At 06:37 AM 11/24/99 -0800, James A Brundage wrote:
>At 01:58 PM 11/23/99, you wrote:
>> The widespread use of positional notation, using
>>so-called Hindu-Arabic digits (represented by the individual numerals 1,3
>>and 4), appears to have only begun to spread in Europe in the quite late
>>Middle Ages, or one might even say the early Renaissance, because of the
>>interest taken in them by commercial people. (It's interesting to me to
>>note a kind of pun on the word "interest" here -- one can compare official
>>attitudes of the Roman Catholic Church toward usury with the desire of
>>embryonic Christian capitalists to lend and borrow with interest rates
> Depends, I suppose, on what you mean by widespread. In the legal MSS that
>I work with, I occasionally find scribes using Arabic numerals from the
>late C12 onwards and by the mid-C13 (at least in some areas) they're
>beginning to become quite common. Certainly people weren't consistent in
>their usage; I've occasionally encountered both Arabic and Roman numerals
>in the same MS. Some of the grand generalizations in this thread leave me
This is very interesting to me. I was going by a number of books written
by historians of mathematics, who led me to state this "grand"
generalization, though I'm always suspicious of grand generalizations
myself. I don't remember without looking back at these books how much
attention the authors paid to earlier uses of positional notation, nor what
their conceptions of "widespread" may have been. In this connection I
wasn't referring to the use of Arabic numerals per se, but rather to their
use in a positional notation. Of course, it may be that Arabic numerals
were customarily used only in connection with positional notatio).
>>There is another thing that can be considered in connection with the use of
>>Roman numerals in the European Middle Ages. Commercial people often relied
>>on the use of abacuses, which rely on a kind of built-in positional
>>representation of numbers, without using an explicit notation or names for
>>the numbers, i.e., no written or even spoken numerals are involved.
> This usage was not necessarily confined to merchants or to abacus users.
>Take a look at the C12 _Dialogus de Scaccario_, where you will find royal
>officials (accountant types, to be sure) employing a positional system
>(modeled on or suggested by the abacus) for reckoning sums due to the royal
>treasury. This kind of accounting notation, in which 134 might be
>represented as . / ... / .... remained in use at least into the C16,
>quite possibly later. Somebody has already mentioned tallies, I think,
>which worked on the same lines. Charles Johnson's introduction to his
>translation of the _Dialogus de Scaccario_ has further information about
>this, as well as additional references.
Ah ha! Again, as you say, this may depend on one's conception of
"widespread". On the other hand, the writers on whom I based my opinion may
simply been unaware of these earlier usages. This is in no way an area in
which I can be considered expert as to the history of mathematics.
> Apologies for butting in--this is not really my field at all, but lawyers
>(as has often been noticed) frequently do get involved with money and
>James A. Brundage
>History & Law
>University of Kansas
><[log in to unmask]>
Yes, my wife and are contemplating establishing a revocable trust of some
sort, and are trying to do it without the usual use of a lawyer, a
procedure which is recommended by a book on estate planning my wife has.
The argument of the book's author is that it's easy to establish such a
trust when you know how (which he explains quite clearly), and lawyers
charge a lot for this service. On the other hand, we will consult our
financial advisors and tax accountant, whom we pay anyway, although here we
think we'll get some free advice from them. But this is undoubtedly more
than you wanted to know about our financial affairs!
Gordon Fisher [log in to unmask]