Subject: [HM] Additive progression in prehistoric mathematics
From: Milo Gardner (milogardner@juno.com)
Date: Wed Mar 22 2000 - 09:32:54 EST
An interesting 'heads up' was dropped into my email box a couple of days
ago, touching on this ancient artifact raises several issues. I strongly
suspect that the sequence of weights were associated with the solution of
the Hultsch-Bruins method to solve RMP 2/p series (even though Bruins
himself may have not suspected).
Note that the first partition A, as some of you may recall as:
n/p = 1/A + (nA - p)/Ap
where A, a highly composite number, selected in the range
p/n < A < p
(after Diophantus the range of A < p was extended to at least 2p, as
Fibonacci
may have copied from others (as recorded in Liber Abaci, 1202 AD).
Any other comments?
Regards to all,
Milo Gardner
------------------------------
Petruso, Karl M.: Additive progression in prehistoric mathematics:
A conjecture. Hist. Math. 12, 101-106 (1985).
The reviewer remarks that ancient systems of numeration show alternating
bundles: in Sumer 6,10, in Greece and Rome, 2,5, in Egypt mostly 10-
bundles. There is no question of a positional system. In addition to
that we have in Egypt two cubit values, the "normal" of 7 palms, the
"royal" of 8 palms, and the Roman "as" was subdivided in 12 unciae,
which had in sets containing a smaller number each their own name. The
present author treats weights found in a shipwreck near Cape Gelidonya:
thirteen weights of which once a multiple of 3,12,31,50,54, twice a
multiple 5 and five times a multiple 7 was approximately present.
Multiplying the unit as experimentally, and thus with the greatest
experimental error determined at 9.3 he suggests a relation with
distorted Fibonacci numbers:
1,2,3,5,8,13,21,34,55,89... assuming a diminishing of 8 to 7, leading to
1,2,3,5,7,12,19,31,50,54. No bothering about the tremendous difference of
54 and 81 he considers this system as warranted. The reviewer computed
the "most probable line" $y=mx.$. on which these shifts from 30-31 etc.
had hardly any influence and found the inclination m to be 9,307..
leading to a set of most probable values in the series 1-3,5,7,12,30,50,55
slightly better represented than the values obtained by the author with
9.3-multiple and shifting to 31 and 54 at two places.. the gap 54-81 not
being explained by the author.
[ E.M.Bruins ]
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