Computing Nth Roots

Another source claimed that Tsai-Yu" would have
had to start with some numbers as large as 105 digits to
get the required precision. I don't know what algorithms he
used, but I suppose they were variations of the long-division
ones for the square and cube roots we learnt in elementary school
and promptly forgot. All 12 roots may then be calculated by taking
various combinations of the square and cube roots.
Ancient calculators were less handicapped by their notations
as might be presumed as computations were done with the help of an
abacus or counting board. Probably no one used Roman, Greek or Babylonian
numerals symbolically.
Pace Aristoxenos and the myth that he meant ET, the ancient
Greeks were perfectly capable of computing 12-tet if they had ever
wanted to. (Ditto for 24, 36, 72, and 144-tets to obtain the other
'shades' of his diatonic and chromatic tetachords.)
Archimedes allegedly invented a mechanical device called
the Mesolabium (in latin) to approximate N-th roots and other
graphical or geometric methods were known.

--John

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