Mike Mahoney suggested, concerning the HM Roman Numeral debate, that:
> ... It would be preferable, I think, to suggest to the teachers that
> they follow the lead of real historians: go to the library and find out
> from the Romans (or at least the cultural sucessors) how that did it.
> Why guess when you don't have to?
My take is: skip over the cultural successors, for they had no real idea
on the cultural basis of Roman Numerals. To Romans, it was more a business
system of numbers; as Stevin set down for base 10 decimal business
calculations, than a scientific one; that Stevin wrote a second book
to detail scientific notation!
Proof lies in the ancient scientific numeral system of Egyptian fractions,
a system that dominated Western thought from at least 2,000 BC to 1200 AD,
when Hindu-Islamic numbers became popularized in Europe (as advocated by
Fibonacci and many others).
To read the scientific texts from time periods that directly precede
or concur with Roman numeral usages, the Akhmim Papyrus from 500 AD is
an excellent starting point. Howard Eves reported the use of an
algebraic identity:
n/p = 1/pr + 1/qr, where r = (p + q)/n
that anyone can confirm was used by Ahmes to compute 2/35 and 2/91
in the Rhind Mathematical Papyrus. More interesting, to me, is the
comparison of ancient scientific texts, such as 11 of 26 lines of
the EMLR, that used the rule:
1/pq = 1/A x A/pq (algebraic identity 1.0)
where p = 8 and A = 5 and 25, suggesting interesting alternatives
(as I may have mentioned previously). To summarize, from line 1 of
the EMLR,
1/8 = 1/10 + 1/40, a series that can be compared by several methods.
However, the ancient EMLR text itself supports algebraic identity 1.0,
with A = 25 used for two out of order series. In addition the RMP
strongly supports algebraic identity 1.0 with all by one 2/pq series,
2/95, that used the general conversion rule, with n = 2,:
n/pq = n/A x A/pq
Note that the RMP used two values for A = (p + 1) and (p + q), the
later agreeing 100% with the Akhmim Papyrus rule pointed out by Eves.
Even more interesting is the scientific notation used by the 300 BC
Hibeh Papyrus, and its n/45 table, which used:
n/pq - 1/A = (nA -pq)/Apq
or,
n/pq = 1/A + (nA -pq)/Apq
where A was mentally selected as a highly composite number, within
the range:
p/n < A < p
such that the aliquot parts, divisors, of A were additively used to
find (nA -pq) values, thereby always exactly partitioning n/pq to
a small set of alternative series. Note that modern mathematicians
would require an LCM type rule to select an optimal series, to
match the historial RMP 2/p series. Did the RMP contain an LCM
rule that may have been used for this purpose, such that
2/p = 1/A + (2A -p)/Ap
was the method used in the RMP as Hultsch suggested in 1895?
I say yes, with red auxiliary numbers being the LCM rule that
Ahmes DID apply, when a set of 10 or more alternatives were
available (and infrequent problem for p < 101).
All of the above points out that even Romans used Egyptian
fractions as their scientific notation, leaving Roman
numeral arithmetic for business transactions, as the
context to understand the Medieval use of Roman numerals.
Regards to all that read and compare the ancient documents
to read Egyptian fractions as number theory, as the 1992
Encyclopedia for the History and Philosophy of Mathematics
(C.S. Roero) strongly suggests is a historical fact.
All that remains is agreeing to the historical number theory
methods used by Ahmes, that later Romans would have understood,
and Medievals, outside of Fibonacci, would not have understood.
Milo Gardner
Sacramento, Calif.