Dear Bella,
Sorry to be so late to take this thread up again, but for some stupid
technical reason (after a temporary absence I had forgotten to resume my
listmail...) I had missed all the med-rel postings since 16 September. So
here I am back again, at least for the moment.
>A belated but heartfelt thank-you for all three responses to my
>dust-board query, which I have been following up. Otfried's lucid
>disentanglement of the abacus (in the modern sense) from the dust-board
>led me to ask why I had got them entangled in the first place, and this
>helped me towards an answer to the problem of the disappearing
>dust-board. It seems to be as much as anything a linguistic difficulty.
>There were two ways of describing the dust-board in antiquity, as an
>'abacus' (a Latinization of Gk. 'abax', genitive 'abakos') or as
>'pulvis' (or the diminutive form 'pulvisculus'). Two successive problems
>arising from this nomenclature: i) since the usual phrasing associated
with the use
>of a dust-board is 'in pulvere', it's often hard to tell whether the
>reference is to a dust-board, unless the context is unambiguous; what
>Cicero called 'pulvis
>eruditus' becomes indistinguishable from ordinary dust [a salutary
>warning to academics against the pride of life?]. ii) once the counter
>abacus is introduced, again it's often hard to tell whether 'abacus' is
>being used in its older or its newer sense.
The few counting-boards extant from Greek and Roman antiquity are all
boards with or for counters, not dustboards. To the best of my knowledge,
written sources from this time don't describe the dustboard as a
counting-board. There is only the Greek name "abakos" to infer that at some
early stage no distinction was made between the dust-board (for drawing and
eventually writing) and the counting-board (for reckoning with counters),
because somehow the name of the first must have been transferred to the
latter. The Latin words "pulvis" and "pulvisculus" will normally refer to
the dust-board (for drawing), not to the counting-board ("calculator"). As
literary motivs, both stand for erudition in geometry (dust-board) and
arithmetic (counting-board).
However, a search of the
>Middle English Dictionary turned up some usefully explicit LME glosses
>for the ignorant. The South English Legendary Life (c. 1300) of St Edmund of
>Abingdon, telling the story I cited before from the canonization
>proceedings, explains what Edmund was doing teaching arithmetic at
>Oxford: 'Arsmetrike is a lore that of figurs al is, And of draughtes as
>me draweth in poudre [=dust], and in numbre iwis' (EETS 236, 2.500,
>lines 227-8). And in the late C14 John of Trevisa, translating Higden's
>_Polychronicon_, which draws here on William of Malmesbury, glosses the
>claim that Gerbert of Aurillac took the 'abacus' [here = arithmetic?]
>from the Saracens, 'Abacus is a table with the whiche schappes beth
>portrayed and i-peynt in poudre, and abacus is a craft of geometrie'.
What Gerbert (although it is possible that he was not the first) imported
from the Arabs in Spain was the counting-board with **numbered** counters
and the specific numerals used to number them. The counters on the ancient
counting-board were **not** numbered, and this ancient counting board was
no longer in use during the early middle ages, but was known only by
hearsay (as in Isidore's _Etymologiae_). It was reintroduced (first as a
new variant with non-decimal columns for monetary units) in the 12th/13th
centuries.
>What both quotations suggest, I think, is that academics at any rate were
using
>dustboards in England into the late Middle Ages, certainly for geometry
>and possibly for other kinds of calculation. But I'd welcome any other
>evidence or comments...
I would not put much weight on the passage in Trevisa, because the
dust-board as an instrument and standard attribute of Geometria was common
knowledge in the tradition of Martianus Capella, but this does not
necessarily mean that it was actually used. Chalk-boards and wax-boards
would have been a much better choice for precise geometric drawings. The
passage in St. Edmunds seems more interesting to me, but it speaks of
"drawing" in dust, not of writing the numbers (or the columns and dots of
an abacus) in dust.
It was only after having written my earlier message that I re-read Paul
Acker, _The Emergence of an Arithmetical Mentality in Middle English
Literature_, in: The Chaucer Review 28,3 (1994), p.293-302, who writes on
p.296:
According to Steele [Robert Steele, _The Earliest Arithmetics
in English_, London 1922 (= EETS, Extra Series 128), p.xv], algo-
ristic reckoning was first carried out using "a board
[covered] with fine sand," similar, apparently, to the Greek sand
tray described by Martianus Capella and Trevisa. Such a board may
be the intended referent in a remarkably early (?a 1200) and
unlikely context, the <i>Ancrene Wisse</i>. Here, in a somewhat
complicated simile, we learn that a covetous man is like a
fire-tender; in the ashes he makes "figures of augrim, as thes
rikeneres doth the habbeth muche to rikenin." These "figures of
augrim" are Arabic numerals, and here we find our earliest known
Middle English reference to them. But in this passage they serve
only to improve the reckoning abilities of the devil's
accountant; such associations with the mercantile world only
added to the taint suffered by arithmetic.
I don't know which sources are adduced by Steele to support his claim, but
if there is only the passage in the Life of St. Edmund in the SEL (written
when algorism had already been diffused in Europe for more than a century)
to support it, I would hesitate to be convinced. There is also Leonardo da
Pisa ('Fibonacci'), who in his _Liber abaci_ (which has nothing to do with
the counting-board, but introduces reckoning with Indo-Arabic numerals, and
was written ca. 1200, revised in the only extant version 1228) frequently
mentions a 'tabula' or 'tabula dealbata' where intermediary results are to
be noted, and this 'tabula' has been interpreted by Andre/ Allard
(_Muhammad Ibn Musa Al-Khwarizmi, Le calcul indien: Algorismus. Histoire
des textes, e/dition critique, traduction et commentaire des plus anciennes
versions latines remanie/es du XIIe sie\cle_, Paris: Blanchard, 1992, p.85;
beware, I am quoting from second hand!) as a dust-board, whereas Heinz
Lu"neburg (see his _Bemerkungen zu Abacus und Algorismus_ at his website:
http://www.mathematik.uni-kl.de/~luene, and see also his _Leonardi Pisani
Liber Abbaci, oder Lesevergnu"gen eines Mathematikers, Darmstadt: BI
Wissenschaftliche Verlagsgesellschaft, 2nd ed. 1993) argues for a wax-board
or chalk-board. I would say that 'tabula dealbata' in fact is very likely
to refer to a chalk board (a wooden board covered with a layer of dried
chalk, to be washed after use and covered with a new layer).
A good place to look for further info on medieval dust-boards might be
Robert Fossier, _Polyptiques et censiers_, Turnhout: Brepols, 1978 (=
Typologie des Sources du Moyen Age, 28).
So much for tonight,
Otfried
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Otfried Lieberknecht, Schoeneberger Str. 11, D-12163 Berlin
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>Many thanks,
>Bella Millett
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