Berossos’ figures constitute a remarkable tribute to the tenacity of ancient priestly traditions, since the Babylonians had normally used base-10 in their mathematical calculations for many centuries. Berossos, however, felt a commitment to honor the ancient heroes whom he was listing in the age-old Sumerian manner.
In attempting to provide a “rational” solution to the problem of large numbers in the antediluvian King List, I have said nothing as to precisely why base-60 squared was employed in the listing.
Scholars who have checked the numbers are satisfied that they have been transcribed accurately, with the result that the issue must now turn on mathematical considerations, as Young has suggested. From a prima facie standpoint it is no longer legitimate to question the numbers themselves, but instead to recognize the possibility that base-60 squared was actually functioning as a mathematical constant.
So little insight has been gained into the theoretical dynamics of Sumerian mathematics that it is impossible to say with certainty what the reason was for employing base-60 squared as a constant, assuming that this was its actual function in the King List, as seems eminently probable.
It was certainly integral to the structure of the various recorded reigns, unlike some constants in modern mathematics that grace an equation but are not indispensable entities. Why base-60 should have been squared in order to perform its function satisfactorily is also problematical. Perhaps, after all, base-60 squared was intended to serve as a symbol of relative power and importance, which the compilers of the ancient Sumerian King List associated with those men whose reigns they recorded.
Regardless of the immediate answers to these queries, it seems clear that base-60 squared should be recognized as an “ideal” constant, which, however, must be factored out once it has been isolated so that it is not reckoned as part of the overall calculation.
In any event, we know that the ancient Sumero-Babylonian sexagesimal system employed at least the following mathematical bases as units: 60° (= 1), which in Akkadian was called ištēn; 60 (to the first power) 1 (= 60), which was called šūšu; 60 (to the second power) 2 (= 3600), which was called šãru; and 60 (to the third power) 3 (= 216,000), which was called šuššārū. The word šãru had a Sumerian antecedent (šár) that means not only “3600” but also “universe.” (See footnote 17 below).
In later times the Greeks put the sexagesimal system to full use, “both in the familiar division of the circumference of the circle into 360 “degrees’ of 60 minutes or 3600 seconds each, and in the division of the radius into units of consecutive sixtieths.” By employing the šãru as the key to unlocking the antediluvian numbers in the Sumerian King List as well as in Berossos, we find ourselves not only discerning “rational” numbers depicting the length of royal reigns in those ancient chronological tables but also walking in the footsteps of noble mathematical tradentes.”
Footnote 17:
O. Neugebauer, The Exact Sciences in Antiquity (2d ed.; New York: Harper, 1957) p. 141. U. Cassuto, A Commentary on the Book of Genesis. Part I: From Adam to Noah (Genesis I-VI 8) (Jerusalem: Magnes, 1961) p. 258, has observed that the 241,200 of the antediluvian Sumerian King List equals one great šãru (šuššārū—i.e., 216,000—plus seven šãru—i.e., 7 χ 3600 or 25,200) and that the 432,000 of Berossos equals 120 šãru (i.e., 120 χ 3600) or two great šãru (= two šuššārū—i.e., 2 χ 216,000).
Footnote 19:
I am deeply indebted to my daughters, C. Felicity Harrison and H. Judith Virta, for reviewing this paper critically, to my son, Graham K. Harrison, for technical advice involving the mathematical analysis, and to Ronald Youngblood for the Sumero-Akkadian and Greek information in the final paragraph and for the references in nn. 17 and 18 (footnote 18 omitted here).
R.K. Harrison, “Reinvestigating the Antediluvian Sumerian King List,” Journal of the Evangelical Theological Society (JETS) 36 / 1 (March 1993), pp. 6-8.
Samizdat
June 18, 2015
Is the šãru the Solution to the Impossibly Long Antediluvian Reigns?
“Regardless of the names, however, it is apparent that when the formula for calculating the actual length of reigns is applied, the figures on Berossos’ list of ancient Sumerian kings are amenable to precisely the same treatment as the original Sumerian King List.
Among all the extant exemplars of the Sumerian King List, the Weld-Blundell prism in the Ashmolean Museum cuneiform collection represents the most extensive version as well as the most complete copy of the King List.
In this depiction, all four sides of the Sumerian King List prism are portrayed.
http://cdli.ox.ac.uk/wiki/doku.php?id=the_sumerian_king_list_sklid=the_sumerian_king_list_skl
This indicates that Berossos was thoroughly familiar with the Sumerian system of computing lengths of reigns, as expressed on the Weld-Blundell prism, and that he was representing the priestly tradition many centuries later in his own configurations.
The revised king list of Berossos is as follows:
Berossos’ figures constitute a remarkable tribute to the tenacity of ancient priestly traditions, since the Babylonians had normally used base-10 in their mathematical calculations for many centuries. Berossos, however, felt a commitment to honor the ancient heroes whom he was listing in the age-old Sumerian manner.
In attempting to provide a “rational” solution to the problem of large numbers in the antediluvian King List, I have said nothing as to precisely why base-60 squared was employed in the listing.
Scholars who have checked the numbers are satisfied that they have been transcribed accurately, with the result that the issue must now turn on mathematical considerations, as Young has suggested. From a prima facie standpoint it is no longer legitimate to question the numbers themselves, but instead to recognize the possibility that base-60 squared was actually functioning as a mathematical constant.
So little insight has been gained into the theoretical dynamics of Sumerian mathematics that it is impossible to say with certainty what the reason was for employing base-60 squared as a constant, assuming that this was its actual function in the King List, as seems eminently probable.
Calculation of the surface area of terrain at Umma, Mesopotamia (Iraq). Ur III Clay tablet (2100 BCE) 7 x 5.8 cm AO 5677, Louvre Museum.
http://www.lessingimages.com/viewimage.asp?i=08020612+&cr=328&cl=1
It was certainly integral to the structure of the various recorded reigns, unlike some constants in modern mathematics that grace an equation but are not indispensable entities. Why base-60 should have been squared in order to perform its function satisfactorily is also problematical. Perhaps, after all, base-60 squared was intended to serve as a symbol of relative power and importance, which the compilers of the ancient Sumerian King List associated with those men whose reigns they recorded.
Regardless of the immediate answers to these queries, it seems clear that base-60 squared should be recognized as an “ideal” constant, which, however, must be factored out once it has been isolated so that it is not reckoned as part of the overall calculation.
In any event, we know that the ancient Sumero-Babylonian sexagesimal system employed at least the following mathematical bases as units: 60° (= 1), which in Akkadian was called ištēn; 60 (to the first power) 1 (= 60), which was called šūšu; 60 (to the second power) 2 (= 3600), which was called šãru; and 60 (to the third power) 3 (= 216,000), which was called šuššārū. The word šãru had a Sumerian antecedent (šár) that means not only “3600” but also “universe.” (See footnote 17 below).
In later times the Greeks put the sexagesimal system to full use, “both in the familiar division of the circumference of the circle into 360 “degrees’ of 60 minutes or 3600 seconds each, and in the division of the radius into units of consecutive sixtieths.” By employing the šãru as the key to unlocking the antediluvian numbers in the Sumerian King List as well as in Berossos, we find ourselves not only discerning “rational” numbers depicting the length of royal reigns in those ancient chronological tables but also walking in the footsteps of noble mathematical tradentes.”
Footnote 17:
O. Neugebauer, The Exact Sciences in Antiquity (2d ed.; New York: Harper, 1957) p. 141. U. Cassuto, A Commentary on the Book of Genesis. Part I: From Adam to Noah (Genesis I-VI 8) (Jerusalem: Magnes, 1961) p. 258, has observed that the 241,200 of the antediluvian Sumerian King List equals one great šãru (šuššārū—i.e., 216,000—plus seven šãru—i.e., 7 χ 3600 or 25,200) and that the 432,000 of Berossos equals 120 šãru (i.e., 120 χ 3600) or two great šãru (= two šuššārū—i.e., 2 χ 216,000).
Footnote 19:
I am deeply indebted to my daughters, C. Felicity Harrison and H. Judith Virta, for reviewing this paper critically, to my son, Graham K. Harrison, for technical advice involving the mathematical analysis, and to Ronald Youngblood for the Sumero-Akkadian and Greek information in the final paragraph and for the references in nn. 17 and 18 (footnote 18 omitted here).
R.K. Harrison, “Reinvestigating the Antediluvian Sumerian King List,” Journal of the Evangelical Theological Society (JETS) 36 / 1 (March 1993), pp. 6-8.
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