THE ARK OF NOAH1. The Book of Genesis reports (6:15) that the Ark of Noah measured 300 x 50 x 30 cubits Interpreters give at best only cursory attention to these figures; the reader can verify that even the most extensive commentaries on Genesis usually do not include even one remark on them. But the writers of the P stratum of the Pentateuch chose them carefully in order to convey an entire series of mathematical significances. Authoritative interpreters agree that the dimensions of the Ark were introduced into the text by the same hand, generally identified with the P writers, that calculated the Deluge as lasting 300 days (150 days of rising water and 150 days of ebbing water). We shall see that these figures were chosen because the height of the water was reckoned as 150 x 150 Roman cubits = 22,500 cubits = 9988 meters, and because the radius of the Earth was calculated as 40 x 40 x 30 x 300 = 14,400,000 cubits = 6,392,422 meters. By these figures I intend to alert the reader to the mathematical implications of the dimensions of the Ark which I am going to elucidate. In reading the current commentaries on the description of the Ark, one finds that they discuss the problems of the windows and of the door and the problem of the wood out of which the Ark was supposedly made. Any person who has attended Sunday school may be expected to be familiar with the issue of the "gopher wood." But the same commentaries ignore the dimensions of the Ark which were presented in the clearest possible way by the P writers. 2. The P authors were active in postExilic times, under Persian rule, when the official lineal standard of the Jews was the cubit of 532.7018 mm., the BabylonianEgyptian royal cubit; however, since the episode of the Deluge took place before the Exile, one could expect a calculation by the socalled cubit of Moses, also known as the Roman cubit, equal to 5/6 of the abovementioned standard, or 443.9182 mm. One is left to wonder which of the two standards was employed in reporting the dimensions of the Ark. It may appear surprising that the P authors, usually accurate and careful in such matters, do not say a word to clarify this point. The explanation is that the P authors cleverly chose figures such that, whether one reckons by one lineal standard or the other, the results are the same on the points that really matter. This intended ambiguity makes clear that what was significant in the Ark was its numerical proportions. 3. Dimensions of the Ark in Roman Cubits Let us examine first the dimensions of the Ark in terms of Roman cubits or cubits of Moses, which was the basic calculation. The measurements of the Ark are:
The volume of the Ark is 450,000 cubic cubits, or 39,366 cubic meters i.e., 39,366,000 liters. The Biblical Ark is about 19 times smaller than the Mesopotamian Ark (exactly 19.2216 times). The height of Noah's ark is only 30 cubits, 1/4 of the height according to Mesopotamian tradition. Possibly there is a metrological explanation of this difference: if a formulation of volume as that found in Berossos was interpreted by making the US equal to 360 natural barley cubits (single stadion or double minute of march) instead of the usual 720 natural barley cubits, the result would be a reduction of volume to one fourth (90^{2}./180^{2} = 1/4). The smaller size is the reason why the biblical Ark is described as a rectangular parellepiped and not as a cube. Those who delineated the biblical Ark wanted to achieve a perimeter of the base of the same nature as that of the Mesopotamian Ark, in spite of the difference in volume. The rectangular shape of the biblical Ark has nothing to do with its supposed resemblance to a ship. The cubic Mesopotamian Ark has a perimeter of one US = 720 natural barley cubits = 480 "great cubits" = 364.50 meters. This was the standard unit of itinerary distance in Mesopotamia; it was considered as equal to 1/5 of a minute of degree of equatorial latitude. The perimeter of the biblical Ark is 700 Roman cubits = 310.7427 meters; it is equal to 1/6 of a minute of degree. Both the Mesopotamians and the Hebrews calculated the basic degree in such a way that a minute of degree is 1859.6591 meters. But they had round figures for computing the subdivisions of the degree in practical everyday calculations. In Mesopotamia the second of degree was computed as 60 natural barley cubits = 30.3750 meters (the exact figure would be 61.2233 cubits or 30.9943 meters), so that its multiple, the US of 720 natural barley cubits was in fact slightly shorter than 1/5 of the standard minute of degree. Among the Hebrews there was used a round figure of 70 Roman cubits = 31.0743 meters for the second of degree (the exact figure would be 69.444 cubits), so that 1/6 of a minute of degree came to 700 cubits with an excess of 1/125. If the second of degree is 70 cubits, the degree would be 252,000 cubits instead of 250,000. The perimeter of both the Mesopotamian Ark and of the biblical Ark was calculated by the practical round values for the subdivision of the standard degree. The surface of the base of the Mesopotamian Ark is 120 x 120 "great cubits" = 1440 square "great cubits." The surface of the base of the biblical Ark is 50 x 300 cubits = 1500 square cubits. In absolute figures the two surfaces are the same except for a difference of a diesis (24/25). The height of the biblical Ark is 30 cubits, that is, in absolute figures, 1/4 of the height of the Mesopotamian Ark, 120 "great cubits." The volume of the biblical Ark is 450,000 cubic cubits. In "great cubits" the volume of the Mesopotamian Ark is 1,728,000 cubic cubits. In absolute figures, the volume of the biblical Ark is 1/4 plus a diesis (25/24). The volume of the Mesopotamian Ark was:
120^{3} "great cubits" 180^{3} natural barley cubits The volume of the biblical Ark was:
76.63094^{3} Roman cubits The Mesopotamian Ark was a cube with sides of 91.1250 meters; the biblical Ark had the volume of a cube with sides of 34.0179 meters. We have seen that the Mesopotamian Ark was a representation of the northern hemisphere on a scale of 1:90,000. The biblical Ark represents the same hemisphere on a scale of 1:240,000. Hence, in principle the volume of the two Arks should relate as 9^{3}./24^{3} = 729/13,824 = 1/18.96296. We have seen that the volume of the biblical Ark is 1/19.2216 that of the Mesopotamian Ark. The volume of the biblical Ark is:
450,000 Roman cubits Taking the Hebrew bath as 66.666 Roman librae (21.600 liters), the volume of the Ark is: 1,822,500 bath. This would be about 600 times the volume of Solomon's Brazen Sea, which is described as being 3000 bath. Perhaps the Ark was intended to be exactly 1,800,000 bath, so that we would have an indication of a bath of 21.870 liters. I have shown that the Mesopotamian Ark was a cube corresponding in volume to a hemisphere with radius of 140 natural barley units. This implies that the radius of the Earth is 140 x 90,000 cubits = 12,600,000 cubits = 17,500 US = 6,378,750 meters. The volume of the biblical Ark indicates that the rectangular shape of this Ark intended to square a hemisphere with radius of 60 cubits. This implies that the radius of the Earth is 60 x 240,000 cubits = 14,400,000 cubits = 6,392,422 meters. This figure for the radius was obtained from a circumference of 90,000,000 Roman cubits by assuming
p = 3 1/8.If we assume
p = 3 1/8, a hemisphere with a radius of 60 cubits has a volume of 450,000 cubic cubits, which is the volume of Noah's Ark. Calculating by the exact value of p, a hemisphere with a volume of 450,000 cubic cubits has a radius of 59.8959 cubits. Multiplying the latter figure by 240,000 we would obtain a radius of the earth of 14,375,000 cubits = 6,381,324 meters. The biblical Ark was not cubic, but it was subdivided into cubical units. The text of Genesis (6:16) refers to this division by reporting that Noah was instructed to make the Ark "with lower, second and third deck." Since the height of the Ark is 30 cubits, this means that there is a partition every 10 cubits; carrying this division all through, the Ark becomes divided into 450 cubes of 10 x 10 x 10 cubits. Since 10 Roman cubits are about 14½ English feet, each cube of the Ark divided at the middle of the height has the volume of an average room in a modern house. The division of the Ark into cubes or cubicles, was such an important element in the tradition that very early there was introduced a misreading into the text of Genesis. The instructions to Noah begin with specifications about the method of construction (6:14):
Make an ark with a frame of solid beams; close it with reeds and coat it inside and outside with asphalt. I have translated the text in accordance with the latest interpretations. The New English Bible (Oxford, 1971) renders the Hebrew of this verse thus:
Make yourself an ark with ribs of cypress; cover it with reeds and coat it inside and out with pitch. The word which is here rendered as "reeds" is qannim, but the Masoretic text of the Old Testament reads qinnim. Very early the letters QNM were read with the vowels of qinnim instead of qannim. This is indicated by the Septuagint translation and by a passage of Philo. The confusion may go back even to the reading of a BabylonianAssyrian text by the biblical writers.^{(107)} In BabylonianAssyrian there are two words of similar sound, but of different meaning. One is the word qanu, which is of Sumerian origin and means "reed, cane, measuring rod" (it is the origin of the English word "cane"); the other is a native BabylonianAssyrian word qanu which means "tying rope, delimited area, enclosed space, nest." The biblical text used or should have used the equivalent of the first word which reads qannim in Hebrew, but it used, or it was read as, qinnim, which is the Hebrew equivalent of the second word. This confusion originated from the circumstance that the division of the Ark into cubicles was an important elements of its structure. The division of the Ark into cubicles was anticipated in a verse which dealt with construction materials. Speiser properly rejects the substitution of qannim for .'us qinnim; I may add that the correctness of the reading qinnim is proved by a passage of Philo (Quaest. in Genesi II.3) which renders it by "compartments." This is the verse, which according to the King James translation mentions "gopher wood." Most likely the Hebrew word gopher must be explained by the BabylonianAssyrian qapsu, "thick, strong, massive, made in proportion." A clue to the right translation of the "gopher wood" phrase, is provided by Septuagint which renders it into Greek as "with tetragonal woods."^{(108)} The Bible states that the Ark is divided into cells, but does not state their number. But since the area of the base is divisible into 6 squares and the Ark has 3 decks according to the Bible, one would expect 18 cells; but the cells must have been the result of a further subdivision by 5, since the commentaries of the Midrash speak of 90 or 900 cells. These statements mean that the mathematical relations implied in the dimensions of the Ark can be made more precise by applying the mathematical procedure of basi. The volume of the Ark can be reduced by 1/90 by removing 5 cubes of 10 x 10 x 10 cubits or be reduced by 1/900 by removing half such a cube. If we remove 1/90 from the volume of the Ark, we obtain a volume of 445,000 cubic cubits, which is the volume of a hemisphere with radius 59.67152 cubits. Multiplying this figure by 240,000, which is the scale of the Ark, we obtain the polar radius of the Earth as: 14,321,164.8 cubits = 6,357,425.7 meters If we remove 1/900 from the volume of the Ark we obtain a volume of 449,500 cubic cubits, which is the volume of a hemisphere with radius 59.8720 cubits, which indicates an equatorial radius of: 14,369,280 cubits = 6,378,784.9 meters. Hence, we can proceed to the following comparison: Biblical Datum Helmert's Estimate Equatorial radius 6,378,784.9 m. 6,378,388.0 m. Polar radius 6,357,425.7 m. 6,356,911.9 m. According to my interpretation
the figures preserved by the Midrashim indicate that the polar flattening
of the Earth was reckoned as: The dimensions of the Ark were so formulated that by simple numerical expressions they indicated dimensions of the Earth that differ from the perfect ones only by trifles. For the sake of comparison I may refer to the figures I have obtained from Chapter LXIV of the Egyptian Book of the Dead:
4. Dimension of the Ark in BabylonianEgyptian royal cubits. As I have stated, the biblical writers did not specify the cubit employed in the construction of the Ark, because the basic results are the same no matter which of the two biblical lineal standards is employed. If the Ark was assumed to have been planned in BabylonianEgyptian royal cubits, all its lineal dimensions would have been increased as 6:5 in relation to those that I have discussed above. The perimeter of the base of the Ark would be 1/5 of minute of degree, like the perimeter of the Mesopotamian Ark. The biblical Ark would have the same perimeter as the Mesopotamian Ark in spite of the fact that its volume would be 1/11 (calculating exactly, 1/11.1235). The calculations prove to be simpler if one reckons by BabylonianEgyptian royal cubits. According to this standard the circumference of the Earth is 75,000,000 cubits, so that, assuming p = 3 1/8, the radius is 12,000,000 cubits. The latter figure relates very simply as 200,000:1 to 60 cubits, which is the radius of a hemisphere having the same volume as the Ark, if p = 3 1/8. Since calculations are simpler in BabylonianEgyptian royal cubits, it may be presumed that the plan of the Ark was formulated in postexilic times. Whoever made these calculations employed the standard currently used in his time, but kept in mind that the Ark must have been measured by the preexilic standard. This conclusions may be taken as giving support to the followers of the documentary theory about the composition of the Pentateuch. According to the followers of this theory the dimensions of the Ark were included in the biblical text by the P writers who were active in postexilic times. The fact that the Ark has long sides of 300 cubits and short sides of 50 cubits can be explained by an Egyptian practice. Usually in the ancient world itinerary distances were calculated by sexagesimal multiples in order to fit into the division of the sphere into degrees, minutes and seconds. Typical of these itinerary measures was the most common Greek stadion of 600 feet. But the Egyptians of the early dynasties adopted septenary units, changing the ordinary cubits of 6 hands into the royal cubit of 7 hands. Following the septenary system of measures, their basic unit of itinerary distance, the khet, was not 300 cubits, but 350 cubits. The Egyptians equated the khet of 350 royal cubits (= 183.4519 meters) to 1/10 of minute of degree. In the period of decadence, when Egypt fell under foreign rule, the Egyptians substituted, or were forced to substitute, their original royal cubit with the BabylonianEgyptian royal cubit, which was somewhat longer (about 1/62), but they continued to calculate khet as 1/10 of minute of degree. The new khet of 350 BabylonianEgyptian royal cubits was 186.4456 meters. Those who calculated the dimensions of Noah's Ark must have been familiar with the fact that 1/10 of minute of degree was 350 of their cubits, an Egyptian khet of their time; and must have been aware that this unit was a septenary unit obtained by adding 1/6 to the standard unit of 300 cubits. Hence, when they decided that the perimeter of Noah's Ark must have been similar to that of the Mesopotamian Ark, which was 2/10 of minute of degree, they set the perimiter of Noah' Ark as 2 khet and divided each khet into 300 and 50 cubits. Calculating in BabylonianEgyptian
royal cubits, the volume of Noah' Ark is: 68,024,444.9 liters 209,951,990.4
Roman librae The volume was such that applying the deduction of 1/90 and 1/900, one arrived at the same results I have considered above. Deducting 1/90, there is obtained the volume of a hemisphere with a radius of 59.67152 cubits, which multiplied by 200,000 indicates a polar radius of 11,934,300 cubits = 6,357,423.1 meters. Deducting 1/900, the radius of the hemisphere is 59.8720 cubits, which multiplied by 200,000 indicates the equatorial radius of 11,974,400 cubits = 6,378,784.4 meters. The results are the same whether one reckons by Roman cubits or by BabylonianEgyptian royal cubits. The only factor to be kept in mind is that in the first case Noah's Ark was a representation of the northern hemisphere on a sace 1:240,000 and in the second case the scale is 1:200,000. The scale of the Mesopotamian Ark is 1:90,000. If the Mesopotamian Ark is taken as a cube with sides of 120 natural barley cubits, its scale is 1:135,000. It could be that the description of Noah' Ark in such a way that it can be measured by both biblical lineal standards, was determined by a desire to follow the Mesopotamian model as closely as possible. The Mesopotamian Ark was so described that it could be conceived either as a cube with sides of 120 "great cubits" or as a cube with sides of 120 natural barley cubits. The excavations of the ziggurat of Babylon, which embodies the dimensions of the Mesopotamian Ark, have revealed that inside the ziggurat with sides of 120 "great cubits" there was a smaller square construction with sides of 120 natural barley cubits.
Later I shall discuss the possible relations between the biblical Ark and the tombs of the kings of the first two dynasties of Egypt, that is, the tombs of the mastaba type that preceded the development of the pyramid in the Third Dynasty. Here, I intend to present some comparisons as to dimensions. For the tomb of the queen who was the consort of Narmer, the founder of the Egyptian state, there are reported dimensions of 53.4 x 26.7 meters, which should be interpreted as: 120 Roman cubits = 53.2702 meters 60 Roman cubits = 26.6351 meters Even closer to the dimensions of the Ark are the dimensions of King Narmer's successor, King Horaha, who is usually listed as the first king of the First Dynasty, which are reported as 48.2 x 22 meters, that is, presumably, 110 Roman cubits = 48.8310 meters 50 Roman cubits = 22.1959 meters The height of these Egyptian buildings can be established only with approximation, but it was in the range of the height of the Biblical Ark. Since the Biblical Ark and even the Mesopotamian Ark, in spite of their rectangular shape, are described as ships, I must mention that next to each Egyptian mastaba tomb there was deposited within a special boatgrave, a ship, the heavenly barque. It has been possible to establish the dimension of the barque of Horaha's coruling queen: reported as 17.75 meters, that is, 40 Roman cubits = 17.7567 meters. 107. That they followed a nonHebrew text is indicated by the circumstance that their account of the Ark contains several words that do not occur anywhere else in the Old Testament or any Hebrew writings. For instance, for asphalt or pitch they use a transliteration of the BabylonianAssyrian kupru, instead of the usual Hebrew word hamar, employed for instance in Gen 11:3; 14:10; Ex 2:3. 108. The translation of the New English Bible which I have quoted, follows the conjecture that the Greeks would have taken their word kypressos, "cypress," from a Semitic word which appears as gopher in Hebrew. This conjecture is certainly false, because the Greek word kypressos has an ending essos, which is typical of Greek words borrowed from the Cretan language; many Greek names of trees and flowers are of Cretan origin.
