Ancient Egyptian Weight Systems

Ancient Egyptian Weight Systems

by

Chandrakant Doshi

email:     jignashi@yahoo.co.uk

Abstract:     

This paper presents an analysis of three of the ancient Egyptian weight systems.

Key words: Weights, Egyptian, ancient, Petrie, Peyem, Qedet, Stater

In "Ancient Weights and Measures", Flinders Petrie¹ lists stone and metal weights recovered from excavations in Egypt. This paper presents an analysis of some of these stone weights which start with catalogue number 2001. These are tabulated in plates 27 to 42 of Petrie’s publication. The metal weights and the stone weights from earlier finds are not included in this analysis.

There are over 2700 stone weight specimens listed in these tables. Of these, about 2100 pieces are purchased items. Just over 600 pieces have provenance².

Petrie classified this collection of stone weights under eight systems. Information in the tables includes details about the material of the specimens, form of each specimen, weight in grains, assigned ratio and the unit value for each specimen. Some of the specimens are damaged. Listed weight for such pieces is corrected by estimating the damage. The amount added to the weight is listed separately in the last column headed "Details". Corrected weights are used in this analysis.

In the tables, the specimens are sorted in an ascending order of the calculated unit value. In the first column is the catalogue number for each specimen. This starts at 2001 to avoid mix up with previously published lists of ancient Egyptian weights. Under each weight system, a few specimen do not have a catalogue number. These appear as an alternative fit with this weight system and full details of such specimen, including the catalogue number, are given under the weight system they are identified to belong. A note in the final column identifies that weight system.

Analysis of the ancient weight systems can be carried out by different methods and this paper examines some of them. These methods are illustrated with three of the ancient Egyptian weight systems. These are: Peyem, Qedet and Stater. The Peyem is the first of the eight systems and has the lowest unit value, Qedet has the most numerous specimens and the Stater weights show the best accuracy of the eight systems.

Log charts

A log chart is a plot of the logarithm of the weight values. The weight list is sorted in ascending order. This way the lowest weight appears on the left of the chart, with heavier weights progressing towards the right.

The logarithm (log for short) of a number is the power to which another number, called the base, is raised. The most common base is 10. Any number can be used as a base. In the ancient weight systems, multiples and submultiples of 2 are fairly standard. Hence  the whole list of weights is expressed as log to base 2 and plotted on a chart. With base 2, logarithms of 2, 4 and 8, for example, are 1, 2 and 3, respectively. A number between the exact multiples of 2 will have its logarithm between the logarithms of those multiples. For any two numbers, the logarithm of smaller number is smaller than that of the larger number.

The log chart displays the weight list in a compressed form. This permits such a large range to be displayed on a reasonable size chart.  Hemmy used a log chart to display a wide range of Harappan weights on a single chart³. He displayed the number of specimens lying within a short range of nominal weight values for the weights from 0.5 grams to 550 grams. This required some processing of weight data before the charts could be prepared. In the charts presented here nothing more than sorting the weight data in an ascending order is required.

In the charts, the vertical scale is  adjusted such that the grids are a unit apart. This means that an increase of 1 on the chart represents a doubling of the weight value. Similarly, a decrease of one represents halving of the value.

A log chart suitably prepared can identify the systematic nature of any weight system and point to different multipliers used in the system. The frequency at any particular value is also easily observable. Such information can be gleaned without processing the weight data. In the end the raw data needs to be processed to gather further information about the make up of the system.

Table of Ratios

A table listing the ratios and frequencies of weights at those ratios is another way of examining the weight system. Petrie assigned a ratio to every weight specimen in each system and then using the weight of the piece, calculated the unit value for each specimen.

Since a specific value is not assigned to the unit of the system, an average of all the unit values is used as the system's unit weight. The nominal weight for each ratio is simply the product of this unit value and the assigned ratio. An additional column lists the logarithm to base 2 of each nominal weight. This should help to identify the plotted points on the log chart.

The frequencies are determined by counting the number of specimens that are allocated to each nominal weight. This is accomplished as follows. Any weight between two adjacent nominal weights A and B will be allocated to A if it is less than (A+B)/2, otherwise it is allocated to B. Continuing this way, all the weights between the lowest and the highest nominal weights are allocated to one or the other weight ratio. The weights lighter than the lowest nominal value are assigned to the lowest ratio. Similarly, those heavier than the highest nominal value are assigned to the highest ratio.

This method of allocation of weights ensures that each specimen belongs to only one ratio and that no specimens are left out.

Histograms

Another way of displaying the characteristic of a weight system is by means of a histogram or a frequency distribution chart. This displays the number of specimens in a given interval of weight. To cover the entire range of the system, the whole collection of weights is first normalised. To do this, each specimen allocated to a ratio or nominal weight is divided by that nominal weight. This will give a value of 1 for specimens that are exactly equal to the nominal weight. Those specimens below nominal weight will return a value less than 1 and those above greater than 1.

The number of pieces within 1% interval of the normalised value is counted. The interval at the centre spans the range from 0.995 to 1.005 of the normalised value and the rest of the range is similarly divided. The count of specimens in each interval is displayed on a chart. This is the histogram or frequency distribution of the weight system.

Weight Systems

Peyem

There are 269 specimens in this weight system. The unit value of these weights ranges from 112.0 grains to 125.2 grains and their average value is 119.6 grains.

Chart 1a is a log plot of these Peyem weights. The binary scale of the weight system, in parts, is apparent, as is the fact that the whole system does not follow the binary scale.

The vertical axis is set to start at 3.90 = log₂(15). The first six steps in the chart, up to 8.90 on the vertical scale, represent a sequence of binary multiples as they all lie on successive grid lines. The steps or flats are made of weights very close in value.

The next flat is below the expected 9.90, indicating a multiplier less than 2 is involved. An inspection of values in Table 1 shows that this multiplier is 1.25. Starting from this step, there are four flats that form another sequence. There are a few odd groups that are not part of this sequence. Of these, two are on the grid line and therefore belong to the sequence starting with the lowest weight. Another single weight, also on the grid line, appears very near the top.

 

By setting the vertical scale to different values, successive flats that form a sequence can be identified. In the chart above, three separate groups that follow a binary sequence can be seen.

Table 1 is constructed using 119.6 grains as the unit of the Peyem system. An examination of the table shows the basic scale of 1, 2, 4, 5 repeating over the next two decades. The expected ratio of 500 is absent and that at 1000 has only one specimen. There are also three fractional ratios of a half, a quarter and an eighth.

There are a few odd ratios: 3, 8, 12, 16, 60 and 1500. Their frequencies are low. They do not form part of any sequence as their decimal multiples are absent. Altogether, 17 specimens out of a collection of 269 do not follow the basic 1, 2, 4, 5 scale.

 

Chart 1b is a histogram of the Peyem system. The set of nominal weights, as listed in Table 1, occurs at the normalised value of 1.00 and there are four values where the frequency is higher than at the nominal weights.  The peak frequency occurs at 0.96 of the normalised value.

  

Qedet

The collection of the Qedet system comprises 861 specimens. The unit value ranges from 135.5 grains to 153.5 grains with the average equal to 144.5 grains.

Chart 2a displays the whole collection on a log scale to base 2. The unit value of 144.5 grains can be seen on the grid line at 7.17. The collections on the following two succeeding grid lines form part of the binary sequence of 1, 2, 4. The presence of a large number of specimens just above the grid line at 9.17 shows a non-binary multiple of 5. Like Peyem, Qedet also follows a basic scale of 1, 2, 4, 5.

All of these multiples can be seen in Table 2b, which also shows the frequencies at different ratios as well as the value to log₂(weight) for different nominal weights.

Again there are fractional ratios, this time a third and a sixth as well as a half, with the fractional ratio of a sixth having a frequency of only one. The collection on the grid line at 6.17 are weights at half the unit value.

Here also there are odd ratios. These are the fractional ratios of a third and a sixth and the ratios 2.5, 3, 6, 8, 25, 30, 150, 250. The frequency at ratio of 25, like that at one third, is surprisingly high. In all, there are 70 of these odd ratios out of the total collection of 861 specimens.

 

Chart 2b is a histogram of the Qedet system. The collection ranges from 0.93 to 1.06 of the normalised values of the weights. The peak frequency occurs at 0.98 of the normalised value.

  

Stater

The Stater system compromises 396 specimens. The unit value of weights ranges from 132.0 grains to 140.3 grains and their average value is 135.3 grains.

Chart 3a is a log plot of these weights. The vertical scale is adjusted so that the value of 7.08 (= log₂(135.3)) falls on a grid line.

 

Table 3 shows the ratios of the Stater weight system. The scale of 1, 2, 5 better describes the system as there are specimens with multiples of tens of this scale up to a ratio of 1000. 

While there are 3 specimens at ratio 4, there are none at ratios 40 and 400. Similarly, ratios 3 and 25, with respectable counts, do not have any at higher multiples. All these ratios, along with the sixth, the third and 8 and 150 are odd to the basic ratio of 1, 2, 5. In all, there are 62 of these odd ratios out of the total collection of 396 specimens.

The Stater system of weights is an odd one out of the three systems being analysed in that its basic scale is different from the other two and it is also the most accurate of the three. This can be seen from the histogram, Chart 3b. This shows the distribution ranging from 0.98 to 1.04 of the normalised values. This in spite of a large number of specimens with odd ratios.

Comments

The analysis of each weight system is based on Petrie's assignment of ratio to each specimen from which a value of the unit of the system is derived. The histogram or frequency distribution of the weight system is based on the average of the derived values of the unit. The frequencies at each ratio listed in the Table of Ratios are obtained from the weight tables. The Log Charts are based solely on the weight of specimens which Petrie determined using a balance and set of grain weights⁴ specially manufactured for the purpose.

A set of weights manufactured to a specific value will display a spread about that specific or intended value. The number of specimens at the intended value is the highest and the distribution of the rest is symmetrical about it. The distribution trails away from the peak. The distribution is shaped like a bell and is known as the Normal distribution.

The frequency distribution for the Peyem and the Qedet systems, Charts 1b and 2b, presents a challenge. The peak frequency is not at the centre of the distribution. Frequencies away from the nominal value do not trail off as expected for a Normal distribution. The frequency distribution of the Stater system, however, is much closer to that expected for a Normal distribution.

These histograms are fairly compact, the range of spread of the weights about the nominal values is mostly less than 5%. The Stater system displays the narrowest range, with most of the collection accommodated within 2% of the nominal values.

Histograms can be examined for slightly different values of the unit weight. For the Peyem and Qedet systems, the distribution will roll along, with the frequencies rippling along the top. No unit value can give a distribution with a single peak frequency. But for the Stater system the matters are different. A slightly higher value of the unit weight is found that produces an almost symmetrical distribution with the peak frequency occurring at the nominal value. Chart 3b1 shows the distribution of the Stater system for two different values of the unit weight: at the average value of 135.3 grains in blue, and at 135.8 grains in red.

All the systems examined here include ternary and quinary ratios as well as the usual binary and decimal ratios found in many of the ancient weight systems. The quinary ratio can be explained as an intermediate stage of a decimal scale. The ternary ratio is not so easily explained. Binary submultiples are a standard feature of each weight system. Also, a few ternary submultiples are present in two of the weight systems examined but none of the systems has any quinary submultiple.

Removing the weights of the ternary scale alters the picture. Both total frequency and distribution are affected. However, reassigning weights of the ternary scale to the nearest ratio applicable alters the frequency distribution only.

Examination of the weight systems presented here is based on weights recovered nearly 100 years ago. Addition of newer finds could alter the picture. A larger collection will give a better value of the unit of the weight system. It should also provide a better view of the ratios that existed in different weight systems.

Acknowledgement

Special  thanks are due to Norman J. Street for providing access to certain books without which this paper would not have been possible.

References:

1.      Ancient Weights and Measures by Flinders Petrie, Department of Egyptology, University College, London. 1926

2.      An Analysis of the Petrie Collection of Weights by A. S. Hemmy in

         The Journal of Egyptian Archaeology, Vol. 23 No. 1 (June 1937), p 42

3.      The Statistical Treatment of Ancient Weights by A. S. Hemmy, in Ancient Egypt, December 1935. Page 86.

4.      Ancient Weights and Measures, op cit, p3

© 2012 Chandrakant Doshi

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