Since the procedure described in chapter 13 involves multiple transformations, it is necessary to estimate the margin of error incurred by it. Most of the effort in this appendix will be spent in finding the mathematical formulation that would allow the calculation of the error. The complex mathematics is due to the abundance of arguments. We therefore recommend collaborating or consulting with a scholar with scientific background when applying it.
As already described in chapter 2, since this process adds information to the second copy based on the reconstruction of the first copy rather than on solid existing fragments, it is prone to larger error. The error derives from the uncertainty of the reconstruction of both copies. Elements of the error may rise from the change in the measurements of the columns: width of lines across columns, change in the number of lines per column, the width of “dummy” columns, etc. However, taking into account additional material details such as the presence of margins (which anchors the margins to a point on the canvas), as well as increasing the number of copies involved in the procedure, may keep the growing error in check and may bestow more validity on the procedure.
As explained in Table 15 in chapter 13 the procedure is:
When expressed mathematically, the conversion from number of lines (LN) to number of letters (LT) depends on the length of a line (LL) and the density of writing (the number of letters per centimeter; D):
LTa = LTb = Da * LLa * LNa
Eventually, when we want to convert the number of missing letters on canvas B again to the number of missing lines, we should run the opposite process, that is, to divide the number of letters (LT) by the density of writing (D) and the length of the lines (LL) of canvas B:
From this equation it is possible to derive an expression of the error of the number of lines (LNb), as a function of the known errors of the other arguments:
While in chapter 13 we demonstrated the placement of three bulks of texts, here we illustrate the procedure for only one of them: placing the text of 4Q418a 17+17b+14a (canvas A) on the canvas of 4Q418 (canvas B). The same process can be carried out for every other bulk of text. The numerical data is taken from the reconstruction in chapter 16.
On the 4Q418a canvas, fragment 17+17b+14a is placed in column XII, one column apart from fragment 18+14b (column XI). Both fragments are placed at the bottom of their columns.
LLa – The length of the lines in 4Q418a column XII is 95.4±13 mm.
LNa – Column XII has 36 lines; subtracting the 3.5 lines from the beginning of fragment 17 to the end of the column leaves 32.5±2.5 missing lines.
D – The writing in both copies 4Q418 and 4Q418a is very similar, and therefore their density could be considered equal. The computed density of writing in 4Q418a is 0.49±0.04 letters per cm.
LLb – The reconstructed length of lines in the columns following the text of frags. 18+14b, when placed in the layout of 4Q418, is 125±5 mm and 115±40 mm respectively.
The number of lines between the two texts in the canvas of 4Q418 is therefore:
Based on the above equation for ΔLNb, the error is 4.9 lines.
This number of lines (24.8) exceeds the number of remaining lines in this column of 4Q418 (see figure 46 in chapter 13). We must therefore have the text spread over two columns, and concurrently add more complication to the calculation of error. The position of the text of 17+17b+14a in the column ii can be calculated based on the width of the two columns and the height of the first column:
Column i in 4Q418 is missing only 19.5±1 lines. The rest of the text (5.3 lines) must be moved to the next column, according to the width of the latter column (reconstructed as 115 mm). The length of 5.3 lines of column i equals 662.5 mm; in turn, this number equals 5.76 lines of column ii.
The error of LNbii is accordingly:
The relative error is thus close to 25% (5.8 out of a total of 24.8 lines). In this particular case, in which the text has to be placed in another column whose reconstruction is not based on preserved fragments, many factors attribute to the potential error. It is therefore much larger than the potential error if the text would have been included in the same column. When the reconstruction continues to further columns, the error might even grow, but eventually it will reach a place with more material information preserved in copy B. At that point it will be possible to validate the reconstruction and reduce the error.