Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secūde partis" file="0028" n="28"/>
                portio / qḋ fuit ꝓbãdū. </s>
                <s xml:id="N12872" xml:space="preserve">Et ſic ptꝫ ↄ̨cluſio gñaliter.</s>
              </p>
              <note position="left" xml:id="N12875" xml:space="preserve">1. correl.
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              ſcḋa ꝓṗe­
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              tas medi­
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              etatꝪ gro­
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              trice.</note>
              <p xml:id="N12881">
                <s xml:id="N12882" xml:space="preserve">¶ Ex hac concluſione ſequitur primo /  conſtitu-
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                tis quatuor terminis in hac medietate ſicut ag-
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                gregatum ex primo et ſecundo ad ſecundū ita ag­
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                gregatū ex tertio et quarto ad quartū vt conſtitu­
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                tis his quatuor terminis .8.4.6.3. ſicut ſe habent
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                8. et .4. ad .4. ita .6. et .3. ad .3. </s>
                <s xml:id="N1288F" xml:space="preserve">Probatur / et ſint q̈-
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                tuor termini in hac medietate geometrica ꝓpor-
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                tionabiles a.b.c.d. / dico /  qualis eſt ꝓportio .ab.
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                ad b. talis eſt .cd. ad d. </s>
                <s xml:id="N12898" xml:space="preserve">Quod probatur ſic et vo­
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                lo /  b. addatur ipſi a. et d. ipſi c. / et arguo ſic / ſicut
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                ſe habet a. ad b. ita c. ad .d. / ergo b. eſt talis pars ali­
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                quota vel partes aliquote et eiuſdem denomina-
                  <lb/>
                tionis reſpectu a. qualis eſt d. reſpectu c. (et proce­
                  <lb/>
                das a maioribus verſus minores) et b. additur ip­
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                ſi a. et d. ipſi .c. / igitur equalem ꝓportionem acqui­
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                rit a. ſupra ſe ſicut c. ſupra ſe </s>
                <s xml:id="N128A9" xml:space="preserve">Patet conſequentia
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                ex correlario vndecime ſuppoſitionis: et eandeꝫ ꝓ­
                  <lb/>
                portionē quã acquiſiuit a. ſupra ſe acquiſiuit pro­
                  <lb/>
                portio ipſius a. ad b. et ſimiliter eam quam acqui­
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                ſiuit c. ſupra ſe acquiſiuit ꝓportio ipſius c. ad d. /
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                vt patet ex probatione none ſuppoſitionis / igitur
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                facta tali acquiſitione qualis eſt proportio .ab.
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                ad b. talis eſt .cd. ad d. / quod fuit ꝓbandum </s>
                <s xml:id="N128BA" xml:space="preserve">Pa-
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                tet conſequentia / quia ꝓportio a. ad b. eſt equa-
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                lis proportioni c. ad d. et equalem ꝓportionem ac­
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                quirunt ille due proportiones / igitur ī fine manēt
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                eq̈les / q2 ſi equalibus eq̈lia addas etc. ſꝫ in fine vna
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                illaꝝ ꝓportionū ē .ab. ad .b. et alia ē .cd. ad d. / er-
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                go proportio .ab. ad .b. eſt equalis ꝓportioni .cd.
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                ad d. </s>
                <s xml:id="N128CB" xml:space="preserve">Eodē mõ ꝓbabis / ſi procedas ad minoribus
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                ad maiores termīos in ꝓportiõe mīoris īeq̈litatꝪ
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                </s>
                <s xml:id="N128D1" xml:space="preserve">Sed eadē hypotheſi retēta gñaliter probat̄̄ corre­
                  <lb/>
                lariū ſic: et volo /  a. diminuat̄̄ ad eq̈litatē b. et c. ad
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                equalitatē d. / et ſic ꝑdēt eq̈les ꝓportiones ex hypo­
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                theſi: deiñ reſiduū ipſius a. acq̇rat ſupra ſeipſū b.
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                et reſiduū c. aq̇rat ipſū d. / et manifeſtum eſt /  ag-
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                gregati ex reſiduo a. et ipſo b. ad ipſū b. et aggre­
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                gati ex reſiduo ipſiꝰ c. et ipſo d. ad ipſū d: eſt eq̈lis
                  <lb/>
                ꝓportio puta dupla: volo igit̄̄ /  aggregatum ex
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                reſiduo ipſius a. et ipſo b. acq̇rat illa quãtitatem
                  <lb/>
                quã deꝑdidit a. ita  maneat aggregatū ex a. et b.
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                et aggregatū ex reſiduo ipſius c. et ipſo d. acq̇rat
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                quãtitatē quã deꝑdidit ipſū c. ita  maneat in fi-
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                ne aggregatū ex c. et d. / et tunc ſeq̇tur /  aggregati
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                ex a. et b. ad ip̄m b. et aggregati ex c. et d. ad ipſum
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                d. ē eadē ꝓportio / qḋ fuit ꝓbãdū. </s>
                <s xml:id="N128F0" xml:space="preserve">Probat̄̄ ↄ̨ña. </s>
                <s xml:id="N128F3" xml:space="preserve">q2
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                illi termini añ acq̇ſitionē quãtitatū deꝑditaꝝ ab
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                ipſo a. et ipſo c. / ſe hēbaut in eadē ꝓportione puta
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                dupla / vt dictū ē: et acq̇ſiuerunt eq̈les ꝓportiones
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                termini maiores illaꝝ ꝓportionū: igit̄̄ iter datos
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                terminos manet eq̈lis ꝓportio: q2 ſi eq̈libꝰ eq̈lia
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                addas etc </s>
                <s xml:id="N12902" xml:space="preserve">Probatur minor: q2 medietates illorū
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                terminorū maiorū eq̈les ꝓportiões acq̇ſiuer̄t: igi­
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                tur et ipſi termini maiores eq̈les proportiões acq̇­
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                ſinerūt / vt ptꝫ ex tertia cõcluſione ſeptimi capitꝪ ṗ­
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                me ꝑtis: et p ↄ̨ñs proportiões quas hñt ad mīores
                  <lb/>
                terminos eq̈les proportiones acq̇ſiuerūt / vt ptꝫ ex
                  <lb/>
                ſuppoſitione huiꝰ.
                  <note position="left" xlink:href="note-0028-01a" xlink:label="note-0028-01" xml:id="N12ACF">
                    <s xml:id="N12AD3" xml:space="preserve">2. correl.
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                    </s>
                    <s xml:id="N12AD7" xml:space="preserve">3. ꝓṗetaſ
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                    medieta­
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                    tꝪ geome­
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                    trice.</s>
                  </note>
                </s>
                <s xml:id="N12916" xml:space="preserve">Et ſic ptꝫ correlariū / qḋ ſit medi­
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                etatis geometrice ſcḋa proṗetas </s>
                <s xml:id="N1291B" xml:space="preserve">¶ Seq̇tur ſeḋo / 
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                in hac medietate cõſtitutis .4. termīs q̈lis ē ꝓpor-
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                tio ṗmi ad m talis ē ꝓportio aggregati ex ṗmo et
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                tertio ad aggregatū ex ſcḋo. et .4. vt cõſtitutis his
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                terīs .12.6.4.2. q̈ĺ ē ꝓportio .12. ad .6. taĺ ē ꝓportõ
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                12. et .4: ad .6. et .2. </s>
                <s xml:id="N12928" xml:space="preserve">Probat̄̄ / ſint .4: terī ī hac medie­
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                tate a.b.c.d. / et dico /  ſic̈ a. ad b. ita aggregatū ex
                  <lb/>
                a. et c. ad aggregatū ex b. et d. </s>
                <s xml:id="N1292F" xml:space="preserve">Qḋ ſic oñdit̄̄ / et .1. ī nū­
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                ris / et volo /  a. acq̇rat c. et b. acq̇rat d. (et ꝓcedo a
                  <lb/>
                maioribꝰ) / et arguit̄̄ ſic ſicut ſe hꝫ a. ad b. ita c. ad d /
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                igr̄ ꝑmutatī ex .4. ↄ̨cĺoe ſicut ſe hꝫ a. ad c. ita b. ad d /
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                et ex ↄ̨ñti ſeq̇r̄ /  c. ē ꝑs aliq̊ta vĺ ꝑtes reſpectu a. eiuſ­
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                dē denoīatiõis ſicut d. reſpectu b. vel eoↄ̈ ſi ꝓpor-
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                tio a. ad c. ſit mīorꝪ īeq̈litatꝪ: et a. aq̇rit c. et b. aq̇rit d /
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                igr̄ q̀lē ꝓportionē aq̇rit nūerꝰ maior hꝰ ꝓportõis q̄
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                ē a. ad b. talē acq̇rit nūerꝰ mīor. </s>
                <s xml:id="N12943" xml:space="preserve">Cõſequētia / ptꝫ ex
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                ſcḋo correlario octaue ſuppõis: g̊ ī fine facta tali
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                acq̇ſitiõe manet eadē ꝓportio ſiue eq̈lis illi q̄ ē īter
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                a: et b. / vt ptꝫ ex correlario decime ſuppõnis / et in fi­
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                ne manet proportio .ac. ad bd. / g̊ proportio .ac. ad
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                .bd. ē equalis ꝓportioni a. ad b. / qḋ fuit probandū
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                </s>
                <s xml:id="N12951" xml:space="preserve">Sed eadē hypotheſi retēta probo gñaliter /  ſicut
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                ſe hꝫ c. ad d. / ita ſe hꝫ aggregatū ex .ac. ad aggrega­
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                tū ex .bd. </s>
                <s xml:id="N12958" xml:space="preserve">Et arguo / ſic ſicut ſe hꝫ a. ad b. ita c. ad d. / g̊
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                ex ↄ̨cluſione ſicut ſe hꝫ a. ad c. ita b. ad d. diminuat̄̄ /
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                igr̄ a. ad equalitatē c. et b. ad equalitatē d. / et ſic ma­
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                nifeſtū ē /  equalē ꝓportionē deꝑdūt a. et b. </s>
                <s xml:id="N12961" xml:space="preserve">Uolo
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                igr̄ /  reſiduū ex a. acq̇rat ſupra ſeipſū c. et reſiduū
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                ex b. ipſū d. et tūc aggregati ex reſiduo a. et ipſo c.
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                ad ipſū c. ē illa proportio q̄ ē aggregati ex r̄ſiduo
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                b. et ipſo d. q2 dupla / vt ↄ̨ſtat: acq̇rat g̊ aggregatū
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                ex reſiduo a. et ip̄o c. quãtitatē quã ꝑdidit a. et ag-
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                gregatū ex reſiduo b: et ip̄o d. quãtitatē quã deꝑdi­
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                dit b. et tūc manifeſtū ē /  ꝓportio aggregati ex re­
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                ſiduo a. et ip̄o c. ad ipſū c. et ꝓportio aggregati ex
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                reſiduo b. et ipſo d. ad ipſū d. eq̈les ꝓportiones ac­
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                q̇rūt q2 medietates maioꝝ termīoꝝ equales ꝓpor­
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                tiões acq̇rūt puta illas quas antea pdiderūt et ſic
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                maiores termī illaꝝ ꝓportionū eq̈les ꝓportiões
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                acq̇r̄t / vt pꝫ ex tertia ↄ̨cĺoe ſeptī capitis ṗme ꝑtꝪ: igr̄
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                īter illos termīos q̇ ſūt iã .ac. et c. et .bd. et b. manet
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                adhuc eq̈lis ꝓportio: et ꝑ ↄ̨ñs ſicut ſe hꝫ aggrega-
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                tū ex a. et c. ad ipſū c. ita ſe hꝫ aggregatū ex b. et d.
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                ad ip̄m d. / igr̄ ex ↄ̨cluſione ſicut ſe hꝫ aggregatū ex
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                a. et c. ad aggregatū ex b. et d. ita ſe hꝫ c. ad d. / quod
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                fuit ꝓbandū. </s>
                <s xml:id="N1298A" xml:space="preserve">Et ſolēt antiq̇ geometre et ſignanter
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                calculator vti hoc correlario ſub his rermīs
                  <note position="right" xlink:href="note-0028-02a" xlink:label="note-0028-02" xml:id="N12AE0" xml:space="preserve">eadē ē ꝓ-
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                  portio di­
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                  uiſoꝝ et ↄ̨­
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                  iūctoruꝫ.</note>
                </s>
                <s xml:id="N12994" xml:space="preserve">Qua­
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                lis ē ꝓportio diuiſoꝝ talis ē ↄ̨iūctoꝝ: vt ſi ſint due
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                ꝓportiões duple: et cõpulet̄̄ terminꝰ maior vniꝰ cū
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                termīo maiore vlteriꝰ: et mīor vniꝰ cū mīore alteri­
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                us īter illos termīos ſic ↄ̨iūctos manebit ꝓportio
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                dupla.
                  <note position="right" xlink:href="note-0028-03a" xlink:label="note-0028-03" xml:id="N12AEC" xml:space="preserve">4. ꝓprie­
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                  tas medi­
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                  etatꝪ geo­
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                  metrice</note>
                </s>
                <s xml:id="N129A6" xml:space="preserve">¶ Seq̇tur .3. /  .4. termīs in hac medietate ↄ̨­
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                ſtitutis: q̈lis ē ꝓportio ſcḋi ad ṗmū talis ē quarti
                  <lb/>
                ad tertiū vt ↄ̨ſtitutis. </s>
                <s xml:id="N129AD" xml:space="preserve">his 4: termīs .8.4.6.3. q̈lis ē
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                ꝓportio .4. ad .8. talis ē .3. ad 6. </s>
                <s xml:id="N129B2" xml:space="preserve">Ptꝫ hoc coreela-
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                riū facile / qm̄ ſꝑ ꝓportiões mīoris īeq̈litatis ſunt
                  <lb/>
                eq̈les īter ſe cū ꝓportiões maioris īeq̈litatis q̇bus
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                corrñdent īter ſe ſūt equales: et eoↄ̈. </s>
                <s xml:id="N129BB" xml:space="preserve">Sicut eī oēs
                  <lb/>
                duple ſūt equales: ita oēs ſubduple ſūt equales: et
                  <lb/>
                ſic̈ oēs ſubtriple ſṫ eq̈les: ita oēs triple igr̄ vĺr ſi ta­
                  <lb/>
                lis ꝓportio fuerit a. ad b. maioris īeq̈litatꝪ q̈lis ē
                  <lb/>
                c. ad d. ↄ̨ñs ē  ꝓportio mīoris īeq̈litatꝪ d. ad c. et
                  <lb/>
                b. ad a ſint eq̈les. </s>
                <s xml:id="N129C8" xml:space="preserve">Et ita ēt ꝓbaſſes ſi a. ad .b. fuiſſꝫ
                  <lb/>
                ꝓportio mīoris īeq̈litatꝪ. </s>
                <s xml:id="N129CD" xml:space="preserve">Et hec ſit .4. ꝓṗetas geo­
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                metrice medietatꝪ. </s>
                <s xml:id="N129D2" xml:space="preserve">¶ Seq̇t̄̄ .4. /  diſpoſitis .4. ter-
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                mīs ſicut ṗmꝰ et ſcḋs ad m et tertiꝰ et quartꝰ ad q̈r
                  <lb/>
                ita ṗmꝰ ad m et tertiꝰ ad q̈rtū / vt ↄ̨ſtitutis his .4.
                  <lb/>
                termīs .8.4.2.1. / q2 .8. et 4. ad .4. ē talis ꝓportio q̈­
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                lis ē .2. et .1. ad .1. / vt pꝫ ex ṗmo correlario huiꝰ ↄ̨clu­
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                ſionis. </s>
                <s xml:id="N129DF" xml:space="preserve">Iõ q̈lis ē ꝓportio ṗmi ad m talis ē terti ad
                  <lb/>
                .4. / vt ↄ̨ſtat. </s>
                <s xml:id="N129E4" xml:space="preserve">Probat̄̄ ṗmo / ī nūeris ſint .4. nūeri a.
                  <lb/>
                b.c.d. et ſicut .ab.. ad .b. ita c. ad .cd. / tūc ḋt correĺm /
                  <lb/>
                 ſicut a. ad b. ita c. ad d. et ſit .a maiꝰ b. et c. maiꝰ d.
                  <lb/>
                et deꝑdat .ab:b. et .cd.d. / et arguit̄̄ ſic ſicut ſe hꝫ .ab.
                  <lb/>
                ad b. ita c.d. ad d. / igr̄ b. ē talis ꝑs aliq̊ta vel ꝑtes
                  <lb/>
                aliq̊te et eiuſdē denoīatiõis reſpectu ipſiꝰ .ab. / q̈lis
                  <lb/>
                ē d. reſpectu .cd. et .ab. ꝑdit b. et .cd. ꝑdit d. / g̊ illi duo
                  <lb/>
                nūeri maiores puta .ab. et .cd. ꝑdūt eq̈les ꝓportio­
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                nes / vt pꝫ ex .1. correĺ. 8. ſuppõis / g̊ ſeq̇t̄̄ /  quãtã ꝓ­
                  <lb/>
                portionē adeq̈te ꝑdit ꝓportio ab. ad b. tãtã ade­
                  <lb/>
                q̈te ꝑdit ꝓportio .cd. ad d. / vt pꝫ ex nona ſuppoſiti­
                  <lb/>
                one: et ille ꝓportões ante erãt equales / vt ponitur
                  <lb/>
                igitur mõ manēt equales: q2 ſi ab equalibus equa­ </s>
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