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I've been informed by James Brundage of evidence that positional notation using so-called Arabic numerals can be founbd in European manuscripts dating from as early as the late 12th century. I wasn't aware, or had forgotten about such use, and in my statement referring to use of such notation becoming "widespread" only in the late Middle Ages or early Renaissance, I was relying on my memories of a number of books which I haven't looked at in some years, and I suppose one could argue about what "widespread" means here, e.g. in connection with everyday financial transactions as contrasted with questions of royal or high level church finances. This is not an area in which I am an expert on history of mathematics, by any means. However, I can well believe that there were relatively early uses of positional notation in Middle Aged Europe in connection with accounting practices, royal finances, etc. Here's a point concerning this which I don't think has been mentioned before. One of the great detractions of the use of Roman numerals in computations in the middle ages was the difficulty of using such algorithms as can be devised for multiplying or dividing using them, as contrasted with adding or subtracting. I expect that multiplying is necessary in connection with interest payments, which appear to have been often enough used, for example, in connection with Jewish bankers and other moneylenders from as early the 11th century if not before. This would certainly make positional notation attractive to those who knew about them, no doubt ultimately from Arabic sources, since multiplication algorithms are considerably simpler using positional notation than they are using Roman numeral notation. Gordon Fisher [log in to unmask] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%