Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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              <pb chead="Secunde partis" file="0057" n="57"/>
              <note position="left" xml:id="N158DE" xml:space="preserve">5. correĺ.
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              Calcu. de
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              lo. elo.</note>
              <p xml:id="N158E6">
                <s xml:id="N158E7" xml:space="preserve">¶ Hinc patet primum notabile calculatoris / quod
                  <lb/>
                ponit in capitulo de loco elementi circa principiū
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                in ſecundo argumento ſub iſta forma. </s>
                <s xml:id="N158EE" xml:space="preserve">Si ſint qua­
                  <lb/>
                tuor termini continuo proportionales arithme-
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                tice: proportio maxima que ſcilicet eſt inter termi­
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                nos duos minores eorum quatuor per plus exce-
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                dit ſecundam proportionem quam iſta ſecunda ex­
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                cedat tertiam que eſt minima illarum trium pro-
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                portionum que ſunt inter illos quatuor terminos</s>
              </p>
              <p xml:id="N158FD">
                <s xml:id="N158FE" xml:space="preserve">Sexta cõcluſio. </s>
                <s xml:id="N15901" xml:space="preserve">Quando aliqua pro­
                  <lb/>
                portio diminuitur per decrementum termini ma-
                  <lb/>
                ioris ſtante minore: tunc ꝓportio illa efficitur mi-
                  <lb/>
                nor per eam proportionem per quam maior ter-
                  <lb/>
                minus efficitur minor, ſiue per eam quam termi-
                  <lb/>
                nus maior deperdit. </s>
                <s xml:id="N1590E" xml:space="preserve">Et quando aliq̈ proportio
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                efficitur minor per crementū minoris termini ſtã-
                  <lb/>
                te maiore: tunc proportio inter illos terminos ef-
                  <lb/>
                ficitur minor per proportione quam acquirit mi-
                  <lb/>
                nor terminus ſiue per quam efficitur maior. </s>
                <s xml:id="N15919" xml:space="preserve">Exē-
                  <lb/>
                plum / vt capta ꝓportione dupla bipedalis ad pe-
                  <lb/>
                dale que efficiatur minor per decrementum bipe-
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                dalis ſtante pedali: proportio illa dupla efficitur
                  <lb/>
                minor per proportionem quam deperdit bipeda-
                  <lb/>
                le. </s>
                <s xml:id="N15926" xml:space="preserve">Sic exēplificabis de alia parte. </s>
                <s xml:id="N15929" xml:space="preserve">Probatur pri­
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                ma pars / ſit a.b. maior terminus: et c. minor inter
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                quos ſit proportio f. et deperdat a.b. proportionē
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                a.b. ad b. ſtante c. / tunc dico /  proportio illa f. effi-
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                citur minor per proportionem a.b. ad b. quã per-
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                dit terminus maior. </s>
                <s xml:id="N15936" xml:space="preserve">Quod probatur ſic / quia ãte
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                tale decrementum termini maioris: proportio a.
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                b. ad c. componitur ex proportione a.b. ad b. et b.
                  <lb/>
                ad c: / et per tale decrementum termini maioris de-
                  <lb/>
                mitur a b. illa proportione f. proportio a.b. ad b. /
                  <lb/>
                igitur proportio illa f. efficitur minor per propor­
                  <lb/>
                tionem a.b. ad b. / quod fuit probandum. </s>
                <s xml:id="N15945" xml:space="preserve">Et ſic ptꝫ
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                prima pars. </s>
                <s xml:id="N1594A" xml:space="preserve">Et eodem modo probabis ſecūdã.
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                  <note position="left" xlink:href="note-0057-01a" xlink:label="note-0057-01" xml:id="N159EC" xml:space="preserve">1. correĺ.</note>
                </s>
                <s xml:id="N15954" xml:space="preserve">¶ Ex quo ſequitur primo /  quando aliqua ꝓpor­
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                tio diminuitur per decremētum maioris termini
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                et crementum minoris: tunc talis proportio effici-
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                tur minor per proportionem compoſitam ex pro-
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                portiõe / quam deperdit maior terminus et ex pro-
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                portione quam acquirit minor. </s>
                <s xml:id="N15961" xml:space="preserve">Patet hoc corre-
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                larium facile ex dictis et concluſione.
                  <note position="left" xlink:href="note-0057-02a" xlink:label="note-0057-02" xml:id="N159F2" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N1596B" xml:space="preserve">¶ Sequitur
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                ſecūdo /  quando aliqua proportio maioris ine-
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                qualitatis diminuitur per crementū vtriuſ ter-
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                mini: ipſa efficitur minor per proportionem per
                  <lb/>
                quam proportio acquiſita minori excedit propor­
                  <lb/>
                tionem acquiſitam maiori. </s>
                <s xml:id="N15978" xml:space="preserve">Probatur / et ſit ꝓpor­
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                tio f. inter b. terminū maiorē et d. minorē et acqui-
                  <lb/>
                rat b. terminus proportionē g. acquirando a. la-
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                titudinem ſupra ſe: et terminus d. acquirat ꝓpor-
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                tionem h. per acquiſitionem c. excedat propor-
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                tio acquiſita ipſi d. proportionem acquiſitã ipſi
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                b. per proportionem e. / tunc dico /  in fine talis cre­
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                menti illorum terminorum proportio inter illos
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                terminos a.b. et c.d. eſt minor proportiõe f. que eſt
                  <lb/>
                inter b. et d. per proportionem e. per quã propor-
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                tio acquiſita termino minori excedit proportio-
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                nem acquiſitam termino maiori. </s>
                <s xml:id="N15991" xml:space="preserve">Quod ſit proba­
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                tur: quoniam ſi quando b. acquirit proportioneꝫ
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                g.d. acquireret tantaꝫ adequate: ſemꝑ inter illos
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                maneret eadem proportio / vt ſepius argutum eſt /
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                ſed modo terminus minor puta d. vltra illã pro-
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                portionem g. quam acquirit terminus maior ac-
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                quirit proportionem e, quieſcente maiori a.b. vl-
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                teriori acquiſitiõe / igitur illa proportio que eſt in
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                fine videlicet / a.b. ad c.d. efficitur minor per ꝓpor-
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                tionem per quã proportio acquiſita termino mi-
                  <cb chead="Capitulum octauū."/>
                nori excedit proportionem acquiſitam termino
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                maiori / quod fuit probandum.
                  <note position="right" xlink:href="note-0057-03a" xlink:label="note-0057-03" xml:id="N159F8" xml:space="preserve">3. correĺ.</note>
                </s>
                <s xml:id="N159B0" xml:space="preserve">¶ Sequitur tertio /
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                 quando aliqua proportio maioris inequalita­
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                tis diminuitur per vtriuſ eius termini decremē­
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                tum: talis proportio efficitur minor per propor-
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                tionem per quam proportio deperdita a maiori
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                termino excedit proportionem deperditam a mi-
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                nori. </s>
                <s xml:id="N159BF" xml:space="preserve">Probatur / ſit a.b.c. maior terminus d.e. mi-
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                nor inter quos ſit f. proportio: et deperdat termi-
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                nus maior proportionem que eſt a.b.c. ad c. et ter-
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                minus minor proportionē d.e. ad e. excedat pro­
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                portio deperdita a termino maiori proportionē
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                deperditam a termino minori per proportionem
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                h. que ſit b.c. ad c. / et tunc dico /  in fine talis decre-
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                menti proportio f. efficitur minor per proportio-
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                nem h. </s>
                <s xml:id="N159D2" xml:space="preserve">Quod ſic probatur / quia ſi quando d.e.
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                perdit proportionē d.e. ad e., a.b.c. perderet pro-
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                portionem a.b.c. ad b.c. / tunc inter tales terminos
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                adhuc manent f. proportio / vt ſepius probatū eſt:
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                ſed modo ipſe terminus maior a.b.c. vltra talem
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                proportionem perdit adhuc proportionem h. que
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                eſt b.c. ad c. / ergo per illam proportioneꝫ h. que eſt
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                b.c. ad c. illa proportio f. efficitur minor / quod fuit
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                probandum. </s>
                <s xml:id="N159E5" xml:space="preserve">Patet igitur correlarium.</s>
              </p>
              <note position="right" xml:id="N159FE">
                <s xml:id="N15A00" xml:space="preserve">4. correĺ.
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                </s>
                <s xml:id="N15A04" xml:space="preserve">Calcu. in
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                capite de
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                aug.</s>
              </note>
              <p xml:id="N15A0B">
                <s xml:id="N15A0C" xml:space="preserve">¶ Sequitur quarto /  ſi ſint duo proportionabi-
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                lia aliqua proportione maioris inequalitatis et
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                proportio inter illa minoratur per vtriuſ mino-
                  <lb/>
                rationem: proportio deperdita a maiori erit ma-
                  <lb/>
                ior proportione deperdita a minori per propor-
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                tionem per quam proportio inter maius et minus
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                fiet minor: hoc eſt per proportionem que deperdi-
                  <lb/>
                tur inter maius et minus. </s>
                <s xml:id="N15A1D" xml:space="preserve">Probatur / ſit proportio
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                f. inter a. terminum maiorem et b. terminum mino­
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                rem et decreſcente tam a. quam b. efficiatur f. pro-
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                portio minor per proportionem h. / tunc dico /  h.
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                eſt proportio per quam proportio deperdita ab
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                a. termino maiore excedit proportionem deperdi­
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                tam a.b. termino minore. </s>
                <s xml:id="N15A2C" xml:space="preserve">Quod ſic ꝓbatur / quo-
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                niam quando aliqua proportio maioris inequa-
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                litatis minoratur. </s>
                <s xml:id="N15A33" xml:space="preserve">per decrementum vtriuſ ex-
                  <lb/>
                tremi: ipſa efficitur minor per proportionem / per
                  <lb/>
                quam proportio deperdita a maiore termino ex-
                  <lb/>
                cedit proportionem deperditam a minori / vt patꝫ
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                ex anteriori correlario: ſed proportio f. que eſt a.
                  <lb/>
                ad b. minoratur decreſcente vtro termino: ergo
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                ſequitur /  ipſa proportio f.a. ad b. efficitur mi-
                  <lb/>
                nor per proportionē per quam proportio deper-
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                dita a termino maiori puta a. excedit proportio-
                  <lb/>
                nem deperditam a minore puta b. ſed illa propor­
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                tio eſt h. ex hypotheſi: igitur proportio h. eſt pro-
                  <lb/>
                portio per quam proportio deperdita a maiori
                  <lb/>
                termino puta a. excedit proportionem deperditã
                  <lb/>
                a minori puta b. / quod fuit probandum. </s>
                <s xml:id="N15A50" xml:space="preserve">Et hec eſt
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                quedam regula et ſuppoſitio quam calculator po­
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                nit in reſponſione ad argumentum / quod facit cõ-
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                tra duas vltimas concluſiões in capitulo de aug-
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                mentatione in opinione prima.</s>
              </p>
              <p xml:id="N15A5B">
                <s xml:id="N15A5C" xml:space="preserve">Septima concluſio. </s>
                <s xml:id="N15A5F" xml:space="preserve">Si aliqua quã-
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                titas maior creſcat reſpectu quantitatis minoris
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                non variate acquirendo ſupra ſe aliquã propor-
                  <lb/>
                tionem: tantam proportionem acquirit ſupra nu­
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                merum minorem hoc eſt ſupra proportionem quã
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                habet ad numerum minorem quantam acquirit
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                ſupra ſe. </s>
                <s xml:id="N15A6E" xml:space="preserve">Et ſi quantitas maior manens maior re­
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                ſpectu quantitatis minoris inuariate deſcreſcat ſi­
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                ue perdat aliquam proportionem: quantam pro-
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                portionem deperdit a ſeipſa tantam deperdit re-
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                ſpectu quantitatis minoris: hoc eſt a proportiõe </s>
              </p>
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