ornamental model, not working models.

The great variability among rod designs and lengths shows that division of the royal cubit into seven parts, and then into four digits, was not sacrosanct. Desire of modern students to place working rods into such rigid criteria is not supported by the evidence, although it may have been a standard in Old Kingdom times. Divisions into six of seven palms, use of rods with digits only, division into 24 or 28 digits, rods with divisions into 12 parts, regular and short rods, and other differences show the flexibility of measures in ancient Egypt. Unfortunately, the evidence is not sufficient to follow these social evolutions, nor can we reliably estimate their origins or causes.

The evidence also shows how futile it is to attempt to assign causes of lengths to human body parts. When digit and palm lengths, and highly contrasting scales, have such variability we can only regard such views of cause as mythical, and deriving from superstitious notions of the past. The designations of palms and digits are merely convenient references to crudely approximate distances that have no connection to actual body parts.

Coincidence with English Inches

Earlier I several times mentioned the coincidence between rod length intervals and integral English inches. This coincidence depends on the length of a royal cubit of 20.625 inches and division into seven palms to give an interval of 2.946 inches, a palm difference of 0.054 (1.36 mm) from 3.0 inches. This was a consequence of the length of the royal cubit being close to 21 inches.

The foreign rods show design that was slightly more than a nominal 21-inch rod length. If we had a measurement standard today of 21 inches, and worked in English units, division into seven palms would make much sense, with an integral 3.0 inches per palm. Six palms would not divide into integral values, nor would four, five, or eight. The reduction to four digits per palm then also causes a coincidence to 0.75 inches. 20.625 divided by 28 yields 0.737, or a difference of 0.013 (0.34 mm) from 0.75. (21 inches divided by 24 does not yield a convenient integral value.)

But the ancient Egyptians did not work in English inches. What was magical about the division into seven palms?

Following is a summary of the evidence for integral English inch units with difference less than two millimeters from integral English value, palm length shown.

 Turin Specimen #1 2.973 -0.027 (-0.69 mm) Turin Specimen #2 2.947 -0.053 (-1.34) Petrie Item #8 2.963 -0.037 (-0.94) Petrie Item #6 2.934 -0.066 (-1.68) Petrie Item #12 3.023 0.023 (0.58)

The odd examples of integral English units are as follows:

6. Turin Specimen #4, Scale B: mean digit length of exactly 0.75 English inches. Four digits of exactly 3.0 English inches.

7. Petrie Item #3: mean digit length of 1.027, 0.68 mm from 1.0 English inches.

8. Turin Specimen #4, Scale C: mean interval length of 1.677, multiplied by three for 5.03 English inches, 0.76 mm from 5.0

We have no cause why these rods would be coincidental to integral English measurement values except as accidental happenstance. (However, see my discussion on the Origin of the Royal Egyptian Cubit.)

Without evidence of how the seven palms created an integral division into some other unit our understanding must remain nebulous. The 28 digit subdivision does not seem to be more than a mathematical convenience.

We could take the view that the coincidence to integral English inch values was due entirely to manufacturing control variation among rod scales. But this would emphasize lack of ability to hold fine control on palm and digit lengths. On the other hand, we know from the Petrie stone standard that such control was possible. Again we are left with a curious puzzle.

The fine subdivisions of the digit found on the ornamental rods has no support from practical rods except the example of the Turin bronze rod, which seems to be a different working system. The bronze rod fine subdivisions run for six divisions and does not increase in number for each digit. This raises the question of the practical usefulness of the increasingly fine subdivisions for each digit on the ornamental rods. How would they be used? Today we design measurement rules with fine subdivision that are fixed throughout the length of the rule. I have in front of me an engineering rule that has six different scales, dividing the foot into parts of 10 to 60 divisions per inch marked off in 12 to 72 parts respectively.

I can easily switch scales according to the multiplier I may use on drawings to represent a construction object. The Turin bronze rod is a good representation of different scales on one rod, similar to what we find in this engineering rule.

If I were to use an Egyptian rod with different but monotonic increasing subdivisions of the digit on one scale I would not be able to lay the rod down on my drawing surface to measure scaled distances except by constantly referring it to digit distances, and then sliding it on the drawing surface to find the subdivision scale I desire. If I were making measurements in the field I would have the same burdensome difficulty.

As an engineer I see the fine scale subdivisions on the ornamental rods as highly impractical, really useless. This impression leads me to speculate that the ornamental rods represent measuring instruments with different scales captured on one piece, as ornamental commemorative. Or they may have been strictly ornamental. Unfortunately, we have no extant representation of these fine subdivision scales except on the ornamental rods.

Evidence from foreign cubit rods confirms Greek traditions that measurement methods were created in Egypt and then borrowed by other societies. This is especially accented by the seemingly sacrosanct Egyptian cubit rod length of 20.625 inches, or thereabouts, preserved in other societies, or with systematic addition to that length.

But another curiosity with Mediterranean measures exists. From Greek monuments, and especially the Parthenon, we know that the Greek foot was 12.145 +/- 0.015 inches. The modern defined Nautical Mile provides 12.152 inches per foot. Since the Nautical Mile is defined according to the circumference of the earth (with due regard for its oblate spheroid shape), and since the Greek foot is close to this value, numerous people have wondered about the coincidence. The Greek foot might represent a geodetic measure. This is related to the Egyptian cubit of 20.625 inches by a ratio of 1.7. The ratio of 1.7 is not a coincidence but can be derived theoretically. Hence, the Greek foot and Egyptian cubit are systematically related, confirming ancient traditions. Refer to my discussions at Origin of the Royal Egyptian Cubit.