A note posted to two discussion groups ... a reply to the following may be necessary :
Thank you (Maria) for the post. Egyptian algebra has been labeled rhetorical algebra by (19th and 20th century) math historians (that only considered additive aspects). Unknown values were rhetorically discussed before hard-to-read scribal shorthand calculations were written down.
RMP 32 is a case in point. The algebra is trivial:
x + (1/3 + 1/4)x =2
is solved today by
(19/12)x = 2
x = 24/19 = 1 + 5/19
Ahmes solved the problem the same way, adding three level numeration aspects (proto-number theory) that otherwise well-informed scholars failed to parse as Ahmes recorded.
One aspect of Ahmes? work scaled 5/18 by 12/12 such that . * = (38 + 19 + 2 + 1)/228 concluded
x = 1 + (1/6 + 1/12 + 1/114 + 1/228
A second aspect scaled the entire equation by 114 scaled
A third aspect included a proof that scaled the remainders to 912.
Greek algebra followed the same scaled arithmetic logic that.recorded rational numbers in exact unit fraction series. .
By the time of Diophantus, indeterminate equation algebra flowered in ways that are explained by the arrival of the Chinese Remainder Theorem (CRT) on the Silk Road. Fibonacci included a medieval version of the CRT in the ?Liber Abaci? that Diophantus would have recognized..
Fibonacci?s arithmetic included exact unit fraction series the used of an algorithm. Medieval number theory scaled rational numbers n/p such that (n/p ? 1/m) = (mn = p)/mp set (mn -p) = 1 as often as possible, a notation introduced by Arab algebra around 800 AD.
Conclusion: Egyptian, Greek, Arab and medieval multiplication included a dual definition that scaled rational numbers and repeated addition, as Maria is discussing.