Alvarus, Thomas, Liber de triplici motu, 1509

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              <p xml:id="N11668">
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                  <pb chead="Prime partis" file="0018" n="18"/>
                duplas: puta quattuor ad duo: et duo ad vnum:
                  <lb/>
                tunc coacerua illos numeros puta quaternarium
                  <lb/>
                binarum et vnitatem et inuenies .7. </s>
                <s xml:id="N116A2" xml:space="preserve">Diuidas igi-
                  <lb/>
                tur corpus in ſeptem ſeptimas: et pro primo ordi­
                  <lb/>
                ne capias quattuor ſeptimas: et pro ſecundo du-
                  <lb/>
                as ſeptimas: et pro vltimo vnam ſeptimam: et ſic
                  <lb/>
                comperies quot partes aliquotas continet quili-
                  <lb/>
                bet illorum ordinū. </s>
                <s xml:id="N116AF" xml:space="preserve">Et iſto modo in qualibet pro­
                  <lb/>
                portione operaberis facile autem hoc demonſtra­
                  <lb/>
                tur ex prima concluſione quoniam ſicut illi tres
                  <lb/>
                ordines cõtinuo ſe habent in proportione dupla
                  <lb/>
                et ſunt partes illius corporis: ita oprtet capere ꝑ­
                  <lb/>
                tes continuo ſe habentes in proportiõe dupla to­
                  <lb/>
                tum corpus abſoluētes eo  oꝑati ſumꝰ artificio</s>
              </p>
              <p xml:id="N116BE">
                <s xml:id="N116BF" xml:space="preserve">Tertia concluſio. </s>
                <s xml:id="N116C2" xml:space="preserve">Alicuius cõtinui
                  <lb/>
                partes aliquota proportionem aliquam rationa­
                  <lb/>
                lem acquirente: proportionē acquiſitam toti inue­
                  <lb/>
                nire. </s>
                <s xml:id="N116CB" xml:space="preserve">vt diuiſio corpore in quin partes aliquo-
                  <lb/>
                tas putas in .5. quintas vna illarum quintarum
                  <lb/>
                acquirente proportionem duplam: inuenire quan­
                  <lb/>
                tam proportionem totum illud corpus proportio­
                  <lb/>
                nē acquirat. </s>
                <s xml:id="N116D6" xml:space="preserve">In illo em̄ caſu illud corpus propor-
                  <lb/>
                tionem ſexquiquintam acquirit: cum acquirat ſu­
                  <lb/>
                pra ſe vnã quintam: hoc eſt tantuꝫ quanta eſt vna
                  <lb/>
                eius quinta. </s>
                <s xml:id="N116DF" xml:space="preserve">Probaemtur hec concluſio / et diuidatur /
                  <lb/>
                a pedale in aliquot partes aliquotas gratia exē-
                  <lb/>
                pli in .7. / et acquirat vna illarum aliquam propor­
                  <lb/>
                tionem rationalem: tunc vel illa proportio acqui­
                  <lb/>
                ſita alicui illarum partium eſt multiplex vel non
                  <lb/>
                multiplex: ſi multiplex tunc aliquotiens vel ſemel
                  <lb/>
                acquirit ſupra ſe tantum quanta ipſa pars eſt. </s>
                <s xml:id="N116EE" xml:space="preserve">et
                  <lb/>
                tot partes equales ſibi quot acquirit ſupra ſe tot
                  <lb/>
                acq̇rit ſupra oēs illas .7. partes aliquotas ī quas
                  <lb/>
                corpus erat diuiſum: et quelibet talis pars acqui­
                  <lb/>
                ſita illi parti eſt equalis cuilibet illarum partium
                  <lb/>
                aliquotarū in quas corpus eſt diuiſum: igitur ille
                  <lb/>
                partes acquiſite vel pars acquiſita eſt vel ſūt eiuſ­
                  <lb/>
                dem denominationis cū parte cui acquiruntur vĺ
                  <lb/>
                acquiritur: et ita ſi ille partes ī quas corpus diui­
                  <lb/>
                debatur ſunt ſeptime: et ille partes acquiſite ſunt
                  <lb/>
                due vel tres vel quattuor / et ſic cõſequenter: totum
                  <lb/>
                illud corpus acquiſiuit duas vel tres vel quatuor
                  <lb/>
                ſeptimas vel ſi eſt vna totum illud corpus acqui-
                  <lb/>
                ſiuit vnam ſeptimam: quo ad inuento: iam patet
                  <lb/>
                quãtam proportionē illud corpus acquiſiuit. </s>
                <s xml:id="N1170D" xml:space="preserve">Si
                  <lb/>
                em̄ acquiſiuit tres tales partes et ille ſūt ſeptime
                  <lb/>
                iam acquiſiuit totum proportionem ſupratripar­
                  <lb/>
                tientem ſeptimas / et ſic habetur propoſitum vbi
                  <lb/>
                pars aliquota proportionem multiplicē acquirit
                  <lb/>
                </s>
                <s xml:id="N11719" xml:space="preserve">Si autem acquirit rationalem nõ multiplicē ma-
                  <lb/>
                nifeſtum eſt /  illa denominatur ab aliqua parte
                  <lb/>
                aliquota vel ab aliquibꝰ partibꝰ aliquotis ade-
                  <lb/>
                quate vel inadequate (non eſt modo cura) ſicut du­
                  <lb/>
                pla ſexquitertia denominatur a numero binario
                  <lb/>
                cum tertia: et ſuprabipartiens tertias ab vnitate
                  <lb/>
                cum duabus tertiis. </s>
                <s xml:id="N11728" xml:space="preserve">Dato igitur /  aliquam talē
                  <lb/>
                proportionem rationalem non multiplicē aliqua
                  <lb/>
                talium partium aliquotarum acquiſiuerit: ad in-
                  <lb/>
                uendiendum quam proportionem acquirit totum
                  <lb/>
                diuidatur quelibet pars aliquota in partes ali-
                  <lb/>
                quotas a quibus denominatur talis proportio / et
                  <lb/>
                tunc coaceruentur omnes ille partes aliquote: et
                  <lb/>
                numerus reſultans indicabit quota pars aliquo­
                  <lb/>
                ta totius eſt aliqua īmo quelibet illarum. </s>
                <s xml:id="N1173B" xml:space="preserve">deinde
                  <lb/>
                illis omnibus addantur ille partes aliquote ac-
                  <lb/>
                quiſite equales eis. </s>
                <s xml:id="N11742" xml:space="preserve">et ſic inuenies quot partes ali­
                  <cb chead="Capitulum octauū."/>
                quotas acquiſiuit totum: et per conſequens qua-
                  <lb/>
                lem proportionem vt ſi in exemplo poſito vna il-
                  <lb/>
                larum ſeptimarum acquirat proportionē ſupra­
                  <lb/>
                bipartientem tertias: et quoniam illa proportio
                  <lb/>
                denominatur ab vno cum duabus tertiis diuida­
                  <lb/>
                tur quelibet ſeptima in tres tertias: et multipliciē­
                  <lb/>
                tur .7. per tria / et reſultabunt .12. et iam ille nume-
                  <lb/>
                rus indicat tibi quamlibet illarum partium eſſe
                  <lb/>
                vnam viceſimam primam: et partes acquiſite ſunt
                  <lb/>
                equales illis quia ſunt tertie vnius ſeptime: et ſūt
                  <lb/>
                due. </s>
                <s xml:id="N1175C" xml:space="preserve">ergo acquiſiuit duas viceſimas primas et ſic
                  <lb/>
                ꝓportionē ſuprabipartiētē viceſimas ṗmas totū
                  <lb/>
                acq̇ſiuit. </s>
                <s xml:id="N11763" xml:space="preserve">Si autē vna illarum ſeptimarū acquirat
                  <lb/>
                duplam ſexquitertiam: diuidas quamlibet ſepti­
                  <lb/>
                mam etiam in tertias: et multiplica ſeptē per tria
                  <lb/>
                et reperies / vt dictum eſt viginti vnum / et quia vna
                  <lb/>
                ſeptima acquiſiuit tantum quanta ipſa eſt puta
                  <lb/>
                vnam ſeptimam totius cuꝫ vna tertia illius ſepti­
                  <lb/>
                me: diuidas etiam illam ſeptimam acquiſitam in
                  <lb/>
                tres partes: et ille tres partes erunt tres viceſime
                  <lb/>
                prime totius / vt conſtat: et totum acquiſiuit illas
                  <lb/>
                tres et cum hoc vnam. </s>
                <s xml:id="N11778" xml:space="preserve">Acquiſiuit igitur quattuor
                  <lb/>
                viceſimas primas: et per conſequens proportionē
                  <lb/>
                ſupraquadripartiētem viceſimas primas. </s>
                <s xml:id="N1177F" xml:space="preserve">Et iſto
                  <lb/>
                modo in omni alia ſpecie proportionis operabe­
                  <lb/>
                ris. </s>
                <s xml:id="N11786" xml:space="preserve">Et ex hoc poteris inuenire proportionem quã
                  <lb/>
                acquirit totum duabus partibus eius aliquotis
                  <lb/>
                nequalibus: ſiue duabus non facientibus vnam:
                  <lb/>
                ſiue pluribus acquirentibus equalem proportio­
                  <lb/>
                nem vel etiam inequalem. </s>
                <s xml:id="N11791" xml:space="preserve">Et conſimiliter cogno­
                  <lb/>
                ſces quam proportionem deperdit totum aliqua
                  <lb/>
                parte eius vel aliquibus partibus aliquotis oli-
                  <lb/>
                quam vel aliquas proportiēes deperdente vel de­
                  <lb/>
                perdentibus.</s>
              </p>
            </div>
            <div xml:id="N1179C" level="3" n="8" type="chapter" type-free="capitulum">
              <head xml:id="N117A1" xml:space="preserve">Capitulum octauum / in quo agi-
                <lb/>
              tur de inuentione proportionis mi-
                <lb/>
              noris inequalitatis et etiam maio-
                <lb/>
              ris reſpectu cuiuſcū numeri ex re-
                <lb/>
              bus diuiſibilibus compoſiti.</head>
              <p xml:id="N117AC">
                <s xml:id="N117AD" xml:space="preserve">PLerum contingit tam in
                  <lb/>
                materia nitenionis difformis quã ꝓ­
                  <lb/>
                portiõis motuum querere proportio­
                  <lb/>
                nem ſubſequialteram vel ſubduplam vel aliquã-
                  <lb/>
                aliam minoris inequalitatis vel etiam maioris
                  <lb/>
                inequalitatis reſpectu numeri non habentis illaꝫ
                  <lb/>
                ſine fratione id eſt diuiſione vnitatis vel vnitatū
                  <lb/>
                talis numeri. </s>
                <s xml:id="N117BE" xml:space="preserve">vt ſi ponat̄̄ /  aliquod mobile per-
                  <lb/>
                tranſeat tripedale ſpacium in hora / tunc mouēs
                  <lb/>
                ſubdupla velocitate tranſit ſubduplum ſpacium
                  <lb/>
                ad tripedale in eodem tēpore. </s>
                <s xml:id="N117C7" xml:space="preserve">Modo non eſt poſ-
                  <lb/>
                ſibile dare ſubduplum ad tripedale ſine fractiõe
                  <lb/>
                vnitatis: quoniam bipedale cum dimidio eſt ſub-
                  <lb/>
                duplum tripedalis. </s>
                <s xml:id="N117D0" xml:space="preserve">Item contingit non nunquaꝫ
                  <lb/>
                querere ſexquialterum reſpectu numeri quinarii:
                  <lb/>
                et illud non poteſt dari ſine fractione vnitatis .7.
                  <lb/>
                enim cum dimidio ad .5. eſt proportio ſexquialte-
                  <lb/>
                ra. </s>
                <s xml:id="N117DB" xml:space="preserve">Quare pro inuentione talis proportionis ma­
                  <lb/>
                ioris aut minoris inequalitatis cum fractione.</s>
              </p>
              <p xml:id="N117E0">
                <s xml:id="N117E1" xml:space="preserve">Suppono primo /  duplex eſt nume­
                  <lb/>
                rus vt ad propoſitum ſufficit quidaꝫ eſt compoſi-
                  <lb/>
                tus ex vnitatibꝰ diuiſibilibꝰ .i. cuius quelibet vni­
                  <lb/>
                tas eſt res diuiſibilis: vt numerus trium pedaliū
                  <lb/>
                quattuor qualitatū .etc̈. alius vero numerus eſt cõ­ </s>
              </p>
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