Egyptian Cubit Rods and Cubits
C 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 Aw@ is Senigalliesi=s difference between the longest and shortest
respective inscribed line intervals. As@ 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.
*********************
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
*********************
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 Aw@ 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, AThe 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.
************************
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
***************************
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.
***************************
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
***************************
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 Aw@ values reported by Senigalliesi with those
reported by Petrie.
Use of the
designators Apalms@ and Adigits@ 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.
***************************
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
***************************
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.
***************************
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.
***************************
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.
***************************
Scale
D:
4 subdivisions of
5.156.
w = 0.091 (2.31
mm)
***************************
***************************
Scale
E:
3 subdivisions of
6.875
w = 0.041 (1.04
mm)
***************************
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.
***************************
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)
***************************
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.
***************************
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
***************************
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.
***************************
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
***************************
***************************
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
***************************
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.