Dear all,

My still somewhat breathless efforts at starting a new dull life beyond
scholarship and academe should rather prevent me from joining your
discussions, as I do not really have the time (nor all the necessary books
at hand) for preparing what I would like to contribute. Yet the recent
thread about numerals  etc. has touched upon several points where I cannot
resist to throw in a few hurried comments. In her last posting on this
subject, Julia Barrow wrote:

>1: Medieval abacuses were not exclusively on the decimal system, but 
>could have different systems concurrently (cf. the English 
>Exchequer).

This seems quite an important observation to me, one I frequently miss in
more generalizing descriptions of this matter as given by certain
historians of mathematics and others. Unfortunately, not much work has been
done on abacuses of the 13th-14th centuries, whereas we are comparably well
informed about the 10th-12th and the 16th- centuries. Nevertheless I
suggest that we distinguish, roughly, the following types and periods:

A) Ancient Greek and Roman abacuses:
With decimal columns and (usually) additional columns for fractions.
Unmarked counters (psephoi, calculi, claviculi) representing the ones,
tens, hundreds, thousands, etc., and -- as regards fractions -- the halves,
[thirds], quarters, [eigths] of an obolus/ounce. Each column accompanied by
an additional column or space where a single counter represents 50% of the
next higher column. In the case of decimal columns this means 'bundling by
fives', that is, to represent 50 you don't put 5 counters into the column
of tens but one single counter into the additonal space of this column.
It's the same principle which rules the Roman numerals (decimals I, X, C,
M, bundled by fives: V, L, D). In the case of the Greek abacus, it seems
that the next higher colum to the thousands (of dragmas) was a non-decimal
column for talents (= 6000 dr.), whereas the columns for fractions of one
dr. (1 dr. = 6 ob.) were for 1, 1/2, 1/4 and 1/8 obolus (inscribed I,
C=hemiobolion, T=tertatemorion, X=chalkos). Hence the Polybian saying that
the courtiers of a king are similar to the counters on a counting-board:
like an abacist the king decides whether a courtier is a 'talent' or a
'chalkos'.

These ancient abacuses where sufficient for additions and subtractions, but
not really helpful if it came to multiplications and divisions. It is
difficult to date their disappearance in late antiquity, when the technique
was still sufficiently widespread to give origin, in Spain around 400 CE,
to the word "calculare". Yet apart from occasional literary references
presumably only quoting what had been found in earlier sources (e.g.
Isidore, Etym. X,43: "Calculator, a calculis, id est lapillis minutis, quos
antiqui in manu tenentes numeros conponebant") there is no direct or
indirect trace of counting-boards in the early middle ages until the late
10th century.

B) The 'monastic' abacus of the 10th-12th centuries:
With decimal columns and (usually) additional columns for fractions. The
counters (called "apices" pars pro toto) are marked by figures 1 thru 9,
thus you need only one single counter for each column. Normally no bundling
of fives. The figures ("apices", "caracteres", "figurae") inscribed on the
counters may in some cases have been Roman or Greek numerals, but usually
they were rather esoteric figures presumed to be of 'Chaldean' origins,
whereas modern scholars have traced them back to the West-Arabic Gobar
numerals (a sort of simplified variant of the Hindu-Arabic numerals). It is
not unlikely that it was Gerbert who imported them from Spain. Their names
were 1=igin, 2=andras, 3=ormis, 4=arbas, 5=quimas, 6=calcis or caltis,
7=zenis, 8=temenias, 9=celentis. They served exclusively for marking/naming
the counters, but were not yet used for writing/reckoning without an
abacus. Zero was known as 'sipos' and written as a circle, but served a
different purpose than 'zero' in the decimal system. The columns were
closed by arcs, and the whole thing was believed to have been invented by
Pythagoras, hence the pars pro toto name "arcus Pythagorei" for this abacus. 

This kind of abacus with marked counters was appropriate also for
multiplications and divisions (although the rules for these have a fame for
being quite hard to learn). It never gained much diffusion, especially not
outside monasteries and schools, and served mainly for scholarly purposes
(nevertheless there is a treatise originated in South Italy in the 11th
century, which recommends a variant of this monastic abacus for the
computus and for the quadrivial sciences, but also gives an example which
may suggest that it was used also for financial reckoning: "quinquaginta
solidi quot sunt denari?"). The monastic abacus and its marked counters
came out of use during the 12th century. Adelard of Bath was one of the
last to write a treatise on it, and the last passing reference that I know
of is in Fibonacci's _Liber abbaci_ (2nd ed. 1228). Scholastic milieus soon
switched to 'algoristic' techniques as described further below, wheras
bankers and the like -- at least outside Italy -- adopted new and simpler
forms of counting-boards with unmarked counters.

C) Financial counting boards and combined variants of the high and later
middle ages:
With unmarked counters (nummi, jetons) and non-decimal (or mixed decimal
and non-decimal) columns usually based on the tradtional Carolingian
measuring system. Early form without, later forms with addtional spaces for
bundling of counters. The early form is attested by Richard Fitz
Nigel of Ely, Royal Treasurer at the Court of the Exchequer under Henry II,
in his _Dialogus de Scaccario_ quoted already by Julia and others (composed
1176-77, containing also some later interpolations). In this case the
columns were seven, the first four of them non-decimal columns representing
nummi (pence), solidi (shillings, with 1 sol. = 12 nummi), librae (pounds,
with 1 lb. = 20 sol.), and vigenae (20 librae), and the last three being
decimal columns representing hundreds, thousands and tenthousands of
librae. There was a rudimentary technique of bundling for the second and
the third column: a silver penny in the second (solidi) equalled 10 solidi,
and a gold coin ("obolus") in the third represented 10 librae. All in all a
very simple device, useful for additions and subtractions, but useless for
multiplications and divisions. Similar counting-boards with, presumably,
more consequent handling of bundling techniques gained wide diffusion in
France, England and Germany from the late 13th century on. They were used
in courts, comptoirs and common households, and the large number of extant
jetons issued by courts, trading companies and other institutions or
corporations offers interesting archeological evidence for this diffusion.
It seems characteristic for this period that the counters were no longer
placed into vertical columns but rather on horizontal lines. It seems also
characteristic that from a certain time on this kind of non-decimal
financial counting board was deviced for additional or optional decimal
use, by dividing the spaces between the horizontal lines into decimal
columns. Thus you could use the same counting-board for financial
non-decimal and for scholarly decimal reckoning. The earliest evidence for
a combined, though still somewhat different, counting board of this kind is
a sort of ruler invented by a certain Johannes de Elsa, a "canonicus et
magister scolarum" at Bordeaux. This ruler served to subdivide horizontal
spaces and additional  bundling spaces, and by renaming these spaces and
intermediate spaces you could switch from financial reckoning "sicut
computant mercatores" (with denarii, solidi, librae, 20 librae, 100 librae)
to decimal reckoning "secundum doctrinam algorismi" (with ones, tens,
hundreds, thousands) or to reckoning with sexagesimal fractions for
astronomical purposes (with signa, gradus, minuta, secunda, tertia).


In the above message, Julia Barrow also wrote:

>2: This is yet another wild generalisation, but I suspect that 
>medieval businessmen may actually have been rather slow to adopt 
>Arabic numerals - I can't talk about Italy, though - Italy may have 
>been different. After all merchants are still using Roman numerals 
>quite often in the early modern period.

Here It seems worthwile to distinguish the use of written numerals
(figurae) for the purpose of *recording* numbers on the one hand, and their
use for *reckoning* on the other. In the domain of *reckoning* (operative
arithmetics), the competition -- if I may say so -- was between
Hindu-Arabic numerals and the counting-board, whereas Roman numerals were
only a system for recording numbers but not appropriate for reckoning
purposes.

As regards the techniques of *reckoning*, Italy was in fact peculiar if
compared to other European countries. Italian bankers, accountants and
merchants already in the 13th century fully adopted the Hindu-Arabic
numerals and the new techniques of reckoning based thereon, while their
French, English and German colleagues at this time were still busy with
learning the equally new art of casting jetons. The history of the word
_abacus_ itself is telling: in Italy, and only there, _ab(b)acus_ and its
vernacular equivalents already by the end of the 12th century had switched
their meaning from 'counting-board, art of reckoning with a counting-board'
to '(art of) reckoning in general or reckoning with Hindu-Arabic numerals
specifically' (see, for instance, the title of Fibonacci's introduction to
the new Hindu-Arabic numerals: _Liber abbaci_).  Judging from the evidence
of literary/documentary sources and from the archeological evidence of
extant Italian jetons we can infer that Italian bankers in the 13th and
14th centuries used abacuses only in foreign countries where these were
still the instruments of common use.

Now as regards the *recording* of numbers, Tom Rees has already pointed out
that Hindu-Arabic numerals can be tampered with more easily than Roman
numerals or numerals spelled out in their linguistic form. This was
precisely the point of what seems to be the earliest extant prohibition,
dating from Florence 1299, and fixing a mute of 20 solidi for moneychangers
who, in their official books of accountancy, use Hindu-Arabic numerals
(i.e. by writing numbers "modo abaci") instead of Roman numerals (writing
numbers "per literam"). Menninger also quotes a Venetian text -- without
giving its date -- where Roman numerals are recommended as follows: "lequal
figure antique solamente si fanno, perche le non si possono cosi facilmente
diffraudare come quelle dell'abaco moderno, lequal con facilita di una sene
[i.e. segno] potria fare un'altra, come quella del nulla, dalla qual sene
potria far un 6 [o] uno 9 e moltre altre si potriano mutare." Until the
18th century or even later, books of accountancy all through Europe
normally use Roman numerals (or a mix of Roman and occasional Hindu-Arabic
numerals) when referring to money or to goods while often adopting, at the
same time, Hindu-Arabic numerals for page numbers or for writing the number
of the year. In such cases we can assume that the reckoning was done in
Hindu-Arabic numerals or by casting jetons whereas the results were still
denoted in  traditional Roman form.

Furthermore, Julia Barrow wrote:

>3: Knowledge of Arabic numerals was for a long time in the Christian 
>west limited to the few people who had attained to a fairly advanced 
>understanding of mathematics, usually a level of education which 
>would not necessarily be available in the schools (at least not, I 
>suspect, in the 12th c. - the 13th c. was probably different) but 
>which would require private tuition from someone already initiated 
>into the system. These people were not merchants and probably had 
>relatively little contact with them. Education is the key to this 
>whole problem. Most school education in the Christian west devoted 
>little attention to arithmetic and would presumably have used Roman 
>numerals. Unlettered people would have used numbering systems (as on 
>tallies) not dissimilar to the Roman system. 

It seems in fact reasonable to distinguish between the 12th and the 13th
century when discussing this point. There were actually two different
though overlapping periods when Hindu-Arabic numerals (or derivates
thereoff) were introduced to the West. The first (the 'abacistic' period,
if we use the term 'abacistic' in a very narrow sense as referring only and
exclusively to the 'monastic' abacus described above) went from the late
10th to the late 12th century, when Gerbert and his followers adopted the
West-arabic Gobar numerals 1-9 marking the counters on the monastic abacus.
But as said before, these "apices" or "caracteres" were not used
independently from the abacus, and so this first period can not really
count as an introduction of Hindu-Arabic numerals to the West.

The second period, the one which introduced the Hindu-Arabic numerals
proper, is the 'algoristic' period, named after Al-Khwarizmi (d. ca. 840)
and his Introduction to the Indian numerals (not to be confounded with his
Algebra).  His Introduction, lost in the Arabic original, was a brief
description of these numerals and of the elementary techniques of using
them for additions, subtractions, multiplications and divisions. It is
extant in various Latin redactions the earliest of which (_Dixit
Algorismi_) seems to have originated in the first half of the 12th century.
The Latin tradition of Al-Khwarizmi was the main source for more widespread
introductions such as Alexander de Villa Dei's _Carmen de algorismo_ (ca.
1220) and Johannes de Sacrobosco's _Algorismus vulgaris_ (ca. 1250). These
were copied frequently, were taught in universities and schools (e.g.
Petrus de Dacia's _Commentum super textum Algorismi_, based on Sacrobosco
and delivered in Paris in 1291), and they came to be translated also into
the vernacular languages (vernacular treatises of this kind abound in
Italy, but they can also be found in England and France, e.g. an
Anglo-Norman 13th century _Algorisme_ in Prose, based on Villa Dei, and,
from the same period and region, an _Argorisme_ in verse, based on
Sacrobosco). From the 13th century on, treatises of this kind where
available a bit everywhere in Europe. Apart from this widespread Latin and
vernacular tradition based on Al-Khwarizmi's Introduction we know also of
independent influences established by cultural contacts or trading contacts
with the Islamic world. A famous case is Fibonacci (Leonardo da Pisa).
Fibonacci's father was a notary (publicus scriba) at the 'dogana' (a sort
of administrative institution) of Pisan merchants in Bougie, in Algeria. He
brought his son with him to Bougie to have the boy educated there in the
art of reckoning with Indian numerals. Fibonacci later travelled also to
Syria, Egypt, Greece, Sicily and Provence and came to be acquainted with
the various techniques of reckoning practiced there.

So I would say, yes, knowledge of the Hindu-Arabic numerals proper (and of
the art of reckoning with them) was relatively scarce in the 12th century,
when knowledge of this kind was still based based on relatively rare Latin
copies of Al-Khwarizmi, or on cultural contacts with the Arabic world or
private tuition as in the case of Fibonacci, whereas this kind of knowledge
gained relatively wide diffusion in the schools of the 13th century, though
still with little impact on financial or mercantile reckoning except in
Italy. Nevertheless it would be interesting to investigate closer than I
myself have done how the new algoristic techniques came to be used
alongside with the counting board, as in the case of Johannes de Elsa who
describes the decimal use of his counting-board (or counting-ruler) with
the words "iuxta doctrinam algorismi".

Btw, my own interest in these worldly matters came from investigating the
question whether abacist and/or algoristic traditions had any impact on
medieval number exegesis and number composition. If anybody is interested
in this particular problem and is not afraid of reading my somewhat baroque
German doctoral prose, I can refer to the relevant chapter of my
_Allegorese und Philologie_ (Stuttgart: Franz Steiner Verlag, 1999, Text
und Kontext 14), chap.5 (electronic PDF version of this chapter available
from me on personal request).

Apologies to all who rightfully think that this message is far too long.

Yours (formerly),

  Otfried

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