Subject: Re: [HM] Binary Numbers
From: Prof. Lueneburg (luene@mathematik.uni-kl.de)
Date: Wed Dec 31 1969 - 18:59:59 EST
Dear Colleagues,
It was known long before Harriot and Leibniz that a natural number can
be decomposed into a sum of different powers of two. For there were
sets of weights consisting of pieces of weight w, 2w, 4w, 8w, ... already
in the 13. century, the so called poids de ville. The oldest known set
dates from the year 1229 and the longest, still existing set of this
century has weights 1/8, 1/4, 1/2, 1, 2, 4, 8 ounces. (G. M. M. Houben,
5000 Years of Weights. Zwolle/Netherlands 1990 ISBN 90-70533-06-5)
Such a set of weights does not represent a positional system, as the
powers of 2 are physically present, but it hints to such a system, if
another positional system is already there. This was the situation in
the 13. century, as the decimal system came into being at that time in
occidental Europe.
Tartaglia mentions systems of weights of that kind. He says that it is
used mainly to weigh gold, silver and condiments, the latter word being
only an approximation to Tartaglia's "speciarie". Leibniz also mentions
this kind of weights in several places. They were often realized as
nested cup weights.
Here the quote from Tartaglia's, General trattato II. fol. 14recto (The
folio in the original carries an incorrect 17.) Venice 1556. The original
text has no accents, i.e., they are not missing because of the ASCII-code.
Anchora questa medesima progressione doppia principiante dalla unita
serue, & si costuma per far li campione per pesare con le bilanze
materiale, che si oprano per pesare oro, argento, oueramente cose di
speciarie di valore, ...
The Anglo-American measures for liquids also follow the powers of two.
Since when are these measures in use? The use of measures for liquids
is, of course, different from the use of weights. Nuggets don't care
about our weighing system, but, yet, we want to weigh them. Milk, however,
is bought by the pint. So, liquid measures, I think, will not lead us
to the knowledge that any number is the sum of different powers of two.
Best regards, Heinz Lueneburg
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