This method presupposes the existence of a conventional rate of seeding, similar tu the Roman rate of 5 modii,pecks to a iugerum. Hence the problem is that of finding what was this conventional rate in Mesopotamia. Since there are many texts that combine the usual definition of areas with the definition by amount of seed, the problem should be easily solvable. It occurs, on the contrary that the essays dealing with this topic are the most abstruse in the field of ancient metrology; a simple problem is made to appear hopeless by scholars who refuse to admit the connection between units of length and units of volume and weight and, hence, cannot make sense of a connection between units of area and units of volume.
What appears shrouded in mystery to followers of the New School becomes transparent if one accepts the method of the Old School. I presume that any of the Old School metrologists could have solved the problem as directly as I have, if they had at their disposal the documents that have become available in the last fifty years.
Thureau-Dangin noticed that the unit of surface, musaru, a square with a side of 12 cubits, is divided into 60 sheqels of 180 grains each, and that this division parallels the division of the qa into 60 sheqels of 180 grains. But he balked at drawing the inescapable conclusion that the rate of seeding is a qa to a musaru.
A clear textual confirmation of this rate is provided by two tablets from Susa published by Father Scheil (RA 35 (1938), 92-103). The tablets contain tables to convert amounts of seed into the corresponding area calculated in the normal way. The tablets do not present any arithmetical difficulty: it is evident that through about 100 entries describing squares, near-squares and circles, the rate of seed to area is always a qa to a musaru, or 75 grains to a square cubit.
The seeding rate is always a qa to the musaru or a figure very close to it. The analysis of the texts proves that the correct rate is 72 grains to the square cubit and that this rate may be adjusted downwards of a diesis or upwards of a leimma for the purpose of obtaining a round figure for the unit of surface used. For instance, the rate a qa to the musaru is a rate increased by a leimma (75 grains to the square cubit). In the case of the Smith Tablet, the rate is once calculated as 108 double qa to the iku, which corresponds to 72 grains, and once as 30 royal qa, which is a diesis less; but we have seen that the discrepancy was compensated by using a correct qa in the first case, whereas the royal qa corresponds to a qa Stereometric netto. This must have been in general the method used to adjust slightly discrepant seeding rates.
The rate of 72 grains to the square cubit suggests how one proceeded to the sowing. In Mesopotamia one sowed by lines using the seeding plow, called absin in Sumerian. One may have spaced the lines half a cubit (about 25 cm.), as one would today, and dropped 36 grains to each cubit of line of seed.
If this interpretation is correct one can answer the question whether the rate was calculated for barley or for wheat. The terminology of the texts is ambiguous since the term seu may refer to both grains, even though most often it refers to barley. Where the volume of 72 grains to the square cubit refers to grains in a concrete sense, the grain which has the volume of 1/180 of a sheqel of 8.1 c.c. is the wheat grain. A sheqel of barley would contain 6/5 more grains; hence, when one sowed barley one would have sown instead of 72 or 75 grains 86.4 or 90 grains to the square cubit. Modern practices indicate that, whether the seed is wheat or barley, the volume to the hectare is substantially the same. The rate of a qa to the musaru can apply to wheat or to barley.
2. The texts are so clear and simple in this matter that when Neugebauer and Sachs in MCT found themselves confronted with the table of Susa, dating from the Old Babylonian period, which indicates a rate of a qa to a musaru and with other texts which indicate with equal certainty a rate of ½ qa to a musaru (except for small fractional adjustments), they should have granted it is a matter of a single and double qa. But Neugebauer finds the existence of a single and a double qa in the Old Babylonian period irreconciliable with his algebraic conception of Mesopotamian mathematics, and hence Neugebauer and Sachs resort to a most amazing procedure. All the texts that calculate the seeding rate by a single qa are grouped together in the Introduction to MCT, without saying a word about the seeding rate. But nobody can doubt that the tables of Susa indicate a rate of a qa to a musaru. Further, together with them Neugebauer and Sachs print the text of two tablets of Yale (NBCT 1913; NBC 8082) from which at a glance one see that 6 2/3 sheqels equal 1/9 musaru, so that the rate is a qa to a musaru, since a qa is composed of 60 sheqels. Having placed these texts at the beginning, without referring to the rate expressed by them, in the last chapter they interpret the texts that reckon the seeding rate by double qa and therefore compute about ½ qa to a musaru.
This second group of texts contradicts Neugebauer and Sachs’ general assumptions also in another way, which is less flagrant but still impressive. They claim that the double cane of 12 cubits, which they call SAR (it can also be called GAR), is the basic unit of length and that other units are calculated in relation to it; whereas in my opinion the double cane is just one of the units used to fit units of length into sexagesimal computation. I have reported that in the practice of measurement one preferred to survey by units of 10 feet or 10 cubits; in the texts in question the unit of surface is an acre with a side of 100 cubits. It is called 100 cubits in the texts, but I call it acre, since it is obviously the amount plowed in a day. Such an acre has a surface of either 2498 square m. (trimmed cubit) or 2563 (natural cubit), and hence is practically identical with the Roman iugerum of 2522 square m. The iugerum is similar in surface to the Egyptian acre, the aroura, a square with a side of 100 Egyptian royal cubits, measuring 2756 square m.; since the iugerum is identical with the Mesopotamian acre, this is one more reason to consider the Roman system of landsurveying to be of Mesopotamian origin; I have noted that the iugerum is calculated sexagesimally as two actus with a side of 120 feet.
Since a musaru has a surface of 144 square cubits and the acre of 10,000, the rate should be 70 qa to an acre (minus 1/225), computing a qa to a musaru. But the figure is rounded up to 72 qa or a PI. This rate is illustrated by the texts quoted by Neugebauer and Sachs at the end of their volume.
One entry of the tablet VAT 7848, makes 768 square cubits equal to 2½ qa, 2 akalu, ½ akalu, 1/10 akalu, plus a smaller fraction which cannot be read; assuming that the missing fraction is 1/20 akalu, the rate is 36 double qa to the acre. Another entry of the same tablet is more incomplete: 1560 square cubits equal 5 qa, 4 akalu, and some other smaller fractions, which Neugebauer and Sachs presume to be 5 grains; this would make for a rate of 33 1/3 qa to the acre. The rate of 36 (double) qa to the acre is also mentioned in a tablet of the Louvre (AO 6848, Nos. 5 and 7). A square with a side of 1 2/3 cubits equals 108 grains, and a square of 12 square cubits equals 466.5 grains. The rate is not exactly the same in the two cases, since in one case it is 38.88 grains to the square cubit and in the second 38.875. The rate of 108 grains to a square with a side of 1 2/3 cubits reveals itself as the one derived directly from the basic rate: 108 grains are 1/100 of 10.800 grains or a qa; hence the rate is a qa to a square with a side of 16 2/3 cubits or 1/6 of 100 cubits (36 qa to the acre).
A wavering between 33 1/3 and 36 indicates that the correct figure was an intermediate one and that one rounded the figure up or down for the sake of convenience. And in fact, if we reckon a qa to a musaru, we should expect a figure of 34 2/3 double qa to an acre since an acre contains 69.555 musaru. If this amount is rounded up to a PI of 36 double qa, we obtain one of the values used. If one starts by considering that a musaru contains 144 square cubits and this figure is rounded up to 150 square feet, by the addition of a leimma, one can calculate mentally that there are 66 2/3 musaru in an acre of 10,000 square cubits; hence the seeding rate of 33 1/3 double qa.
A calculation by acre is found in a Neo-Babylonian tablet from Nippur (CBS 8529), which I have already mentioned because in it the cubit is divided into 34 fingers. The reckonings that follow indicate that the cubit is the barley cubit. A square with a side of 100 cubits equals 5 sata, 3 qa, and 3 1/3 akalu. Here the saton is a unit of 6 double qa; the amount of seed is 33 1/3 qa, as in the preceding text.
It is possible to explain why we find a cubit divided into 24 fingers instead of 30, as is usual in Mesopotamia. In the matter of seeding rates it is necessary to calculate by seed; since there are 72 grains to the square cubit, it is expedient to divide the cubit into 24 fingers. If my hypothesis that one sowed 36 grains to the cubit of sowing line is correct, this division would have been more desirable. On the other side when the rate is calculated as 75 grains to the square cubit, it is more expedient to divide the cubit sexagesimally into 30 fingers of 5 six-fingers.
There are ziqqurats with a surface of an acre, which indicates that the acre was a standard unit of a surface. But as the Tower of Babel indicates, a more common unit of surface was the iku, a square with a side of 120 cubits or 180 barley cubits. According to the Smith Tablet the amount of seed either is 30 royal qa or 180 normal (double) qa. I have pointed out that the rate by normal qa is a diesis larger than the rate by royal qa, since 30 x 3 3/8 = 101.25 and that this was compensated by the circumstance that the normal qa was of the correct variety. But taking the figures at face value and converting them to units calculated by musaru, considering that the iku contains 225 musaru, we find that 108 double qa or 216 qa correspond to 225 musaru, so that the rate is a leimma less than the rate of a qa to a musaru (25/24 216 = 225). The rate is that of 72 grains to the square cubit I have mentioned before; the rate is that of 33 1/3 double qa to the acre I have considered. But the rate of 30 royal qa to 225 musaru corresponds to a lower rate of 67.2 grains to the musaru.
3. Weissbach in his extensive commentaries to the Smith Tablet cannot accept the existence of a royal qa, cube with an edge of 6 fingers of great cubit, in spite of the clear wording of the text, and as a result he is faced with the problem of explaining how the amount of seed is once said to be 30 qa and another time 108 qa. he solved the difficulty by drawing a distinction between an older and a newer rate of seeding. It has become one of the dogmas of the New School that at times fields were sown much more thinly that usual, even though no one has attempted to explain why and how one followed this peculiar technique of husbandry. Walter Schwenzner interprets the texts in which an iku corresponds to 30 royal qa, as implying a rate of seed of 27.14 liters to the hectare (about one sixth of what one would expect).
Mrs Lewy, who, although trained to accept the dogmas of the New School, does not believe that the Mesopotamians lived in a world of strange and misty practices, has strained her ingenuity in order to find an explanation for the theory of thin sowing. She begins by granting that biological facts must have been the same today as in ancient Mesopotamia and that on principle one should expect a rate of seeding similar to the modern one. If the rate of seeding is much less, a part of the land must have been disused: consequently she evolves the hypothesis that in Mesopotamia lines of seed were spaced about 75 cm. instead of about 25 cm., as one would expect according to modern practices. In order to explain why the lines were so spaced. she introduces another equally ingenuous hypothesis: rotation was not practised by letting some fields lie fallow, but by sowing grain only in one of three possible parallel lines. Each year one would have sown grain on a different line, so that two thirds of the land would remain unused.
Mrs Lewy thinks that some passages of the Mishnah (Treatise Kilaim) describe a similar technique of tilling the land, but in reality the quoted texts prescribe the space to be left between fields with different crops, in order to observe scrupulosly the biblical prescription against sowing diverse kinds in the same field (a rule against miscegenation). She seems to be on firmer grounds when she quotes a body of instruction for farmers, preserved in three different tablets (one is WB 170) and interpreted by Benno Landsberger. The latter, however, avers that the text is not too clear to him and that his interpretation, by which the lines of grain would be spaced 75 cm. leads to a figure which is too great. Landsberger’s tentative interpretation of the Sumerian text is taken as final by Mrs Lewy because it agrees with the doctrines of the New School.
The relevant passage reads (line 21, OEC I, Plate 33):
Landsberger interprets ab-sin, seed plow, as referring to the interval between the lines, and understads that 8 lines shall occupy the width of a GAR or 12 cubits. Neither Landsberger nor Mrs. Lewy seem aware that the same phrase occurs four times in a Sumerian tablet published by Hilprecht (BE III No. 92) and cogently explained by Father Deimel in his essay on Sumerian units of surface (Orientalia 4, 1924, 36). The ideogram GAR means a sexagesimal multiple and hence it can refer to the double cane of 12 cubits, but in this particular case it means 60 musaru. The passage signifies: the rate of seeding (absin) is 8 qa for each (taam) GAR or 60 musaru.
The musaru of this formula is the usual one with a side of 12 barley cubits, whereas the qa is a royal qa equal to 33/8 double normal qa. If this amount is converted into royal qa at the rate 33/8:1, the result is 8.9633, or at the rate 31/3:1, the result is 9. If the amount is reduced by a leimma, according to a basic rate of 72 grains to the square cubit, the result is 8.6 royal qa. But the text studied by Landsberger calculates 8 royal qa to 60 musaru; this rate is in a perfect proportion with the rate of 30 royal qa to the iku, since 8:60 = 30:225.
As a last point, one should consider why the unit of 60 musaru should be important. There are 69.555 musaru in an acre, hence the unit of 60 musaru is a sexagesimal unit very close to an acre. Calculating one qa to the musaru, the amount of seed is 5 sata of 12 normal qa to the unit so that the rate is similar in form to that of 5 modii to the iugerum.
4. All the rates mentioned in the texts I have considered, can be summed up in the following table. Double qa are converted into single qa.
In discussing the units of area I have not mentioned whether they were calculated by a natural or by a trimmed cubit. It appears that one could overlook this difference in calculating areas. This would cause a discrepancy 35:36, which is the one just noted. This discrepancy between areas calculated by the trimmed cubit and by the natural cubit may explain the reckoning of a tablet of Uruk (Warka, No. 25). The surface of a house is 1 sheqel and is divided into six- sections of which one is 1/4 sheqel 5 grains, while the others are 25 grains each; the total is 175 grains, with a discrepancy 35:36, since there are 180 grains in a sheqel.
The flexibility of the seeding rate explains why it was necessary to state specifically which one is used. A tablet of the relations between areas and seed is included in the Smith Tablet; boundary stones list not only the dimensions of the fields they enclose, but also the seeding rate by which they are measured.
This survey of seeding rates proves that Sachs’ interpretation of Problem 10 of tablet BM 85196 is unacceptable. According to his interpretation, the trapezoidal side of the through has a surface of 14/36 square cubits and corresponds to 43 3/4 grains of seed. There is no rate of seeding that fits this relation. According to MCT, the rate of seeding would be about 36 grains to the square cubit, since no distinction is made between asingle and a double qa.
5. The fullest rate for an acre is 72 qa (36 double qa). Since the acre has the same surface as the iugerum it is easy to compare with the Roman rate of 5 modii of 16 sextarii, or 80 sextarii. The Roman sextarius is a basic pint which is 10/9 of the reduced pint or qa of correct form. It follows that the Roman rate is 88 qa or 5/4 of the Mesopotamian rate. The explanation of this is that in Mesopotamia one sowed by the line, whereas in Italy one sowed by casting. The nature of the seed does not affect the rate of seeding as much as the method of sowing. The Roman rate of 43.20 liters of wheat to a iugerum corresponds. to 171.28 liters to the hectare; whereas the Mesopotamian rate of sowing by the line is about 136 liters. Buth figures are reasonable by modern standards. But Pliny and Columella report that there were good soils on which one could use 4 modii; in such a case the rate would be the same as in Mesopotamia. This suggests that the tradition of the Mesopotamian rate was well preserved in Rome. On the other side tablets of Nuzi provide the apparently contradictory information that 4 imeru of land were sown with 5 imeru of seed. Here one used the Roman rate; possibly in this case the sowing was by casting.
It is to be noted that the Romans also used a iugerum castrense, a square with a side of 180 feet; the amount of seed was 5 modii castrenses. Since the two iugera are in a relation 8:9, the modius castrensis is a unit of 18 sextarii. This peck is the one used in the Edict on Prices of Diocletian, but it is called sporimos modios, sowing peck, in Greek, suggesting that its main use was to calculate seeding rates. The modius castrensis has the volume of the most typical Mesopotamian saton, that of 10 double qa.
The iugerum castrense has a side of 53.28 m. and it is therefore identical to the Mesopotamian acre if the trimmed cubit is calculated as having 32 fingers, instead of the usual 30. Such a cubit has a length of 532.7 mm. It is the occurrence of this irregular cubit that caused Oppert to conclude that the Mesopotamian cubit measures 533 mm. As I have reported, Problem 10 of tablet BM 85196 makes use of this cubit. The difference in area between the normal acre and the acre so increased, is as 30 squared:32 squared= 900:1024. This difference well corresponds to the difference between Rate I of 72 qa to the acre and Rate III, which would be 63 qa or 63 plus a komma (900:1024 =63.292:72). Hence it seems that, since the calculation of areas by amount of seed was most common, one changed the units of surface to make them fit to the variations of the seeding rate. The iugerum castrense would be the perfect equivalent of the Mesopotamian acre calculated by a cubit of 32 fingers. I have noted that in Egypt too one must have adapted the aroura to the dimension of this acre, since there are Hellenistic measuring rods calculated not for an Egyptian royal cubit of 525 mm. but for a cubit of 533 mm.