Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N140E1" level="3" n="6" type="chapter" type-free="capitulum">
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                <s xml:id="N14A74" xml:space="preserve">
                  <pb chead="Secunde partis" file="0048" n="48"/>
                nūeris reperirent̄̄ irratiõales ꝓportiões: vt ſatis
                  <lb/>
                cõſtat ītelligēti. </s>
                <s xml:id="N14A80" xml:space="preserve">Et ſic ptꝫ correlariū.
                  <note position="left" xlink:href="note-0048-01a" xlink:label="note-0048-01" xml:id="N14B33" xml:space="preserve">5. correĺ.</note>
                </s>
                <s xml:id="N14A88" xml:space="preserve">¶ Sequit̄̄ q̇n­
                  <lb/>
                to:  ꝓpoſita q̈uis ꝓportiõe ratiõali: nõ difficile ē
                  <lb/>
                īueſtigare et ſcire an habeat ꝓportionē rõnalē ſub
                  <lb/>
                multiplicē: an aliquã aliã rationalē minoris ineq̈­
                  <lb/>
                litatꝪ: vt ꝓpoſita ꝓportiõe dupla īueſtigare et ſci­
                  <lb/>
                re poterimꝰ an habeat ſubduplã: ſubtriplã: ſubq̈-
                  <lb/>
                druplã rationalē .etc̈. nec ne: cõſiderando primū ex
                  <lb/>
                doctrina vndecime ↄ̨cluſiõis: an habeat medieta-
                  <lb/>
                tem: tertiã: quartã: quintã rationales: et cõperien-
                  <lb/>
                tes  nõ: dicemus ipſam nõ habere ſubtriplam:
                  <lb/>
                ſubquadruplã .etc̈. rationales. </s>
                <s xml:id="N14A9F" xml:space="preserve">Et eadem ratione
                  <lb/>
                dicemꝰ ipſam nõ habere ſubſexq̇tertiã rationalē:
                  <lb/>
                q2 nõ habet ꝓportionē cõpoſitã ex tribus quartis
                  <lb/>
                eius rationalibus: nec ſubſexquialterã rationalē:
                  <lb/>
                q2 nõ habet ꝓportionē compoſitã ex duabus ter-
                  <lb/>
                tiis eius rationalibus. </s>
                <s xml:id="N14AAC" xml:space="preserve">Et ſic in omnibus aliis di­
                  <lb/>
                ces. </s>
                <s xml:id="N14AB1" xml:space="preserve">Demonſtratio huius correlarii innititur huic
                  <lb/>
                baſi et fundamento /  nun̄ aliqua ꝓportio ratio­
                  <lb/>
                nalis cõponitur adequate ex vna rationali et vna
                  <lb/>
                irrationali. </s>
                <s xml:id="N14ABA" xml:space="preserve">Applica tu demonſtrationē. </s>
                <s xml:id="N14ABD" xml:space="preserve">Iſto mo­
                  <lb/>
                do inquirere debes an habet ſubſuprapartientē
                  <lb/>
                rationalē aut ſub multiplicē ſubſuprapartientem
                  <lb/>
                rationalē: aut ſub multiplicē ſubſuꝑparticularē:
                  <lb/>
                īueſtigando et inquirendo ex cõcluſione vndecima
                  <lb/>
                an talis ꝓportio rationalis ꝓpoſita habeat par­
                  <lb/>
                tem aliquotã rationalē vel partes a qua vel a qui­
                  <lb/>
                bus denominatur dicta ꝓportio minoris inequa­
                  <lb/>
                litatis: et ſi ſic aſcribenda eſt ei talis ꝓportio mi-
                  <lb/>
                noris inequalitatis rationalis: ſin minus: aſſeren­
                  <lb/>
                dum eſt ipſam nõ habere talē ꝓportionē minoris
                  <lb/>
                inequalitatis rationalē. </s>
                <s xml:id="N14AD6" xml:space="preserve">Patet igit̄̄ correlarium.
                  <lb/>
                </s>
                <s xml:id="N14ADA" xml:space="preserve">Profundius em̄ velle illud demonſtrare eſt ipſuꝫ
                  <lb/>
                tenebris īuoluere.
                  <note position="left" xlink:href="note-0048-02a" xlink:label="note-0048-02" xml:id="N14B39" xml:space="preserve">6. correĺ.</note>
                </s>
                <s xml:id="N14AE4" xml:space="preserve">¶ Sequitur ſexto per modum
                  <lb/>
                epilopi oīm eoꝝ / que preſenti capite digeſta ſunt:
                  <lb/>
                 quauis ꝓportione rationali ꝓpoſita: ſcire po-
                  <lb/>
                terimus an habeat aliquã ꝓportionē rationalem
                  <lb/>
                maioris inequalitatis ad ſeipſam et minoris ine-
                  <lb/>
                qualitatis: et quas habeat: et quas nõ. </s>
                <s xml:id="N14AF1" xml:space="preserve">Et hoc ca-
                  <lb/>
                put diligenter conſidera quoniã ex eo pendet fer-
                  <lb/>
                me vniuerſalis huiꝰ materie īquiſitio: et ſuprema
                  <lb/>
                eius difficultas. </s>
                <s xml:id="N14AFA" xml:space="preserve">¶ His adde /  doctrina huius ca-
                  <lb/>
                pitis habita: ꝓpoſita aliqua certa velocitate ꝓ-
                  <lb/>
                ueniente ab aliqua ꝓportione rationali nota: iu-
                  <lb/>
                dicare poterꝪ de quacū alia velocitate a quauis
                  <lb/>
                alia ꝓportiõe ꝓueniente cõmenſurabiles ſint. </s>
                <s xml:id="N14B05" xml:space="preserve">nec
                  <lb/>
                ne. </s>
                <s xml:id="N14B0A" xml:space="preserve">Item ꝓpoſita quauis velocitate ꝓueniente ab
                  <lb/>
                aliqua ꝓportione ratiõali nota: ſcire de quacū
                  <lb/>
                alia velocitate date velocitati cõmenſurabili a q̈
                  <lb/>
                ꝓportiõe ꝓueniat: ratiõali vcꝫ vĺ irrationali / q̊ ex
                  <lb/>
                his ſcito et ſequētibꝰ: particulariꝰ ſcire poteris ex
                  <lb/>
                qua rationali vel irrationali ꝓueniat ſpecifice.</s>
              </p>
            </div>
            <div xml:id="N14B3F" level="3" n="7" type="chapter" type-free="capitulum">
              <head xml:id="N14B44" xml:space="preserve">Capitum ſeptimū / in quo agitur de medie
                <lb/>
              rei inuentione et proportione proportionuꝫ
                <lb/>
              rationalis et irrationalis.</head>
              <p xml:id="N14B4B">
                <s xml:id="N14B4C" xml:space="preserve">AD habendam aliqualē noti-
                  <lb/>
                ciã de ꝓportiõe ꝓportiõis rationalis et
                  <lb/>
                irrationalis et duarū irrationaliū ſit.</s>
              </p>
              <p xml:id="N14B53">
                <s xml:id="N14B54" xml:space="preserve">Prima ſuppoſitio. </s>
                <s xml:id="N14B57" xml:space="preserve">Oīs numerus ha­
                  <lb/>
                bet numerū ad ſe duplū, triplū, quadruplū, et ſic
                  <lb/>
                in infinitū: aſcēdendo per ſpecies ꝓportionis mul­
                  <lb/>
                tiplicis. </s>
                <s xml:id="N14B60" xml:space="preserve">Iſta ſuppoſitio patet ex ſe / qm̄ dato vno
                  <lb/>
                numero ex duabus vnitatibus adequate cõpoſito
                  <lb/>
                dabitur vnus alter compoſitus ex quatuor: et ille
                  <lb/>
                erit duplus: et alter ex ſex: et erit triplus: et alter ex
                  <lb/>
                octo: et erit quadrupus: et ſic ſine termino.</s>
              </p>
              <p xml:id="N14B6B">
                <s xml:id="N14B6C" xml:space="preserve">Secunda ſuppoſitio. </s>
                <s xml:id="N14B6F" xml:space="preserve">Omnis nume­
                  <lb/>
                rus rerum diuiſibiliū ſiue quantitas habet cuius
                  <cb chead="Capitulū ſeptimū."/>
                cū denominationis aliquam partem aliquotaꝫ
                  <lb/>
                cum fractione vel ſine fractione. </s>
                <s xml:id="N14B79" xml:space="preserve">Uolo dicere /  ſi-
                  <lb/>
                gnato quocun numero rerū diuiſibiliū talis nu­
                  <lb/>
                merus habet medietatē tertiam, quartam, quin-
                  <lb/>
                tam, ſextam, ſeptimam, et ſic in infinitū. </s>
                <s xml:id="N14B82" xml:space="preserve">Proba-
                  <lb/>
                tur: quia capto numero duodenario ille habet me­
                  <lb/>
                dietatem, puta numerum ſenariū: habet numerū
                  <lb/>
                quaternariū pro tertia, ternariū pro quarta, pro
                  <lb/>
                quinta vero habet numerū cū fractione, ad quam
                  <lb/>
                fractionē inueniendã oportet duodecim per quī
                  <lb/>
                diuidere: et exibit binariꝰ cū duabꝰ q̇ntis iuxta do-
                  <lb/>
                ctrinã ſuperiꝰ poſitã octauo capite ṗme partꝪ. </s>
                <s xml:id="N14B93" xml:space="preserve">Et
                  <lb/>
                ſic operãdū eſt in cuiꝰ vis alteriꝰ ꝑtꝪ aliq̊te īuētiõe.</s>
              </p>
              <p xml:id="N14B98">
                <s xml:id="N14B99" xml:space="preserve">Tertia ſuppoſitio. </s>
                <s xml:id="N14B9C" xml:space="preserve">Supra quēcū
                  <lb/>
                numerū rerum diuiſibiliū contingit dare numeꝝ
                  <lb/>
                continentē ipſum et medietatē: et alium continentē
                  <lb/>
                ipſum et vnam tertiam, et duas tertias: aut tres
                  <lb/>
                quartas: et ſic de qnibuſcun aliis partibus ali-
                  <lb/>
                quotis. </s>
                <s xml:id="N14BA9" xml:space="preserve">Patet / qm̄ ad dandū numerū continentē
                  <lb/>
                ipſum et medietatē ſufficit addere illi medietatem
                  <lb/>
                ſui: et ad dandum numerū continentē ipſum et du-
                  <lb/>
                as tertias ſufficit ei addere illas duas tertias: vt
                  <lb/>
                patet ex ſe aſpicienti in numeris. </s>
                <s xml:id="N14BB4" xml:space="preserve">Quomodo autē
                  <lb/>
                tales partes īueniant̄̄ p̄cedēs ſuppoſitio declarat</s>
              </p>
              <p xml:id="N14BB9">
                <s xml:id="N14BBA" xml:space="preserve">Quarta ſuppoſitio. </s>
                <s xml:id="N14BBD" xml:space="preserve">Quodlibet con-
                  <lb/>
                tinuū eſt duplū ad ſuã medietatē: triplū ad tertiã:
                  <lb/>
                quadruplū ad quartã: ſexquialterū ad duas ter-
                  <lb/>
                tias: et ſic de qualibet alia ſpecie ꝓportionis. </s>
                <s xml:id="N14BC6" xml:space="preserve">Pa­
                  <lb/>
                tet hec ſuppoſitio ex diffinitionibus terminorum.</s>
              </p>
              <p xml:id="N14BCB">
                <s xml:id="N14BCC" xml:space="preserve">Quinta ſuppoſitio. </s>
                <s xml:id="N14BCF" xml:space="preserve">Omnis ꝓportio
                  <lb/>
                habet medietatē: tertiam: quartã: et ſic in infinitū.
                  <lb/>
                </s>
                <s xml:id="N14BD5" xml:space="preserve">Probatur hec ſuppoſitio / q2 oīs quantitas cõti-
                  <lb/>
                nua: et quodlibet cõtinuo ſucceſſiue diminuibile eſt
                  <lb/>
                huiuſmodi et oīs ꝓportio eſt quantitas continua
                  <lb/>
                aut cõtinuo partibiliter diminuibilis (et diſtribu-
                  <lb/>
                at ly omnis pro generibus ſingulorum more ma-
                  <lb/>
                themathicorum) / igitur propoſitum.</s>
              </p>
              <p xml:id="N14BE2">
                <s xml:id="N14BE3" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N14BE6" xml:space="preserve">Si aliq̄ due quã-
                  <lb/>
                titates cõtinue ſe habeant in aliqua proportione
                  <lb/>
                ratiõali vel irratiõali: dabilis eſt vna tertia qua-
                  <lb/>
                libet illarū maior que ſe habeat in eadē ꝓportiõe
                  <lb/>
                ad maiorē illaꝝ. </s>
                <s xml:id="N14BF1" xml:space="preserve">vt ſi .4. et .2. ſe habeãt in aliqua ꝓ­
                  <lb/>
                portione dabilis eſt alter numerus puta .8. qui in
                  <lb/>
                eadem ꝓportione ſe habeat ad .4. et ſi diameter a.
                  <lb/>
                ſe habeat in aliqua ꝓportione ad coſtã b. dabilis
                  <lb/>
                eſt vna alia quãtitas puta c. que ſe habet in eadeꝫ
                  <lb/>
                ꝓportione ad b. </s>
                <s xml:id="N14BFE" xml:space="preserve">Patet hec ſuppoſitio ex ſe.</s>
              </p>
              <p xml:id="N14C01">
                <s xml:id="N14C02" xml:space="preserve">His poſitis ſit prima cõcluſio. </s>
                <s xml:id="N14C05" xml:space="preserve">Que-
                  <lb/>
                libet ꝓportio ratiõalis in q̈libet ꝓportiõe multi-
                  <lb/>
                plici ab aliq̈ ratiõali excedit̄̄. </s>
                <s xml:id="N14C0C" xml:space="preserve">Hoc eſt q̈libet ꝓpor-
                  <lb/>
                tio ratiõalis hꝫ ꝓportionē duplã: triplã: q̈druplã
                  <lb/>
                et ſic in īfinitū rõnales. </s>
                <s xml:id="N14C13" xml:space="preserve">Probat̄̄ hec ↄ̨cĺo / qm̄ ſi illa
                  <lb/>
                ꝓportio fuerit mĺtiplex manifeſtū ē /  ad nūeꝝ eiꝰ
                  <lb/>
                maiorē dabit̄̄ aliq̇s nūerꝰ ſe hñs in eadē ꝓportiõe /
                  <lb/>
                ad illū ſicut ille partes hꝫ ad minorē / vt ptꝫ ex ṗma ſup­
                  <lb/>
                poſitiõe: et tūc illiꝰ ad minimū erit ꝓportio dupla
                  <lb/>
                ad ꝓportionē medii ad minimū: qm̄ illa cõponit̄̄
                  <lb/>
                ex duabꝰ eq̈libꝰ illi: et ſi addat̄̄ q̈rtꝰ nūerꝰ ſe hñs in
                  <lb/>
                eadē ꝓportione ad tertiū in qua tertius ſe habet
                  <lb/>
                ad ſecundū: ſicut poteſt fieri ex prima ſuppoſitiõe:
                  <lb/>
                iã ꝓportio illius ad minimū erit tripla ad ꝓpor-
                  <lb/>
                tionē ſcḋi ad minimū: et cū poſſint ſic addi infiniti
                  <lb/>
                ṫmini ↄ̨tinuo ꝓportiõabiles illa ꝓportiõe mĺtipli­
                  <lb/>
                ci / vt ptꝫ ex ṗma ſuppõe: ſequit̄̄ /  ad illã ꝓportionē
                  <lb/>
                dabit̄̄ ꝓportio dupla, tripla, q̈drupla, et ſic ī īfini­
                  <lb/>
                tū. </s>
                <s xml:id="N14C32" xml:space="preserve">Ptꝫ ↄ̨ña ex octaua ↄ̨cĺiõe p̄cedētꝪ capitꝪ </s>
                <s xml:id="N14C35" xml:space="preserve">Si o
                  <lb/>
                illa ſit ſuꝑparticĺarꝪ ad maximū extremū eiꝰ adde­ </s>
              </p>
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