Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N15216" level="3" n="8" type="chapter" type-free="capitulum">
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                <s xml:id="N153CE" xml:space="preserve">
                  <pb chead="Secunde partis" file="0052" n="52"/>
                ¶ Ex hac concluſione ſequitur /  ſi
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                aliq̈ latitudo maior puta a. vniformiṫ cõtinuo in
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                aliquo tēpore deperdat aliquam partē ſui: et vna
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                alia latitudo minor puta b. deperdat cõtinuo vni-
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                formiter in tanto tēpore, maiori, vel minori (non
                  <lb/>
                curo) tantã partē adequate ſui: maior ꝓportio eſt
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                inter latitudinē minorem in medio inſtanti prime
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                medietatis tēporis in quo ipſa diminuitur et ſeip­
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                ſam in medio inſtanti ſecūde medietatis eiuſdē tē­
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                poris: quã īter latitudinē maiorē in inſtãti medio
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                prime medietatis tēporis / in quo ipſa diminuitur
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                et ſeipſaꝫ in inſtãti medio ſecūde medietatꝪ eiuſdē
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                tēporis. </s>
                <s xml:id="N153E9" xml:space="preserve">Exemplū / vt capta latitudine .12. graduū
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                et .8. graduū: et diminuatur latitudo: 12. graduuꝫ
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                in hora cõtinuo vniformiter, deperdendo adequa­
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                te quatuor gradus. </s>
                <s xml:id="N153F2" xml:space="preserve">et in tanto tēpore vel maiori vĺ
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                minori (nõ curo) cõtinuo vniformiter deperdat la-
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                titudo .8. graduū etiã quatuor gradus adequate:
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                tunc ipſius latitudinis minoris in inſtanti medio
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                ṗme medietatꝪ tꝑis in quo ipſa diminuit̄̄ ad ipſã
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                in inſtãti medio ſecūde medietatis eiuſdē tēporis
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                eſt maior ꝓportio: quã inter latitudinē maiorē in
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                inſtanti medio prime medietatis temporis in quo
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                diminuitur et ſeipſam in inſtanti medio ſecūde me­
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                dietatis eiuſdē tēporis. </s>
                <s xml:id="N15407" xml:space="preserve">Nam illa eſt ꝓportio ſu-
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                prabipartiens quintas puta .7. ad .5. hec vero eſt
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                ſuprabipartiens nonas puta .11. ad .9. </s>
                <s xml:id="N1540E" xml:space="preserve">Modo illa
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                maior eſt hac / vt conſtat ex predictis. </s>
                <s xml:id="N15413" xml:space="preserve">Hoc correla-
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                riū eandē cū cõcluſione petit demonſtrationē: qm̄
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                ipſa latitudo maior ab inſtanti medio prime me-
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                dietatis tēporis in quo diminuitur vſ ad inſtãs
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                mediū ſecunde medietatis eiuſdē tēporis tantam
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                latitudinē deperdit adequate: quantam latitudo
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                minor perdit ab inſtanti medio prime medietatis
                  <lb/>
                tēporis in quo diminuitur vſ ad inſtans mediū
                  <lb/>
                ſecūde medietatis eiuſdē tēporis: q2 illa tempora
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                ſunt medietates totaliū tēpoꝝ / vt conſtat in quibꝰ
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                deperduntur medietates latitudinū deꝑdendarū
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                adequate / igit̄̄ maiorē ꝓportionē deꝑdit minor la­
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                titudo in tali tēpore: quã maior in tꝑe correſpõdē­
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                ti. </s>
                <s xml:id="N15430" xml:space="preserve">Patet hec ↄ̨ña ex ſcḋa parte octaue ſuppoſiti-
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                onis p̄allegate: et ꝓportio deꝑdita ab aliqua lati­
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                tudine in aliquo tꝑe eſt ꝓportio īter eandē latitu-
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                dinē in prīcipio talis tꝑis et ſeipſã in fine / vt patet /
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                ergo maior eſt ꝓportio inter minorē latitudinē in
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                inſtãti medio prime medietatis temporis in quo
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                diminuit̄̄ ad ſeipſam in in inſtanti medio ſcḋe me­
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                dietatis tꝑis eiuſdē: quã īter latitudinē maiorē in
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                inſtãti medio ṗme medietatꝪ tꝑis in quo diminuit̄̄
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                et ſeipſã in inſtãti medio ſcḋe medietatis eiuſdem
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                tꝑis / quod fuit ꝓbandū. </s>
                <s xml:id="N15447" xml:space="preserve">Patet igitur correlariū.
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                </s>
                <s xml:id="N14DA2" xml:space="preserve">¶ Ex quo ſequitur ſecundo /  ſi latitudo motus a.
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                maior et b. minor diminuantur vniformiter cõti-
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                nue in tempore equali vel inequali perdendo ade-
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                quate equalem latitudinem: maior eſt proportio
                  <lb/>
                inter motum b. in principio temporis in quo ipſe
                  <lb/>
                diminuitur et ſeipſum in fine talis temporis: quã
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                inter motum a. in principio temporis in quo ipſe
                  <lb/>
                diminuitur et ſeipſum in fine eiuſdem temporis: et
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                ſimiliter maior eſt ꝓportio inter motum b. in inſtã­
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                ti medio prime medietatis temporis in quo ipſe
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                diminuitur et ſeipſum in inſtanti medio ſecunde.
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                medietatis eiuſdem temporis: quam inter motuꝫ
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                a. in inſtanti medio prime medietatis temporis ī
                  <lb/>
                quo ipſe diminuitur et ſeipſum in inſtanti medio
                  <lb/>
                ſecunde medietatis eiuſdem temporis. </s>
                <s xml:id="N14DC1" xml:space="preserve">Prima
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                pars huius auxilio concluſionis precedentis oſtē­
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                ditur et ſecunda ex correlario facile ſuam demon­
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                ſtrationem aſſumit.
                  <note position="left" xlink:href="note-0050-01a" xlink:label="note-0050-01" xml:id="N14DDA" xml:space="preserve">calcu. de
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                  mo. loca.</note>
                </s>
                <s xml:id="N14DCF" xml:space="preserve">Et hoc correlarium eſt quar-
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                tum ſuppoſitum calculatoris ī capite de motu lo­
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                cali cõcluſione .38. / quod ponit ſub his verbis.</s>
              </p>
              <p xml:id="N14DE2">
                <s xml:id="N14DE3" xml:space="preserve">Omniū duarū latitudinum equalium extenſiue et
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                inique intenſarum maior eſt proportio gradꝰ me­
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                dii medietatis intenſioris in latitudine remiſſio-
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                ri ad graduꝫ medium medietatis remiſſioris eiuſ­
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                dem latitudinis / quam eſt proportio graduum me­
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                diorum medietatum latitudinis remiſſioris.</s>
              </p>
              <p xml:id="N14DF0">
                <s xml:id="N14DF1" xml:space="preserve">Quas auteꝫ vocat latitudines extenſiue equales
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                vide ibi. </s>
                <s xml:id="N14DF6" xml:space="preserve">Et ex hoc probatur etiam regula / quã po­
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                nit calculator in capite eodem ſoluendo argumen­
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                tum factum contra .33. concluſionem / quam ibi nõ
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                probat: ſed ipſa facile oſtenditur ex hac concluſio­
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                ne et ſuo correlario / hoc addito /  in omni latitu-
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                dine vniformiter difformi partium equalium ex-
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                trema equaliter ſeſe excedunt: quia de talibus la­
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                titudinibus intelligitur regula eius.</s>
              </p>
              <p xml:id="N14E07">
                <s xml:id="N14E08" xml:space="preserve">Secunda concluſio. </s>
                <s xml:id="N14E0B" xml:space="preserve">Quando inter
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                aliquos terminos eſt ꝓportio maioris inequali-
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                tatis, et maior illorum terminorum acquirit ali-
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                quam proportionem ſtante minore inuariato: vel
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                minor terminus deperdit aliquam ꝓportionem ī­
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                uariato maiore: proportio inter illos terminos
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                augmentantur. </s>
                <s xml:id="N14E1A" xml:space="preserve">Probatur / et ſint b. terminus ma­
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                ior et .cd. minor inter quos ſit ꝓportio f. / et acqui-
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                rat terminus b. vnam ꝓportionem que ſit .ab. ad
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                b. / tunc dico /  proportio f. auget̄̄ ceteris aliis ma­
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                nentibus paribus. </s>
                <s xml:id="N14E25" xml:space="preserve">Item ſi .cd. perdat ꝓportionē /
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                que eſt .cd. ad d. proportio f. augmentatur. </s>
                <s xml:id="N14E2A" xml:space="preserve">Pri-
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                mum probatur / quia quando b. acquirit propor-
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                tionē que eſt .ab. ad b. ceteris manentibus paribꝰ
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                ipſi ꝓportioni f. que eſt b. ad .cd. / additur ꝓportio
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                .ab. ad b. / ergo ſequitur /  ipſa ꝓportio f. augetur
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                </s>
                <s xml:id="N14E36" xml:space="preserve">Patet hec conſequentia ex ſecunda ſuppoſitio­
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                ne huius. </s>
                <s xml:id="N14E3B" xml:space="preserve">Secunda pars ſimiliter oſtenditur: quo­
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                niam quando terminus minor .cd. perdit ꝓportio­
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                nem que eſt: cd. ad d. ꝓportioni f. que eſt b. ad .cd. /
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                additur ꝓportio que eſt .cd. ad d. / quoniam in fine
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                totalis ꝓportio componitur ex proportione b. ad
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                .cd. et .cd. ad d. / ergo proportioni f. que eſt b. ad .cd.
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                fuit addita ꝓportio que eſt .cd. ad d. / ergo ꝓportio
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                f. fuit augmentata. </s>
                <s xml:id="N14E4C" xml:space="preserve">Patet hec conſequentia ex ſe­
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                cunda ſuppoſitione preallegata. </s>
                <s xml:id="N14E51" xml:space="preserve">Et ſic patet con-
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                cluſio.
                  <note position="left" xlink:href="note-0050-02a" xlink:label="note-0050-02" xml:id="N14EED" xml:space="preserve">1. correl.</note>
                </s>
                <s xml:id="N14E5B" xml:space="preserve">¶ Ex hac concluſione ſequitur primo / 
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                cum inter aliquos terminos eſt ꝓportio maioris
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                inequalitatis: et vtro creſcente maiorem propor­
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                tionem acquirit maior terminus quam minor / tūc
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                ꝓportio inter datos terminos augetur. </s>
                <s xml:id="N14E66" xml:space="preserve">Proba-
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                tur / ſint duo termini .abc. maior: de. minor: et ſit ꝓ­
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                portio c. ad .e.f et ꝓportio .abc. ad c. / excedat pro-
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                portionē .de. ad e. per proportionē que eſt .abc. ad
                  <cb chead="Capitulum ſextum"/>
                bc. / et acquirat e. ꝓportionem .de. ad e. et c. ꝓportio­
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                nem que eſt .abc. ad c. / et tūc dico /  proportio f. au­
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                getur. </s>
                <s xml:id="N14E76" xml:space="preserve">Quod ſic ꝓbatur / quia ſi c. acquireret adeq̈­
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                te tantam proportionem quanta eſt .de. ad e. quaꝫ
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                acquirit e. adhuc inter illos terminos maneret ꝓ­
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                portio f. / vt patet ex correlario decime ſuppoſitio­
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                nis ſecundi capitis huius partis: ſed modo c. ter-
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                minus maior acquirit vltra proportionem quam
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                acquirit terminus minor proportioneꝫ q̄ eſt .abc.
                  <lb/>
                ad .bc. / ergo ꝓportioni f. que eſt .bc. ad .de. / additur
                  <lb/>
                proportio .abc. ad .bc. / et per conſequens ꝓportio
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                f. augetur / quod fuit probandum. </s>
                <s xml:id="N14E8B" xml:space="preserve">Patet conſeq̄n-
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                tia ex ſecunda ſuppoſitione. </s>
                <s xml:id="N14E90" xml:space="preserve">Patet igitur correla­
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                rium.
                  <note position="right" xlink:href="note-0050-03a" xlink:label="note-0050-03" xml:id="N14EF3" xml:space="preserve">2. correl.</note>
                </s>
                <s xml:id="N14E9A" xml:space="preserve">¶ Sequitur ſecundo /  datis duobus termi­
                  <lb/>
                nis inter quos eſt ꝓportio maioris inequalitatis
                  <lb/>
                et diminuatur vter terminus: minore maiorem
                  <lb/>
                proportionem deperdente quam maior ꝓportio ī­
                  <lb/>
                ter datos terminos augetur. </s>
                <s xml:id="N14EA5" xml:space="preserve">Probatur / ſint .ab.
                  <lb/>
                terminꝰ maior: et .cde. minor. </s>
                <s xml:id="N14EAA" xml:space="preserve">et ſit inter .ab. et .cde.
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                ꝓportio f. / et deperdat .ab. ꝓportionem que eſt .ab.
                  <lb/>
                ad b. et .cde. deperdat ꝓportionem que eſt .cde. ad
                  <lb/>
                e. / excedat proportio .cde. ad e. ꝓportionem .ab.
                  <lb/>
                ad b. per proportionem .cde. ad .de. / et tunc dico / 
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                tali decremento facto in vtro illorum termino-
                  <lb/>
                rum ꝓportio f. augetur. </s>
                <s xml:id="N14EB9" xml:space="preserve">Quod ſic probatur. </s>
                <s xml:id="N14EBC" xml:space="preserve">quo-
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                niam ſi .ab. terminus maior et .cde. terminus mi-
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                nor equalem proportioneꝫ deperderent puta .ab. ꝓ­
                  <lb/>
                portionem que eſt .ab. ad b. et .cde. ꝓportionē que
                  <lb/>
                eſt .cde. ad .de. / tunc adhuc maneret ꝓportio f. / vt pa­
                  <lb/>
                tet ex ſecunda parte decime ſuppoſitionis. </s>
                <s xml:id="N14EC9" xml:space="preserve">ſecun-
                  <lb/>
                di capitis huius: ſed modo vltra illam proportio­
                  <lb/>
                nem adhuc minor terminus deperdit ꝓportioneꝫ
                  <lb/>
                .de. ad e. / ergo ſequitur /  ipſi proportioni f. addi-
                  <lb/>
                tur ꝓportio .de. ad e. et ſic ꝓportio illa f. auget̄̄ / qḋ
                  <lb/>
                fuit probandum.
                  <note position="right" xlink:href="note-0050-04a" xlink:label="note-0050-04" xml:id="N14EF9" xml:space="preserve">3. correl.</note>
                </s>
                <s xml:id="N14EDB" xml:space="preserve">¶ Sequitur tertio /  quãdo duo
                  <lb/>
                termini ſe habent in proportione maioris īequa-
                  <lb/>
                litatis: et minor perdit aliquam ꝓportionē et ma­
                  <lb/>
                ior acquirit: ꝓportio inter illos terminos auge-
                  <lb/>
                tur. </s>
                <s xml:id="N14EE6" xml:space="preserve">Patet correlarium ex concluſione.</s>
              </p>
              <p xml:id="N14EFF">
                <s xml:id="N14F00" xml:space="preserve">Tertia concluſio. </s>
                <s xml:id="N14F03" xml:space="preserve">Qnando inter ali-
                  <lb/>
                quos terminos eſt ꝓportio maioris īequalitatis
                  <lb/>
                et maior illorum diminuitur ſtante minore: vel mi­
                  <lb/>
                nor augetur ſtante maiore: proportio inter illos
                  <lb/>
                terminos diminuitur. </s>
                <s xml:id="N14F0E" xml:space="preserve">Probatur prima pars: et
                  <lb/>
                ſit proportio f. inter .ab. maiorem terminum et c.
                  <lb/>
                minorem: et ſtante c. deperdat .ab. ꝓportionem q̄
                  <lb/>
                eſt .ab. ad b. / quam deperdit deperdendo a. parteꝫ
                  <lb/>
                ſui: tunc dico /  proportio f. diminuitur. </s>
                <s xml:id="N14F19" xml:space="preserve">Quod ſic
                  <lb/>
                probatur / quia a ꝓportione f. demitur aliqua ꝓ-
                  <lb/>
                portio puta proportio que eſt .ab. ad b. / igitur pro­
                  <lb/>
                portio f. diminuitur. </s>
                <s xml:id="N14F22" xml:space="preserve">Patet cõſequentia ex quar-
                  <lb/>
                ta ſuppoſitione: et antecedens probatur / quia ꝓ-
                  <lb/>
                portio f. componitur ex ꝓportione .ab. ad b. et b.
                  <lb/>
                ad c. in principio diminutionis / vt patet ex ſuperi­
                  <lb/>
                us dictis capite quarto huius: et ex illa prpportio­
                  <lb/>
                ne f. non manet niſi ꝓportio b. ad c. / igitur propor­
                  <lb/>
                tio f. perdit proportionem que eſt .ab. ad b. / qḋ fuit
                  <lb/>
                probandum. </s>
                <s xml:id="N14F33" xml:space="preserve">Secunda pars probatur: et ſint duo
                  <lb/>
                termini ſe habentes in proportione maioris ineq̈­
                  <lb/>
                litatis a. maior et c. minor inter quos eſt f. propor­
                  <lb/>
                tio: et acquirat c. terminus minor aliquam ꝓpor-
                  <lb/>
                tionem acquirendo b. ſupra ſe: ip̄o aggregato ex
                  <lb/>
                .bc. manente minore ipſo a. </s>
                <s xml:id="N14F40" xml:space="preserve">(Hoc enim ſupponit
                  <lb/>
                concluſio) et maneat a. inuariatum / tunc dico /
                  <lb/>
                 proportio f. diminuitur. </s>
                <s xml:id="N14F47" xml:space="preserve">Quod ſic probatur:
                  <lb/>
                quia ꝓportio f. in principio componitur ex pro-
                  <lb/>
                portione a. ad .bc. et ex ꝓportione .bc. ad c. / vt cõſtat
                  <lb/>
                et in fine talis augmentationionis termini mino-
                  <lb/>
                ris: ꝓportio illa manet p̄ciſe proportio a. ad .bc. / </s>
              </p>
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