BAYERISCHE AKADEMIE DER WISSENSCHAFTEN

PHILOSOPHISCH-HISTORISCHE KLASSE

A B H A N D L U N G E N . N E U E FOLGE, HEFT 99

Scire litteras Forschungen zum mittelalterlichen Geistesleben

Herausgegeben

von

Sigrid Krämer und Michael Bernhard

Sonderdmck

M O N C H E N 1988

VERLAG DER BAYERISCHEN AKADEMIE DER WISSENSCHAFTEN IN KOMMISSION BEI DER C.H.BECK'SCHEN VERLAGSBUCHHANDLUNG MONCHEN

MICHAEL IiERNHAlll l

GLOSSEN ZUR ARI'TtlMETIK IIES BOETHIUS

Der Lcidcncr Codes Voss.lat.fol. 70"' enthält auf den Ulättci-ii 74r-78v ciiic aiii Aiifaiig fragincntarischc Saiiiiiiluiig voii Glosscii zur Arithiiictik. Auf die Haiidschrift machte iiiicli I'rof Uischoff aufmcrksaiii, der sie aii dcii Aiifaiig des 11. Jhs. oder iiiöglicherwcisc noch an das Eiidc des 10. Jhs. datiert. Dic Glossen siiid als fortlaufciidcr Tcs t ohiic Auszcicliiiung der Lciii- iiiata gcscliricbcii, auf die sie sich bcziclicii. Ilic Übcrlicfcriiiig ist iiiclit siiigiilär. Eiii großer Teil des Iiicr vorliaiidcncii Textes fiiidct sicli iii Haiidscliriftcii von Uoctliius, I lc iiistitiitioiic aritliiiic- tica Iiauptsäclilich als Marginal- und zu111 gcriiiscrcii ?eil als Intcrliiicarglosscii wicdcr. Wie bei Glosscii üblicl„ wcclisclt der Ucstarid voii Ilandscbrift zu Ifaiidschrift. Folgciidc Codiccs iiiit idcntisclicii Glosscii sind mir bislicr bekannt gcwordcii: Wicii cpv .iO (s. X) ciitliält fast alle der iiii Lcidciisis voriiaiidciicii Glosscii, darüber hinaus zalilrcichc wcitcrc; I>aris, DN lat. 7184 ciitliält cinc Koinincriticruiig s. X, die iiiclit iiiit dein Lcidciicr 'Text idciitisch ist, sowie Glosscii aus dein 14. Jli., die ciiicii großcii Teil des vorlicgciidcn Materials ciitlialtcii. Der l'arisiniis lat. 11241 ciitliält iicbcti aiidcrcni ciiicii klciiicii Teil der Lcidciicr Glosscii. Clm 18764 ciitliält iicbcii aiidcrcti Glosscii ciii cinzcliics dazugcbuiidciics Blatt (fol. 5). das iiiit iiiehrcrcii Glosscii bcscliric- bcii ist. Von dicseii siiid ciriigc iiii Vossianus wicdcrziiiiiidcii.

Dic Zusaiiiii~ciistcllung iin Vossiaiius bcschräiikt sich auf Glosscii voll allgcincincrcm Iiitcrcs- sc und vcroicidct iin Wesciitliclicii die rciii iiiathciiiatisclicii Erläutcruiigcii, die iii dcii Uocthiiis- Ilandscliriftcii zalilrcicli vorliandcii siiid. llalicr siiid aiicli die Lciiiniata sehr unglcicliinäßig über den Tes t verteilt. Bcriicksichtigt werden vor allein philosopliisclic, astroiiomisclic, gcotnetri- sclic und inusikalisclic Erörtcruiigcii, zu deiicii dic Erklärung griccliisclicr 'lcrriiini iiiid Naiiicii tritt. Bcsoiidcrc Bcachtuiig vcrdiciit die Glosse zu p.S0,7, wclchc die Gciicra der Musik crläii- tcrt: „Sccuiiduin vocatur diatoiiiciiiii, q ~ i o d cst durissiiiiuiii, ct hoc apud Srottos ct Urittoncs".

Uiiiiiittclbar auf die Glosscii zur Aritliiiictik folgt ciii kiirzcr Tcs t iiiit dciii Iiicipit , ,Quid sit tonus". Es handelt sicli Iiicrbci um ciiic CIossc ziir Iiistitritio iiiusica des Boctliius, die in nichrcrcri Haiidschriftcii vcrtrctcn ist, aber aiicii ciiizclii übcrhclcrt wird'. Eiiic Sichtinig des gcsaiiiten Glossciiniatcrials zur Uoctliiaiiisclic~i Aritlirrictik ist angcsiclits der Meiigc der crhaltc- iicii Handschriftcii' ciric ka~irn zu bcwältigciidc Aufizabc. Um wciiigstcris den liihalt cincs Teils dicscs Materials bekannt zu iiiaclicn, gcbc icli hier den Tes t der Lcidciicr Handschrift (L). Offcnsichtliclic Abschrcibcfclilcr habe icli aus den Haiidschriften Wicii 50 (W), Müiiclicii clin 18761 (M) und Paris 7181 (P) vcrbcsscrt, oliiic aridcrc Lesarten wcitcr zii bcrücksichtigcii. Für dcii Boctliius-Tcxt vcrwcisc icli auf die Seiten- und Zcilcnzählunc bei Fricdlcin'.

I Siclic K. A. de Meyicr. Codirei b'osiiniii Loiirii J (Codices i>iariiisciipti XIII) (Leiden 1973). S. 119. ' 2. B. in Aiiiun 40B. I'aris 7200. 7297, 1027.5 iiiid Wicii 2269; ciiizcln in Siiisrcl 1007S9i. ' Eint Listc dcr cilialtciieii Haiidsrliiiftea gibt. M. Masi. Borilzioi, Nirrtihcr 'i'lziirnry: A Trariinrioii ?Tiii? Dc iriiliiiiiioiir

oriilwierird (Ariisicrdaiii 1983). S. 5S4,3. Folgeiidc uoii inii ciiigcscl~aic Haiidscliiifteii sind iiiir arideicii Glossaturcii veischcii: Brüsscl 5144, Einsiedel~i 358, Madrid 9088, Mailaiid C 128 i i i f , Müiicha~ clni .?517, 6405 und 14401. Oafoid Balliol Coll. 306. Osford Corpus Cliiisti Co l l 118 und 224. Paris 6639, 7185, 10251. 11241, 13009, !4!6% 17858 und NA 3044. Wien 55. Niclit glossicit sind dic Maiidsclirifrcii: Ccsei,a Plut X X V I 1, Loiidoii Lambctli lJal. 67, Müiiclic~i clni 13021. I'arir 7181, 11242, St. Gallcn Vadiaiia 296, Wiai 2269. ' A. M. T. S. Boctii dc iiisririirioi>c aiithnictica cd. C;. Friedlciii (Leipzig 1867).

<;losscii zii i Aiicliiiiciik dcc Boetiiiiis 25

tica prior est digiiitatc tribus artibiis, sicut ct gciius prius cst spctic' quia piiira coiitiiict i i i sc. Diaieiiai.o~i (11,12), id cst dc qnatuor soiiis dicitiir vcl cordis iiioduiatio. Dya ciiiiii, qiiaiido (pcr) y scribitur. dito, qiiaiido pcr iota, dc vcl cx sigiiificat. Uiidc diatcssaroii dc qiiatiior sotiis dicitur; quac inodulatio fit iii iiiusica, cum iiiaior soiius iiiiiiorciii tcrtia partc pracccdit. Iii aritliii~ctica vcro cpitrica vcl scsquitcrtia dicitur proportio, ut quatuor ad trcs coinparati. Vciiiuiit ciiiiii cx ariclimctica iiiusicac proportioiics. Diapeilt~ (11,12) dc qiiiiiquc. Iii qiiiiiquc ciiiin cordis fit illc soiius iii musica, iibi soiius soiiiiiii iiicdictatc siipcrat, qiiod iii aritlinictica scsqiialtcra ct scscupia atquc Iiciiiiolia proportio dicitur. I-lciiiiolioii coiiposituiii cst cx licmis ct oloii, id cst sciiiis ct totuiii, iit II l 'bd diios. Maior ciiiiii iiuiiicriis iriiiiorcm totuni iiuirieruiii Iiabct iii sc ct iiisupcr cius incdictatcm. Dialiaso /11,13) iiitcrprctatur dc oniiiibus, ct est cx diayciitc et diatcs- saroix. Oiinics ciiiiii iniiiorcs coliigit in sc proportioncs. Vocatur aiitciii dupia in aritliiiictica ct diapasoii iii iiiusica. Ej,o~'qdorrs (11,20) dicitur quasi siipcr VIII" vcl supcroctaviis, uhiciiiiiqiic ciiini corda cordain siipcrat iii noiia partc, sivc iii acuiliiiic. sivc iii gravitatc. Epogdoiis dicitur ct iii aritliiiictica scsquioctavus, ut sunt VII11 ad VIII. Tonus dicitiir iii iiiiisica, vox pcrfccta. Spherira (11,22) ct astronoiiiica iiiiiiiii est. Iiitcr astroiioinicaiii ct astroironiiaiii bcatus Isidoriis diffcrciitiain facit ct dicit, quia astroiiomia iioii proliibctur disci, qiiia iiihil Iiabct iii sc, quod rci vcritas repudiet. Astroiioiiiica habct quacdaiii, qiiac proliibciitur disci, qiiia iiiatlicmatica ac coiistcllatio sccuiiduni qiiaiii diccbaiit iiiatliciiiatica jiciiituraiii Iioiiiiiiuin coiistarc in illa subsi- stit. In ailroiioifiia eiliiir iiuruli j11,21) suiit scilicct scptciil plaiictaruiii. I~laiies graccc, Iatiiic dicitiir crraiis. Hinc plaiictac dicuiitur stcilac crraiitcs, quac aliqiiaiido cuiii solc, aliquaiido (coiitra solciii) gradiuntur; inodo cum niuiido, iiiodo coiitra miiiidiiiii fcruiitur; iiiodo siirit statioirariac practcr solciii ct luiiaiii. Splieliliii (11,Z.j) vocat anibituiii totius iiiundi vcl orhciii sydcrum. Ce~ziuiiiii !11,25) dicuiit gracci puiictiiiii circa quciii volvitiir circiiius. Cciitruiii auteiii dicit mcdictatcm cacli. Pamileli rivirrli (11,23/ siiiit, sicuti suiit V zoriac, quarriiii duac ~iiiiiio frigorc squalciit: A scptcnitrioiiali partc uiia, altcram ab aiistrali, ct suiit iiriiabitabilcs. Iluac tciiipcratc. uiia turrcns cst, quac cst iii austro, pcr qiiaiii so1 currit, idcoqiic iniiabitabilis. Est ctiaiii orizoii circulus, qiii intcrprctatiir liiiiitans, co quod cacl~iiii ihi vidctur a tcrra sciuii#i. Asciii (11,26) vocat iincarn iiitcllcccualcni, quac a ~ o l o usqiic ad poluiii, id cst a scptciiitrioiiali plaga iisquc ad notium vci austriiium polum tciidit. D u o ciiiiii poii suilt: scptciiitrioiialis ct australis. Taii ciiiiii modo caeluiii iii traiisvcrsum posituiii cst. E: Iiabct scptcintrio stcllaiii, qiiac dicitiir polus; aiistcr qooquc siiiiilitcr, scd iioii apparct, quia infra tcrrani cst. Iii his duahos stcllis totuiii vcrtitur firmanicntuiii, ct ipsc polus cciitruiii vocatur. Oiiinis soniis iiiotuiii liabct, ct ubicuniquc cst iiiotus, soiius subscquitiir, qui scinpcr e s iiiotu fit. Uiidc dicciis: iiiortrc ipit rcic,bvoiiir- nrroririfr (12,2), tangit musicaiii caclcstcin, quae, quia inotuiii habct ct soniiiii, sciiiper ciiiiii voliibili lapsu vcrgitur, dc quo iiiasiiiia iiiodiilatio proccdit. Cuiiis nicdiiiiri dicitur obtiiicrc sol. Est ciiiiii iiicdius iii armonia caelcsti, sicut nicsc in cordis. Naiii taiitus illic soiius cst taiitcquc duiccdiiiis, iit bcatus Isidoriis dicit: U t si Iiuinaiiac aurcs valcrciit iilaiii aiidirc, ab oriciite in

1.2 occidciitcm traiisircnt scquciitcs inotuiii splicrc cacicstis. l>r-iiiietiaiii i i n l ~ r l n i ? ~ (12,141 vocat illain informein iiiatcriaiii, qiiac fuit iiiitiiim 1111"' clcmciitoruiii. Ratio (12,l.i) cst mciitis iiitiiitus vcl aspcctus.

Priiirii~de rscii~piar (12,16) dicitur fiiissc iiumcrus iii aiiiiiio roiidiroric, id cst i i i dispositioiic cius, qiiia, diftiiiituni crat ab co quasi pcr iiunicruiii qiiot faccrct crcaturas. Moiiri (12,18) solis qiiadri- farius cst: ab ortu ad iiicridiciii, iiidc ad occasuin. Ab occasu ad mcridicm sub tcrra, iiidc rursus ad ortuin. Ceiiqire roiri~eiirio (12,22) scilicct suiiipta cst. iii quot Iioris volvitur, scilicct in XXIIII"'.

i Niiiiicri~s roiijposii~re (12,22) cst iioii taiiicii 1.x dii>rucic. Niliil cniiii cx divcrsis coniponi potcst, sicut viiiiini ct aqria iioii possuiit faccrc ct coiiipoiicrc viiiiiiii. Ncquc cx similibus cst coilipositus, quia cx similibiis quoquc iioii aliqiiid componitiir. Quid ciiiiii coiiipoiii potcst cx auro ct argciito? Niliil, quia sciiipcr crit auriiin. I>robatiiin crgo cst, non cssc compositiiiii iiuiiicruiii cx dissiiiiili- bus artibus, id cst gramatica, dialcctica, rctliorica; scd iicqiic cx siniilibus, id cst gcomctrica,

1 0 inusica ct astroiioiiiia. Ergo cx ceijlso (12,251 cotiipoiiitiir. Ei-iiili e r y , iiu,iievos puiiiia qiine iriiigaiit (13,1). Prima, id cst iiiitia ct fiiiidamciita atquc priiicipia,

qirae coi~steiit iei i iperquepeii irni ieaiit (13,2). Nuiiicrus ciiiiii aetcriius cst ct iiiii~utabilis; ~ i t ciiim dicit Augustiiius: Aiite potcst ficri, ut tcrra sursiiiii; cacluiii dcorsuiii vcrtatiir, aiitcqiiain trcs iioii sirit trcs ct qliattuor quattilor.

1; Ipia (13,4), fi~iidamcnta iiumcroruiii, diiiiiiiilia CI roiiipnrici~di poiei7iia /7.?,4/, id cst paria atqiic iiiiparia, quac dissiiiiilia qiiidciii, scd uiiuiii possuiit coinpoiii iiuincruiii. I'nu i?ii~iicrrri eei ceriiiidiiiii 1.1

IJyta,qoraiii, qrii iub radcriz divisionl, poiesi iii iiinxiriin pari>isiiiiiaquc diiiidi (13, 19). 111 iiiaxiiiias partcs par iiurncriis dividitiir, cum iii diias irrcdictatcs partitiir. Ncc cst pars ulla, quac miiior sit iiicdictatc. Cuiii crgo i i ~ duas iiicdictatcs niasiniac partcs paris iiuincri siiiit' simul ctiani parvis-

20 siiiia quacdaiii suiit, id cst particio ct divisio, qiiac taiitiim iir dunbus partibiis facta cst. Vcrbi gratia: XVI par iiuiiierus. t i i r crgo in niasiiiia dividitiir, scilicct sccuiidiiiii suiiinram partiiirii in VI11"' ct VIII'", ct in miniina. scilicct scciiiidiirii iiunieriiiii partiiiiii. l luac cniin taiitiiiiiinodo partcs suiit, quac iii oiiiiii divisioiic iniiiiiiiac siirit, quia oiniiis iiiiincriis priiiio iii duas partcs, iiidc iii trcs; si tcrtia~ii liabct, dciiide autcni iii qiiattuor partitiir, quoad pcrvciiiatur ad cius

23 ullitatclil siilliiitcr. ,bl<i.siiiin (13,Zij in q)atiii cst i n iiiagnitiidiiic divisi iiiiiiicri. l'ari,ii<iiiin qitaiiiiiate (13,21) scilicct divisionis. qiiia adliiic scmcl divisio facta cst. Scoriidriiii isti~rrriir iiiioririii ~iirr~errriii (13,22/, id cst crciiiciiti ct iiiiiiiiiutioiiis, [oiitrariae j~nssioi~ei

(13,223. 13assio cst coiitrarius seiisus. Idco dicit coiitrarias. qiiia, quo iiiaiiis est spatiuiii, id cst 30 si~i~iirra, CO iiiiiior cst divisio, quac dicitiir qiiaiititas; ct qiio iiiaior cst quaiititas, co minus cst

spatiuin. Vcrbi gratia: Si dividas VlIl iii 1111 ct 1111, inaxiiiiuiii quidciii spatiuiii cst quatuor, iiiiiiinia vcro qiiaiititas, id cst divisio. Non ci~iiii iiisi scmcl cst diiisuiii. Itcruiii quattiior si dividas iii I1 ct 11, spatiurn; id cst suiiitiia siipradicta irnmiiriita cst, sct qua~ititas, id cst divisio, aucta. Siiiiilitcr si duo iii I ct I partiaris, spatiuiii niiiiirniiiri cst, quantitas vcro maxiiiia. Non

35 cniiii VIII iniiius spatiiim est quani uiius, iicquc inaior qiiaiititas. id cst divisio, quaiii dividcrc V111 octics. Dntiic (14,2j id cst propositus. Subtilius adliiic, quod dictum cst. pcrspicicndum cst. Ncccssc cst cniiii, ut lector animuin ad pracccdciitcm Icctioiiciii coiivcrtat. Nam cx illa hcc cx partc pcridcnt. Aj4axima (13,21) iii spaiio, hoc cst iii inagnitudinc, quia quotics postca fcccris

<:losscii ziii Aiithilictik drs lioerliiur 27

divisioiicin, toticiis iiiiiiiies istdaiii inagiiitudiiicm nuiiicri. I'ari~issiiiin 113,21) in qirnniiinte, scilicct divisionis, quia sciiicl divisio facta cst seii~iidrii~i roritiaiins isior-uiti 'qiiolcrriiii pas.cioiics (13,ZZ). Mag- nitudiiiis scilicct ct ii~ultitudiiiis. quia iiiiiltitudo dcbuit aliquod crcincntuin accipcrc. iiiagiiitiido vcro dccrcsccrc. I'assio illata cst, qiiia facta scctioiic non apparct coruiii proprictas. iit iii inultitii- diiic cssct crcmcntuiir ct in iiiagiiitudiiic dctri~nciitiiiii. Scd ciim facta cst scctio niinicri. iii 5

niuititudiiic iion apparct iiisi unitas, i i i iiiagiiitiidiiic vcro ipsuiii iioiiicii remaiisit $1-aiiditas, qiiia iii iiiaxiiiia facta cst illa divisio. Spntiiriii (14, 11) iiaiiiquc appcllat iiiagiiitiidiiic~ii, qiiaiii disit supi-a a fiiiita iiichoari quantitatc ct iiitiiiitissiiiias scctioiics (quaiititatis dividi). Qiraiiiiiniriii (14,14) autciii vocat iiiultitudiiiciil siipcrius dictani ab 1111o progrcdi ct ad iiifitiitatcm protcii- dcrc. I O

1.7 Alti.ii~secirs (I6,10), id cst cx utraqiic partc vc1 c rcgioiic. 1.9 Ciiipi rioi?iiiic ipso i iocnbii loqiiepaii iei.pnr- iiii,eiiioiiii.; iiiiri eiiniii qirairiiiaie (17, 18). Nota liic: Noiiicii

quodainii~odo ad corporalcs. vocabuliiiii ad incorporalcs pcrtiiict. Atquc liic utriiiiiqiic poiiit, quia ct iiiiincri corporibiis iuiigiiiiti~r, iiiidc ct iioiiiinautiir, ct iiicorporalcs sunt natural uiidc vocaiit~ir. 15

Qi<od < > i i i i i ~ i j~aifes eiiis iioiiiiiie (18, 11, scilicct qiio iioiniiiaiitii~~, ei qiinrititni~~ (18, I), id cst divisio- iic, par-itei. pares (18,l) siiiit. Sciciiduiii. qiiod iicrfcctus iiuiiicriis taiituiiiinodo iiitcr (]iiattii»r tcsscras iiii~ncroruin, scilicct iiitra dcnariaiii sciiariiis pcrfcctus cst; intra cciitc~inriaiii XXVIII. Iiitra iiiillciiariam CCC:CXCVI. Iiitra dccciii inillciinriaiii pcrfcctus cst VIII.CXXVII1. I-li taii- tuin suis partibus rcparantiir iii propriairi suiiiniarii. Iilird aijteili iioii iiiiiiiinn ioiisidci-aiioiir dijiiiririi ?o rst, q u d ooiiiis ciirs p a n /18,12), id cst dispositionis vcl pavis ii~iilicri, ob iliin ynr-t< qiinr iiit~n ipsr<iir iiiriiieiiriii /18,13), id est dcii»iiiiiiatiir. Vcrbi gratia: Sit siiii?iiia paritcr paris iiiiincri CXXVIII, nicdia cius pars LXIIII"' siint; ct Iiacc pars dciioiiiiiiata cst ab iriinyal-te qirae iirii-a ipsiriii iii~iiieirriii ~ . r t (Ih',l.ij, id cst a binario. Bis ciiiin LXlIII colligiint CXXVIII. Advcrbia iianiquc iiiirncri dciio- i i i i i ia t im~ni ostciiduiit. Idciriqiic biiiariiiiii pars cst LXIIII"". Sie ct dc cctcris partibus. Quarta 25

pars supradictc suniiiiac XXXII suiit. ct Iiacc qiioquc pars dciioiiiiirat~ir pcr ad\-crbiiiin ab iiiia 11ai.r~ ciusdcin iiuincri propositi, id cst a qiiatcriiario i i t quatcr XXX'21" CXXVIII"'. Et idciii quatcrnarius XXXII" pars cst ciiisdciri iiuiiicri.

fiiitniiiqire sniiiiiiniii (18,Iij, id cst collcctioiiciii iiuincri. qiraiititiltis, id cst iiiultitlidiiiis. iiicliidii, qiioin pai' est alter riiriiiei.i~s pniiiei- pnr-ii iilirrs (18,16). id cst niinicri, qiii caiidciii parrciii coiiiiiict 30 qiiniitiinlis (18,li). Vcrbi gratia: Si quarta pars fucrit cuiusciiiiiqiic paritcr paris ~ ~ ~ i i i i c r i , ipsa quarta pars pcr quatcriiariuiri_ cx qlio dciioiniiiatiir, totain su~iiinaiii iiicludit, iit cst Iioc: Poiio LXIIII; cius quarta pars XVI, quac quarta pars pcr quatcriiariiiin inultiplicata, supradictaiii sumiiiarn collicit, id cst LXIIII. Siiiiilitcr cactcrc partcs pcr ipsas, cx quibus dciiomiiiaiitur, multiplicatac supradictain suiiimaiii coiicludunt. Et bis oinniiio dicit' quotacuinquc fucrit pars 35

totius multitudiiiis, id cst sivc sccuiida sivc quarta sivc alia quaclibct pcr ipsani, cx qua dciioini- nata cst, iiiultiplicata totain nunieri surriiiiarri coniprchciidit.

Q~raiiiitatemqire coiiveitiiirr (19,20) id cst inultiplicatioricm. Rursus qua~iiitate (19,29.;j, id cst 1.17 divisioiic. E t sepius hic pro tali poiiitur scnsu. Erastoteiics (33,20) grecus fuit arithmcticus. 1.19 Iizfrodiritionis (39,18) id cst doctriiic, qua iiitroducitur aliquis ad ititima scicntiac. 10

Ceiztiii1aiius2ijias (40,261 a poctis siiic iiorniiic profcrtur, qui quiiiquaginta capita fcrtur habuis- sc. Geiio (40,27) dicitur fiiissc gigas tricorpor.

Monsirosiriii (40,28) cst, quod siibito extra naturam apparct. Quidaiii autcm dicunt inonstriio-

7-8 Cf. I,1 (9.16) am C' 1.1 (9,14) 17-18 Cf 1.20 (42.3-5)

8 quaiititatis dividij qiiaiiti diuirii L dividi M I3 Aiq~ie j A d q ~ i c L 23 parr] pias L 40 id est] cst L

suiii, scd potius dicciiduni cst i~ioiistrosuiii. Siibi.ipr!ii (41,1) id cst fcfcllit. G e i i o !40,27) rcx fiiit I-Iispaniac, qui idco dicitur tricorpor ftiissc, quia tribus iiisulis inipcravit.

Ad quciii IElcrculcs cuiii acrca olla ct bicipiti canc vciiissc dicitiiu. Cuiii acrca olla o b acratas iiaucs 5 ct procliuin iiavalc; cuiii bicipiti caiic? qiiia iiavali ct tcri-cstri proclio p~uriiiiuiii volvit.

Cii/o,>s (41,3j gigas fiiit iiioiiotalariius. id cst unioculris, quciii Iiabuit iii fi-oiitc, quod idco fiiigitur. qiiia priidciitissimus fiiit ct sedis sapicntiac iii ccrcbro cst. Euiidciii autcin oculuin Ulixcs dicitur tciicbrassc. qiioiiiaiii illo pi-udciitior fiiit. Ciii-tati is (11,3/ dicitiir truiictatus. Uiidc ludci curti diciiiitur, id cst iii pal-tc pudciida truiicati. I>nssiiii (42,2j vulgo. Bis e i~ i i l i fi-es l / ~ I ~ j ä ~ i i i i i i , I,?()

i i i qui linbei iirioiir qiiideiii n i e dciiori i i i iotnii~ pai-teiii id est sexfaii i . 71.13 ivero scii! i idi i i i i i l icdiemtni i , seaii iduii l

di ial i toici i i (43,j-8). quia bis dicitiir: trcs, trcs, ct pcr lioc dualitas ibi cst. AI iiei-o diio (43,8), scilicct ibi Iiabct scrriildiirii conrcriiarioiieiii id cir icr i i i idr ! i~ i teuinri ir i l i (43:8), qiii cst coaccruatus cx uiio ct diiobus, quia diciiiius tcr duo. Tali iiiodo Iiahct sciiarius diio scciiiiduin tcrtiaiii partcrii, qiiod cst biiiariiis.

15 (47,9) Ilisci-iiiicii propric vocatur aciis iiiaior, quo capilli iriiilicriiiii scgrcgaiitur; uiidc discri- miiiari diciniiis scgrcgarc.

H ~ i i i i o l i o ( j O , 18) Iiabciis diiiiidiiiiii ct totuni. Hcmi ciiini scmis diciiiiiis, Iioloii totuiii. Perctrii7- 1,2426

jy i i icc (52,26j parciii taigcntcs. Tesa i i i r ('5.?,4): ordiiictiir. Notand~ii i i iii figiira (i3,iO): quod ab iiiio iisqiic ad X priiiia uriitas vocaturl a X usquc ad C sccuiida uiiitas vocatur, quoi~iain pcracto

20 dciiario iiuiiicro rursus qiiasi ab iiiiitatc rcpctitiir dicciido uiidcciiii, duodccim ct cctcra. 'Tertia vcro iiiiitas a C usqiic ad iiiillc dicituu. t l is tribus unitatibos oiiiiiis iiuiiicrus consistit. Ergo ultcrius si aclit quis iiiimcrarc, ncccssc cst, iit Iiac uiiitatcs scinpcr rcpctantui-, ut duo iiiilia, tria iiiilia ct cctci-a. Notaiiduiii: di io ni l t i i ln i - i i i i i i t<, iqoi lo i - i ! i i i n i ig i i l i (56,281. Scilicct Iioc iii figura cst 1.27

coiisidcraiiduiii ubi suiit I ct C, c rcgioiie positi, qui coiicl~idiiiit aiigiilos tctragoiioruiii iii sc. 1-11, 25 si pcr iiivicci~i ~iiultiplicciit~ir. taiitaiii s~iriiiiiaiii faciuiit; qiiaiita cst, si iiiiiltiplicctiir sccuin

iiiiusquisquc dciiarius, qui iii rcliquis angiilis biiiis cst coiistitiitus, augciido dccics dcccm. In utraquc cniin multiplicatioiic cciiteiiz ' iilis -. cst.

I 'r ipw rns l~ jn ln i i i />rci i i tatei~i (.57,3). Rcstat iiobis docciitibiis dc spcticbiis inacqualitatis. Resini 1,3?

j66,?j, Iioc cst supcrcst. ii-adei-e j66,6), id cst supcrcst, ut tradainus. Prqfir i idicsii i~ail i (66, i ) : i i i obscurissiiiiaiii ct subtilissiiiiain dis i ip l i i ia i i i , qiiniii pcrfi i ieat (66,6), cuiii cogiiita fucrit ad oiliiieili

i iai i irf i i , v i i i i (66,6j, id cst ad pciictrationcm ct iiitcllcctiim pliisicac; id est iiaturalis philosophiac. Et qiine pel-firieat nd o i i i i i ~ i i i d ~ i ~ i t a t e i i i reririil (66,6j, qiiae vcstigatur pcr caiii yliilosophiani quac appcllatur pliisica, quia iriliil in rcbus crcatis sinc iiumcro coiisistit. E t Iiac discipliiia, quam traditiiriis cst; dcpi-ciiciiditur. quoinodo nianavcriiiit oiiiiics inacqualitatcs ab acqualitatc, quac

35 sciiipcr cadciii cst ct iii iiullo sibi divcrsa cst; siciit oiniiis crcatura proccssit ab ca natura, quac cst crcatris. Oinnis ciiim iiatura ant cvcatrix cst aiit crcata. Crcatrix dcus, crcata' quaccuriiq~ic ab co facta cst. Iii co cst similitudo, quia cxordiuiii Iiabciit oiniics crcaturac a dco. Non taiiicii ab cius iiatiira, scd ab cius opificio, sicut istc oiiiiics iiiacqualitatcs iiiaiiant cx cqiialitatc, qirae ad oriiiicin

i i i i i i ilatiiizle reiiiiiiqir? inie,oritoieiii i i insi i i ia i -a i io i iepcr t i i~cn i . (66,6j Oiiiriis iiatura, quaiiidiu sc in suo 40 statt1 conscr\.at, iiitcgra cst ct iiicorrupta sicut oinnis acqiialitas, aut postquaiii cxorbitavcrit ct

3-5 CC .Myrliogi. 11, l i ? ; Scivius. Coiiioi. in Vcrg. Acii. VII.662

5 ca,,c] P ,,avc L l I At vcio] Ad vcio L 17 I>cisrii~,gciires. pariiiii taligciircs. pcistiiiigciitcs. piriiiii iaiigeiircs L I I r 1. -31 i d crt2/ est L 32 q u x ] i'riidi<~iit qiiciii L

<;losseii zur Aritliiiierik des Uociliim 29

dcviavcrit a suo principio ad vitia dcflcctciis, quasi ab acqualitatc ad inacqualitatciii labitur, quac fiiic iioii claiiditur.

Friictirs (66,8j pro utilitatc poiiitur. Uiidc ccoiitra iiifructuosos inutilcs diciiiiiis ct iiifructuosc advcrbiuni.

Boizitas dfffinita (66,8) cst proptcrca, quod iioii cst coiiiposita scd simpics. lpsc spetialitcr 5

dicitur bonitas dcus nullaiii coti~positionciii rccipicns. Er boriitas dci cadeiis siib scierifiotri (66,9j substantiac huiiiaiiac, iit possit intcliigcrc iiicoiiipositurn esse dcuiii ct siiiiplicciii. Ailiiiio 166,9j vcro seniper pevceptihiiis e.st secuiidum iiostrain capacitatciii ct sccuiiduin moduiii nostrum; ct iinitabiiis (66, ZO), iit 110s conforincnius ipsi boiiitati, ut iion siinus rnutabilis sicuti luiia, quia stultus ut luiia mittatur, scd adhcrcaiiius illi scnipcr, iic absorbciitc cius boiiitatc consuiiiinctur iii 10

nobis improbitas. 1ii1itabiii.i (66,ZOj ctiaiii iii hoc dici potcst, qiiia ct iios, postqiiani cssc ccpimus, sciiipcr sunius in csscntia, liabcrnus coiiim~iiiioiicm. 111 pcrpctuitatc iioii liabciniis; quia illc scinpcr fuit, nos cx tciliporc cocpimus.

Derove sirhstaiztiaeperyefira (66,1Oj, quia cius siibstantiac iicc accidit aliqiiid pcr auginciitum, iicc dccidit pcr dctriincntuin, quia dcus 11on cst obiiosiiis teiiipori iit 110s. OR170. lpsa boiiitas, I 5

quac pcr iiaturaiii cst prima, pcrpctua cst dccorc substaiitiac siiae, iit iioii sit capax iiiutabilitatis. Ita cnim dcus cst boiius, ut sit ctiani botiitas. Ncquc est dco aliud cssc boiiuin ct aliud habcrc boiiitatcm. Aliud esse dciiiii ct aliiid cssc boiiiim.

lt~fiizitirni vcro iliaiitiac dedeciis est (66,ll) iiuliis propriis nixum; ii~fii~itiiriz, quia vidctiir composi- tun1 cx divcrsis affcctibus. Ncc capit aliquaiido tiiiciii sacictatis cxccutio. Niriiis priiicijiiis i?ixiri?i 20

(66,12j, quia iiialitia, quac coiitraria cst boiiitati, siiic iiiitio cst. Si ciiiiii rnaluiii iiiitiurn liabcrct, ct aliquid esset; scd quia non liabuit iiiiciuiii, iiiliil cst ct niliil aliitd cst iiisi absciitia boiii; vcl idco cst sinc principio, quia dcus iioii cst auctor cuiiislibct mali, iicc dcus fccit aliquando maluni; sed natirra (66, IS), id cst pcr iiaturaiii, sci:iprr evraiis quasi si8iilo aliqrro optitiiac.fi~iirac iiiiplrssn (66,131, quia malitia vidctur imitari boiiitatcni. Noii taincii cst bonitas. Coiistaiitia niagiia rcs cst. Iiuic 25

sigillo optiniac figiirac quasi impressa. Est qiiacdaiii rcs, quac vidctur similitudiiicni Iiabcrc coiistantiac ct vcrtitur in siipcrbaiii libcrtatciii, duin yiiis vidctur Iiabcrc coiistaiitiaiii. E t practcr- rilissa incnsura cxccdit ct traiisit in s~ipcrbaim libcrtatcin, yuac rioii rcciyit disciplinam. Ita dc aliis rcbus, quac vidciitur Iiabcrc similitudiiicin virtutuiil. S~ii i t tamcii vitia proptcrca, quia csccduiit modum. Et ex ilio erroris~flucfii retiii~tiiv j66,14), id cst, ab ipsa diffinitioiic, quac siinplcx cst, 30

rctii~ctur ipsa malitia crroris. Fluctii cst crroris iiiiitabilitatc, quia crror iioii cst solidus, scd cffluit huc illucquc ct iinprobata placcnt. Scd licct iiatiira cxorbitct ct dcvict a suo priiicipio, taii~cii ad suaui bonitatcm rcdigitur, quia a borio priiicipio crcata fuit. Quicquid ciiiiii iiobis iiialuiii vidctur cssc, dco bonurri cst. Na171 niiniaiti cupiditaic~ii (66,15): Cuiu vidcatiir cssc cupiditas bonus affcctus pcr dcsidcriuin, si exccdat iiiodiini, fit vitium. Iraeqire iizrriodicai~i ~ffier:atioireiti 166,Zj) id 35

cst iiitemporatain dissolutioncin. Modicuin diciinus tciiipcratuin, sccuiidiim proprictatcin Lati- iii eloquii. Imque ii7i~iodicatit (66,151: Coiiccssit, quod naturalc crat, scd iioliiit dcclinarc iii vitiiiin ipsuni affcctuiii.

Pirra iiztcll~ycntia roboratirs (66,161, id cst vidciis. quid sit agcndiini, quid vitaiiduiii. As t r i i i~ i~ (66,17), id cst: Siciit habcnis rcteritis in maiiibus cqucs rcstriiigit iiiinodcratuiii cursiim cqiii 40

quando vult, ita ctiain illain cffrcriatioiicm irac ct iiitciiipcraiitiam astririsit ai~iiiius quasi aliquis rcctor ct aliquis aurica, ct iiitra mctas coiitinct discipliiiac, quia adliuc cst iii diffiiiitionc boiii

30 M i c l i a c l i'rciiiiiard

priIleipii. Ei /ins qiiodaiiiniodo ir iaeqi<ali iai isfori~ias (66,171, id cst, qiiac fiiiiit divcrsis affcctibiis incitate. ieiiiperaia bi,riitnir ioirriiiirit (66,18), quia iiicqualitas intcmperaiis cst ct ipsain iiiacquali- tatciii rcdigit rcctoi- aiiiiiius; iiiacqualitatcm, qiiac boiia cst. lsta vitia quasi quacdaiii inacqualita- tcs suiit. Per acqualitatciii virtus dcsigiiatur ct boiiitas. quac dcus cst. Notaiidiirn: Ex his ifiiiirr

5 secrrndirrit j~re~epli i70rii.i odiiieiii iiid~os priiiiiii71 iiarci iiiiiiiipliies (67,1), id cst scc~iiiduiii qiiod iii

quadrata figura docuit iriultipliccs pcr ordiiiciii iiasci, priiiio quidcni duplos, dciildc iriplos (67,3), sic ct cctcris pcr ordiiiciii. Tribirs Itis prrieptis (68,211: Notaiiduiii: irieiiirrr /68,22), id cst tribus disciplinis iiuiiicroriiiii, quos quodaiiiinodo praccipicndo iubct poliere, dicciis: Sir11 (69,2) vcl dispoiiaiitur ii-es irri~iii7i. Notaiiduiii: ldco ai7iiiios iii~~redici~iiiiiii (72,18) dicit, scilicct aiiiinos legeil-

10 tiuiii, quia istc prirnus libcr pcnc proptcr iiitroductioticni scquciitis libri pracccssit, quatiiius lcctor Iioc libro qiiiddam iiiforniatus, quatinus faciliiis iicxuosas qucstioiics iriatliciiiaticc in scqueiitibos pciictraret.

FINIT I)E LIUlIO I. INCII'IT 1)E 11. Eleriieitla (77,Q dicta quasi clcvaiiiciita vcl cliiiiciita a vcrbo cliiiio, id cst furiio, vcl ctiam, ut alii l1.1

1.5 voluiit, ilciiieiita. qiioiiiaiii ilcii rnatcria. Ipsaqiic eleiiiciita vocat priinordia ct iiiicia qiiacdaiii. Ui, id cst qucinadiiiodu~ii, qiioi7iaiii ai-tiiuiaris iiorir ciciiiciiia srrnf iiiierfle (77,8j: Articularcm vo- cciii liic appellat quac scrihi potcst, qiiia iioii rcsolvcrctiir in littcras, nisi scribi possct.

Soi?iis cst iii iiiiisica (77 , f l ) vox acqiialitcr porrccta, quod ~iiiiiiiiiuiii cst iii miisica. E t ctim iuiiguiitiir soiii acutus ct gravis, faciuiit toll~liil. l'oiius vcro cst vox C duobiis soilis coilsistclis.

2ii Notaiiduiii iiai?iqire ut ail: E.x iiiii~ri terraqiie anii~ia ~qi~qiiiri?iiii. er (qi7i (77,13). f4oc cxciiip~uiii siiic autorc protulit, quasi iiotissiiiius cssct ct auctor ct cxcnipiuiii. Est ciiiiii vcrsos Lucrctii. I<s aere, id cst aniiiia.

114arfiiii~ (77,15), id cst fiiic ct priticipio. Ub i ciiiiii fiiiis, ibi priiicipiuin, ct ibi fiiiis, iibi priiicipiuin cst. Solirrinii (78,23): pcritiiiii ct iiiultuiii iiiibiituin. Sollaii ciiiiil dicitur multurn vcl

2.i totiiiii. 1~xprpri.ieiiiciii (78,24): approbaiitcni. Eliqirabiiiiii~ /79,1.?): rcsolvcii>iis ct quasi vi cxpriinc- iiius, qiiod propriuiii cst huic vcrbo significarc. Ei? j80,i): in, iiiiisiia: id cst iiiusicae, ioii: artis, 11.2

ilieorcriia (80,i): spcculatioiiciii. Oiri~ei/iii (80,6) appcllavit q~iciidani libruiii l'lato. uhi gciiituraiii iiiuiidi ct aiiimac, sicut ci

visiiiii cst; cxposuit. Oinic ciiiin graccc dicitur aiiiiiia. 30 Maius iiitcrvalluiii distciiia, iiiiiius iiitcrvalluiii sistciiia apud iiiusicos iioniiiiatur. Ilii eniiii

(80,7), id cst ariiionica discipliiia, iiihci71itr esferiderc III vci 1111 sesqiralteror (80,7), scilicct iit poiiaiiius ordiiiatini scsqualteros, ct dicaiiius; quis sit priiiius, quis sccuiidus. Siinilitcr ct iii scsqliitcrtiis coiisidcratur. ldco dicit: lirodiir(>rr ci rsieridcre 180,8), quia tuiic cxtciidimus, ciiiii ab uiio scsqualtcro pcrvciiiiiiiis iisquc ad iiiultos sesqualtcros. Aniioriicac /80,7), id cst iiiusicac.

3.5 Ariiioiiica ciiini dicitur orriiiis coiisoiiaiitia a tcrtio gciicrc quod ariiioiiicoii dicitur, quod cst suavius cactcris. l'rimuiii ciiiiii dicitur croiiiaticuiii. quod cst dissolutissiiiiuiii, ct coiistat in caiitilciiis feiniiiariirn. Nam croina graccc color. Undc croinatic~irii, id cst colorabilc ct variuin proptcr dissolutioiiciii. Sccuiid~iiii vocatur diatoiiic~iin, quod cst diirissiinuiii, ct hoc apud Scot- tos ct Urittoiics. Tcrtiuiii dicitur ariiioiiiciirn, quod cst dulcissiiiiuiii ct cx duobus cornposituiri.

40 Idcoquc oiiinis coiisoiiaiitia ariiioiiia dicitur. Ii?feraci (80,13) id cst iiifruct~ioso. Notaiidum: iri qriotieiisii~iiiqrrc rrltiiiiirs iiirjnenis ii~i~ei~iiiri, qiri ioco dirpiiiis ab rii?ita/e sit pnr, iaiis sii,

ir1 iii niedieia/e(s) dividi xecnriqrre iioii poiir (8i , i8) . I-ioc cst: illc iiuiiicrus angularis, iii q u a pcr singulos diiplos scsqualtcri tcrrriiiiaiitui- ct fiiiiuiitiir, qui ctiam ab iiiiiinie par cssc dicitur, quia

45 quoto loco duplus ab uiiitatc cst, co loco aiigularis diiplici subiectus est, talis cssc dcbct, ut in

20 1,iicrctiiis 1.716 21 Ex aerej V . l ' i icdlciil, Appc i i r . : aiiitiia ci i u p m ivrr"tii1 i. acrc 11.1

(;lcsseii zur .4iitliiiicrik dcs Bocrliiiir 31

iiicdictatcs dividi rioii possit. Quia Iiacc cst Causa, quur talis iiuiiicrus iiltimus sit scsqualter, co quod incdictatc carct, quain si Iiabcrct, possct ad altcrum iiiaiorcin comparari, qiii cuni totiini Iiabcrct ct iiicdiurn, ac pcr Iioc nori cssct aiigularis ct ultiiiius. Sciciiduiii autcin, quia latitiido harum figuraruiii ita cst vidciida: -, loiigitiido ita: I. Et sicut priiiius ordo duplicis scciinduiii longitudincni quidcm duplus cst, ira iii cadciii figiira oiiiiics scquciitcs ordiiics ad loiigitiidiiicin 5

dupli suiit. Sic ctiaiii qui tripli suiit in priiiia loiigitudiiic, sccuiiduiii caiidciii qiioquc loiigitudi- iieiii rcliqui scqucntcs tripli suiit. Eodciii ctiaiii iiiodo iii rcliqiiis.

11.4 Plai~a diriieiuiu [86,15) cst iiuiiicroruiii, quac fit in loiigitiidiiie ct Iatitodiiic, iit bis biiii. Cybiiiii 186,16) dicunt gracci solidiiiii. Uiidc iiuinerus vocatur cybus, qui iii altitudiiicim, latitudiiiciii ct loiigitudiricm cxtciiditur, iit bis biiii bis. i(l

Spherici /86,17) siirit iiriineri circularcs, qui siciit circulus a quo piiiicto iiiclioatiir in cuiidciii cxit, ita a quo iiuiiicro acqualitcr ii~ultiplicaiitiir iii altitiidiiic, latitudiirc ct loiigitudiiic, siiinma iiuiiieri iiiiiltiplicati in euiidciii cxit. U t ciiin dico quinquics qiiiiii qiiiiiqiiics, sicut a quiiiario fit multiplicatio, ita suiniiia ii-~ultiplicationis iii quiiiariiiiii dcsiiiit. Suiit ciiiin CXXV.

Pyraiiiis (86,17/ graccc dicitur acdificiuiii infra spatiosum, supra acuiiiiiiatiini, sicut soleiit cssc i i

absidc in quibus saiictoruiii Corpora qiiicsciiiit. Uiidc nuiiicri pyraiiiidac dicuiitur, qui supcr basirn latcra fuiidantur, scd iii altit~idiiicni cxilciu porrigiintitr. Laieriili j86,17) siint, qui iii iiiodum latcruni acqualciii liabciit Iongitudinciii ct latitudiiiciii, scd iii iiiiiiiniaiii cxtciidiiiitur altitudiiiciii, ut sunt tcr tcriii bis. T i q i ~ l i (86, l i j vcro suiit, qiii loiigitiidinciii iiiagiiaiii iii

moduin tigiioruin habciit, scd altitudiiicin ct latitudiriciii miiiorcm, ut quatcr biiii bis. 20

Ciiiiei (86,181 autcin suiit, qui niilla Icgc constriiiguiitur, scd iiiodo loiigiorcs, iiiodo latiorcs, ct iiiodo altiorcs pro libitu foriiiaiitis cxistiiiit.

Ii~teriinllriiii /87,18j vocat spatiuiii qiiodlibct, vcl loiicitudiiiis, vcl latitiidiiiis, vcl altitudinis. ii.3 Esplicata cot~,qerier (90,l.j): id cst digcsta iiiul(ti)ttido. Diictus 170,15/: Ordo .

Il.h;iS Dissipatiir (92,4): dividitur. Libraia /lO.?,1): taxata ct trucinata. I-'roi,niiciit /107,1.3): iitilitcr 25

vciiicnt. l'rovcnirc aliquaiido pro utilitatc poiiitur. I KOAOYPON(110,26) dicitur quasi KOAON OYPON, id cst mcnbruin caiidac. Ciirta ciiiiii rcs

dicitur, cui aliquid dc cauda aufcrtur. KOAON ciiiiii iiicnbruiii, YI'A cauda vocatur graccc. AYKOAOYPON [111,1): bis curta. Tricoiiiroii [111,3): tcr curta. Iir~ctirr 1112,Ij): constringitur

11.25 ct coliigitur. Notaiidum: ei i i i sirpeiius iiioiiifraiuiii cii tot rriiitatiiiii est /11.3,2), qiiia, quot uiiitatcs 30

tctragonus Iiabct iii latcrc, to t uiiitatcs ct cubus, qui cx CO iiascitur, pcr siiigula Iiabct latcra. Notandiiiii: Qiiorirrti siiiplirs (113,3), scilicct aiigiiliis, siib trihiis ciu~itlodi coritiiieii~r ( 1 1.?,4/, scilicct tctragoiiis, quia trcs tctragoiii coiitiiiclit uiiiim a11goliin-I.

Sca1o~o.s jl14,6) graccc asccnsio, quia scalc gracce asceiidcrc dicitur. Spiieiiisciiri 1114,9) graccc cuiicus. Supra Ioriiciriii /114,17): marc. Alcxaiidcr magiius quaiidaiii coiididit civitatcm, quain cx 35

suo iioniinc castra Alcxaiidri vocavit. Iuxta qiiain crant quacdaiir arrtlac /111,17) in basi latiorcs, supcrius vcro acutiorcs, at quaruiii siiiiilitudiiicm boiiiisci 1114,16) vocaiitiir similitcr iiuiiicri multiplicati. Parallrlepili~di jl15,I) sunt numcri, qirorirrii nlterrintiiii poiiifiir lnfitiido 11 15,4/, iit bis

11.26 tcriii bis. E T E P O N graccc altcrutii diciiiius; uiidc eferoriiicltis 1115,X) dicuritur partc altcra lon- giorcs. Notatidum: er iluilr quid<,in ilflcieiliri [115,23), id cst: tantuiii dixissc sufficiat, quia cactcra 40

nionstrata sunt iii liac figura. Notaiiduni: Eaildeiii aiiieiii i?ntiiraiii ci sciiiper siOi siitiiloii roiisoiiiert- 11.27 ieviqrie iiiillain aliaiii nisi prii~ievaiii irt'qeiicraiaiiiqire suhsiaiitiaril /117,3), id cst dciim. Ccrtc licct ad

uiiitatis sirnpliciratciii possit pcrtiiicrc. Av,oiriiieitiiiiii [117,12/: apyrobatio. Notaiiduiii: Aiiildius /117,15): Scilicct arguiiiciituni cst, qiiod dicturus suiii. Nota Iiic vcrsiculuiii: ncfas suiii, illuiri

11,78 invciiias, ut solct. Naiiiqirc si iiitiiiii sone(faiirins iiel si seiifpl rriliriii seiiiel /120,23). Docide.< id cst asievi~s

Glosscii riii Ariiliiiictik des Doctliiiis 33

scsqualtcra proportionc vcl in scsqiiitcrtia vcl in sesqiiiquarta t>el nd aiis,ioi?oii~icas sirbtilifates (137,4), quia c s iiuincris constaiit, vel ndgcoi?tctriiae roi?sideratioi?is uirii (137,5), quac talis cst, ut in oiniiibus iiumcris, quos iiitucniur, sivc in maioribus sivc in iiiiiioribus, acqualis sit proportio; i,el ad i,etevirin lettioi2iri~t ii~tell~qeiitiaiii (1.37,5). Vetcrcs ciiiin iii talibus spcculatioiii(bu)s vcrsati siiiit.

U& aiitei?2 com~izrrizitev d!Jjiiriaiizirs (137,10) liacc ct cctcra, id cst, ut noii soliiiii dc proportionali- i

tatibus dicanius, scd ctiaiii dc diffcrciitiis proportioiiuiii, qiiia paiilo supcrius dc proportionalita- tc dixit. Hic autcm ctiani dc diffcrciitiis dictiirus cst.

Secirridiri?~ qiiailtitateiii qrroqiie iiuiiiei~i (1.?8,7), id cst diffcrciitiani. quain inter sc discrcpant; ct qtranturn iinirs a drioliris i7iiilor est (138,Y). Hoc sccuiiduin aiitiquos disit i?iir?or a diiobirs, quod nos diciinus iiiiiior duobus. if i

81.41 Coi?fissae /138,26), id cst iiotac ct scitac. Coiifcssa rcs est, quac ab onitiibiis scitiir. C:onfessc medictates (139,2), hoc cst probate, quia illud confitcinur ct in publicuin offcriinus, qiiod criidi- toruin iuditio corriprobatiir. Coi?fissac (138,26) et itotae et qiiae pe~i~~i ieni i? t ad siieritiaiil Pytqyore (13Y,1), qui iiistituit riinari physicani; ;,CI Plafoiiis (139,1), qui quiiitus siicccssor fuit I'ytagorc. Plafo stirdiosissiniiis Pyiacqo7e (139,16), id cst scctc illius ciiiulator ct scquas. 15

lride deceni ileriibvoviriri particiilc (1.39,lYj: Quidain diciint dccciii digitos Iiic dcsigiiassc pcdiini vcl inaiiuum, alii dicutit decciii ossa capitis, quorutii quatuor ccrcbruin coiitinciit, qiiatuor iii

11.43 crucis modum coticsa palatuin tciicnt, diio vcro masillis cotistaiit duabus. 'l'alihirs (142,6) id cst sigiiis tractutii a pcdis vcstigio, qiiod duiii fit, signutii rcliiiqiiitur pedis.

Al~dircet (142,8): avcrtct, alictiabit. 10

11.45 Statirs (149,5) diciinus dispositioncs ct ordiiics rcriim. Statuta qiioquc orditiata atquc disposita diciinus. E t status populi Roiiiaiii trifariain craiit divisi in sciiatorcs, niilitcs ct plcbciri. Iiis statibus trcs medictatcs mathcrnaticac coinparat.

11.46 Cosniopeia Tiiizci (149,22), id cst factnra iiiiiiidi, quac in libro Platoiiis, qui Tiincus vocatur, contiiietur. Notaiidurri: Eo loio, qcri dicitirr i?irpiialis (151,23j vidclicct. quia ibi loquitur dc iiiiptiis 25

Mcrcurii ct l'liylologiac. Ex pevsoiin riirisaniri~ (151,24), quas ir~trodiriit clliasi ad lovc i i~ ibi loquctitcs.

1i.49 Ariiionica incdictas jieonieti-icnc iiicdictatis copiita es1 (158,18/, qiiia sicot gcot~ictrica easdctn habct proportioiics in otiiiiibiis, ct in iiiaioribus ct in ininoribus, ita ct ariiioiiica incdictas, qiias liabct proportioncs sub cstrcinis tcrmitiis. casdctii iiabct sub diffcrciitiis tcriiiinoruiri. .?(I

E t gcornctricam nicdictatcin iybriri~ diciii?t (l78,19), qitia iii oiniiibus cubis potest cotisidcrari. Naiii koybus pcr oiiiiiia latera acqualitcr iiiiiltiplicatur. E t gconictrica iiicdictas pcr acqualcs

ii,.io proportioncs fit scmpcr vcl iti niultiplicibus vcl supcrparticularibus ct cctcris. 'ios ariteinprnestare deiiciizirs (160,8), id cst operam darc atquc studcrc.

Dato calaiito (160,9), id cst proposito ad tnodulaiiduni. M

Vel cirnt dirahiis altvinserus proterisis iordiiis j160,12j intcrduin canitur tibiis ct calamo, iiitcrdiim caiiitur cordis. Sumit sirnilitudiiiciii ab liis, qui calanio caiiuiit. Suniit ctiam ab liis, qui iicrvis rcdduiit soiium, lioc cst cordis, quia cordac cx ncrvis aliquando fiuiit.

Vel adstri~l~endo tenuilt, vel verilittendo gratiat (160,13), quia, quo cxtciisior corda fuerit, co acutior cst c t tcnuiorcln einittat sonunl. Q u o autcin rcmissa ct iaxata fucrit, Co grossiorcm 40

gravioretiiquc souum facit. Et iniriorrrm tcvrizinonr;?~ yvopovlio iilaior es1 i/ia ioiitparatioi?e, qlrae iizter ;?laiores rzrrmeros ioiztinetr~r (161,11), quia X X V collati ad X""' duplicciii scsqualtcrani faciunt.

25 cf. Mairiri>i~s Capcllr, I>c oiipriis Pliilologiac c t Mciciiili

Coiitinct ciiim niaior iiuiiierus bis niiiiorciii ct iiiiiioris mcdictatctii, crgo duplcs scsqiialter cst. Rursus XL, qui cst maior numcrus, coiitiiict incdiuiii iiiiiiorcin ct iiiinoris trcs partcs, lioc cst X X V ct insupcr XV. Erit crgo supcrtriparticiis.

Qrribiis partihu, iriaioris n inaiore iilediirs uincitur, eisderii parfibrrs iriirioris prn?ferit miiiovem (162,9), i id cst X; tribus quiiitis partibus iiiinoris nuiiicri, id cst tribus binariis. Naiii biiiarius quinta pars

est dciiarii ut quiiiquics duo X, ct ipsi duo tcr ducti, faciuiit scx, quo iiicdius minorcin traiisit. Siris vcro iiori eisdeiif partibus i ~ e l a niaiore 1iii7citrri (162,ll) ct supra iiicdictatc sua, id cst

octoriario. Miiiorciii vcro iiiiincruiii, id cst X, viiicit illc scdciiarius quarta partc sua, id est 1111, ct octava ctiani Parte, id cst biiiario.

io E t in iiinioi-ibrrs teriilii7is iiinior eit proporfio, iii iifiiioribiir iiiiiior (162, 12), quia XL ad XVl duplus scsqualter cst, XVI ad X supcrtriparticiis quiiita cst.

Ei~rrdatiirs (161,17): maiiifestius; qiraiii posirri~iirs (l@,lO), id cst (quaiituni possuiiius). 11..51;.52

f i i i ~ ~ ~ ~ ~ r a i x c n t k mirsici iitodrzlaiiiii~is (l69,13), quia invciiitnr in his iiuincris coiisoiiatitiaruin di- 11.54

vcrsitas coiiiprcliciisa: diatcssaroii, diapcutc, diapasori, bisdiapasoii. Et divcrsa siiiit tciiipcra- 1.5 menta, cuiii fiuiit istc iiiodulatioiics, ct istc colliguiitur consoiiaiitiac. llivcrsitas cst, cum ostcn-

ditur diatcssaroii coiistarc duobus toiiis ct scinitoiiio, diapciitc tribiis tonis ct scinitoiiio. 13iapa- soii ostcnditur coiistarc CS qiiiiiquc toiiis ct sciiiitoiiiis duobus, ct Iiacc varictas rcpcritur ct ostcriditur iii liis nuiiicris, ut tot varietatcs coiisoiia~itiaruiii rctiiicaiit hi iiuiiicri, qui asscripti sunt. Etciiim pcrfcctius iiuiusniodi mcdictatc iiiliil potcrit invciiiri, quia niliil potcst adici in

20 corporibus supra hacc intcrvalla ct supra loiigitiidiiiciii, latitudiiicrn (er altitudincni). Toiirrs (172,l) graecc, sonus latinc dicitur. Et quod cst uiiitas iii iiiimcro, Iioc cst sotius iii

musica. E t sicut nihil miiius cst unitatc ili iiuiiicro, ita sono iiiiiii niinus cst in musica csccpto scinitoiiio, quod cst iiicoiiiprcliciisibiIc oniiiihus.

Solo epqdoirs /172,6) est sola uiiitas.

I bis] M'P I i i L 1 scsqiialter] iVP scsqualtciar L 3 XXV] xxxv L 6 deiiaiii] mir. im biiiaiii L 7 Suis] Si vis L

12 qilaiiiiiiii possiiriios] W o i a L 20 supra loii~iiiidi>icni. laritudiiiciii ci nlriiiidiiiciii] WP supci loiigirildinc laritiidiiicii~ L