   # Stereometric Texts

1. The name of stereometric texts is given to cuneiform tablets that formulate in cubic measures the value of the main units of volume. Scholars have realized that these tablets share common characteristics, but have never defined what this common element is. The common element is the calculation of length by a sexagesimal computation in which the unit of six-finger becomes a basic one, taking the place usually taken by the foot or the cubit. The six-finger unit has as submultiple the tenth-of-finger and as multiple the SAR or double cane of 12 cubits. I have reported that this is one of the two types of sexagesimal computation of lengths compared in the table of Larsa. As a result of this computation the basic unit of volume is the double qa conceived as a cube with an edge of a six-finger; this double qa is called simply qa in these texts.

The stereometric texts constitute the main portion of the cuneiform tablets published by Neugebauer. All the texts I shall mention, with the exception of the Smith Tablet, are included in the three volumes of Mathematische Keilschrifttexte (MKT) or in Mathematical Cuneiform Texts (MCT); the second work has been written by Neugebauer in collaboration with Abraham Sachs.

The realization attained by Lehmann-Haupt in 1888 that there are texts in which the qa is calculated as a cube with an edge of 6 fingers marked the transition from the first to second generation in the study of Mesopotamian metrics. That the qa is calculated as a cube having a volume of 216 cubic fingers, was accepted as an evident and enlightening fact up to the moment the new school declared ex cathedra that it is an error to assume that the ancients conceived of volumes as cubes of units of length. The stereometric texts were considered direct and unequivocal, but those who went proclaiming the new dispensation in metrology, as for instance Weissbach, never bothered to examine them.

Thureau-Dangin in his last comprehensive treatment of Mesopotamian metrics, published in 1909, still stated that the qa is a cube with an edge of a six-finger. But after this date he gave in to the general trend of scholarship. Since the opposition to the old metrology was directed against any aspect of ancient metrics that could remind of the French metric system, Thureau-Dangin compromised by rejecting the notion that volumes were conceived as cubes, but preserved the connection of volume and weight with length. In the rhetoric of Rev. Johns, it was argued that it is preposterous to conceive the qa as a cube because if the mina is the weight of a pot full of water, the pot would be cylindrical and not cubical. Superficially there is some value in this argument, since ancient measuring vessels were cylindrical and not cubical; the old school could be considered remiss for having neglected this knotty point, except for the investigation of Father Grienberger in 1605. An explanation is necessary and I have provided it; however, it is apparent that the argument of the new school is superficial since the French liter is defined as a cube, but liter vessels have always been legally defined and manufactured as cylinders. One should have answered to the new school by a further deepening of the investigations according to the method of the school, but the latter had lost its vitality in an unfriendly atmosphere, so that Thureau-Dangin retreated by rejecting the image of the cube. He did not reject the link between length and volume, but interpreted the stereometric texts not as describing cubes or prisms, but as describing cylinders: but these are cylinders in which the volume is obtained by squaring the diameter and multiplying the result by the height. Any person conversant with mathematics would realize that these cylinders are cylinders only in a verbal sense, but the new school was not concerned with mathematics, so that by this concession Thureau-Dangin satisfied the opposition. It is apparent that in the new climate of metrological studies one is concerned with questions of faith and not of objective science: the position of Thureau-Dangin may be compared to that of a theologian who would declare that he believes in the actual fire of Hell, but as a fire sui generis that does not involve heat or combustion.

In order to maintain his position Thureau-Dangin was forced to take liberties with the arithmetical interpretation of the texts, and at times he found himself enmeshed in such imbroglios that he declared his own results to be impossible. Neugebauer has built his reputation as an expert in the interpretation of cuneiform rekronings by exposing the distortions of Thureau-Dangin. But, even though Neugebauer has treated Thureau-Dangin as a schoolboy who had to learn from him how to read sexagesimal computations, the latter was higly skillful and endowed with real insight into metrological problems. He had the weakness of his scholarly virtues: he had a mind open to facts and open to the criticism of his colleagues. He steadily tried to include more material and more data in his metrological research; in his response to new facts, and to new opinion, he was always ready to revise earlier positions, but in the process he became so open to argument that he fluttered like a reed in the wind, changing his opinion from article to article.

The case of Thureau-Dangin is similar to that of Segrè, who did not have the persistency to maintain the position of the old school when confronted with the opposition of the academic world. In his essay of 1909, which is his most comprehensive study of metrology, Thureau-Dangin totally subscribes to the principle and methods of the old school, as the title of the essay itself indicates: but in the other articles written earlier and later he tried impossibe compromises with the dogmas of the new school. Thureau-Dangin had a knowledge of the Sumerian language that was equalled only by Father Scheil, but further he had an unusual breadth of information and a variety of skills, and could have been one of the major historians of his time. But a man who wants to open new ways for truth must have the power to fight the forces of superstition, as Oppert had proved. Thureau-Dangins father as a liberal Catholic in the Third Republic had met the opposition of clericals and anticlericals, but obtained recognition as the authoritative historian of the July Monarchy. In spite of his membership in the Academie Française, he was a lonely man, not only intellectually, but also in his personal life. Science suffered because his son was not willing to pay the same price.

Thureau-Dangin sacrificed the logic of mathematics to what appeared to him accepted opinion. He operated with the opposite bias of Neugebauer who has proceeded with an irresistible drive, because early in his career he formulated a consistent and simplified scheme and then rejected all data that could require revisions, reformulations, qualifications, or doubts. In my opinion the confused but rich interpretations of Thureau-Dangin in the long run contribute more to the progress of science than those of Neugebauer, which are formally clear and mathematically correct, but ignore the several forms of the qa and related units, ignore fractional adjustments, ignore texts and other metrological evidence that may alter and complicate the picture.

2. In his criticism of Thureau-Dangin, Neugebauer made an important contribution to science: in the first and second volume of MKT he wondered whether the stereometric texts dealt wich cylinders or with prisms, but in the third volume positively concluded that it is a matter of cubes or quadrangular prisms. Thureau-Dangin accepted this conclusion, which actually was the one he had followed up to the age of forty, but probably was too advanced in his life to revise his interpretations.

In the third volume of MKT, Neugebauer appeared as agreeing with Lehmann-Haupts conclusion that the qa is a cube with an edge of 6 fingers and has a volume of 216 cubic fingers, but balked at identifying himself with the old school. Like Thureau-Dangin he thought that at least the image of the cube, so openly identified in the modern mind with the French metric system, is a ghost that must be exorcised. At the moment he seemed tobe acepting beacep accepting the old methodology, he stressed the notion that the fundamental unit of volume is the one he calls volume-SAR, which is not a cube but a slice of a cube.

In MCT, which he published together with Sachs in 1945, Neugebauer eases out of his predicament by never mentioning the cube with an edge of a six-finger and referring all figures to the volume-SAR. Volumes are not conceived as cubes, but are all derived from the units of surface given the height of a cubit. Since one of the units of surface is the musaru, square with an edge of a SAR or double cane of 12 cubits, the volume-SAR of 144 cubic cubits (a surface-SAR with a height of a cubit) acquires paramount importance in Neugebauers construction. That there were units of volume so conceived had been already stated by Oppert in 1889, and Oppert is the only metrologist of the old school ever mentioned by Neugebauer. But that there are units so conceived is not particularly surprising since the Mesopotamian system of volumes and weights essentially starts from the kurru, the cubic cubit, and hence one can expect to find units conceived as slices of the kurru or as a series of kurru put next to each other on a surface. What is peculiar to Neugebauer is that he asserts as a general principle that volumes were conceived by giving a height to units of surface and not as cubes.

The procedure by which one considers as volumes the surface of a musaru or an iku (square with side of 120 cubits or 10 canes) given the height of a cubit, has practical value in the calculations of the amount of earth to be moved in the digging of canals, which usually have a depth such as a cubit or two cubits. And in fact these units units occur in the many texts dealing with the excavation of canals. The notion that the unit of volume iku is connected with the digging of canals is supported by the fact that the Sumerian equivalent eki means” water-earch, embankment, canal.”

My disagreement with Neugebuaer and Sachs concerns the emphasis to the given to these facts. They reject the notion of units conceived as cubes and state “The units of volume (*Yes originate from the units of area by multiplication with a height of a cubit” (p. 5), and, after this, reference is never made to the existence of units conceived as cubes or near-cubes. Nowhere it is mentioned that the qa has a volume of 216 cubic fingers, as it was granted in MKT. Texts that would indicate the existence of a qa as a cube of 216 cubic fingers, are interpreted as fractions of the volume-SAR or musaru. In my opinion they achieved this by complicating unnecessarily the meaning of some texts; I will have the occasion to deal with these texts in detail, but here I can stress one general issue. Cuneiform computations are such that one never knows which degree of sexagesimal multiples or submultiples is involved: 3/60; 3; 3.60; 3.602; 3.603; etc. are represented by the same sign. It is up to the interpreter to decide from the context which sexagesimal degree is intended; my contention is that Neugebauer and Sachs introduce unnecessarily extra sexagesimal degrees in the multiplications and extra sexagesimal degrees in the divisions. They themselves grant that the volume-SAR is the sexagesimal multiple of the great kurru, which is a cube with an edge of two feet (a cube that contains 300 double qa with a six-finger edge, less a discrepancy komma or 81/80), so that by lowering the figure by one sexagesimal degree we are again in the field of cubes. The next sexagesimal submultiple is the saton of 10 normal single qa or 5 double qa, which is a cube with an edge of 10 fingers plus a discrepancy komma. Hence, when Neugebauer and Sachs calculate the double qaas 1/5.60.60 of volume SAR, they merely say that there are 5 double qa in a saton.

From the scientific point of view whether the basic units were calculated as cubes or as slices of cubes does not affect in any way the issue whether units of volume and units of length were related. Hence one could wonder why in MCT Neugebauer and Sachs should engage in a tour de force in order to refer all calculations of volume to the volume SAR and ignore the existence of units calculated as cubes. This quibbling is the equivalent of Thureau-Dangins theory that the units of volume were not calculated as cubes, but as cylinders, even though these are cylinders of an unusual nature such that the volume is calculated like that of prims, by squaring the width of the base and multiplying it by the height. Whether one accepts Thureau-Dangins theory or that of Neugebauer and Sachs, there remains the fact that the units of volume are a function of the unit of length, but as long as the dreaded image of the cube is not openly suggested, the academic world may think that the dogmas of the new school are not flouted. One may wonder whether in the history of theology there could be found worse examples of cavilling verbal distinctions.

In his “Rejoinder” to Mrs. Lewy, Sachs did not clarify himself on the main issue of his agreement or disagreement with the old or the new school; on the second point raised by her agred with her that “It is perfectly true that the relation used in Old-Babylonian mathematical texts can be expressed as a qa is the cube with an edge of 6 fingers,” but remained ambiguous on the question whether this is a correct or an adjusted value. Considering that the last is the element of the controversy that is most pregnant of consequences, one is left wondering why, if Lehmann-Haupts notion that the qa is a cube with an edge of 6 fingers is “perfectly true,” in MCT Sachs and Neugebauer should go to great pains to prove that instead the qa must be calculated as 1/5.60.60 volume-SAR. Utque leves Proteus se tenuabit in undas.

I shall have occasion to interpret Problems No. 22, 24, and 30 of the Great Mathematical Tablet, which according to Neugebuaer and Sachs are the main evidence that units of volume were conceived as fractions of the volume-SAR, and to show that they can be more easily interpreted without any reference to this unit: but after the above-mentioned declaration of Sachs I marvel whether it is necessary to disprove their interpretation

4. My contention is that the people of Mesopotamia conceived of units of volumes as cubes, whether they were as small as a shekel or as large as a cubic iku. When units had a size such that they could not be constructed as cubes with an edge measured in round numbers, they conceived of them as near-cubes, that is, as cubes increased or decreased in height.

The main illustration of this point is provided by a tablet of Yale (YBC 4669) in which the qa is described as a cube with an edge of a six-finger. The multiples and the submultiples of the qa are described as near-cubes. There is no disagreement about the litteral meaning of this tablet, but one has not drawn from it the inescapable conclusion that one aimed at defining all units of volume as cubes. I have explained that the mathematical concept of near-cube is introduced in ancient mathematics when the cubic root of a number cannot be expressed as an integer; in such a case the cube is decomposed into a square of a number multiplied by another number as close as possible to the first number. The fact that in this tablet volumes are expressed either by cubes or by near-cubes is enough by itself to destroy all refutations of the old school.

A massiqtu of 60 qa has a base of 24 x 24 fingers and a height of 22½ (that is, 24 minus a diesis). If the unit had been a perfect cube, the volume would have been 64 qa or a PI. A sutu simid of 30 qa, that is, an artaba, has a base of 23 x 23 fingers and a height of 20. A. sutu of 10 qa has a base of 12 x 12 and a height of 15; if the unit had been a perfect cube of 12 fingers the sutu would have been a saton of 8 qa, as it is occasionally mentioned in the texts. The qa is a cube with an edge of 6 fingers; this is the only unit calculated as a perfect cube, and this indicates that one fixed the volume of the qa in such a way that this basic unit at least would be a perfect cube. A half qa has a base of 4½ x 4½ fingers and a height of 5 1/3: a third of qa has a base if 4 x 4 fingers and a height of 4½; a sixth of qa has a base of 3 x 3 fingers and a height of 4. Finally a sixtieth of qa or a sheqel has a base of 1 x 1 finger and a height of 3½.

Only the last figure is approximate, since a qa of 216 cubic fingers should be divided into 60 sheqels of 3.6 cubic fingers. None of the commentators has noticed this discrepancy; a discrepancy 36/35 is the difference between the qa Stereometric brutto of 518.4 c.c. and the correct qa brutto of 504. The tablet indicates that when one came to the sheqel, the unit used to weigh the media of exchange, one reckoned by referring to the basic sheqel of 9 grams: here the unit is a double sheqel of 18 grams reduced of a diesis; it is equal to two sheqels of 8.4 grams. By indirection one gathers that the other units are units brutto calculated by the natural cubit. Hence all units are Stereometric brutto, increased of a diesis in relation to the correct units. The first unit of the tablet, the massiqtu is equal to 64 correct (double) qa.

One can also gather another datum not noticed by Thureau-Dangin and Neugebauer: the cubit in this case is divided into 24 fingers. The unit of 60 qa called massiqtu has a base of 24 x 24 fingers, which prove to be a natural barley cubit. If one calculated a perfect cube with an edge of a trimmed barley cubit, one would obtain the same volume. Hence, it appears that in this tablet one combined a calculation by the cube of the six-finger, as typical of stereometric texts, with the more conventional calculation of the basic unit as the cube of the cubit.

Another illustration of a unit of measurement described as a near-cube is contained in problem No. 24 of the Great Mathematical Tablet of the British Museum. This problem sets the question: “There is a sutu full of grain. How much must I descend to remove a qa?” Since the width is 2|0 and the height is 2|30, the answer is 15. The calculation is made in tenth-of-finger, which are called fingers in the text, as it happens in some other texts; the existence of this terminology has been recognized by Thureau-Dangin, but he has failed to see that it applies in this case. This sutu is a near-cube measuring 120 x 120 x 150 tenth-of-finger (12 x 12 x 15 fingers); in order to take out a qa (1/10 of sutu) one must skim the measure to the depth of 15 tenth-of-finger (1½ finger). Thureau-Dangin understands the text as describing a cylinder with a diameter of 2 fingers and a height of 150 fingers, so that he himself exclaims ce sont là des dimensions absolument invraisembables. This text illustrates how one arrived practically at the notion of near-cube. The ancients had the habit of heaping measures to adjust their volume when necessary; at times the heaping was obtained by providing the measure with a collar (*xeijc” *) and filling up this collar level, and this may have given the idea of increasing the height of a cube by a fraction. On the other hand many units of measurement were calculated according to the amount of food or drink consumed in a given period of time, and one lowered the level so much each day by lifting one ration; this would give the idea of cubes decreased in height of a fraction

5. Neugebuaer and Sachs relate length to volume through the volume-SAR, but they are not willing to relate the volume to weight. They dissociate the volume of the qa from the volume of the mina; this dissociation is emphasized by considering the double qa as the only qa and considering the single mina as the only mina. The reason for this dissociation is the desire to simplify problems by ignoring the issue of fractional adjustments. As I have said there is only one published sample of qa, the vase of Entemena, whereas there are hundreds of samples of mina; these samples indicate that the mina can be simple or double and that it exists in several forms. Further, weights and other evidence indicate that the mina can be either of the normal or of the Euboic variety. There are as many varieties and forms of qa as there are varieties and forms of mina. Neugebauer and Sachs dispose of the proplem of the mina by saying that the mina “corresponds to 500 grams (approximately 1 pound)” ; by this statement, which treats as identical a value of 500 and 453 grams, the entire problem of the special forms of the mina is obliterated, as the normal mina of 486 grams is confused with the Euboic mina of 405. Having glossed over the problem of the mina, they imply that the only qa is the double qa, but they do not commit themselves as to its volume, except to say that it is 5.60.60 of the volume SAR, which in substance means 216 cubic fingers. But since they do not commit themselves as to the length of the cubit, by saying that it has a length of about 500 mm., the volume of the qa remains uncertain. By so doing they avoid the difficulties faced by Thureau-Dangin, since by ignoring fractional adjustments they can offer an interpretation of the texts which is consistent in a superficial way.

In order to grasp the implications of this procedure it is necessary to follow a lively controversy that developed in scholarly journals in 1944. I have already touched upon this controversy from the angle of the exchange between Mrs. Lewy and Sachs; now I shall consider the disagreement between Segrè and Sachs that involves the same general issues, but goes to specific points of textual interpretation. In 1944 Neugebaer and Sachs had completed the manuscript of MCT: in this work they take the position there is only kind of qa: in the interpretation of the texts they imply that there were variations in the volume of the qa, but they do not account mathematically for them, limiting themselves to imply that the qa had a volume of about a liter. Nothing is said about the unit of weight and its relation with volume and length. In general Neugebauer and Sachs seem to agree with Weissbachs contention that Mesopotamian units were fixed only with aproximation; Weissbach had formulated his position in order to reject Lehmann-Haupts theory of fractional adjustments and they neglect fractional adjustments in the interpretation of the texts.

When the manuscript of MCT was about to be entrusted to a publisher, Segrè, who upon coming to the United States had renounced his allegiance to the new school which he had reluctantly accepted under the influence of Gaetano De Sanctis, published an article examining Mesopotamian metrics strictly according to the method of the old school (“Babylonian, Assyrian and Persian Measures”, JAOS, 64 (1944), 73-81). Segrè brought into light the older studies on the special forms of the mina and the qa and made several steps towards the right solution of the issues that had been left open by these studies. He remarked that the qa calculated as a cube with an edge of 6 fingers and contained 125 times in a cubic cubit, is a double unit, corresponding to the weight of two single minai, and that the double qa varies between 975 and 1050 c.c. (these values are very close to the mine of 972 and 1036.8 c.c.), following the variation of the length of the cubit from 496 to 508 mm. (I distinguish a trimmed cubit of 499.4 and a natural cubit of 506.25)

Segrè further concluded that there is another qa (the one I call Euboic) and that this qa is used in the computation of Problems 6 and 10 of tablet BM 85196. He did not succeed in finding the right value of this qa, because he did not examine directly the text and limited himself to follow the interpretations presentated by Thureau-Dangin and Neugebauer; but he sensed the right solution by indicating that the use of a cubit different from the usual one is involved (it is the great cubit) Since Segrès and a cubit of 32 barley fingers article was cutting the ground from under the forthcoming volume written by Neugebauer and Sachs, the latter undertook the task of replying to Segrè in an article by the title “Some Metrological Problems in Old-Babylonian Mathematical Texts” (BASOR, No. 96 (1944), 29-39).

In MCT Neugebauer for the first times specifically committed himself on questions of metrology, even though he tried to reduce to a minimum these commitments. This laid bare the weaknesses of his metrology, which are the result of his general interpretation of Mesopotamian mathematics. Some of the weaknesses of his metrology were called to attention by the polemic note of Mrs. Lewy, which I have discussed; but the article of Segrè, written without knowlege of MCT, makes these weaknesses quite evident by contrast. Neugebauer began with the assuption that Mesopotamian mathematics, which he calls Babylonian since they would be a creation of the Semitic mind and not of the Sumerians, have an “algebraic” character. Whereas according to my interpretation Mesopotamian mathematics originate out of concrete problems such as metrics, for Neugebauer Babylonian metrics are nothing but an aspect of a mathematical structure conceived algebraically. For this reason he never connects metrology with any concrete archeological material, such as sample weights or the dimensions of buildings. Because he wants to reduce units of measure to algebraic entities, he must deny the existence of several varieties of the same unit, because this would shift the problem from algebra to arithmetic. In my opinion the system of measures was conceived before the development of the sexagesimal system, but the structure of measures gave the idea of a sexagesimal system. Once a consistant system of sexagesimal computation was developed, the Mesopotamians were confronted with the problem of fitting the units of measure into a strictly sexagesimal system. The Tablet of Larsa presents two different methods to fit the units of length into sexagesimal computation; but the very division of the cubit into 30 fingers is an example of this process, so that Neugebauer is compelled to gloss over those cases in which there is preserved the original division of the cubit into 24 fingers. Mrs Lewy with a brilliant insight observed that the interpretation of the qa (actually a double qa) as a cube with an edge of 6 fingers was developed at a second moment and gives a value (which I call Stereometric form) that does not correspond exactly to the original value (which I call correct form) Sachs was forced to dismiss this valuable insight, which I have exploited to the full in my treatment of the special forms of the mina and the qa, and to deny in general the calculation of the qa as a cube with an edge of 6 fingers, which had been taken as starting point in Segrès article. The “Rejoinder” of Sachs on the question of the qa conceived as a cube with an edge of 6 fingers, is weak and uncertain: in substance it says that such an interpretation is in agreement with the mathematics of the texts, but that there would have not been any practical advantage, in calculating the qa as a cube This is a peculiar argument, since through the history of metrics the advantage of conceiving units of volume as cubes has been apparent; according to Sachs argument the French metric system would have no pratical advantage. But we are dealing wich a concrete historical problem, and the issue is whether the Mesopotamias did or did not conceive of the qa as a cube. I have shown that they did, because in fitting the units of length into the sexagesimal computation they had conceived of the six-finger as a basic unit. Concerning their interest in conceiving units of volume as cubes, I have shown that they were so much interested in it that when, for reasons of arithmetic, they could not reduce a unit of volume to a cube they reduced it to a near-cube.

One must ask the question why, whereas Neugebauer in MKT had come to grant that the qa was calculated as cube with an edge of 6 fingers, later he should take together with Sachs the position that the only link between length and volume takes place through the volume-SAR. Sachs in answer to Segrè defends “the value for metrological discussion of the only relation which is solidly attested in the Old-Babylonian mathematical texts between measures of capacity and measures of volume.” In his “Rejoinder” to Mrs. Lewy, he affirms: “In proceeding from a set of linear measures, it is necessary to calculate the volume in terms of the SAR system, the fundamental unit of which is not a cube, but a parallelepiped.” This last statement makes clear that Neugebauer and Sachs are concerned with conforming with the ideas of the new school: hence, they reduce the link between length and volume (leaving completely in the dark the link between volume and weight) to a single link which would take place in such uncommon a unit as the volume-SAR and in a unit which is conceived as a parallelepiped which is wide and low and resembles as little as possible a cube. But, besides this concern with conforming as much as possible to the dogmas of the new school, there is a reason that has to do with the inner structure of Neugebauers interpretation of Mesopotamian mathematics. If one grants that the units of volume can be fitted into sexagesimal computation in two different ways, then one cannot conceive of the metric units as algebraic entities that were applied to concrete problems as a second step. In MKT Neugebauer came to grant, in conflict with the new school, that the qa was conceived as cube with an edge of six fingers, but later, having realized that there are units based on a different cube, units that he calls the SAR system, he had to deny that the first interpretation was correct. Sachs, in his behalf, attacked Segrè and Mrs. Lewy for saying what Neugebauer had said earlier. Concerning the so-called SAR system, I may observe that it is improper to take as starting point a unit called SAR since this term means a sexagesimal multiple, but in any case the so-called volume-SAR is nothing but the sexagesimal multiple of the greater kurru, a cube with an edge of two Mesopotamian feet; this unit has as sexagesimal submultiple the saton of 10 qa. Meissner has published a tablet from the Old-Babylonian period (VAT 2596) which is fundamental for the study of units of volume and has included in his publication a commentary written by Lehmann-Haupt (Beiträge zum altbabylonischen Privatrecht, Leipzig 1893, 98-101). This tablet presents a table of symbols for the qa and its submultiples and multiples up to the enormous amount of 216,000 kurru; the table has to be compared with the Table of Larsa. In the Table of Larsa units of length are organized sexagesimally according to two different methods: one method begins with the cubit which has as submultiple the half-finger and as multiple the subban of 60 cubits, and so on; the other method begins with the six-finger which has as submultiple the tenth-of-finger and as multiples the double cane of 12 cubits (GAR or SAR) and the US of 720 cubits, and so on. In the tablet VAT 2596, there are two methods of fitting the units of volume into sexagesimal computation: one begins with the qa and has as submultiple 1/60 of qa or a sheqel; the other begins with 10 qa or a saton and has as submultiple 1/6 of qa. This second system in which the multiples are the greater kurru and the units Neugebauer and Sachs call volume-SAR, is the system they call SAR system. The preconceived notion of the nature of Mesopotamian mathematics in Neugebauer is such that he cannot accept the existence of two alternative systems of arranging sexagesimally the units of measure: hence, in MCT, having accepted the existence of a system based on the saton and the greater kurru, he had to deny the existence of a system based on the basic unit.

For the same reason Neugebauer has to deny the existence of a single and a double qa (and other single or double units), the existence of a normal qa and a smaller qa which I call Euboic, the existence of the special forms of qa and related units, at least for the formative period of Mesopotamian mathematics, which he identifies with the Old-Babylonian period. He is willing to admit the existence of varieties of units for a later period, since in such a case one can say that a structure originally conceived algebraically was applied to the solution of concrete problems. Because of his presuppositions Neugebauer had always been uneasy about the group of documents which he calls British Museum texts, of which the most important are the Great Mathematical Tablet (BM 85194) and the similar tablet BM 85196, since they rebeal how Mesopotamian mathematics are closely related to problems of measurement in the construction of buildings, canals, embankments, and in the construction of measuring vessels. In MKT II, p. 53, he argues that these documents belong to the Kassite period or are close to it; but this position has become difficult to defend. In MCT Thureau-Dangin dated them, as they should be, in the Old-Babylonian period and properly put them at the head of his edition of mathematical texts as fundamental ones. Neugebauers main argument for dating later these documents is that Problems 6 and 10 of tablet BM 85196 mention a saton of 12 qa instead of the usual 10 and according to him this saton of 12 qa would be first documented in the Kassite period, that is, the middle of the second millennium. The argument in substance is circular: one should date later a group of tablets that according to style belong to the Old-Babylonian period, because one of them mention a unit of which there is no evidence for this period, but the tablet in question most openly provide such evidence. Implied in this there is the undemonstrated assumption that in the Old-Babylonian period units existed in only one type; it is up to Neugebauer to prove that the metrology of the Old-Babylonian period is different from that of the successive period, since all the evidence submitted up to now indicates an amazing persistence of the same system of metrology throughout the history of literate Mesopotamian culture. On the question of the sutu of 12 qa Thureau-Dangin properly answered that it is attestée dès les temps de la première dynastie babylonienne.

Segrè in his article not only revived all the arguments of the old school, but assigned paramount significance to Problems 6 and 10 of tablet BM 85196. He contributed to the clarification of their metrology by understanding that they mention a saton of 12 qa instead of 10, because they reckon by Euboic qa. Thereby Segrè not only stressed the existence of the saton of 12 qa in the Old-Babylonian period, but also the existence of the Euboic qa, whereas a great effort is made in MCT to obliterate the existence of this unit. In answering to Segrès interpretation, Sachs states that it “cannot be taken seriously because it assumes metrological relations for which there is not the slightest evidence from the texts of the Old-Babylonian period.” This is a forceful and sweeping statement which has no factual. basis, until Sachs himself proves that the evidence submitted up to now has not the slightest value. Sachs realized that he must prove that Problems 6 and 10, at least, do not constitute evidence, since it is not longer possible to date tablet BM 85196 later than the Old-Babylonian period.

Concerning Problem 6, Sachs states: “Although I cannot give a satisfactory explanation of No. 6 in BM 85196, I am convinced that Thureau-Dangins interpretation points in the wrong direction.” Nothing is added to justify such an assurance, in rejecting the evidence of a text that mentions three times a saton of 12 qa.

Concerning Problem 10, Sachs proposes a new reading of lines 20 and 21 that would change most of the figures in them; probably it is the most drastic emendation of the edition of a mathematical tablet ever suggested.

The purpose of the emendation is to eliminate two references to a saton of 12 qa; in order to achieve this he reads the sign for “saton” as meaning “half,” since this sign has also this meaning, and changes the number 12 once to 13 and once to 14. He questions the apograph prepared by Sidney Smith on the basis of the photograph printed with it, but the photograph is printed in such a way that it cannot be read at the points in question. I must presume that Thureau-Dangin, who published the photograph, saw the original of it and could not read any glaring discrepancy between it and the apograph; he was greatly perplexed by this text and would have suggested emendations if he had found them possible. Neugebauer edited the text with the same reading in MKT.

It would be desirable to examine the original of the tablet, but the main issue can be settled on substantial grounds: Sachs interpretation of the text must be rejected for the following reasons:

a) Problem 10 is parallel in substance and wording to Problem 6 to which the interpretation offered by Sachs cannot be applied.

b) The new reading has the purpose of proving that Problem 10 does not calculate volumes in cubic cubits and fractions thereof, but in units based on the seeding rate, such as the sheqel-area and the grain-area. However, these units are never used in all other problems of the tablet, which clearly reckons by cubic cubits and fractions thereof.

c) According to Sachs interpretation the units of volume used in this text are obtained by taking the area seeded by a sheqel or grain of seed-grain and giving to it the height of a cubit. I shall discuss these units of area in the section “Seeding Rates,” and there it will appear that if these units have the surface assigned to them by Sachs, they imply the use of a qa far different from the qa of about a liter Sachs claims is the only one used in Old-Babylonian documents. The entire effort of Sachs in rereading and reinterpreting this text aims at proving that there was only one type of qa, but his owm interpretation, if valid, proves the contrary.

In his summation Sachs presents arguments that are valid for his psychology, but have no objective force. His cumbersome interpretation of Problem 10 should be preferred because it eliminates:

“the use of this problem for the determination of the qa,”

“one of the two mathematical problems offered in evidence for an alleged Old-Babylonian sutu of 12 qa.”

But in reality he would have eliminated only one document, and the documents about the existence of a qa equal to 5/6 of normal qa are much more than two

.The refutation of Segrè and of all the evidence collected by the old school and by Thureau-Dangin concerning the single and the double qa, the qa I call Euboic, the saton of 12 qa, the special forms of the qa, the calculation of the qa as a cube, is such a contrived and improbable argument that it must not have convinced the writer himself. For, in the Introduction MCT (p. 5) one reads this sentence: “The volume of barley, oil, etc. was measured by special units.” This brief sentence that may pass unnoticed to those who are not conversant with metrological writings, destroys the entire argument of the book as far as metrics are concerned: that there was only one type of qa and saton, and that there were not those varieties determined by the specific gravity of barley, wheat and oil, as claimed by other scholars since Lehmann-Haupts essay in 1888. One should not engage in guessing games, but on the other side one must presume to be living in a rational universe; hence one is bound to wonder whether this all-embracing declaration is an escape clause added after the appearance of Segrès article. Neugebauer should explain how this statement can be reconciled with the “algebraic” nature of Old-Babylonian metrics.

It is obvious that if there are units adjusted according to the specific gravity of wheat and barley, there will be units related as 5:6. Hence, the main argument used in the refutation of Segrè collapses. Unfortunately there were Greek scholars who were incensed by Segrès espousal of the old school, but did not know how to refute him, being totally ignorant of metrology; they took at face value Sachs loud but totally unsupported declaration that Segrès argument cannot be taken seriously. When one reads closely one sees that Sachs limits his argument to the Old-Babylonian period, and Segrè was not particularly concerned with this period buth with the general structure of Babylonian, Assyrian and Persian metrology. Even according to Neugebauer the saton of 12 qa is documented for the immediately following period, so that his reservations would apply at most to a period of two to three centuries. But the remark made by Sachs was used to preclude to Segrè any academic connection, with the result of demoralizing him had discouraging him from arguing further on an issue on which he stood for the correct scientific approach.

In answering to Mrs. Lewy and to Segrè in behalf of Neugebauer and himself, Sachs has shifted ground with such pliancy that one risks losing sight of the problems that need clarification. Furthermore, as I have reported, in the Introduction to MCT there is a sentence that puts the metrological meaning of the entire work in question. For this reason I shall stress three paramount issues on which Neugebauer and Sachs interpretation of the texts cannot be accepted and on which they did not retract themselves, at least outspokenly. In my opinion, the stereometric texts indicate that:

a) there is a qa calculated as 216 cubic fingers by the great cubit, that is, the royal qa.

b) the normal qa can be calculated by the great cubit and in such a case is defined as 60 cubic fingers.

c) there is a qa, the Euboic one in my terminology, which is 5/6 of normal qa and, hence, calculated as 180 cubic fingers by the barley cubit and 50 cubic fingers by the great cubit.

6. The existence of the royal qa is documented most clearly by the Smith Tablet, a document Neugebauer does not discuss. I have already dealt with the metrology of this text, but I shall cover the ground again in order to clarify specifically the issue of the royal qa. The Tablet gives the area of the basis of the Tower of Babel first according to the barley cubit and then according to the great cubit. In the first case the area is 225 musaru (15 x 15 double canes) or 2 1/4 iku and corresponds to 108 qa of seed; in the second case the area is one iku (10 x 10 double canes) and corresponds to 30 qa of seed. As the Tablet states, in the second case both the area and the volume of seed are calculated “by the great cubit.” Since the great cubit is 1½ barley cubit, the royal qa should be (11/2)3 =3 3/8 normal qa; instead we find a relation 1:3 3/5 between the two amounts of seed, with a discrepancy diesis (3 3/8 . 16/15 = 3 3/5). This indicates that one calculated the normal qa in its correct form and the royal qa in a form which is 3 3/8 the normal qa Stereometric brutto. This means that the royal qa was calculated as a cube with an edge of 6 fingers of natural great cubit, that is, as 216 cubic fingers; all lengths of the Smith Tablet are of the natural version, as shown by the excavation of the Tower of Babel. But when one converted the royal qa into the normal qa, one did not compare 216 cubic fingers by the great cubit with 216 cubic fingers by the barley cubit, since this would have been awkward; one compared 216 cubic fingers of great cubit with 60 cubic fingers by the same cubit. But a normal qa of 60 cubic fingers by the great cubit proves to be of the correct form (double of qa of 486 c.c.) One compared a unit of the form Stereometric brutto with a unit of the correct form; the two forms differ of a diesis. In practice this procedure made the reckoning of the Smith Tablet extremely simple: when the unit measures 216 cubic fingers the total volume is 30 qa; when the unit measures 60 cubic fingers the total volume is 108 qa.

The foremost illustration of the normal qa calculated as 60 cubic fingers of great cubit is provided by Problem 22 of the Great Mathematical Tablet. I read the text as follows:

A massiqtu. The width (is) 4. (It measures)

1 (qa) of grain. What is the depth, I repeat,
the depth? Square the width, obtain 16. Divide
(60) by 16, obtain 3|45. The depth (is) 3|45.
(Such is) the procedure.

Thureau-Dangin in his first interpretation assumed, as I do, that the text deals with the volume of the qa measured in fingers; but following his general approach assumed that the vessel was a cylinder, arriving at a volume of 45 cubic fingers, a figure that he considered unlikely. Later (TMB No. 66) he did not commit himself to any specific interpretation . Neugebauer at first (MKT I, 181) agreed that the text deals with the volume of the qa, but rightly observed that it is not a cylinder but a prism; later (MKT III, 54) he understood that a unit of 60 qa was measured in units of six-finger. He would read the text as follows:

A massiqtu. The width is 4 (six-finger).
(It measures) 1 (massiqtu) of grain.

The only argument for this interpretation is that the term massiqtu used in a technical sense refers to a cubic cubit unit of 60 double qa. But the term massiqtu, from the root of sheqel, may also be used in the general sense of “measuring vessel,” since practically all names of measuring vessels can be used in a general sense; for instance, in the mentioned tablet YBC 4669, which is Neugebauers main authority for assuming that massiqtu means a unit of 60 qa, the term akalu, which strictly speaking refers to a tenth of qa, is used to describe vessels of several different sizes. Against Neugebauers interpretation I may remark that in all other texts the phrase “1 of grain” means “1 qa of grain,” as it was pointed out by Thureau-Dangin, with whom Neugebauer originally agreed. Even Neugebauer agrees that the calculation is in qa, since the statement “Divide (60) by 16, obtain 3|45,” can only mean in his terms “Divide (60 qa) by 16 (qa of surface), obtain 3|45 (six-fingers).” His interpretation introduces too many entites not mentioned in the text: the qa, the qa of surface, the length of a six-finger. One must also note that he presented this interpretation in a period in which he believed that the qa was conceived as cube with an edge of a six-finger. Later he denied that the qa was so conceived; at the end of this chapter I shall present a case in which he dismisses the calculation by qa-surface as a proper interpretation of a text.

My interpretation assumes the more usual calculation by fingers and presumes only the fact that one knew by heart that a qa is equal to 60 cubic fingers of great cubit. The text describes a near-cube with a basis of 4 x 4 fingers and a height of 3 3/4. If it were a perfect cube the qa would be 64 cubic fingers with an excess of a diesis; in such a case instead of a double correct qa one would obtain a double qa Stereometric brutto.

7. Neugebauer and Sachs have shown that a brick measuring 15 x 10 x 5 fingers was considered a typical brick, but they have not explained why such a brick should have been considered typical. The explanation is that the brick has the volume of a royal qa: 750 cubic fingers by the barley cubit equal 216 cubic fingers by the great cubit. However, the brick is calculated with an excess of a leimma over the volume of the qa, for the purpose of obtaining a brick measured by be convenient figures 15 x 10 x 5 barley fingers. Deducting a leimma the volume of a brick becomes 720 cubic fingers. Since 720 = 216 x 3 1/3, the relation between royal qa and normal qa is calculated as 1:3 1/3, a relation frequently used instead of the less convenient relation 1:3 3/8, with a resulting discrepancy komma. The dimensions of the brick are such that 6 occupy the area of a square cubit and 36 have the volume of a cubic cubit. Below it will appear that these bricks were counted by the dozen, and that a dozen of bricks is as much as a man can carry. Calculating by the great cubit each brick is a royal qa with an excess of a leimma, so that 120 bricks, or 10 dozens, make exactly a cube of great cubit

There are two texts that indicate that one was concerned with the problem of computing easily the weight of bricks for the purpose of transportation. Neugebauer and Sachs have not realized that these texts deal with the issue of specific gravity. The specific gravity of brick is not calculated directly in relation to water, but in relation to barley. The reason for this procedure is that in Mesopotamia one usually calculated by kurru, cubic cubit, which is the volume of barley weighing an imeru, cubic foot of water. The imeru is the weight a man can carry for a short distance and the normal load of an ass. The key to the reckoning is that a brick of a volume of a royal qa has a volume of 3 1/3 normal qa (leaving fractional adjustments out of consideration), and the same figure expresses in royal qa the volume of barley weighing as much as a brick. Since barley has an assumed specific gravity 66..., this reckoning implies that the specific gravity of brick is 2 1/9; but taking the discrepancy leimma into account the specific gravity is 2.2. If the relation between royal qa and normal qa were to be calculated by the more precise relation 1: 3 3/8, instead of 1:3 1/3, the specific gravity would be 2.25. This is the correct specific gravity of brick.

The first of the two texts I shall consider mentions a SAR of bricks. Thureau-Dangin has properly gathered that the term means 60 dozens of bricks and the bricks are counted by the dozen, but he has not explained why a dozen of bricks appears as a unit. The explained from is that 12 bricks weigh an imeru; hence an ass can transport a dozen of bricks 1/3 a hod carrier can carry a dozen bricks the mason. A royal qa equal 3 1/3 normal qa; a brick of 3 1/3 normal qa has a density in barley calculated by the factor 3 1/3, so that a brick equals in weight 10 qa of barley. A dozen bricks equal 120 qa of barley or a kurru, taking the discrepancy leimma into account, since a kurru is equal to 125 qa.

Problem 30 of the Great Mathematical Tablet reads:

If a ship carries 1 SAR of bricks, how
much barley will it carry? As for yuo,
41|40 is the volume of brick. Multiply
41|40 by O 5; 3|28 20 is the volume, (that
is), 3 1|3 qa 8 1|3 sheqels (is) 1 brick.
Multiply 3 |28|20 by 12 sixties; obtain
41/40, obtain 8 kurru |1|40 of barley. (Such
is) the procedure.

I have translated the text without the small but consequential emendations introduced by Neugebauer and Sachs (MCT, 96) because they are not necessary to my interpretation. Neugebauer and Sachs have obscured the reckoning by assuming that 41|40 means that a brick has a volume which is 41 2/3: 60.60.60 of the unit they call the volume-SAR. They explain 5 as indicating that a qa is contained 5 x 60 x 60 times in the volume-SAR. This text is a main argument for their thesis that volumes were calculated by referring them to the volume-SAR; but their interpretation even though arithmetically carrect in a complicated formmas, nothing to do with the problem discussed in this text. In order to stress the notion that 5 is a fixed coefficient they observe that after the number there is the sign GAN; but Thureau-Dangin who so read the sign stated that it “mest tout a fait inexplicable)” The sign must be read EKI and is simply a sign of punctuation, dividing the number 5 from the following number,: EKI is used here as one most commonly used E to separate numbers.

The reckoning contains an adjustment of1/24. By leaving out the discrepancy leimma, the text reads most plainly:

If a shipcarries 1 SAR of bricks, how much
barley will it carry? As for you, 40 is the
volume of brick. Multiply 40 by 0|5; 3|20
is the volume, (that is ), 3 1/3 qa (is ) I
brick. Multiply 3|20 by 12 sixties; obtain
40|0, obtain 8 kurru of barley.

The reckoning could not be more simple. One begins with the volume in qa of 12 bricks, and this amount is multiplied by 0|5, that is, divided by 12 (all divisions are performed by a multiplication by the inverse). The result is the volume of a brick, which multiplied by 12 sixties gives the total volume of the bricks. The only point that needs attention is that the kurru used in the final step appears to be the great kurru of 300 qa (cube which an edge of 40 fingers or 2 barley feet). The total volume of 720 bricks is 41|40 or 2400 qa equal to 8 kurru. The calculation is so simple that it is tautological; hence, it must have a further purpose. In fact the text speaks of a kurru of barley and not of bricks: the kurru of the final result is not the great kurru, but the greatest kurru of 1000 qa (cube with an edge of 60 fingers or 2 barley cubits). Since 300:1000 = 1:3 1/3, the relation between the two kurru corresponds to the relation between the density of brick and the density of barley.

In summation, the text aims at demonstrating the validity of one of the practical calculation by igi-gub, a term that Neugebauer in the spirit of modern mathematics renders as” fixed coefficient,” but that I would rather render more literally as” tested result, formula.” Igi-gub is the equivalent of the Greek lovgo” as used in logistic. Calculations by igi-gub were most common, and there were tables listing together igi-gub of all sorts. In this case the igi-gub is 5: if one takes the volume of the bricks expressed in sixties of qa and divides it by 5, one obtains the corresponding volume of barley expressed in greatest kurru. The text proves the validity of the formula in reference to the typical brick equal to a royal qa, but the formula applies to any type of brick.

In the calculation there are employed three types kurru. Thirty-six bricks make the usual kurru of 125 qa; this kurru is referred to by the term SAR of bricks which means 60 kurru. The volume of bricks is also calculated as 8 great kurru; this is the kurru of 300 qa, equal to 900 bricks. The volume of barley equivalent to the SAR of bricks is 8 greatest kurru of 1000 qa. As one can see, between the kurru and the great kurru there is a relation which corresponds to specific gravity of brick in relation to water, that is, 2|1|2 minus a diesis. Between the greater kurru and the greatest kurru there is the relation 1:3 1/3 which occurs between the density of brick and the density of barley

Considering the importance of bricks in the life of Mesopotamia it is not surprising that the units of volume and weight were so well adjusted the problem of measuring and transporting bricks.

Neugebauer and Sachs recognize that a small tablet of Yake (YBC 7284) is very similar in object and method of calculation to the one just discussed, but they cannot explain its meaning at all (MCT, 97). They have not recognized that the calculation deals with specific gravity, even though the reference to it is most patent in this case. One face of the tablet reads:

One brick.
What is the weight?
Its weight is 8 1/3 minai.
On the other face one finds the explanation:

41|40
8|20
igi-gub 12

Further down, in larger characters, there is written:
1|3|40. This means 63 2/3.

It is evident that the specific gravity of brick is calculated as 2.4. The brick has a volume of 750 cubic fingers; according to a specific gravity 2.4 it corresponds to 1800 cubic fingers or 8 1|3 minai of 216 cubic fingers of water. But on the other side of the tablet there is written in large characters the figure 63 2|3; this indicates that the mina is of the correct form, so that 63 2 |3 sheqels of this mina are necessary to make a mina of the form Stereometric brutto. The stereometric calculation gives a result is in minai stereometric brutto, but the result taken as being in correct minai, with the consequence that the specific gravity computed as 2.4 become 2.25 by shifting to a mina which is a diesis lighter.

The calculation is performed by taking the volume of 12 bricks which is 41 qa |40 and multiplying it by the inverse of 12 which is 0|5. The result is 8 minai|20. One should divide by 12, number of the bricks, and multiply by 2.4, the specific gravity; the two operations are combined into one, so that one divides by 5. In practice all that a mason has to know is the igi-gub 12: by taking the volume of 12 bricks and dividing it by 5, the inverse od 12, he obtains the weight in minai of a brick. The qa are of the form Stereometric brutto and the minai of the correct form, as the text warns by the figure 62 2/3. The demontration is based on the typical brick, but the formula is valid for any brick.

8. Problems 6 and 10 of the tablet BM 85196 deal with the Euboic qa; in one case this qa is calculated as 50 cubic fingers by the great cubit and in the other case it is calculated as 180 cubic fingers by the barley cubit. In both cases the figure used in calculating the volume of the normal qa has been reduced by 1/6 in order to obtain the Euboic qa. When the Euboic qa is calculated as 50 cubic fingers by the great cubit it is of the carrect form; when it is calculated as 180 cubic fingers by the barley cubit, it is of the Stereometric form.

Volume. Width 30. height 40. What is the
volume? As for you, square 30, obtain 15|0.
Multiply 15|0 by the factor 0|5; obtaing 1|15.
The saton (is) 12 (qa) of grain, I repeat the
saton (is) 12 (qa) of grain. Multiply 1|15,
the surface (of the basis) by 40 (which is)
the height; obtain 50|0. (It is) sixty sata
of grain; I repeat the saton (is) 12 (qa) of
grain. Such is the procedure.

The mathematical principle of the calculation is simple; a near-cube of 30 x 30 40 fingers has a volume of 26, 000 cubic fingers; since the Euboic qa has a volume of 50 cubic fingers by the great cubit, the volume is 720 qa or 60 sata. The saton contains 12 Euboic qa. instead of 10 normal qa; the text reminds three times of this fact. This repeated reminder is necessary because of a procedure by which one calculates the volume as if it was the volume of a qa and obtains the volume of a saton. Apparently one was expected to know by heart that 50 cubic fingers is the volume of a Euboic qa, but not to know the volume of a saton in cubic fingers. Hence, in the calculation one first obtains the surface of the basis; then before multyplying it by the height, one divides it by 12 (number of the qa in a saton), by multiplying it by 0|5, the inverse of 12. The result is evaluated as if one had calculated qa and is applied to sata

Thureau-Dangin interpreted this text a referring to a cylinder and arrived at a volume of 75 cubic fingers for the qa, a value about which he had doubts. Neugebauer who usually corrects Thureau-Dangins interpretation of prisms as cylinders, in this case (MKT II, p. 53) interpreted the text as referring to a cylinder and understood the division by 12, as being a division by 4¹ with ¹ = 3. The ideogram GAM which I render as with” may mean” circumference, “but is commomly used for rectilinear dimensions: here, it means the” depth” from front to back.

***The clarification of the meaning of Problem 6 allows to understand the similar Problem 10. In this text the calculation is by barley feet, so that the Euboic qa is calculated as 180 cubic fingers. The procedure is the same as in Problem 6, with only one slight complication added, since here the prins has two trapezoidal sides. Hence, the first step is to average the two sides of the trapeze; the division by 12 (multiplication by 0 5) is ntroduced after this first step. The following steps are as in Problem.6.

Translated literally Problem 10 reads as follows:

A trough. The top (is) 3|20 cubits,
the bottom 2|30,
the depth 2/3 of cubit.

What is the volume?

As for you, add 3|20 and 2|30; obtain 5|50. Fraction 5|50 by half, obtain 2|55. Multiply by 0|5 (which is the side, obtain 0|14|35. One sixth (when cal-culating) 12 (qa to) a saton of grain, Irepeat one fourth (when calculating) 12 (qa to) a saton of grain. Multiply by 0|40 which is the height; obtain 0|9|43|20. (It is)
29 (sata) of grain. Such is the procedure.

*****

The length of the side is not mentioned, since its one cubit, and it is not necessary to multiply by 1.In the development of the problem the side appears as 0|5, since the multiplication by 1 is combined with the division by 12. Thureau-Dangin and Neugebauer agree in this interpretation. If the side is calculated as 1 cubit, it follows logically that the other dimensions are also in cubits; but the inference is not necessary since the text clearly states, that the dimensions are in cubits. But Sachs follows an interpretation first offered by Thureau-Dangin and later rejected by him (TMB, No. 86): by eliminating the wordl ’ ‘cubit” from the text, dimensions such as” the top is 3|20 cubits” are understood as meaning” the top is 0|3|20 double canes” that is, 2/3 of cubit.

Reading the text as it stands, the trough has a front in form of trapeze, with lengths of 3 1/3 and 2½ cubits and a height of 2/3 cubit. The side is one cubit long. One can see that the volume of 70/36 cubic cubits, or 2 cubic cubits minus a discrepancy of 1/36. If one neglects this discrepancy, the trough contains 2 cubic cubits. The reason for the discrepancy is that the cubit used here is not the usual cubit of 1 1`/2 barley foot or 30 fingers, but a cubit of 32 fingers.

The cube of this cubit is a kurru which is usually calculated as 300 normal qa and 360 Euboic qa or, calculating by double units as it is the rule in stereometric texts, 150 normal qa and 180 Euboic qa. (cf. Thureau-Dangin, TMB, p. 41 n.30. The use of this different cubit has been sensed by Segre. But the cube of 32 is 32, 768, whereas 180 Euboic qa of 180 cubic fingers have a volume of 32.400 cubic fingers; by a reduction of 1/36 one obtains 31, 857.75. The reduction is slightly excessive and at the ent the writer carrects the approximation by giving the answer 29, instead of 29.166...But the writer, while trying to offer an exact reckoning, is thinkind of 2 kurru having the volume of 2 cubic cubits, and hence rounds the figures to” one fourth” of square cubit and to” one sixth” of cubic cubit. The adjustment 35 /36 indicates that the trough was calculated by Euboic qa of the form Stereometric brutto, as it was necessary in a stereometric reckoning, but was intended to be equal to 30 sata and 360 Euboic qa of the correct form brutto.

The meaning of the tablet is the following, analyzing it step by step The two sides of the trapeze are averaged by being added and divided by 2. The product is divided by 12 (multiplied by 0|5). At this point the writer is guilty of a lapsus and saya” one sixth”, but he carrects himself by repeating the sentence with” one fourth” of square cubit; he lapsed because he was thinking of the final result which is” one sixth” of cubic cubit. The surface which is 0|14|35 or one fourth minus 1/36, is multiplied by the height (here called depth since werare dealing with a trough 0|40 or 2/3 of cubit. The resulting volume 0|9|43|20 or one sixth, minus 1/36. If the result were 0|10 or one sixth, the answer would be 30 sata; instead it is something less which is compute as 29 sata (sexactly it should be 29.166...). As I have said the essence of the procedure is that by introducing a division by 12. one obtains the result in sata, by using the value of teh qa with which pople were more familiar.

The origin of the procedure by which one calculates in qa in order to obtain the volume in sata, is to be found in the mentioned tablet VAT 2596, in which there are combined two different methods of formulatin sexagesimally the submultiples and the multiples of the qa and the kurru: one method begins at the level of the qa, whereas the other method begins at the level of the saton

9. The desire to gloss over the fundamental datum that the normal qa was conceived stereometrically as cube with an edge of a six-finger, has caused Neugebauer and Sahs (MCT, p.57) to miss the explanation of a problem contained in tablet YBC 8600. It is a question of the “thickness,” that is, the surface of the section, of a log having the circumference of a cubit. The surface is to be measured in qa; the qa of surface is a square having a six-finger as side. The formula used to calculated the surface is c2/4 II with II = 3 1/8, so that the circumference is equared and divided by 12.5. The circumference is calculated as 5, that is, as 5 six-finger, since the six-finger is the edge of the qa. By squaring 5 and dividing the resulting 25 by 12.5, one obtain 2 qa of surface (72 square fingers). The problem is simple, once one recognizes the existence of the unit qa of surface. Since one calculates by qa, the unit of length is the six-finger: the length of a cubit is understood as 5.

The first part of the tablet reads;
1 cubit is the circumference of a log.
How thick was it?
As for you, multiply 5 by 5. 25 by the
igi-gub 0|4|48 will produce 2. 2 qa
(is) the thickness of the wood.

The rest of the tablet presents the inverse calculation, deriving the circumference from the known surface of 2 qa.

This problem makes evident that one was so used to conceive the qa as cube with an edge of 6 fingers, that one used the qa also as unit of surface and as a unit of a length. In the same spirit the term iku applies to a length of 120 cubits, to a square with a side of 120 cubits, and to cube with an edge of 120 cubits. It is because of their refusal to recognize that the qa was conceived as a cube, that Neugebauer and Sachs fail to understand this text. Their only pertinent remark is the incidental observation that the text seems to indicate a reckoning by II = 3 1/8; they rejected this interpretation as” unwarranted,” but later, in 1952, Neugebauer accepted the contention of Bruins that in Mesopotamia one calculated II as 3 1/8. In my opinion, such a reckoning is normal application of the used of the barley cubit which is equal to 27 basis fingers.   