Appendix 3 Margin of Error for Placing the Fragments on the Canvas

Authors:
Jonathan Ben-Dov
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Asaf Gayer
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Eshbal Ratzon
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The ultimate purpose of the procedure suggested in this book is creating a digital canvas for every scroll. In this section we calculate the error that may be incurred when a fragment is placed using the Stegemann method, as described in chapter 12. The distances within the canvas (i.e., the coordinates for the placement of each fragment) should be expressed by means of a mathematical equation, from which the error can be derived.1

As was explained in chapter 12, according to the Stegemann Method, a reconstruction begins with an anchor fragment, around which the circumference of the scroll is known. Other fragments, belonging to consecutive layers, are placed in relation to that fragment. The distances between consecutive layers of the scroll increase or decrease. Most scholars who have used the Stegemann method calculate the sum of these distances, layer after layer, and treat them mathematically as a series of concentric circles with an increasing diameter and circumference. While the scroll is closer to a spiral than to a series of concentric circles, the concentric circles approximation is easier and more practical for DSS scholars, and the difference between the error caused by the two mathematical approximations is negligible.2 We thus prefer to use the concentric circles approximation.

A concentric circles approximation means that the circumference of each consecutive layer grows linearly.3 We can therefore regard the distances as an arithmetical series, and calculate the distance of a certain layer from the known circumference as the sum of all distances of the missing layers in between. The formula for the sum of arithmetical series can be used for this purpose. The first term of the series will be the known circumference, and from that point the distances will increase or decrease.

Let us call Sn the distance of a fragment n from the anchor fragment (to be precise, the distance of a specific point on fragment n, whose damage pattern recurs in other turns of the scroll, from the same damage pattern in the anchor fragment). The equation for the sum of an arithmetical series is:

where

• . This is the sum of the series, which in turn equals the position of fragment n in relation to the anchor fragment;

• a1 is the first term of the series (i.e., the first known length of a turn, derived from the anchor fragment);

• d is the difference between successive terms ai and ai−1 for each i = 2, …, n;

• an = a1 + (n − 1)d : this is the nth term.

In the case of concentric circles of a scroll,

• Sn= the distance between the same damage pattern on the anchor fragment and fragment n;

• n = the number of layers between a1 and an;

• d = the difference in circumference between two consecutive turns;

• a1= the known circumference;

• an= the circumference of the scroll between the discussed fragment and the one preceding it.

Based on the formula of the distance, we can now compute its error. The potential error resulting from material reconstruction in the Stegemann method is comprised of three factors:

1. The error for the first reconstructed or measured circumference (Δa1). In the optimal case of a fragment containing at least one complete turn of the scroll, the error will arise only from the potential error in measurement and from the changes that the fragment may have gone through during the years (such as shrinkage, cracks, twists, etc.). If, however, the first turn of the scroll is not known but rather reconstructed based on parallels and other information, an additional error should be taken into account, its exact magnitude to be estimated by the editors to the best of their knowledge. Factors to be considered are the width of the intercolumnar margin that may vary within the same scroll and the width of the columns if they are reconstructed. If the columns are reconstructed using a font, the experiment described in chapter 10 may help in determining the margin of error for the width of the column. The basic factors described above, i.e., scaling, background removal, and fragment restoration, can also create some error; if not negligible, they too should be taken into account.

2. The next factor to be considered is the error incurred by the different consecutive damage distances (Δd). Stegemann’s assumption that this difference is constant is a mere approximation, which would have been justified only if the thickness of the skin was even throughout the scroll, and if the tightness of the roll was even throughout. However, empirical examination of the distances between recurrent damages in complete scrolls proves this assumption to be wrong, as the change of distance can vary immensely. Even within one and the same scroll it may vary between less than a millimeter and over a centimeter.4

We derive our estimation of this error from measurements carried out on 11Q5 (= 11QPsa), a scroll that was preserved comparatively intact. One of the damage patterns of this scroll comprises holes, created by a worm eating from the outside through all layers of the scroll to its middle. The distances between sequential wormholes give excellent indication for the circumference of every turn, and can be measured with very good precision. The situation of 11Q5 is thus outstandingly favorable for carrying out these measurements. The growth of the circumference in this scroll between subsequent turns varies between 0.15 cm and 0.6 cm, with an average of 0.4 cm and a standard deviation of 0.1 cm.5 Therefore, even in this favorably preserved scroll the results are far from stable. 11Q5 is the only well-preserved scroll with wormholes, and accordingly there is no way to measure the growth of circumference with sufficient precision in other scrolls. We will therefore use the standard deviation of the differences of 11Q5 as the error (Δd) for other scrolls, too.

A deviation of 0.1 cm per turn is rather meager, but the errors add up as we continue to shift away from the known circumference into the calculations of more and more fragments in the periphery of the scroll. Moreover, even the first circumference itself usually bears some error.

3. Finally, it is not always certain how many layers (n) are missing between the anchor fragment and the fragment under discussion, when placed using the Stegemann Method. If all fragments were preserved in one wad, the number of layers is certain, but if they are preserved in several wads or separately, this number is reconstructed based on other considerations, and the measure of uncertainty should be stated. We mark this number Δn.

Δd is the error for d. It equals 0.1 cm. Δn is the error for n, and Δa1 is the error for a1. As is customary in the sciences, ΔSn (= the error for the position of the nth fragment) depends on all of these values as well as on n, and equals:

If an argument is computed from other values, one should substitute it with them, in order to avoid using correlated arguments. Let us present an absolute value for this formula in a theoretical example. In this case:

the known circumference at one given turn of the scroll is 10 cm ± 2 cm;

we estimate the difference between the circumference of every consecutive turn as 0.3 cm ± 0.1 cm;

the number of turns is known for certain;

We seek to compute the relative position of the tenth fragment.

The calculation runs as follows:

This means that the tenth fragment will be 113.5 cm away from the fragment around which the circumference is known. We compute the error as follows:

The margin of error is 20.5 cm, hence the tenth fragment will be between 93 and 134 cm (113.5 ± 20.5) away from the original fragment. In this case, the error comes to approximately 18%.

Let us now examine a different case, in which the number of turns is not certain, and may vary between 9–11 layers. Thus Δn = 1. The computation of the error will be:

As expected, in this case the error is larger and the tenth fragment can be placed in the range between 89.3 and 137.7 cm (113.5 ± 24.2) away from the original fragment. In this case, the error comes to approximately 21%.

This large error may be reduced if additional types of information serve as anchors for the reconstruction, such as the amount of text in the intermediate space or other textual data, as described in chapters 9–10.

The distance of a fragment from the anchor fragment gives the length of the scroll between these fragments. A scholar may also be interested in the distance between two consecutive fragments, whose placement is based on the same anchor fragment using the Stegemann Method. In this case, the distance is:

an = a1 + (n − 1)d

The error is based on the same parameters specified above:

Again, it is important to avoid correlated arguments. The equations for an, Δan, Sn, and ΔSn will be used in chapter 16 in the reconstruction of 4Q418a.

1

As the known factors for every scroll vary, estimating the error with empirical means may be complicated and irrelevant for every specific scroll. We therefore chose to compute the error theoretically.

2

Ratzon and Dershowitz, “The Length of a Scroll.”

3

For a mathematical proof of this conclusion see Stoll, “Die Schriftrollen vom Toten Meer,” 205–18.

4

Ratzon and Dershowitz, “The Length of a Scroll.”

5

Ratzon and Dershowitz, “The Length of a Scroll.”

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Material and Digital Reconstruction of Fragmentary Dead Sea Scrolls

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