Aiutf^nietic^ Pars I. Cap.VIL 235 



1 1. Operatio -continctur his rcgiilis.i ./« diiiifionc 

 qturittir^ cjuoties < iitiifor in mtmero verticali fue fupra- 

 fripto contineatm', x.Si plures frit mt<i dtttifri^^de fni- 

 ftra tantiim inflitn. UHr cjHxflio , ejuoties infuperiori conti- 

 neutnr.^.Tot ftnt vperattones,ijitot fint notie in diuiden- 

 do reli<]H£iadiiitii^7ittate,fdiHtfor iarnfit fubfcripttis:vt 



14 

 Htcftnt tresoperAtiones.^.Quotmfempervnica tantitm 

 fiiuru expritnttur. 'I^ltinejuarn igitur nurncrns nouenario 

 rnaior in Oitotiente fcribi poteit : licet diuifor ftpiits 

 atiam nottics m diuidcndo conttneatur, j .Quotns per fn- 

 ^ttlas diuiforis notat debet rnulttpltcari : produHum infrk 

 fcrtbi ora diitidendofui/trahi. 6.Hoc rnultipltcationis (^ 

 diitifionis negotitim toties fiequentandttm, ^HOtfunt notA 

 m dtuidendo numero reliqtt& : donec nttUus in numero di- 

 uidendofipcrfit , in tjuem dinifor pojjit promoueri. 7. Si 

 in aHcjit.i promotionc diuifor ftterit rnaior diuidendo^cri- 

 benda ell ciphra ad locum Quoti , rnox^^ delendus diui- 

 for,<j)' ad fecjHctitem locum promoiiendus. %.SirefidHHm-> 

 posiftbtraBionemftterttrnaittsdittifore , error eSi com- 

 rnifits. c).Si quid poHperaHarn totarn diuifionemfiiperft, 

 illud ad dextrarn Quoti per modtim fiaElionis fcrtben- 

 diim ftterit , itavt ftperne reftduttm dittifonis , infirne 

 AUtern diuifir collocetm\ Exemplafunt : 



4444^ 



,a^ 



T^n-'^- 



III. Probatio : cjiia inftituitur per rmtltiplicatio- 

 nern. Si narnqtie dittifor rj- Quottts infe multiplicentur, 

 (3" prodtitlo refiidumn dmtfiionts addattir,prodibit ,fi res 

 tcnefe habet , numerus dtuidendus. igy^lij exploranty 

 per abiedionem nouenarij , hoc modo. Klouenarius 

 abjicittir exdittiferedr Quoto,/]uotiespoteB, (j refidna 

 ponnntitr inftperiore (^ inftriore crttcis partc_^. Rcfidita 

 tjh m f ducHntter ^ c^ ex fk^lo iftoatijue refiduo dtnifio- 

 viffi cjuod fuertt,rHrfius abjiciHntttr cfjCfttoties ficri potefl . 

 Quodfi refiduo ifti d^ttale ftterit refiduum numeri diui^ 

 dendi jprcbabile esl operatianem bonam ejfe. e.gr. 



48^ 



M5. 



o 



DiuiHo calculis peragicnr hoc modo. tn diuifonti^ 

 ' \ defcendenditm eli a dexti latere in Imtim contrk tjuam 

 'fiiLium cFl in multipltcat.ione. Diitidendus ita<jHe collo- 

 tetur in dcxtro latere : poftea digitus applicetur infniftro 

 latere tjlt line£ , (j* infia. coUocetur Ouotiens : vt ji 1 60. 

 dmidcndafintper \o.ap pofito digito lineafecttndx, nitm- 

 mus in ftmmk pofitus denarium effcit , sjmponendus 

 iuxta digittim : deindt' proxirnus , (jtiinariiim efficiens, 

 transfirendits ad Jpatiti m /aititmfub digito.Tertius fimili 

 raitone,tranffofiio digrtOjponendus in infirna llnea. 



R E G V L i€. 



/. 'Ditiifio eFi apjylicatio datijpatij ad totam reUam: 

 ynde dicifiolet iTafctCcXtiy comparatio,Wff Geornetrica. 



Sicutduorum latcr um mutiius in fe Huftus.conficit fi^uram 

 tcvft.mgulam in niult iplic.uionc : fic in diuifionc > diuiticndfs 

 eftvcluti reflanguli aica , duiiror vtlut longitudinis latiw, 

 lioc cft,vnum coruil 1 , qul rtiT-anJ^ulum comptclit ndunt , coi 

 altcrum latiiudinis latus ortlio!;oiialitrr iungcndbm, in cjuod 

 piius duinum , i(i.i'tium fibi congrucns capiat : atquc id iatus 

 lcpr.Tfciuat >Quotv;s ;■ quippc qood indiuiformi duflu;,di- 

 «idcndum fcmpcc tcllituat. Piuifio itaciuc cft .iccommodatio 

 lateris ad daiuni latus : vndc rcL^ane^ulum^paiallclogram- 

 Jnum dcfciibi qucat . Vnd.c patct.vlum diuifi^nis rfie, vt quo- 

 tics in inaiorc minl >r continc.uur , manifcftiim fi.ir. Ita v.gr. 

 numcii dati in nni Itiplicationc j & 5,vcl 5 Sc i f"nt latcra i 

 numcrus t^ii.rlitus 15 vcl i( c[\ .irc-a. lii diuifionc diuidcn- 

 rfus cft aica, vt n vcl 1 5, diuifoi cft Litus vnum, vt 5 vd }» 

 l]uotus cft latus ali.crumj vt 5 iii vtroijuc diagiammate. 



D 



fi 



If 



J J 



Qttottts efi pars diuidendi, a diitifore denomi 



Qijotu» 



JI. 



rtata. 



Nam fi numerus numcnim metiatur, mcnfus liabet partcm 

 mcnfori cognominem , & reciprocc : vt quoniam 48 pcr 6. 

 diuidi pofTuncaio datum numcrum fextam habere partem, 

 id cft 8 : atque hic eft quotus.a diuifore nomcn liabcns.C.-cte- 

 riim pars numcri hoc loco cft numerus minor.maiorem men- 

 furant,hoceft,conftitucnsvel auferens : vt i. funt pars tcrtia 

 de 6.5 pars dimidiade 10. Qiii vero non folus aliquotics.fed 

 cum vno pluribiilve affumptus, numerum componit.is totius 

 funtpartcs: vt i.fuat partes quinarij,fcil.dui quinta;.Binaiius 

 cnim non folus , fcd afTumpto tcrnario quinaiium conftituit. 

 Mi/i®' itaque Eudidi fuerit pats quota : Mi^ii, pars quanta. 



/ / /. Simplicis Quoti diuifio , ex mtiltiplicationis slmpfic!* 

 abaco efi depromenda. '^°" **'- 



' 111(10 vnae 



Nam per ^uos numerus ftr.pfer cos quoque diiiiditur. Atque P^"^"''-' ' 

 lioc cft diuifionis quoddam rudimentum. Vt fexics fcptcm 

 funt 41. Quarc eundera numerum diuidct 7. per 6. & 6 diui- 

 det 4t.pcr 7. Item fcptics j.funt ff j. Idco cundem diuidet 9. 

 &quotus crit y.vel eum diuidet 7. &quotus etit 9. Scd hoc 

 pcricquamur plunhus.jibacus ditilfientt Pythagoricus eft du- 

 plc.x : diuifioBis cxadar,& non-exa(ftx. Abacut diuifionis e.t:- 

 aciA rurLus eft duplcx. Etfrimo quidem c multiplicationibus^ 

 notarum pcr fe aigumtntamut enthymematice dc diuilTonc 

 carundem fecundiim fe : e.g. 



I tacit I per i : ergo i diuidit 1 per i. 



1 facit 4 pcr 1 : crgo 1 diuidit 4 per 1. 



3 facit 9 pct 5 : etgo j diuidit 9 pei 5 &Ci 

 Deinde c fingularum notarum cum fingulit maiotibus multi- 

 plicationc gcmino fyllogifmo gcmina diuifio concluditur .• 

 c.g. 1 facit 6 pcr y, ergo i diuidic 6 pcr 5 j &contra j diuidic 

 6pcr 1. 



a facit S pei- 4:ergo i diuidit S per 4;& 4 diuidit 8 pci i &c. 



3 facit 15 pet 5: ergo j diuidit 15 pet 5 > & 5 diuidic 15 

 per 3; & fic pcrgcndo vfque ad noucm. 



Arque vt hunc abacum faciliiis mcmorijc mandcs , conli- 

 deia non cantiim notarum ordinem , fcd ctiam fyllogifmo- 

 tum , de vnaquaque numetotum fcrie paulatim vnitate defi- 

 cientem.Nam de 1 funt 7,de 3 fex,de 4quinque,de 5 quatuor, 

 de 6 tres, de 7 duo , dc S vnicus : tum his oppofiti , totidera. 

 Abactn diuifiotiii non.exaH/i fic habet. 



1 in 3 continetur femel,& fupeicft l ; in 5 bis,& fupereft n 

 in 7 tcr,& ftipcreft i, in 9 quater,& fupercft i &c. 



5 in 4 continctur femel , & fupercft i -, in 5 femcl , & fuper- 

 eftii in 7 bis,& fupercft i; in 8 bis,& fupctcftiiin 10 ter,3c 

 f upcreft I , in 1 1 tcr,& fupcreft r &c. 



4 in 5 continetut femcl, & fupeteft ii in 6 femcl, & fupor- 

 eft J. iin 7 fcmel, & fupcrcft j &c. 



5 in 6 continctur fcmel, & fupcicft l; in 7 femel, & fupet- 

 cftiiin 8 femcl,& fupcreft jrin 9 fcmcl,& fupetcft 4&r;. 



Ita peragi potcft ad 6.7.8.9.&C. 



Atque hic Pythagorcus duplcx diuifionls abacus cft, quem 

 affecutus difcipulus diuifionem quamlibct facilcm habcbit. 



tF. 'Diitifia peragitur tribits ; Jniientione Quott, 

 Mttltiplicationt ,iy-Subtra^ione : i tjmbus prirnnm eft 

 principale opns. 



Etcnim diuifionis vis in copofitaeft, vt fciacurquoties in- 

 ferior contihcatur in fupcriore. Quoco igitut inucnto.diuifio 

 tacito mcDtls conccptupcrarfla cflc vidctur. Scd quando ui- 

 Ucntus cli quotus , ncccllc cft vt fciatur", quotics diuifor fupc- 

 liorcm fccct.atnueabcodcduciqucat. Id vt oculis fubjiciatur, 

 multiplicatiouc & fubduckionc cft opus. 



Dilli/io 



quouodrt' 

 (><-ug.«iut I 



y. 7)iuifionis cornpendta funt dito. 



I . .*)»■ dihifi>r in ciphrat definat, oferatio folU figntficantibK'' 

 peragitur : cifhris interim »d dtxtram extrtrnu diuidendi 

 notis fubitRh -. rt 



3<74894 I 4567 I 



3 8 000 I * " I 



Quid.-]uid autcm fupra ciphras illas pofituitl dicptebendir 

 ^BtifaiUjliuiiJont iicdundabit,& cor.lhtuct fui.1ioDcm. ^ 



V 4 *• Si dlnt 



tsmpen- 

 nit (^ux / 



