We defend here our conclusion that the quantities written in šār are meant to be and have to be taken seriously. A thousand years after the Ark Tablet was written these numbers in Gilgamesh XI became fanciful expressions to convey great magnitude, although an intervening scribe might have back-calculated certain measures in the attempt to reconcile textual variants. The important issue – which crops up several times in the Ark Tablet text – is that quantities of a raw material are given solely as totals with the standard units only implied. Here we have found that out of the two likely choices – sūtu or gur – for the standard volume measure behind numbers in šār, only sūtu gives meaningful results.
Given the shape and size of the Ark’s basketwork hull, we substantiate this claim about the accuracy and nature of the numbers by comparing the amount of material that should be used in such a construction – which we call VCALCULATED – with the amount given in the text – which we call VGIVEN
To perform this calculation we need two additional items of information. The first is the thickness of the woven rope basketry. Although this is not given in the cuneiform text, a clue comes from the (partly restored) ‘ropes … for [a boat]’ (line 10), implying, very plausibly, a type of rope peculiar to boat-making whose thickness was no doubt standard. The text also tells us the ropes were to be made by someone other than the shipwright, presumably a ‘boat-rope’ artisan, who would manufacture rope of a type independent of the size of coracle it was for. We assume that the thickness of rope used to make a guffa was always independent of its size, so that the width of the rope ordered would not scale up with the final size of the boat. Indeed, the descriptions of traditional guffas show that their structural rigidity depends on the internal framework, so that the basket is just a skin of convenient material to support the applied waterproofing layer. Assyrian sculptures mentioned in Chapter 6 prefer skins to rope for the hulls of their guffas.
This means the basket body of the Ark is constructed using standard materials and techniques, and, although it is nearly seventy metres across, the walls are still seen as having the same thickness as conventional-sized coracles. The most likely standard width of rope used is one finger, which is supported by early photographs of Iraqi guffas (e.g. ‘Building the peculiar round boat …’), which show that the rope used was about one finger in thickness. This is supported by another calculation below about bitumen.
A new Iraqi coracle has just been finished. The rope’s thickness can be seen to be roughly the thickness of a finger (or a toe). High-quality stereo photographs of this type from the 1920s preserve important information which often is otherwise unobtainable. Close-up of modern reed basket-work.
(picture acknowledgement app.1)
The second piece of information we need is the cross-sectional curve of the walls. These should have an outward camber at the base to resist hydrostatic pressure, and this is what is seen on the photographs of actual guffas. There the curvature of the walls is seen to lie somewhere between a straight-sided cylinder and the semicircle of the outer half of a torus (ring-doughnut). Therefore, we believe that it would not be far off the mark to assume it was exactly halfway in-between, and approximate the curvature by a semi-ellipse whose width is a quarter of its height. This means the walls of our reconstructed Ark – the sides of which are one nindan high – bulge out from the base by one-quarter of a nindan at their maximum diameter, thus:
As is approximately true of real guffas, the walls are symmetrical about the mid-transverse plane, meaning the Ark would look the same if it was turned over, top-to-bottom. The important corollary to this is that the area of the roof is identical to the area of the base.
Rope calculations
The first step for rope volume used is to calculate the total surface area ‘A’ of the vessel. This is area of the base ‘B’, plus the area of the roof ‘R’, plus the area of the walls ‘W’.
We are given B = 12,960,000 fingers2, and have assumed that R = B. To calculate the area of the walls W we need Pappas’s First Centroid Theorem: The surface area W of a surface of revolution generated by rotating a plane curve about an axis external to it and in the same plane is equal to the product of the arc length L of the curve and the distance D travelled by its centroid (centre of gravity): W = L × D.
Here, the plane curve is the semi-elliptical shape of the walls, and its length is just half of the circumference of the full ellipse of which it is a part. Calculating the circumference of a general ellipse is a nightmare of complexity, but for the specific case we use here – one whose major axis of length ‘a’ is twice as long as its minor axis – we have a neat formula to use called Ramanujan’s Approximation which is correct to three places of decimals:
Ramanujan’s Approximation
Here, a is simply the height of the walls, 360 fingers, and we are only interested in half of the circumference, which gives us L ≈ ½ × 2.422 × 360 = 436 fingers.
The other component we now need is D, the length travelled by the centroid as the semi-ellipse is rotated to form the walls of the Ark. This is the length of a circle swept out by a radius equal to that of the base of the Ark plus the additional distance from the edge of the base to the centroid. We know the base of the Ark is a circle of area one ikû, so (from ‘Area = π × Radius2’) we can calculate its radius ‘r’ to be:
r = √(Base Area B/π) = √(12,960,000/π) ≈ 2,031 fingers (working to the nearest whole finger.)
The distance ‘d’ of the centroid of a semi-elliptical arc to the axis of the ellipse is given by:
Using the familiar rule for circles ‘Circumference = 2π × Radius’, we are now in a position to calculate D, the circumference of the circle travelled by the centroid:
D = 2π × (r + d) = 2π × 2,088 fingers ≈ 13,119 fingers.
Finally, this gives the area of the walls W as:
W = L × D = 436 fingers × 13,119 fingers ≈ 5,719,880 fingers2;
giving the total area of the Ark as:
A = B + R + W = 12,960,000 + 12,960,000 + 5,719,880 ≈ 31,639,880 fingers2
We now assume that the ropes are whipped tightly enough to each other that they are densely packed and their cross-section can be taken as square with negligible error. Similarly, since the basket is very thin compared to its area, we can calculate its volume by just multiplying its area by its thickness of one finger, again with negligible error.
Thus our calculated volume (VCALCULATED) of rope needed to make the basketwork of the Ark is:
VCALCULATED = 1 finger (thickness) × 31,639,880 fingers2 = 31,639,8800 fingers3 or, dividing by 2,160 to give units of sūtu:
VCALCULATED = 14,648 sūtu.
The given volume (VGIVEN) of rope according to Enki is:
VGIVEN = 14,430 sūtu
which differs from our calculated figure by just a little under 1½ per cent. This is a striking result, and we take it as evidence to support assumption that the quantities given in the Ark Tablet are factual.
We can work out the length of rope represented by VCALCULATED by dividing it by the assumed cross-sectional area of the rope:
Length of Rope = 31,639,880 fingers3/1 finger2 = 31,639,880 fingers = 527 km.
As pointed out earlier, this is roughly the distance from London to Edinburgh!
The Babylonian reckoning
The very closeness of the figures VCALCULATED and VGIVEN leads one to question how the Babylonians might have made their calculation of the quantity needed.
We believe the answer lies in the fact that one ikû is defined as an area equivalent to that of a square of ten nindan × ten nindan, thus making it easy to visualise the area in terms of such a square. This proposition seems to us reinforced by Enki’s actually saying:
Draw out the boat that you will make
On a circular plan;
Let her length and breadth be equal,
especially given the circle-in-its-square school diagram illustrated on this page above.
The Babylonians found it difficult to do accurate arithmet
ic involving circular measures due to their imprecise value for π. If we assume that for the sake of ease of calculation they visualised the one-ikû base of the Ark as a square, then the walls will now be four panels, each ten nindan long by one nindan high, and this would be topped off by a square roof identical to the base. A trivial calculation of the area of this shallow biscuit-tin shape allows us to give the volume of material needed to make it by multiplying it by one finger thickness, as is done for the Ark above. If we call the volume ‘VSQUARE’ we find:
VSQUARE = 14,400 sūtu.
This is four šār exactly, a difference of 0.2 per cent from VGIVEN!
When first encountered, the ‘plus 30’ in the figure VGIVEN seems like an insignificant if not inexplicable quantity, but the above calculation underscores its critical importance, for without it, it could be argued that the intention was to make a square-based vessel, but the extra thirty sūtu shows this cannot be the case. However, the ‘square-based’ method was almost certainly how the Babylonian scribes ‘back engineered’ their figure for the volume involved given the shape. We can see this by doing the calculation for the volume of fibre needed for a circular-based vessel with straight vertical sides – a cylinder. As a circle has the smallest circumference which encloses a given area, the length of these walls will be less than the ‘square based’ value, resulting in an overall volume smaller than VSQUARE by about 2 per cent. As we saw from our figure for VCALCULATED, the extra area provided by the bulge in the walls slightly overcompensates for this 2 per cent, and empirical knowledge of this may have led the Babylonians to formulate a rule of thumb for such volume calculations of the type, ‘Calculate the volume for a square-based vessel then add an extra bit on’
The ‘extra bit’ is what we believe the role of the thirty sūtu in VGIVEN’s ‘4 šār + 30’ to be. Whether or not such a procedure as this was actually used by the ancient Mesopotamian shipwrights, it is easy to see how it would have been useful in the typical scribal tasks of calculating the amount of rope needed to manufacture a particular size of vessel, as well as the quantity of bitumen needed to waterproof it.
The obvious question which then follows is how did they arrive at a number for that ‘extra bit’? For the Ark this figure is ‘30 sūtu’, so a natural assumption is that this is thirty times some real amount used for regular guffas. One way of pursuing this idea is to apply the above techniques to a guffa whose diameter is thirty times smaller than that of the Ark.
The diameter of this craft would then be:
4,062 fingers/30 = 135.4 fingers,
that is, a little over two metres. The walls of the Ark would not scale down in the same way, as their height is determined by practicality, as must have been true for the different sizes of guffas. The ‘square-based’ version of this would obviously have walls 10 nindanu/30 = 120 fingers long. We can now check what height of wall would give a difference (extra bit) of one sūtu between the round guffa and its square-based approximation, and see if this would be a practical size for this boat.
A slightly more involved calculation shows this height to be 34.4 fingers, about 58 centimetres. That is, this mini-Ark would have a diameter about four times the height of its walls, a proportion which seems reasonably safe and practical for a boat ferrying goods and people in calm water. Indeed, the photographs of traditional guffas being built show boats with very similar dimensions.
Given the simplicity of Enki’s exhortation to build a boat ‘as big as a field’, it seems unlikely that this measurement is seen as being a regular guffa scaled up by a factor of 900 (= 302). However, this is possibly how the figures were arrived at in the scribal exegesis of the story. It is known that boats of the period came in standard sizes thought to be related to their cargo capacity, and it may have been either noticed or calculated that some measures for the Ark could be derived from those of a standard boat of one-thirtieth the diameter of the Ark.
3. FITTING THE INTERNAL FRAMEWORK
In parallel with the description of building a traditional guffa given in Hornell, the next stage of construction comes in where the main structural framework is fitted (lines 13 and 14). These are called ribs on the Ark Tablet, and are simply described as being ‘set in’, with no clue as to the exact process or their arrangement, or even of the material from which they are made.
The only hard rib information concerns dimensions: the length is given as ten nindan (sixty metres), while they are ‘as thick as a parsiktu-vessel’. A parsiktu is a volume unit equal to sixty qa, deriving from the name of the wooden vessel used to measure out grain in approximately sixty qa amounts. That thickness is meant here rules out understanding parsiktu in this instance in its common meaning as a volume unit. It must refer to the seldom-mentioned measuring vessel itself. As explained, we take its usage in this context as hyperbole corresponding to our ‘as thick as a barrel’, designed to be an awe-inspiring superlative showing on the spot how much bigger the ribs of the Ark are when compared to those of a normal-sized vessel. Clearly we are meant to understand some approximate size from this statement, so the obvious question here is ‘how thick is a barrel?’
A traditional square grain measure from Japan. Most such objects seem to be round.
(picture acknowledgement app.5)
Traditional grain measures come in a variety of sizes and shapes, the most common being a squat cylinder whose width is about the same as its height. If as a working model we take this to be the shape of a parsiktu with an interior volume of 60 qa and stout walls 2 fingers thick, then the width across its mouth would be about 29.5 fingers, or 49 cm. However, given the lack of evidence for cooperage in Old Babylonian times, it seems much more likely that the shape of vessel used as a grain measure would be a simple box shape, like that shown in the above picture.
Only one known cuneiform text actually quotes the size of a parsiktu-vessel, and then only hypothetically. Significantly for the composition of the Ark Tablet, this is a school tablet with a problem in which the schoolchild has to calculate the depth of a 60-qa parsiktu-vessel which is four unspecified units ‘across’. Since they don’t mention ‘sides’ as they usually do in such problems, this is is likely to be a square-topped box, with the ‘across’ being the diagonal from corner to opposite corner. The units can really only be ‘stacked hands’ of ten fingers. Of course the problem does not take into account the thickness of the walls of a real measuring box, but if we again estimate this to be two fingers then an elementary calculation (40/√2) tells us its width along each side is 32.3 fingers, or 54 centimetres (and, solving the schoolboy problem, 18.2 fingers deep if you include the assumed thickness of the walls).
The ‘60 qa’ parsiktu-vessel reconstructed from a school problem text.
This is not so far from the quoted estimate for a cylindrical measuring vessel, and means we can take ‘as thick as a parsiktu’ to mean roughly one cubit (∼fifty centimetres) thick no matter what the parsiktu’s shape. The fact that the ribs were not described as one cubit thick indicates the use of the term parsiktu as an informal and easy-to-grasp literary device rather than an exact measure. The ribs of the Ark are thus ten nindan long and about thirty fingers wide.
As to their cross-sectional shape the cuneiform text is silent, but this is no doubt implicit in the name ‘rib’, which must have had a technical usage in boat-building. We can work out all we need to know from the corresponding elements in the traditional guffa, described by Hornell as ‘lathes’, meaning they have a thin rectangular cross-section. They are made of a resilient wood, and are sown into the basketwork of the hull under tension as the main source of rigidity in this structure. They run from the gunwales down the walls and across the base of the boat, but they are not all directed at the centre. Instead, each one of a series is offset from the angle of the wall so that they run parallel across the base to one side of the centre. These ribs are then interwoven with a second series set at 90o to the first, like so:
Plan view of the Ark with two series of ribs
set in at 90o.
As more of these pairs of series at 90o are set in around the circumference, they not only strengthen the walls but build up a floor structure as well, which is later reinforced by pouring bitumen between the ribs. The scheme above uses six ribs from our total of thirty, so another four such sets need to be laid in, each rotated around the circumference by 360o/5 = 72o with respect to each other. Hornell tells us that the number used on the largest of the traditional guffas was twelve to sixteen, so the Ark uses about twice as many.
The curved walls have been shown above to be about 436 fingers long from top to bottom, so each 10-nindan-long rib will run down the wall and then approximately 8½ nindan along the base of the boat. The gap between ribs at the wall will be a quite large one of seven metres or so.
As these ribs in a normal-size guffa are thin springy strips of wood, the implication is that giant ones here are also intended to be wooden. Although there are no trees from the Ancient Near East of sufficient size for these to be carved from one piece, plank-sized sections can be scarf-jointed together, and, if the resulting ribs also had a shallow enough depth, they would be sufficiently flexible to interlace like the regular lathes. Given the comparative fragility of the basket walls, however, it seems improbable that such ribs could be fitted without damaging the hull unless they were pre-shaped into long, laid-back ‘J’ shapes.
Importantly, unlike the thickness of the shell of the boat (and, as we shall see later, its waterproofing) – where no concession was made to its exaggerated size – these structural elements do scale up in comparison to those of a regular guffa, in both size and number. The practical aspects of handling such huge structures seem to have had no interest to the authors, and no information is given as to how or with what they were to be installed into the hull.
The Ark Before Noah Page 31
