source file: mills2.txt Date: Sat, 14 Oct 1995 08:30:29 -0700 Subject: Computing Nth Roots From: "John H. Chalmers" 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 Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 14 Oct 1995 19:08 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA20542; Sat, 14 Oct 1995 10:08:24 -0700 Date: Sat, 14 Oct 1995 10:08:24 -0700 Message-Id: <951014170639_71670.2576_HHB40-4@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu