**Comparison of Ancient Weight Systems**

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by

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Chandrakant Doshi

*Abstract: This paper presents a comparison of Harappan and Egyptian weight systems using a detailed analysis that reveals the reason for the differences among these systems.*

In two earlier papers
the weights from the ancient civilizations of Harappa and Egypt were examined.
Harappan weights were shown to be more accurate than the Egyptian Peyem and
Qedet weight systems while the Staters showed comparatively similar accuracy to those of the Harappan weights. It
is the purpose of this paper to explore further these weight systems.

The
Harappan weights from three excavation sites were analysed in "Ancient
Weights from South Asian Civilization

^{1}". The paper on the Egyptian weights is titled "Ancient Egyptian Weight Systems^{2}" and is based on the collection of stone weights in Petrie’s Ancient Weights and Measures^{3}.
For the purpose of carrying out a detailed
examination of these systems, a range of weights lying within 5% of the unit
weight and its multiples, ie. the nominal values, is selected. Frequency
distribution of the weights for each ratio is prepared and displayed on a
single chart. Such individual frequency distributions are best displayed using
line charts as the distribution for each ratio shows up clearly. The composite frequency
distribution, shown in previous papers, is a sum of individual distributions at
each normalised value. A composite distribution chart for the limited range
being examined in this paper is also included for easy reference.

The charts of the four systems analysed are
given below. The horizontal axis represents normalised values. The vertical
axis represents the frequency. Individual distribution charts are plotted in
different colours to help identify the nominal values represented. Only the
nominal values with sufficient number of specimens to show up on the chart are plotted.

Total collection of weight pieces in each
system is tabulated in Table 1. Included in the same table are the number of
specimens within 5% of nominal weights as well as the number that make up the charts
of the individual frequency distributions for each of the systems.

Harappan weight system

Image 1A shows the composite frequency
distribution for the Harappan weights that falls within +/- 5% of nominal
weight values. A chart of the total collection of 697 specimens of Harappan
weights is to be found in the earlier paper

^{1}.
The individual frequency distribution is
shown in Image 1B. The nominal weights represented in the chart range from 1.703
to 136.25 grams. This represents 533 specimens out of 546 pieces.

Three of the weight systems of ancient
Egypt, i.e. Peyem, Qedet, and Stater are examined similarly and the individual
and composite frequency distribution charts are given below.

The composite distribution of 262 out of 269 Peyem weights is shown in Image 2A. The range of nominal weights in the individual frequency distribution, shown in Image 2B, is from 30 to 12010 grains.

The composite distribution of 791 out of
861 Qedet weights is shown in Image 3A. The range of nominal weights in the individual
frequency distribution, shown in Image 3B, is from 72.25 to 7225 grains.

The composite distribution of 374 out of
396 Stater weights is shown in Image 4A. The range of nominal weights in the
individual frequency distribution, shown in Image 4B, is from 67.7 to 6766.4
grains.

Harappan weight system

Individual frequency distributions of the Harappan
weight system all peak at the nominal values. They are fairly symmetrical about
these peaks. The collection includes both the damaged and unfinished specimens,
as noted for some of the specimens. This probably accounts for the slight asymmetry
of the distribution observed in some cases.

There are also two
small peaks, one on either side of the nominal values. The small peak on the
left of the nominal values is likely to be the result of damaged specimens,
whereas that to the right of the nominal values likely due to the unfinished
specimens. It is strange that the peaks on the left, at 0.97 of the normalised
values, are all coincident. Those to the right, occurring from 1.03 to 1.05 of
the normalised values, are spread out, as one would expect.

It is certainly remarkable that the peaks of all the
individual weight ratios of the Harappan system are concurrent at the nominal
values considering the weights were recovered from three different locations,
all a great distance apart.

Egyptian weight systems

The Egyptian weight systems all show
multiple peaks. As many as four peaks are observed for the Qedet and Peyem
systems and these are distributed across +/- 5% of the nominal values. While
many peaks are concurrent at certain points, there is no single point where
peaks of all the ratios occur.

For the Stater weight system a few of the
ratios have a single peak and the rest have a double peak. More than 2 peaks
are not observed for any ratio. Single peaks are concurrent at the nominal
values and the double peaks are separately concurrent and appear on either side
of the nominal values.

Named location weights

In the table of weights, Petrie

^{3}has recorded the find locations of some of the specimens. An examination of these weights shows that a large majority are within 5% of the nominal values but very few are exactly at the ratios of the system. Nor are multiples exact. None of the specimens from a single location represent a complete set of nominal weights for the system represented.
Images 5, 6 and 7 display the information of these
specimens. The images comprise two charts locked together, sharing the title
and the horizontal axis. The upper chart displays the weight of each specimen,
expressed as log

_{2}(weight). The lower chart represents the normalized value of each specimen. The horizontal axis represents the item number, on a list, of these weights. The list is sorted first alphabetically and then by value.
The line in off-white in the upper chart connects all
the weights in the list while the segments in colour connect all the specimens
found at a single location. The legend on the right identifies the location.
Only the locations with sufficient number of specimens are so identified in the
chart.

Individual points only are plotted in the lower chart.
All the specimens are plotted in off-white to start with and where sufficient
number of specimens are present at a named location, these are plotted over in
colours, identifying their location.

The charts are locked together such that a vertical line through a point on one chart connects to the same specimen on the other chart.

An inspection of these charts shows that the
individual frequency distributions for some of the locations occupy the full
range of normalized values observed with the whole collection while it is much
reduced for others. However, such reduced spreads do not line up in a
systematic way to support any view of improved accuracy among these specimens.

An examination of the shapes of the weight specimens of
the two systems may provide some clues regarding the different levels of
accuracy.

Harappan weights occur in five different shapes. These
are cubical, truncated sphere, barrel, cylinder and conical. Each of these
shapes has at least one flat face. The weight of any specimen can be adjusted
by grinding the flat face.

The cubical shapes are mostly cuboid, with a few
cubes. With six faces all at right angles, it is an easy shape to achieve.

There is one weight list from Mohenjo Daro

^{5}that includes not only the shape but dimensions of each specimen and an examination of these show that all the cubical shapes have at least one parallel set of square faces. Either of these faces can be ground to adjust the weight of the specimen. Shape of the square face would be preserved, only the separation of the square faces would change as the weight of the piece is reduced. This is not the case with the truncated sphere and barrel shapes. Grounding only one of the faces would alter the symmetry of the shape. This can be tested by measurements of these two shapes. If the flat faces are symmetrical about the equatorial plane then a special effort must have been made in manufacturing these weight pieces.
Egyptian weights

^{3}come in a large number of shapes and not all have a flat face. Those with a flat face could be ground down to adjust the weight where necessary. If the manufacturing did not require a close control of the final weight, then the shape of the piece may not matter much. Perhaps this is the reason why there are many oddly shaped specimens.
A system of weights with multiples based on a specific
unit value is expected to have the largest number of specimens at those values.
The rest of the specimens should, in general, be equally distributed about those
values. The closeness of the distribution about the peak determines the
accuracy of the manufactured weights.

Of the four weight systems examined, the Harappan
system comes closest to the expected distribution. These weights could have been calibrated at some central
location, in which case all the specimens have to be transported there.
Alternatively, they could have been calibrated at the location they were found.
This would require, at each location, accurate sets of standard weights as well
as balances capable of weighing to high precision.

Of the Egyptian systems examined, the Stater system
has a few of the ratios close to the expected distribution, the remainder
showing a split distribution. The Peyem and Qedet systems, on the other hand,
show multiple peaks.

The Egyptian weight standard may have changed over the
long period of usage and if the recovered weights are a collection manufactured
under different standards then the observed distribution would be the result.
Both the Peyem and Qedet specimens are confined within 5% of the nominal
weights and this could be possibly because they were manufactured to no better
than 5% accuracy. A
specimen within this range could end at any value and the occurrence of peaks would
then be purely a matter of chance. A different collection would show different
arrangement of peaks.

This
detailed analysis of the weights points to an exercise of control in their
manufacture. Control implies organisation. It is strongest for the Harappan systems
and covers the whole area of the Harappan civilization. The Egyptian weight
systems demonstrate a varying degree of control in their manufacture.

Harappan
civilization has left no textual documentation to learn more about this
organization. The Egyptian civilization has left plenty. A search through those
might prove beneficial regarding the organization pertaining to the manufacture
of their weight systems.

References:

1.
This paper can be viewed
here: https://jignashi.blogspot.co.uk/

A copy is also
available on Academia: https://www.academia.edu/4846613/Ancient_Weights_from_South_Asian_civilization

2.
This paper can be viewed here: https://jignashi.blogspot.co.uk/search?updated-min=2013-01-01T00:00:00Z&updated-max=2014-01-01T00:00:00Z&max-results=1 Ancient_Egyptian_Weight_Systems

A copy is also available on
Academia: https://www.academia.edu/4846620/Ancient_Egyptian_Weight_Systems

3.
Ancient Weights and Measures by
Sir Flinders Petrie

Department of
Egyptology, University College, London, 1926

4.
Ancient Weights from South
Asian Civilization, op cit

Chart 1

5.
Further Excavations at Mohenjo
Daro, E. J. H Mackay, 1938, pp
607-612

6.
Ancient Egyptian Weight
Systems, op cit, Charts 1b, 2b and 3b.

© 2016 Chandrakant Doshi