バビロニア数学における容量単位silàの特殊用法とVAT8522の解釈について

抄録

A Babylonian capacity unit "silà" was also used as a unit of "thickness" of a log. Neugebauer and Sachs almost clarified the meaning of the silà through a study of YBC 4669, 8600, that is, "x silà of a log" means a capacity of a circular cylinder whose height is 6 šu-si But the contents of VAT 8522 Vs.I which treats the thickness of a log remains obscure despite their efforts. I find a clue to the solution to the calculation in VAT 8522 Vs.I in the expression of relation 1 silà=(6 šu-si)³=(0;l ninda)³ and the number 1;20 which is hidden in line (6a) of the text. Neugebauer suggested that 1;20 was a "normalizing constant" and I regard this as a proportional constant between the area of a square and the area of its inscribed circle in case of a certain equivalent transformation of a prism into a circular cylinder The process of the calculation made by a Babylonian scribe in a roundabout way is as follows. In the first place he takes the cubic roots of 1,4 silà and 8 silà, and gets 4 dal,2 dal respectively (unit; 6 šu-si). Namely he transforms each circular cylinder into a cube retaining the same volume. Consequently "dal" is a side of the cube and not a diameter here. Next after taking the average of the dais, an assumed circular cylinder which is inscribed in the averaged cube is introduced and the area its base is calculated by a usual formula. This "whole area" is multiplied by 6,40 ( = 5,0 * 1;20) to get the true volume. At this point Babylonian "normalization of a log" has been completed. The last calculation, which is omitted in the text is, in my judgement, as follows. By dividing the volume by the area of the base of the normalized log, the length of the log is obtained, and then by multiplying it by 9/10 the length of the log which should be cut down is obtained.