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24
MarekOtisk
intofourpartsbyparallellines.Thesymbolsoftheindividualdecimalscales
are,accordingtothetradition,writteninthetoppart.Ingeneral,thisdraw-
ingservesprimarilyasanillustrationofthetext,intendedforthepurpose
ofclarificationforthereaderofthetreatise,and,inallprobability,itdoes
notrepresentanyrealcomputinginstrumentusableformathematicalopera-
tions
41
.
TheVaticanabacus[V]seemsinmanywaystohavebeeninspiredbythe
pseudo-Boethiusabacus,anditconstitutespartofaconcludingmathematic-
abacistinsertioninacollectionofscientifictexts(Vatican,Lat.644,f.77v1
78).Althoughthemanuscriptdatesbacktothe10
th
century,itslastfolios
withtheabacusdrawingcomefromthe11
th
century
42
.Furthermore,thisaba-
cus(seeplate5),similarlyto[G],statesnamesandsymbolsoftheghubar
numerals(includingthesiposassistantmarkwhichispushedintoonecolumn
togetherwiththenumbernine,probablytoeliminatethepossibilityofconfus-
ingitwithzeroorten),anditisdedicatedtocalculationsuptohundredsof
billions(i.e.10
11
),thusconsistingofonly12columns.Itseemsthattheauthor
ofthedrawingstruggledalittle(similarlytotheauthorof[P]),sincehesub-
sequentlyaddeda13
th
columnforcalculationswithfractions,asevidencedin
thedrawing.Theusualarchesroofingthecolumnsare,unusually,writteninto
theindividualcolumnsandrepeatedthreetimes.Incertainways,thisaba-
cusseemstobethemostcomplex.Apartfromcolumns,whichincludemarks
fortheindividualdecimalscales(thoughthecommondivisionwiththehelp
ofthelettersC,D,andS/Mismissing,similarlyto[G]),andapartfromsym-
bolsaswellasnamesforArabicnumerals,italsoincludesthehexameter
onGerbertofReimsandhisinfluenceontheuseofnewnumeralsamong
Latinscholars.Furthermore,itcontainssymbols,names,andmutualratiosof
fractions.Thehexameterandfractionsarealsoincludedin[B],thoughin
thisparticularcase,thesymbolsarepartiallydifferentand,moreimportantly,
thesiliquaisincorrectlystated(accordingto[V],itrepresents1/2ofcalcus,
whileusuallyitwouldrepresentthevalueof3/4ofcalcus,sinceitis1/6of
scripulus,andcalcusis1/8ofscripulus).Thus,thevalueofsescunciaisstat-
edincorrectlyaswell.Itis,therefore,possibletoassumethattheindividual
whodrewthisabacushadonlypartialknowledgeofthemathematicalissues
described.
41
Similarcasecanbefoundinmathematicalmanuscriptfromthe11
th
century(Mont-
pellier,H491,f.76r,resp.f.79r1v),includingBernelius’stextLiberabaciwithexamples
ofabacisticcalculationsandRomanfractions.Allthisisthentransferredintoaccompanying
illustrationsinformofabacistictables.
42
Details,incl.photocopyandreferencestoothereditionsinM.Folkerts:FrüheDar-
stellungen…,pp.29130,resp.p.42;orIdem:Thenamesandforms…,p.253.
