Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Prime partis" file="0019" n="19"/>
                poſitus ex vnitatibus indiuiſibilibus vt numerus
                  <lb/>
                5. punctorū .5. intelligentiarum et .10. animarū ra­
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                tionalium. </s>
                <s xml:id="N117F5" xml:space="preserve">Hec ſuppoſitio ex ſe patet.</s>
              </p>
              <p xml:id="N117F8">
                <s xml:id="N117F9" xml:space="preserve">Secunda ſuppoſitio. </s>
                <s xml:id="N117FC" xml:space="preserve">Nõ oīs nume­
                  <lb/>
                rus habet ſubduplū. </s>
                <s xml:id="N11801" xml:space="preserve">nec oīs habet ſubtriplum: et
                  <lb/>
                ſic conſequenter. </s>
                <s xml:id="N11806" xml:space="preserve">Probatur / quoniã aliquis nume­
                  <lb/>
                rus puta rerum indiuiſibiliū cuiuſmodi: eſt nūerꝰ
                  <lb/>
                ternarius angelorū nõ poteſt diuidi in duo equa-
                  <lb/>
                lia: igitur nõ habet ſubduplū: nec in quatuor par­
                  <lb/>
                tes equales: et ſic non habet ſubquadruplum: et ſic
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                probatur de aliis / igitur ſuppoſitio vera.</s>
              </p>
              <p xml:id="N11813">
                <s xml:id="N11814" xml:space="preserve">Tertia ſuppoſitio </s>
                <s xml:id="N11817" xml:space="preserve">Oīs numerus re­
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                rum diuiſibiliū habet ſubduplū ſubtriplū: et vni-
                  <lb/>
                uerſaliter oēm proportioneꝫ minoris inequalita-
                  <lb/>
                tis: et etiaꝫ maioris aut habere poteſt. </s>
                <s xml:id="N11820" xml:space="preserve">Probatio
                  <lb/>
                huius ſuppoſitionis: quia talis numerus poteſt
                  <lb/>
                diuidi in duo equalia cū ſit numerus rerū diuiſi-
                  <lb/>
                bilium et tria equalia et in .4. et in 5. / et ſic in infini-
                  <lb/>
                tum </s>
                <s xml:id="N1182B" xml:space="preserve">Quare dabitur quilibet numerus habēs pro­
                  <lb/>
                portionē minoris inequalitatis ad ipſum: et etiaꝫ
                  <lb/>
                maioris. </s>
                <s xml:id="N11832" xml:space="preserve">Nam ad ſui medietatē habebit propor­
                  <lb/>
                tionem duplã: ad tertiam triplã: ad duas tertias
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                ſexquialteram: et ſic in infinitum.</s>
              </p>
              <p xml:id="N11839">
                <s xml:id="N1183A" xml:space="preserve">Quarta ſuppoſitio </s>
                <s xml:id="N1183D" xml:space="preserve">Ad diuidendum
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                numerū aliquem per alterum ſiue maiorē, ſiue mi­
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                norem, ſiue equalem, ſiue oporteat vti fractione,
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                ſiue nõ: diuidenda eſt quelibet vnitas numeri diui­
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                dendi in tot partes aliquotas quotus eſt numerꝰ
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                per quem fit diuiſio: et dande ſunt tot partes illa­
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                rum cuilibet vnitati numeri ꝑ quē fit diuiſio quo-
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                tus eſt numerus diuidendus: et ſic quelibet vnitas
                  <lb/>
                habebit equaliter. </s>
                <s xml:id="N11850" xml:space="preserve">Exemplū / vt ſi velis diuidere nu­
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                merū quinariū per numeꝝ ternariū: vt puta quī
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                gradus in tres partes equales: vel quin denari­
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                os per tres homines: diuidas quãlibet vnitatem
                  <lb/>
                numeri quinarii ī tres partes aliquotas: puta in
                  <lb/>
                tres tertias quia numerus per quem fit diuiſio eſt
                  <lb/>
                ternarius: deinde da quin tertias culibet vnita­
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                ti ternarii: quia numerus diuidēdus eſt quinariꝰ
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                </s>
                <s xml:id="N11862" xml:space="preserve">Item ſi velis diuidere tria per quin: q2 numerus
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                per quē fit diuiſio eſt quinarius: diuidas quãlibet
                  <lb/>
                vnitatē numeri ternarii diuidēdi in quī partes
                  <lb/>
                equales. </s>
                <s xml:id="N1186B" xml:space="preserve">puta in quī quītas et q2 numerus diui-
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                dendus eſt ternarius: da cuilibet tres quintas: et
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                quilibet illorū quī habebit equaliter. </s>
                <s xml:id="N11872" xml:space="preserve">Probat̄̄
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                hec ſuppoſitio / qm̄ ſic diuendo cuilibet equaliter
                  <lb/>
                datur / vt patet ex ſe et nichil manet: ergo illa diui­
                  <lb/>
                ſio eſt cõpleta: et modus diuidendi ſufficiens: et per
                  <lb/>
                cõſequens ſuppoſitio vera. </s>
                <s xml:id="N1187D" xml:space="preserve">Probatur minor / qm̄
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                quando tria diuiditur per quin gratia exempli
                  <lb/>
                oportet iuxta tenorē ſuppoſitionis diuidere quã­
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                libet vnitatē numeri ternarii in quī partes equa­
                  <lb/>
                les. </s>
                <s xml:id="N11888" xml:space="preserve">et ſic erunt partes ille, ter, quin: et per conſe­
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                quēs quīquies tres partes adequate / vt patꝫ: erūt
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                igitur ibi quī ternarii illarū partiū adequate et
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                datur cuilibet vnitati quinarii numeri vnꝰ terna­
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                rius: igitur nullus ternarius manet / qm̄ illi terna­
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                rii et vnitates numeri quinarii ſunt numero equa­
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                les: igitur tunc nichil manet diuidendū. </s>
                <s xml:id="N11897" xml:space="preserve">Et ſic pro­
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                babis de quibuſcū aliis numeris quorum vnus
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                per alterum diuiditur: ſequitur igitur ſuppoſitio</s>
              </p>
              <p xml:id="N1189E">
                <s xml:id="N1189F" xml:space="preserve">His ſuppoſitis pono talem regulam
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                </s>
                <s xml:id="N118A3" xml:space="preserve">Ad diuidendum numerum ſe habentem in qua vo­
                  <cb chead="Capitulum octauū."/>
                lueris proportione minoris inequalitatis ad quē­
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                cū numerum volueris capias in numeris duos
                  <lb/>
                numeros ſe habentes in tali proportione: et diui-
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                das numerum reſpectu cuiꝰ queris numerū ſe ha-
                  <lb/>
                bentem in proportione minoris inequalitatis in
                  <lb/>
                tot partes equales quotus eſt numerus maior ta­
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                lis proportionis: et ex his capias tot illarū par­
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                tium quotus eſt numerus minor dicte proportio-
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                nis. </s>
                <s xml:id="N118B9" xml:space="preserve">Et ſic inuenies propoſitum. </s>
                <s xml:id="N118BC" xml:space="preserve">Hoc facili mõſtra­
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                tur exemplo: vt ſi vis inuenire numerū ſe habentē
                  <lb/>
                in proportione ſubſexquitertia reſpectu numeri
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                quinarii in rebus diuiſibilibus (quoniã in indiui­
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                ſibilibus nõ eſt poſſibile / vt patet ex primis duabꝰ
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                ſuppoſitionibus) capias in nūeris .4. et .3. qui ſūt
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                numeri ſe habentes in proporſitione ſexquitertia
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                et numerus maior eſt quaternariꝰ: diuidas nume-
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                rum quinariū reſpectu cuius queris ſubſexquiter­
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                tium numerum in quattuor partes equales: et hãc
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                diuiſionem facies per quarte ſuppoſionis docu­
                  <lb/>
                mentū: et q2 nūerus mīor eſt ternariꝰ capias tres
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                quartas quinarii: et illarum trium quartarū ad
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                illum numerum quinarium qui componitur ade-
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                quate ex quattuor talibꝰ eſt proportio ſubſexqui­
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                tertia. </s>
                <s xml:id="N118DD" xml:space="preserve">Et iſto modo in omībus aliis operaberis
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                </s>
                <s xml:id="N118E1" xml:space="preserve">Patet hec regula quoniã / tunc talis numerus ſe
                  <lb/>
                habebit ad illas ſuas partes aliquotas ſicut ſe
                  <lb/>
                habent nūeri proportionis queſite / vt conſtat: igit̄̄
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                illo modo oportet operari ad inueniēdū id quod
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                docet regula: et per cõſequens regula vera.</s>
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              <p xml:id="N118EC">
                <s xml:id="N118ED" xml:space="preserve">Secunda regula. </s>
                <s xml:id="N118F0" xml:space="preserve">Ad inueniendum
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                numerū ſe habentem in proportione maioris ine­
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                qualitatis ad quem volueris numerū: et in quacū­
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                 libuerit proportione: capias in numeris duos
                  <lb/>
                numeros ſe habentes in tali proportione: et diui­
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                das numerū reſpectu cuius queris numerū ſe ha-
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                bentē in illa proportione maioris inequalitatis
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                in tot partes equales quotus eſt numerus minor
                  <lb/>
                talis proportionis: et tunc illi numero minori ſic
                  <lb/>
                diuiſio addas tot equales partes partibus diui­
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                ſionis quot ſunt per quas numerus maior talis
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                proportionis excedit minorē. </s>
                <s xml:id="N11909" xml:space="preserve">et tunc numerus re-
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                ſultans ex nnmero minori et illa additione eſt nu­
                  <lb/>
                merus ſe habens ad numerū ſic diuiſuꝫ in prppor­
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                tione data maioris inequalitatis. </s>
                <s xml:id="N11912" xml:space="preserve">Hoc facile de-
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                clarabit exemplū </s>
                <s xml:id="N11917" xml:space="preserve">Si em̄ velis īuenire numeꝝ ſex­
                  <lb/>
                quialterū ad numerū quinariū in rebus diuiſibi-
                  <lb/>
                libus (in īdiuiſibilibus em̄ id nequit fieri / vt dictū
                  <lb/>
                eſt) capias in numeris duos numeros ſe habētes
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                in proportione ſexquialtera: vt puta .2. et .3: et q2
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                numerus minor eſt binarius diuidas numeꝝ qui­
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                narium reſpectu cuius queris numerum ſexquial­
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                terum in duas partes equales quod fiet ſecūdum
                  <lb/>
                documentum quarte ſuppoſitionis. </s>
                <s xml:id="N1192A" xml:space="preserve">Oportt em̄
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                tunc diuidere .5. per .2. et quia ternarius numerus
                  <lb/>
                maior talis proportionis excedit numerum bina­
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                rium minorem numerum talis proportionis per
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                vnam vnitatem adequate: addas ſupra numeruꝫ
                  <lb/>
                quinariū vnam de illis partibus duabus in quas
                  <lb/>
                iam diuiſus eſt quinarius puta medietateꝫ ipſius
                  <lb/>
                quinarii: tūc aggregatum ex quinario et illa par­
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                te ſe habet ad quinarium in proportione data pu­
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                ta ſexquialtera. </s>
                <s xml:id="N1193F" xml:space="preserve">Patet hec regula ſicut ſuperior
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                </s>
                <s xml:id="N11943" xml:space="preserve">Applica probationem. </s>
                <s xml:id="N11946" xml:space="preserve">Et hec breuiter de prima
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                parte huius operis introductionis gratia dicta
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                ſufficiant.</s>
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