# Units of Volume & Weight

1. In dealing with the units of length, I have mentioned that the weight of the Mesopotamian mina as documented by sample weights, appears to be a rather uncertain quantity. Mommsen was the first to suggest that the Mesopotamian mina existed in two different forms, and Brandis formally maintained that there were several forms of mina. Lehmann-Haupt tried to determine the mathematical rationale for the existence of the several forms, but failed. I shall try to demostrate which is the true mathematical rationale. The new school, of course, denies that there is a mathematical basis for the existence of the several forms; the varieties of the Mesopotamian mina are the major argument in support of the contention that the ancients in general did not know precision in matter of measurement standards.

Up to the coming of the new school, it was accepted that in Mesopotamia the units of volume and weight are the same, that the Mesopotamian mina is the weight of water contained in the Mesopotamian pint, called qa in Akkadian and sìla in Sumerian. Thureau-Dangin so stated still in his major work of metrology, written in 1909. I may note that the Mesopotamian identification of the mina and the pint is at variance with the Greek practice: in the Attic system the pint is the basic pint of 540 c.c. , whereas the mina is either 5/6 or 450grams (mina brutto), or 4/5 or 432 grams (mina netto); similarly in the pheidonian system to the reduced pint of 486 c.c. there corresponds an Euboic mina of 5/6 or 405 grams. In the Mesopotamian system all the units that compose the pheidonian system in Greece can be both pint and mina. Followers of the new school deny on principle that the mina and the qa coincide, but they have never entered the details of the problem. Neugebauer and Sachs grant that the volume of the qa is in some way connected with the unit of length; but deny the connection between the qa and the mina, without however specifying their reasons. Everyone, including Weissbach, agrees that the mina and the qa can both be single or double. Neugebauer and Sachs are alone in claiming that the only qa is the one usually considered double and the only mina is the one usually considered single; they go to great trouble to maintain this point, and apparently their purpose is to prove thereby that units of weight are not connected with units of volume. Both supporters of the old school and of the new school agree that the mina and the qa existed in two varieties related as 6:5 or 5:4; I call the larger form normal and the lesser form Euboic, since this is the name it had in Greece. Here again Neugebauer and Sachs stand alone in denying the exitence of the lesser mina and lesser qa, at least in the Old-Babylonian period.

The Euboic mina or qa varies between 405 and 432 grams or c.c. , according to my estimation. The normal mina or qa has a form that I call the correct one, of 486 grams or c.c. , and increased forms up to 518 4, always according to my estimation The problem of the several forms of mina and qa has its exact counterpart in Greek metrology; this was seen by the scholars of the old school who passed from one problem to the other, but the new school denies any connection between the value of Mesopotamian units and that of Greek units. Several of the old investigators of Mesopotamian units began by considering questions of Greek or Roman monetary weights; but today this link is totally denied, with the result that numismatists cannot solve basic issues in the history of the Athenian drachma or the Roman denarius, because they cannot classify the several weight standards.

In general scholars of Mesopotamian metrics agree with my figures for the Euboic mina or qa, but they have not made a great effort to define this unit; assuming correctly that one can derive from it the Euboic unit by taking 5/6. Thureau-Dangin at times gives the figure of 406 grams for the Euboic mina, but at other times considers. a higher figure, as 410. Meissner in 1920 evaluated the qa around 410 c.c. and in 1936 around 840 c.c. (a double unit).

Whereas there are hundreds of samples of mina, only one sample of qa is known. and this is a sample of Euboic qa. The silver vase dedicated by the Sumerian ruler Entemena contains, by the appraisal of Léon Heuzey, 4.07 liters to the neck and 415 filled through the neck; Thureau-Dangin understood that the capacity is a saton of 10 Euboic qa, but could not decide whether one should reckon by the first or the second figure. On the basis of what can be deduced from Greco-Roman practices, I am of the opinion that one shaped the vase so that it would contain one unit filled to the neck, 10 Euboic correct qa of 405 cc. , and another unit when filled to the maximum, 10 Euboic qa of 415, a form I call Stereometric netto.

2. The effort of scholars has been directed toward the classification of the several forms of the normal mina. Petrie, who was a good statistician and a keen observer of facts, if not always sound in his deductions because of the superstitious notions he had inherited from his pyramidite father, clearly grasped that the mina exists in several forms which cannot be whisked away by assuming that the sample weights are approximate, and built upon this fact the theory that ancient weights in general have forms varying a maximum and a minimum, the strong and the weak forms. This theory does not explain anything and begs the question, but the figures he set as a minimum and a maximum provide an excellent starting point: he sets them at 486 and 516, whereas my conclusion will be that the correct form is 486 and the heaviest form is 518.4. I have reported how Oppert, when only a few texts and a few weights were known, set the limits at 495 and 518. Wilfrid Airy who tried to take an entirely fresh look at the question by considering in isolation the lion weights in good state of preservation at the British Museum, found the smallest sample to be 483 grams and the largest to be 518. A perusal of the extensive and intricate literature on the subject reveals that scholars are substantially in agreement on the limits of the variations; they disagree only about the values that must be chosen as significant within the range.

The area of disagreement can be restricted to the reason for the variations. Weissbach took the position that all that there is to it is that the people of Mesopotamia did not know how to establish standards with precision; his efforts were directed towards the demonstration of this assertion. His position was consequent, since he believed that volumes and weights were not connected with lengths and were not mathematically defined. He agrees, however, with the prevailing view about the limits of the variations. Lehmann-Haupt on the other side tried to solve the problem according to the methods of the old school; he suggested that there was a correct form of mina directly related to the unit of length and then there were special forms (Sonderformen) increased by specific fractions. Neugebauer has followed Weissbach in asserting that the value of the mina and the qa was only approximately set, but he seems to go far beyond him in allowing for imprecision; he states that the qa is about a liter (1000 c.c.) and that the mina is about an English pound (453 grams), indicating that he is thinking of a deviation of about 10% plus or minus, and in fact in the interpretation of mathematical texts he treats as one computation of volumes qa differing as 5:6. Nevertheless at the same time Neugebauer maintains that the volume of the qa is somehow linked with the unit of length. Mrs Hildegard Lewy has tried to render logical Neugebauer’s position by claiming that the geometrical definition of the units was adopted at a second time and falls within the approximate range of the volumes and weights, which to begin with were not set mathematically.

Sutzu, who believed that weights should be studied independently of the problem of units of length and volume, argued that by the mere analysis of the weights one can conclude that there were highly defined values and that within the range of the variations of the Mesopotamian mina one can ascertain the adoption of specific values. Most scholars have come to agree with him in some way or another. Concerning the normal Mesopotamian mina it is accepted by most metrologists that there is a form of 504 grams; this view was expressed by Hultsch in his authoritative manual, and dissent has limited itself to some occasional statement that the value is 502 and more positive assertions that the value should be calculated some fraction of gram above 506 grams (I shall show why the form of 504 at times occurs as 506.25).

In general those who have intestigated the sample weights with care have concluded that one can set specific values within the range of the normal mina. The difficulty is a statistical one: if there are several forms of mina, one risks averaging together specimens belonging to diferent forms. Most scholars have followed the method of deciding more or lees by instinct which were the forms and then averaging together the specimes assigned to each form. Bielaiew reviewed all the previous endeavours and concluded that their shortcoming was the one I have just stated. He tried to follow a method conforming to the best statistical science and determined that there is a concentration of weights in the following areas: 491.14; 502 72; and 511.20. His conclusions are a positive indication that I have followed, but they cannot be accepted as final because his sample was not large enough to justify the application of a merely statistical method. The figure of 491.14 is obtained by averaging specimens belonging to the form of 486 and the form of 495–497.663. Here the empirical analysis was influenced by the figure 491.17 arrived at by Lehmann-Haupt on theoretical grounds; this proves that one may use statistics to arrive at results set a priori. Furthermore, Bielaiew followed a statistical method that ignores an important trait of the subject matter; specimens that are particularly heavy have to be given more significance. It is accepted by numismatists that if one intends to determine the weight of the drachma, pieces of four-drachmai provide a more reliable datum than those of of two-drachmai or one drachma; a most reliable figure would be derived from the fractions of drachma. Bielaiew considered a set of weights of the British Museum all coming from Uruk, all in excellent state of preservation, and all ascribable to the same period (about 2000 B. C.), and found instances of three different forms of mina. But actually in this set of weights there is one piece representing a mina (496 grams), then there is a third of mina (that would give a mina of 491.7) and two sixths (that would give a mina of 491.4 and 502.2), whereas the rest is made of twelfths and twentieths that give more discrepant figures from 484.8 to 511.4. The best way to interpret this material is to say that it indicates a mina of 496 grams. Sutzu properly stressed that great significance must be given to the Lion of Susa, which is composed of 240 minai weighing 506.429 grams, and to the Lion of Dur-Sharrukin composed of 120 minal of 502.525 grams. To these one can add a stone duck-weight found in the Temple Esagila of Babylon with the inscription “one right talent” and weighing 29.68 kg (mina of 4962/3); it is regrettable that Koldewey does not provide a transcription of the inscription and does not indicate to which period the weight belongs.

Many metrologists concur in the opinion that the several forms of Mesopotamian mina account for the standards of the Persian coinage and of coinages influences by it. The numismatist Kurt Regling ascertained that whereas Persian silver coins were minted by a mina of 504, the similar coins of King Chroisos of Lydia were minted by a mina of 486. To conclude the review of the most significant opinions expressed on this subject, I may point out that Lehmann-Haupt, on grounds that are partly theoretical and partly empirical, set the following four values;

491.17  504.82     511.825  15.73

Viedebantt in a work of bitter polemic against Lehmann-Haupt stressed the empirical approach and concluded for the existence of the following four values;

489.5  502  513  521

The only worthy remark concerns the first value: Viedebantt properly observes that Lehmann-Haupt had obtained the value 489.5 from the sample weights. and had increased the figure in order to make it in some way agree with the volume of the qa as a cube with an edge of 6 fingers. In may opinion the values 489.5 and 502 are substantially correct, but have been slightly shifted by including in the average samples belonging to a third and intermediary form of 495–4972/3 grams. The values 513 and 521 are introduced by Viedebantt merely to find fault with Lehmann-Haupt and they are based on two samples. In spite of all his thunder against Lehmann-Haupt, Viedebantt merely accepted the four forms set by the latter and carped about the exact value of them.

In order to facilitate the comparison I state here my conclusions which are based on the use also of different and new types of evidence, such as the bronze bar of Nippur and the cuneiform mathematical texts. I distinguish four forms:

486   495–497.663  504-506.25   518.4

As one can see my third form agrees with the opinion of the generality of scholars. I consider that all samples heavier than this one belong to a single form. My most radical innovation is the introduction of a form of 495-4972/3, but this form is clearly represented by the sample weights

3. In order to prove that sample weights strongly indicate specific values, so that they force a substantial agreement among scholars in spite of their methodological quarrels, I shall quote the opinions of Weissbach and Unger. For Weissbach weights are not determined by a mathematical relation to the units of length and hence it is pointless to ask for precise values. He stated that the ancients could not fix standards with accuracy and quoted as evidence that a person cannot tell the difference between a weight in one hand and a weight in the other hand if the difference does not exceed 61:64; apparently the ancients did not employ scales in establishing their weight standards. Weissbach tried to reduce the problem of special form to nonexistence by proclaiming that one should consider as evidence only those sample weights that state the number of the units and their name; weights without inscription or with the mere inscription of a number should not be considered. This means that if one finds an object having the traditional shape of a weight, as a lion or an astragalos, and it weighs a talent, for instace, it should not be considered as evidence; if an object that appears to be a weight bears the inscription “five’’ and weighs five times some mina, it should be considered as evidence only if supported by fully inscribed samples. Further, no conclusions should be drawn from the weight of coins, since coins are too discrepant in weight. Numismatists would take exception to this conclusion, since even though coins of the same type may manifestly differ in weight, averages indicate usually a nicely established standard, or foot as the numismatists say. In the ancient minting practice one did not measure carefully each coin, but one did so measure the mina or the libra of gold and silver from which so many coins had to be struck. This general impeachment of numismatic evidence was brought forth by Weissbach in order to refute the mentioned opinion of Regling who from the coins has argued for the existence of a form of 486 and one of 504.

Weissbach added that one should discount the statement of Greek authors about Babylonian and Persian units, and that one cannot quote the conclusions arrived at by Böckh, Brandis and Mommsen because they are “secondary sources. “ In spite of all these strictures he could not cause the problem to evaporate and concluded that one should distinquish between an “older” mina of 490.8 grams and a “newer” mina of 505. He grants also that in spite of this terminology the two forms were used concurrently.

evertheless he came to agree with the interpretations of the old school. If one puts together the conclusions arrived at by Weissbach and Unger, one finds a strong confirmation of the existence of the four forms I have presented.

The study of Unger proved the opposite of what he intended to demostrate; the forms of the Sumerian period account also for weights of the Assyrian and the Neo-Babylonian period. They also account for the weights of Persian coins, not to mention Greek and Roman coins. The best expression of this persistence of the same standard through the millennia is offered by a weight of the British Museum (No. 91005) which bears the inscription:

One true mina, property of Marduk, King of the
Gods, copy of a weight which Nebuchadnezzar,
King of Babylon, son of Nabopalassar, King of
Babylon, according to a copy of a weight of
Shulgi, a previous king, placed.

Shulgi (ca.2000 B. C.) is well known as one of the most active rulers of the Third Dynasty of Ur; his strenuous effort to unify and centralize the realm was expressed also in a concern with weight standards. Apparently the Sumerians in the last effort to preserve their supremacy stressed what was their particular field of achievement. The copy of the weight of Shulgi amounts to 978.3 grams, and hence is the double of a mina of 489.15. This weight is only slightly heavier than the theoretical weight of the correct normal mina which I calculate as 486 grams. It is to be noted that when Calif Al-Mamun sent to Charlemagne a sample of this mina, the sample must have weighed 488 or 489 grams. The standard kept at Troyes, which, because of the fair of this city, came to be considered through Europe the authoritative sample, indicated a libra of 489.5 grams. The French pound at the moment of the adoption of the metric system was calculated as 489.506 grams. Such is the persistence of the weight standards through the millennia! The English pound should be 15/16 of this weight, but was reduced by about two grams to make it equal to 100 Byzantine solidi.

4. In order to explain the several forms of mina, Lehmann-Haupt formulated the theory of norms. According to this theory weights exist in a basic norm and in increased norms. He distinguished three increased norms and formulated the following scheme:

correct mina            491.17 grams
increased by 1/36   504.82
increased by 1/24   511.64
increased by ˝0     515.73

The theory of norms developed by Lehmann-Haupt has no theoretical basis, but it was developed merely to justify the variety of forms. It has the appearance of a theory, but in fact it is a statement of empirical facts. Lehmann-haupt was right in noticing the existence of the discrepancy 24:25 (which I call discrepancy leimma), but did not explain the reason for this discrepancy. I have shown that the structure of the metric system explains, and in the eyes of the ancients justified, the use three discrepancies: the discrepancy diesis (15:16). the discrepancy leimma (24:25), and the discrepancy komma (80:81). The pattern of discrepancies I have formulated explains the facts Lehmann-Haupt had roughly grasped in this theory of norms. These discrepancies are of great importance in world culture, because musical scales were derived from the structure of the units of volume. As I have stated, the fact that both Greek and Chinese musical scales treat discrepancies, that is, accidents, in the same way indicate a common derivation from Mesopotamia.

If one calculates the correct mina as 486 grams, as it should, one can see that the unit increased by 1/24 is that of 506.25, which I call correct mina brutto. Lehmann-Haupt set the value of the correct mina too high, at 491.17 grams, and was forced to assume the existence of a mina 511.64, since he expected to find a unit increased by 1/24. But since unquestionably there is a unit of 504-506.25 grams, he explained it by an increase of 1/36. But the only evidence for the existence of such an increase is the passage of Aelianus that implies that the Babylonian mina is equal to 72 Attic drachmai, whereas other Greek authors speak of 70 drachmai. But, as Mommsen concluded, the passage of Aelianus indicates mina which is 1/35 (the figure 1/36 of Lehmann-Haupt is inexact) heavier than the mina of 504 grams. If one multiplies an Attic drachma of 4.32 grams by 70, one obtains a mina of 504; and if one multiplies it by 72, one obtains 518.4 grams This last mina is equal to a correct mina increased by 1/15, and not by 1/20 as Lehmann-Haupt claimed.

One can draw the following scheme;

correct mina            486 grams
increased by 1/24   506.25
increased by 1/15   518.4

By formulating the theory of norms, Lehmann-Haupt violated a principle of the old school which he was trying to defend: the principle that all units of volume and weight must be cubes of units of length or fractions thereof. According to his scheme only the correct mina is related to the unit of length. He put himself in a weak position in the face of the attacks of the new school, because in a way he accepted their position. The article “Talente” published posthumously in 1956 in RE, is an attempt to defend this scheme, while defending at the same time the method of the old school.

In my opinion, the only way to defend the old school against the paroxysm of irrationality that has developed in the study of ancient metrics is to deepen the method of the old school and to remain strictly faithful to it. Hence one must start from the units of length. The greatest handicap of Lehmann-Haupt was that he had not succeeded in calculating the exact value of the Mesopotamian cubit. As I have said before, he drew the false conclusion that the double mina equal to 216 cubic fingers was the correct form.

On the basis of my calculation of the length of the barley cubit, the problem of the special forms can be easily solved. A volume of 216 cubic fingers is the cube of 1/5 of cubit: calculating by the natural cubit it is 1037.97 grams for a double mina (single mina of 518.986), and calculating by a trimmed cubit it is 996.45 grams for a double mina (single mina 498.226). But from the stereometric texts one gathers that the first mina was calculated as 518.4 grams which makes it a diesis more than the correct mina of 486 (16/15 of 486 = 518.4) and a leimma less than the normal pint of 540 c.c. (24/25 of 540 = 518.4). I call this unit normal mina Stereometric brutto; the corresponding netto unit is a mina or qa of 497.664 grams or c.c.

The following distribution of the units results:

correct mina             486.00
Stereometric netto    497.66
correct mina brutto   504.00
Stereometric brutto   518.50

But the correct form brutto is considered equal to a basic pint reduced by leimma with the result that at times the correct form brutto is calculated as 506.25.

The mina of 486 is justified by assuming a reduttion by the calculation pondo olei: a basic pint of 540 c.c. filled with oil weighs a mina of 486 grams according to an assumed specific gravity of oil 9/10. But it was known that really olive oil and similar vegetable oils have a specific gravity which is rather 11/12; therefore at times one used this more exact calculation that has the advantage of fitting into duodecimal reckoning. The reduced pint (Alexandrine pint) is usually 486 c.c. but at times it is calculated as 495 c.c. (11/12 of 540 c.c.); I call this unit Thasian pint because it was used in the metric system of the island of Thasos, a system adopted by the former allies of Athens after their revolt. From the sample weights it appears that the form Stereometric netto of 499.66 grams was identified with the Thasian form of 496.

The rationale for the value of the several forms of mina or qa, is expressed by the following table:

Basic Relations

sheqels

Basic pint 540 60 1

Stereometric brutto 518.4 24/15 1 16/15

Correct brutto 506.25 15/16 25/24

Correct brutto 504. 56 14/15 35/36

Stereometric netto 497.66 24/25

Stereometric netto 495 55 11/12

Correct 486 54 9/10 15/16 1

By reducing the forms of the normal mina or qa by 1/6 one obtains the corresponding forms of the Euboic mina or qa.

 Basic sheqels Stereometric brutto 432 48 Correct brutto 423 47 Stereometric netto 414 46 correct 405 45

Calculating by basic sheqels one can draw the following scheme:

 45 sheqels 405 grams correct Euboic 46 414 Euboic Stereometric netto 47 423 correct Euboic brutto 48 432 Euboic Stereometric brutto (Attic monetary) (50 450 basic mina) 54 486 correct normal 55 495 normal Stereometric netto (Thasian) 56 504 correct normal brutto (60 540 basic pint)

There remains to be explained why the Mesopotamians developed a system that entails four different forms of mina or qa. The explanation has to be found in the structure of the units of length in order to make them fit into the sexagesimal system. This process of adaptation is illustrated by the tablet of Larsa (Senkereh) discovered by Sir Willian Loftus in the ruins of Larsa in 1854. This tablet is written in very minute characters and apparently was intended to be carried around as ready reference. On one side there is a tablet of squares and cubes; since the other side is a tablet of units of length, it follows that these squares and cubes were to be applied to the units of length. The tablet of units of length presents in two separate columms, for ready conversion, two different ways of calculating sexagesimally the units of length. According to one way, the starting point is the half of finger, so that the cubit of 30 fingers is the first sexagesimal multiple; the following multiple is the subban of 60 cubits, and so on. In the other columm the starting point is the tenth-of-finger, the first multiple is the six-finger unit, and the next multiple is the SAR or double cane of 12 cubits, followed by the double stadion of 720 cubits. The second sexagesimal method of calculating the units of length is the one we find employed in stereometric tablets.

Since stereometric tablets do not use the cubit as unit of length, they calculate the units of volume and weigth by the six-finger unit. This is the reason for the existence of a qa of 216 cubic fingers, the qa I call Stereometric. Since both the Stereometric qa netto and the Stereometric qa brutto are larger than the correct qa, there was introduced a brutto variety of the latter which has a value intermediary between the two Stereometric qa.

5. The scheme I have reconstructed of the several forms of mina, allows to understand the chapters of Herodotos (III, 89-96) dealing with the tributes paid by the several provinces of the Persian Empire. It is not true that Herodotos was guilty of errors or that the test of the manuscrits must be radically emended, as suggested by the many scholars who have commented on this text. The only difficulty is due to the fact than Herodotos calls Euboic mina both the unit of 405 grams and that of 432 grams, that is, both the correct form and the form Stereometric brutto. When Herodotos quotes the Persian accounts he refers to a correct form of 405, because this was the form used in the Persian Empire, but when he explains the Persian units to his Greek public he refers them to the form Stereometric brutto of 432 grams which was the best known form of the Greek mina, being the basis of the Attic coinage.

The nineteen provinces of the Persian Empire that contribute silver pay a tribute of 7740 talents. This amount is expressed in talents based on a normal mina of 504 grams, which Herodotos calls Babylonian; the figure of 7740 Babylonian talents is converted into 9632 talents based on a Euboic mina of 405 grams. To the amount of 9632 talents there must be added the income serived from Lake Moeries since Herodotos, in listing the tribute paid by the Egypt, states: “700 talents, besides the money derived from Lake Moeris. “ In chapter III, 149 he specifies that the income from Lake Moeris is one talent a day for half of the year and 20 minai for the rest of the year; the total is 240 or 2422/3 talents (360 or 365 days). Since in this case he does not specify which talents are used, he must mean the talents based on the Euboic mina, the unit with which his public is familiar. By adding the tribute from Lake Moeris, the total tribute of nineteen provinces comes to a few talents less than 9880 talents; one must assume that Herodotos rounds up the total to the next decimal figure,. since he himself states that he has rounded the minor amounts (III, 95) and since it is clear that this computation is by tens. The twentieth province contributes 360 talents of gold powder that are calculated as Euboic talents of silver at the rate 1:13, as specifically stated, producing an account figure of 4680 talents. The total amount of the tributes is 14560 Euboic talents.

Gold powder which does not contain an alloy is converted into silver coins at the rate 1:13, but gold coins are converted at the rate 1:12.44. When one pays in silver coins, he must deliver 10 silver coins for a gold coin; the silver coins are struck by a mina of 504 grams, but calculated as if they were struck on a mina of 405 grams. The Greek rate of conversion of coins is also 1:12.44, but it is calculated by making 70 drachma of 4.32 grams of silver equal to 6 gold coins of 4.05 grams.

6. Given the several forms of the mina there is a certain flexibility in the multiples of the mina. For this reason I shall present here only the general structure of the multiples; from this general statement it should be easy to determine which multiples are specifically used in each set of documents.

The sexagesimal multiple of the mina or qa is the artaba, cube of the artabic foot (60 x 486 grams or c.c. = 29.160). The artaba is called simid in the texts. Since the mina or qa are often double there is a double artaba called massiqtu. The sexagesimal multiple of the form Stereometric brutto contains 64 minai or qa of the correct form. The multiple of 64 correct units is the cube of the wheat foot: the natural wheat foot of 318.15 mm. forms a cube of 32,385 c.c. and is almost exactly 64 minai Stereometric brutto (64 x 518.4 = 33,177.6); the trimmed wheat foot of 314.52 mm. forms a cube of 31.090 c.c. which is equal to 64 correct minai and to 60 minai Stereometric brutto (64 x 486 =31.104). The division of the cube into 64 parts is geometrically much easier than the division of the cube into 60 parts. The unit of 64 minai is called Babylonian talent by Greek authors. The Babylonian talent equals 72 Euboic minai Stereometric brutto (72 x 432 grams = 31,104).

There is a unit of 72 minai called PI (to be read as the equivalent of the Hebrew ephah). This unit is the cube of the barley foot. The natural barley foot of 337.5 mm. forms a cube 38.443 c.c. which is practically equal to 80 correct minai (80 x 486 = 38.880) and to 90 Euboic minal Stereometric brutto. The trimmed barley foot of 332.94 forms a cube of 36.905 c.c. which is practically equal to 90 correct Euboic minai (90 x 405 = 36.450) and to 75 correct normal minai; it is equal to an artaba increased of 1/4 and, hence, it represents the volume of the monthly ration of barley, whereas the artaba is the monthly ration of wheat.

The load is called meru “ass, ass’ load,” in Mesopotamia. If the imeru is composed of 3 artabai it amounts to 180 minai of 486, or 87,480 grams (basic load netto). But there were slightly different kinds of imeru. In principle the load is the cube of the basic cubit of 444 or 450 mm. , but probably in Mesopotamian practice it was calculated as the cube of 27 fingers of barley cubit or 9/10 of barley cubit, that is, as the cube of 449.460 or of 455.625 mm.

The imeru is the cube of the basic cubit and represents the basic load, load of an ass or load that can be carried by a man for a short distance. But in Mesopotamia most often one calculated by the kurru (Sum gur) which is the volume of barley weighing an imeru. The kurru is the cube of the barley cubit. According to the variety of cubit used, trimmed or natural, one obtains:

Kurru netto 125.524 cc.
Kurru brutto 129.746 cc.

The kurru netto is almost exactly 1˝ times an imeru brutto of 90.000 grams. The imeru brutto of 91.125 grams is equal to 180 minai of 506.25 grams.

The kurru brutto appears calculated as 129.600 c.c. , because this makes it equal to 240 basic pints of 540 c.c. , to 14,400 (120 x 120 = 602 x 4) basic sheqels of 9 grams, and to 300 Euboic minai Stereometric netto of 432 grams.

There is also another kurru which is equal to the cube of 2 barley feet. This great kurru brutto, cube with an edge of 675.0 mm. , amounts to 307.546 c.c. and can be calculated as 600 normal minai Stereometric brutto (exactly 518.4 x 600 = 311.040). The variety netto is the cube of 666 mm. , 295.408 c.c. , and can be calculated as 600 normal minai Stereometric netto (exactly 600 x 495 = 297.000). These units have the advantage that they can be calculated as 10 artabai.

There is also a third form of kurru which is equal to a cube with an edge of 2 barley cubits. I call this unit the greatest kurru. The advantage of this unit is that of being equal to 1000 double normal minai or qa and to 300 royal minai or qa. The brutto version of this unit is 1,037,970 and hence 200 normal minai Stereometric brutto; the netto version equals 200 normal minai Stereometric netto.

Another important unit is the peck, the saton, called sutu in Akkadian and Ban in Sumerian. The reason for the existence of this unit is that of breaking into two steps of 6 and 10 the calculation of the sexagesimal multiple of the mina or qa. For the same reason the mina is divided into 10 akalu of 6 sheqels. There are sata of 6 minai and of 10 minai; or qa can be double there are sata of 12 and 20 minai. The process of duplication leads to the formation of a saton of 24 normal minai. The shift from a saton of 10 minai to a saton of 12 minai is facilitated by the fact that 10 normal minai are equa to 12 Euboic minai.

There is also a saton of 8 minai; the reason for the existence of this unit is the artaba of 64 minai which can be most easily divided into 8 sata of 8 minai.

The use of the saton is more easily illustrated by its derivations in the metric systems of other countries. This unit, which is the peck, is called seah or midah, “measure” in Hebrew. In Greek it is called savton, mevtron and movdio”, The Latin name modius has a Latin dress but it is derived from an Aramaic form môdiå. In the Roman period there was used a modius of 16 basic pints (sextarii), which is the typical roman modius; this unit corresponds in type to the Mesopotamian sutu of 8 double qa. There is also a modius cumulatus of 24 sextarii, but this unit, as the name indicates, is considered merely a heaped form of a unit called modius xystus, “shaven peck” of 22 or 213/5 sextarii. This last unit is equal to 24 Alexandrine sextarii; this unit is the perfect equivalent of a Mesopotamian sutu of 12 double qa. The reckoning by reduced pints, Alexandrine sextarii or qa, occurs also in the form of modius castrensis which is equal to 18 basic pints or 20 reduced pints; this unit is the perfect equivalent of the sutu of 10 double qa.

7. There are also units calculated by the great cubit (1˝ barley cubit). There is a qa of 216 cubic fingers, called royal ( Sum. lugal), which is the counterpart of the qa Stereometric, calculated as a cube having for edge the six-finger unit.

In all the texts I have considered the royal qa is always based on the natural version of the great cubit, and measures 3503.16 c.c. By definition it is equal to 33/8 double qa Stereometric brutto, since (11/2)3 =33/8. It is practically equal to 3.6 double correct qa (7.2 x 486 = 3499.2).

The royal qa is equal to 4 double Euboic qa Stereometric brutto, except for a discrepancy komma, since 33/8 x 6/5 =162/40 = 4 plus 1/80. In other words, the royal qa would be equal to 8 Attic monetary minai, if these measured 437.4 grams, instead of 432. But the discrepancy komma is often disregarded.

By forming the cube of the great cubit one obtains a kurru of 437,820 c.c. which is equal to 900 correct normal qa or minal (900 x 486 = 437,400).

The great advatage of the calculation by the great cubit is that cube of the finger is equal to 16.218 grams and hence is almost exactly a double sheqel of 16.2, calculating the sheqel by the correct mina of 486 grams. I have stated that the kurru which is the cube of the natural barley cubit would amount to 129.746 c.c. , but it is actually calculated as 129.600 c.c.; by making the same adjustment in the volumes derived from the great cubit, the cubic finger is exactly a double sheqel, since 16.2:129,600 = 16.218:129,746. Since the cubic finger of the great cubit is a double sheqel, one can calculate easily the volume of the double correct mina as 60 cubic fingers. The correct Euboic mina is equal to 25 cubic fingers or 50 cubic fingers when double. Hence, the great cubit allows to calculate stereometrically the correct units.

The best illutration of the use of the royal qa is provided by the Smith Tablet. When the amount of seed is calculated by the barley cubit it is 108 qa,. but when the amount of seed is calculated by the great cubit it is 30 qa. The relation is 30:108 =1:33/5, and, since a royal is equal to 3.6 double correct qa, it follows that the amount of seed corresponding to the basis of the Tower of Babel is calculated as 108 double correct qa or 216 correct qa. It is equal to 3 PI of 72 qa each, whereas the amount of seed by the great cubit is a simid of 30 royal qa (royal qa are double units).

As a result, there was unearthed a bronze bar about four English feet long and weighing about 100 pounds, marked by divisional lines all across its length, and the excavators did not consider it their task to call attention to it. The bar was shipped to the storage of the Imperial Ottoman Museum of Constantinople where it was noticed about twenty years later by Unger, but he too did not fully realize its significance.

From 1911 to 1918 Unger was curator of Oriental antiquities at this museum and in this capacity he proceeded to classify the Mesopotamian weights, mostly from Nippur. In the process he came across the bar but did not realize that is a standard of weight and merely as a minor detail reported that it weighs 4˝ kg. He noticed only that the bar has a length of four feet and took it as a standard of length. He calculated the length of each foot so as to confirm the theory of Father Deimel that in Mesopotamia there was a cubit of 518 mm. This cubit of 30 fingers would correspond to a foot of 16 fingers measuring 276.856 mm. As I have stated this contention is based on the doctrine of the new school that the rule of Gudea is composed of 16 fingers, and on Father Deimel’s figures for the rule of Gudea based on an enlarged photograph. It was considered a great triumph by the new school to be able to prove that the cubit was of shifting value and had been of 518 mm. at Nippur in Sumerian times, whereas it was at least 10 mm. shorter in Neo-Babylonian times.

The bronze bar of Nippur is first of all a standard of weight. The iron bar of the Heraion of Argos is also a standard of weight, but it was given a length of four feet. The bronze bar of Nippur is a standard of length of four feet: the reason of this length is the same: in indicating the volume of a cube one measured the perimeter of four sides of the cube, treating it as if one were measuring a cylinder for which one reports the circumference. The quadruple units of length indicating the volume of a cube, were called stereometric units or stereav. The bar appears calculated by a slightly shortened trimmed oil foot, for the reason that the cube of this foot gives a weight which is half the weight of the bar. The bar indicates the weight in two ways: directly and through the unit of length.

Unger reports that the bar has a length of 1103.5 mm., but since it is rounded at the two ends it is difficult to estimate the total length directly. However, the bar is divided by transversal lines into segments of a variable number of fingers so that, by examining the distribution of the divisional lines, it is possible to calculate the intended length of the total rule. The divisional lines are marked by an indentation in the metal and do not appers to have been cut with great precision, accorsing to Unger’s report. But their intended length can be calculated easily as the following table indicates:

 Number of fingers Actual Length Intended Length Difference 15 256.0 mm 257.625 –1.5605 257.5605 4 67.0 68.700 –1.6828 68.6828 12 209.5 206.100 +3.400 206.0484 14 241.5 240.450 +1.050 240.3898 4 67.5 68.700 -1.200 68.6828 3 53.0 51.525 +1.475 51.5121 12 209.0 206.100 +2.900 64 1103.5 1099.200 +5.700 1098.9248

I have calculated the finger as 17.175 mm. As I have stated, the measurement of the two end segments is the most uncertain datum because the two extremities of the bar are rounded. In spite of the roughness of the markings and the approximation of Unger’s data, the intended figures can be calculated to the one thousandth of millimiter, because of the tight mathematical structure of the bar. Calculating with all exactitude the foot of the bar is 274.1314 and the finger is 17.1707 mm.

By cubing a foot of 274.8 mm., that is, constructing a cube the perimeter of which is equal to the length of the bar, one obtains a weight of 20.751 grams, half of the weight of the bar.

The total bar is a standard of weight; this is indicated also by the fact that at one end the bar is increased by an extra blob of metal. Apparently the smiths who made the bar were not as skillful as those who made the bar of the Argive Heraion who succeeded in producing a bar of metal of the proper weight with a regular rectangular section; the Sumerian smiths of a much earlier age had to increase the thickness at one end. The excavation of Khafajah have revealed silver bracelets of approximately the same historical period to which a blob of metal is added to make them of the proper weight.

The total bar has a weight of 41˝ kg. according to Unger; this indicates that it was calculated as 96 Euboic stereometric brutto minai of 432 grams, or 41,472 grams. The bar may also have been calculated as 80 normal minai Stereometric netto of 518.4 or as 100 Euboic minai Stereometric netto of 414.72, but the calculation that follows indicates that the unit used as the main one is the Euboic mina Stereometric brutto.

As stated earlier the length of the bar is the perimeter of a cube weighing half the weight of the bar. The foot of 274.8 appears to be a reduced version of the trimmed oil foot of 277.347 mm.

The division of the bar into segments can be explained according to the principles of the old school: the several segments must be edges of cubes of a given volume. The following table accounts for the division into segments.

 Number of fingers Length in mm. Cube in c.c. 432 g. Number of minai Exact value by minai of Discrepancy in grams 3 51.525 1,367.9 3.2 1,382.4 –14.5 4 68.700 3,242.4 7.5 3,240.0 +2.4 12 206.100 8,754.5 20 8,640.0 +94.5 14 240.450 13,901.6 32 13,825.0 +79.1 15 257.625 17,099.4 40 17,280.0 –180.6

It is possible to understand why the sivisions of the bar are inaccurately marked even though they were calculated with great precision: they are not direct indications of length, but merely suggest the number of fingers that would form the edge of cubes containing a round number of minai. Somebody solved a mathematical conundrum dividing the length of 4 feet into a number of segments which are the edge of cubes of standard contents. The bar was almost certainly preserved in a temple as measuring standards usually were, but this particular one must have been considered as somehow magical for embodying some sort of mathematical of wonder.

This bar, which tentatively may be dated in the early Sumerian period, indicates how groundless is the prevailing doctrine that the ancients were not concerned with precise standards and that in any case did not possess scales that would allow them to assess weights with precision

Unfortunately we do not know anything about the place where the bar was found. Unger reports that it was found 8. m. below ground: if this means 8 m. below the level set as 0 by the excavators it may mean at the level of the foot of the pre-Sargonic ziqqurat. But objects such as the bronze bar were preserved through the centuries, and at times through the millennia, being transferred from one temple to another. At times conquerors remove them from one city to another. The bar may be as old as the beginning of the third millennium. A chemical analysis could allow its comparison with other more precisely dated bronze objects.

From the metrological point of view one can note that the bar seems to indicate both a mina calculated as Stereometric netto and as Stereometric brutto. This suggests that at the time the bar was cast one had already conceived of the double mina as a cube with an edge of 6 barley fingers. The total bar is equal to half an imeru, 82,944 grams, This is with a very minor adjustment the basic load of 91,125 reduced by 1/10 or 82,012.5. This clearly indicates that the units of the bar belong to the system of reduced units, the system the Greeks called pheidonian and which is the typical system of Mesopotamia. In the Attic system, which conforms to the basic system, the mina of 432 grams is a netto unit and hence has a multiple of 50 minai, whereas it is the mina of 450 grams that is contained 48 times in the Attic monetary talent. I have observed that the key to the success of the Attic system is that it was based on the mina of 432 grams which belongs both to the Attic and to the Euboic system. Considering this, it is remarkable that the bar of Nippur stresses the importance of the unit of 432 grams. It is abstractly conceivable that in Mesopotamia too the unit of 432 grams was used as a transition from the basic system to the reduced system. If the bar belongs to the period in which the reduced system was introduced, then it would be really the equivalent of the bar the of Archive Heraion, that is, the symbol of the introduction of the pheidonian system in Greece.

9. A most important task would be the classification of the weihghts of Mesopotamia for the purpose of dettermining when system of units I have described was introduced. Since I have shown that the several forms of the Mesopotamian mina result from an adaptation of the units to a computation thoroughly sexagesimal, the date of the adoption of the system of measures would be an indication of the date of the adoption of sexagesimal reckoning.

Miss Margaret Cross has tried to classify weights from strata IV to VIII at Tepe Gawra (Sargonid age to Protoliterate period). Four specimens bear the marking of he number of the units:

 No Stratum Marking Weight Sheqel Mina 1 IV 10 82,29 8.29 497.4 2 VI 2 16.68 8.34 500.4 3 VI 2 17.70 8.85 531.0 4 VI 8 65.50 8.19 491.4

A mina of the form Stereometrical netto of 497.60 grams is strongly indicated by tree specimens. Miss Gross is surprised by No. 3 and suggests that on the surfase of the specimen, which is of porous stone, perhaps one can see some sort of line indicating a fraction: she proposes that the marking be read as 21/10 sheqels, and not as 2 sheqels as a first reading had indicated. But one must object that a specimen of 21/10 sheqels would be most unlikely. In my opinion this specimen represents two basic sheqels of 9 grams according to a mina of 540 grams

In classifying the objects that are not marked, Miss Cross had to face the problem of deciding which objects are to be considered as weights. At times the shape is highly indicative, but at other times it is ambiguous: hence, she followed the procedure of classifying as weight the stone objects that have a weight conforming to a known unit. As known values of the mina she accepted those formulated by Bielaiew. This procedure is dangerous, but in part unavoidable. I have pointed out the shortcomings of Bielaiew’s typology, but one must grant that Miss accepted one the best classifications available. One wonders, nevertheless, whether she missed any samples of the basic sheqels of 9 grams. She discarted as weight an ovoid of basalt (Stratum VIII) because it has a weight of 1368 grams, but she mentioned it because it has all the appearance of a weight. I see in it a rather accurate sample of 2˝ minai of 540 grams (exactly 1350 grams) or of 3 minai of 450 grams.

Particular significance must be given to four specimens she decided to classify as weights even though they do not fit the forms of mina formulated by Bielaiew:

 No Stratum Weight (g.) Sheqels Weight of Sheqel (g.) Mina 5 V 18 grams 2 9 grams 540 6 V 21 2 1/3 9 540 7 VI 2.9 1/3 8.7 522 8 V 5.9 2/3 8.85 531

The two heavier pieces definitely indicate a sheqel of 9 grams; one cannot expect precision from specimens as small as the other two, but these too may represent a sheqel of 9 grams.

Apparently the form of the mina used in literate Mesopotamia made its first appearance in the Early Dynastic Period. Two specimens from Stratum VII weighing 8.41 grams indicated a normal monetary mina of 504.6. Miss Cross found that eleven specimens (Nos.9–19) from stratum VI indicated a sheqel between 8.05 and 8.34 grams (normal mina varying between 483 and 500.4); possibly they represent the correct form of 486 and the form Stereometric netto of 495–497.6.

Arthur J. Tobler has continued the highly valuable research of Miss Cross, applying the same principle of classification to the specimens found in the earlier strata of Tepe Gawra. He has presented the following results:

 No Stratum Weight (g.) Sheqels Weight of Sheqel (g.) Mina 1 IX 66.81 8 8.35 501.0 2 IX 45.70 5˝ 8.31 498.6 3 XIII ? 47.45 6 7.93 475.8 4 XVIII 11.43 1˝ 7.62 457.2 5 IX 3.56 6 XI 30.79 4 7.70 462.0 7 X 19.36 2˝ 7.75 465.0 8 XIII 5.83 3/4 7.77 466.2 9 XII 8.71 1 8.71 522.6

No 5. must be excluded from the analysis because it is too small and because Tobler himself observes it may be a bead. One must add a weight of 24.0 grams which, in Tobler’s own words, is the object that most specifically has the shape of a weight (crouching animal) and which he rejected because it cannot be explained according to Bielaiew’s types of mina

Tobler sees a clear indication of the sheqel of 8.4 grams in No. 1 and No. 2 (mina of 504 grams), but the existence of a unit representing the unusual amount of 5˝ sheqels is doubtful. In his opinion No. 9. would represent an irregular sheqel of 8.71 grams. The rest would represent a sheqel oscillating widely between 7.62 and 7.93. grams. If such a sheqel existed, one wonders why No. 2 should not be interpreted as representing six such sheqels. Tobler has rendered a great service to metrology by taking the unusual step for an archeologist of paying attention to weights of this early period. But his interpretation is valuable only in a negative sense: it proves that the form of the mina of literate Mesopotamia do not fit the specimens.

In analyzing this material I would take as a starting point the doctrine of Hultsch that the sheqel of 9 or 9.1125 grams is at the root of the system of weights. Assuming a half-sheqel of 4.56grams, No. 2 would be 10 half-sheqels, and No. 3 would indicate the existence of a unit brutto increased by a diesis and hence 4.75. With these half-sheqels related as 24:25, one could explain the specimens listed by Tobler.

 No Actual weight Half-sheqels netto Half-sheqels brutto Intended Weight Crouching animal 24.00 5 23.75 1 66.81 14 66.50 2 45.70 10 45.60 3 47.45 10 47.50 4 11.43 2˝ 11.40 6 30.79 7 31.192 7 19.36 4 19.04 8 5.83 11/4 5.93 9 8.71 2 8.55

If my interpretation is valid the samples would be particularly accurate, being all correct within half a gram. Since these little weights must have been used to weigh precious metals, the occurrence of specimens of 7 and 14 sheqels (Stratum IX) suggests a relation 1:7 between gold and silver, as the one documented for the Third Dynasty of Ur, one thousand years later.

The specimens of Stratum VIII which have been studied by Miss Cross are to be grouped together with the specimens of earlier strata studied by Tobler. I have already remarked that Miss Cross did not consider a weight an object that has the appearance of a weight because it weighs 1368 grams. This could be considered equal to 300 basic half-sheqels netto of 4.56 grams: it would be a perfect sample of the half-sheqel indicated by the samples presented by Tobler.

Miss Cross classified as weight another specimen of Stratum VII (No. 23) weight 9.60 grams, but considered it an aberrant sample. It appears that this is a sample of two half-shekels brutto (exactly 9.50 grams) according to the standard of the earlier strata.

If my interpretation of the sample weights is correct, at Tepe Gawra from the beginning of the Obeid Period to the end of the Protoliterate Period (Stratum VIII) one used a system of weights based on the basic sheqel of the form increased by a komma (9.1125 grams instead of 9) together with a variety brutto of the same weight. The unit taken as reference point seems to have been the half-sheqel. From the limited data available one can tentatively gather that the multiples were decimal, but a specimen of 300 half-sheqels suggests a sexagesimal factor

A radical break in the organization of the weights. takes place with Stratum VII (beginning of the Early Dynastic Period) with the appearance of the forms of the mina of later Mesopotamia. But one finds also specimens of the basic sheqel of 9 grams: this agrees with the mina of 432 grams indicated by the bronze bar of Nippur. This mina can be calculated as 48 basic sheqels, besides being an Euboic mina Stereometric brutto

My hypothesis that in very early Mesopotamia one used a basic sheqel of 9.12 grams may be supported by a finding of the early agricultural village of Tall-I-Bakun in Iran. The excavators did not expect to find weights, but reported that an alabaster cone that they were inclined to classify as a game pawn weighs 9.185 grams: they must have had in mind the possibility that it was a weight since they tested it. But unfortunately they did not weigh a bigger similar object because it was chipped; if it could be weighed, making the proper allowance for the missing fragment, and it were to be found that it amounts to a multiple of the smaller piece, then we could know that the two alabaster cones are weights. At the present moment one can only say that at Tall-I-Bakun in Stratum AIII (equivalent to the Obeid Period in Mesopotamia) there was found an object that could be a weight and could represent a sheqel equal to the sheqel of Tepe Gawra

X

THE DIMENSIONS of buildings indicate a general use of the basic foot and of the artabic foot up to the close of the Protoliterate Period: these are the two feet used to construct units of the basic system. The basic foot occurs always in the trimmed variety, indicating that the talent was calculated netto as 48 basic pints or 25,920 c.c. or grams (Attic monetary talent). The same talent can be divided into 60 minai of 432 grams (Attic minai netto or Euboic mina Stereometric brutto). A talent equal to 50 such minai has for edge a length intermediary between the natural and the trimmed oil foot: this could explain the frequent occurrence of the two varieties of oil foot in the Protoliterate Period. When the unit of 432 grams comes to be considered a unit netto, one shifts from the basic units to the reduced units. The bronze bar of Nippur calculated by a shortened oil foot may be a document of this shift. To the Euboic mina of 432 grams there corresponds a normal mina of 518.4 grams, and this may have led to the realization that the double normal mina can be constructed as a cube with an edge of 6 fingers of barley cubit. The first certain occurrence of buildings calculated by the barley cubit is in the Early Dynastic Period: this would confirm the hypothesis, based on the analysis of weights, that the typical metric system of Mesopotamia was introduced at the beginning of this period. The oil foot seems to have been used in the transition from basic units to reduced units; since the oil foot is equal to 15 fingers of basic foot this may have given the idea of dividing the barley cubit into 30 fingers.