Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N130F7" level="3" n="4" type="chapter" type-free="capitulum">
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                <s xml:id="N1354A" xml:space="preserve">
                  <pb chead="Prime partis" file="0035" n="35"/>
                duplã: manent igitur in eadē ꝓportione / vt ptꝫ ex
                  <lb/>
                correlario decime ſuppoſitiõis ſecūdi capitꝪ huiꝰ
                  <lb/>
                īueniatur / igitur mediū inter illos duos numeros
                  <lb/>
                et īueniētur due ꝓportiones tnequales in quas di­
                  <lb/>
                uiditur ꝓportio inter illos duos numeros / vt pre-
                  <lb/>
                oſtenſum eſt. </s>
                <s xml:id="N13560" xml:space="preserve">Patet igitur vniuerſaliter concluſio
                  <lb/>
                  <note position="left" xlink:href="note-0035-01a" xlink:label="note-0035-01" xml:id="N13701" xml:space="preserve">Primuꝫ
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N1356A" xml:space="preserve">¶ Ex qua ſequitur primo /  quelibet proportio in
                  <lb/>
                infinitas ꝓportiones ſecari valet in numeris ſine
                  <lb/>
                vnitatis fractione: et capio ly infinitas ſyncathe-
                  <lb/>
                goreumatice. </s>
                <s xml:id="N13573" xml:space="preserve">Probatur / qm̄ capta ꝓportione a.
                  <lb/>
                in numeris manifeſtū eſt /  illi numeri ſaltē ꝑ vni-
                  <lb/>
                tatē diſtabūt / hoc eſt ſaltē maior excedit minorē ꝑ
                  <lb/>
                vnitatē que vnitas eſt pars aliquota minoris: du­
                  <lb/>
                pletur igitur vter illoꝝ numeroꝝ: et ſequitur / 
                  <lb/>
                adhuc inter illos numeros duplatos manet ꝓpor­
                  <lb/>
                tio a. / vt paulo ãte deductū eſt: igitur iam exceſſus
                  <lb/>
                erit in duplo maior: q2 erit pars aliquota eiuſdē
                  <lb/>
                denomīationis numeri in duplo maioris: igitur
                  <lb/>
                iam ibi inter illos duos numeros reperietur vnꝰ
                  <lb/>
                numerus medius vt ſuperiꝰ oſtenſum eſt: et ꝑ cõſe-
                  <lb/>
                quens due ꝓportiones inequales in quas diuidit̄̄
                  <lb/>
                talis ꝓportio. </s>
                <s xml:id="N1358E" xml:space="preserve">Iteꝝ duplent̄̄ illi numeri īter quos
                  <lb/>
                eſt ꝓportio a. et iam inter eos īuenientur tres nu-
                  <lb/>
                meri intermedii et ſic erūt quatuor ꝓportiões in-
                  <lb/>
                termedie. </s>
                <s xml:id="N13597" xml:space="preserve">Et ſi tertio duplentur illi numeri īueni-
                  <lb/>
                entur ſeptē numeri intermedii: et ſic erūt .8. ꝓpor-
                  <lb/>
                tiones: et ſic in infinitū duplando ſemꝑ numeros.
                  <lb/>
                </s>
                <s xml:id="N1359F" xml:space="preserve">Data igit̄̄ quã volueris ꝓportione ipſa vel ſibi e-
                  <lb/>
                qualis (quod ꝓ eodē reputo) in infinitas ꝓportio-
                  <lb/>
                nes ſecari valet: quod fuit oſtendendū. </s>
                <s xml:id="N135A6" xml:space="preserve">Et ſicut ꝓ-
                  <lb/>
                batur in numeris: ita et facilius ꝓbabitur in quã-
                  <lb/>
                titatibus. </s>
                <s xml:id="N135AD" xml:space="preserve">Et ſicut ꝓbatur capiēdo primos nume-
                  <lb/>
                ros excedentes ſe vnitate: ita per locū a maiori ꝓ-
                  <lb/>
                babitur capiendo numeros excedētes ſe numero:
                  <lb/>
                vt ſatis conſtat. </s>
                <s xml:id="N135B6" xml:space="preserve">Patet igit̄̄ correlariū.
                  <note position="left" xlink:href="note-0035-02a" xlink:label="note-0035-02" xml:id="N1370B" xml:space="preserve">Secūduꝫ
                    <lb/>
                  correlar̄.</note>
                </s>
                <s xml:id="N135BE" xml:space="preserve">¶ Sequit̄̄
                  <lb/>
                ſecūdo /  capitis tribꝰ terminis cõtinuo ꝓportio-
                  <lb/>
                nabilibus arithmetice: et captis aliis tribus ſic ſe
                  <lb/>
                habentibꝰ /  qualis eſt ꝓportio inter duos maio-
                  <lb/>
                res primi ternarii: talis ſit inter duos maiores ſe­
                  <lb/>
                cūdi ternarii: et qualis inter duos numeros primi
                  <lb/>
                ternarii: talis etiã ſit inter duos minores ſecundi
                  <lb/>
                ternarii: tūc termini ſecūdi ternarii ſunt ꝓportio-
                  <lb/>
                nabiles arithmetice: ſicut et termini ṗmi ternarii:
                  <lb/>
                vt captis his tribus terminis .4.3.2. qui ſunt pro-
                  <lb/>
                portiõabiles arithmetice: dico /  iſti .3. termini .8.
                  <lb/>
                6.4. ſunt etiã arithmetice proportionabiles: qm̄
                  <lb/>
                qualis eſt ꝓportio inter .4. et .3. talis eſt inter .8. et
                  <lb/>
                6. et qualis inter .3. et .2. talis inter .6. et .4. / vt patꝫ
                  <lb/>
                </s>
                <s xml:id="N135DC" xml:space="preserve">Probatur / ſint tres termini a.b.c. ꝓportiõabiles
                  <lb/>
                arithmetice: et ſint alii trrs d.e.f. et ſit inter d. et e.
                  <lb/>
                talis ꝓportio qualis inter a. et b. et inter e. et f. q̈lis
                  <lb/>
                inter b. et c. </s>
                <s xml:id="N135E5" xml:space="preserve">Et tunc dico /  d.e.f. ſunt tres termini
                  <lb/>
                ꝓportionabiles arithmetice: </s>
                <s xml:id="N135EA" xml:space="preserve">Ad quod probandū
                  <lb/>
                volo /  exceſſus quo a. excedit b. ſit g. et quo b. exce­
                  <lb/>
                dit c. ſit h. equalis g. / vt oportet: et exceſſus q̊ d. exce­
                  <lb/>
                dit e. ſit i. et quo e. excedit f. ſit k. / et manifeſtū eſt /  g.
                  <lb/>
                eſt tota pars aliquota ipſiꝰ b. vel tote partes q̊ta
                  <lb/>
                vel quote i. eſt ipſiꝰ e. et eiuſdē denominationis: et
                  <lb/>
                h. eſt tota pars vel tote partes aliquote et eiuſdeꝫ
                  <lb/>
                denomīationis reſpectu c. ſicut k. reſpectu f. / vt ptꝫ
                  <lb/>
                ex probatione quarte ſuppoſitionis ſecūdi capi-
                  <lb/>
                tis huiꝰ. </s>
                <s xml:id="N135FF" xml:space="preserve">Quo ſuppoſito arguit̄̄ ſic / i. quod eſt ex-
                  <lb/>
                ceſſus inter d. et e. eſt equale ipſi k. / quod eſt exceſſus
                  <lb/>
                inter e. et f. / igit̄̄ illi tres termini d.e.f. ſunt ꝓporti-
                  <lb/>
                onabiles arithmetice. </s>
                <s xml:id="N13608" xml:space="preserve">Cõſequentia ptꝫ manifeſte:
                  <lb/>
                et arguit̄̄ antecedens: q2 ſicut ſe habet b. ad .c. ita e.
                  <lb/>
                ad f. / igit̄̄ ſicut ſe habet b. ad e. ita c. ad f. </s>
                <s xml:id="N1360F" xml:space="preserve">Patet cõ-
                  <lb/>
                ſequentia ex ſecūda cõcluſione tertii capitis huiꝰ:
                  <lb/>
                et ex ↄ̨ſequenti ſicut ſe habet b. ad e. ita c. ad f. puta
                  <cb chead="Capitulū quartū."/>
                in l. ꝓportione / igitur g. ſe habet ad i. in l. ꝓporti-
                  <lb/>
                one et h. ad k. etiã in l. ꝓportione. </s>
                <s xml:id="N1361B" xml:space="preserve">Patet cõſequen­
                  <lb/>
                tia ex vndecima ſuppoſitione ſecūdi capitis huiꝰ:
                  <lb/>
                ille em̄ ſunt partes aliquote eiuſdē denoīationis
                  <lb/>
                numeroꝝ ſe habentiū in l. ꝓportione: et vltra g. ſe
                  <lb/>
                habet ad i. in l. ꝓportiõe: et h. ad k. etiã in l. pro-
                  <lb/>
                portione: igit̄̄ ſicut ſe habet g. ad h. ita i. ad k. </s>
                <s xml:id="N13628" xml:space="preserve">Ptꝫ
                  <lb/>
                per locū a. ꝑmutata proportione: ſed g. et h. ſe ha-
                  <lb/>
                bent in proportione equalitatis: igit̄̄ i. et k. / qḋ fuit
                  <lb/>
                probandñ. </s>
                <s xml:id="N13631" xml:space="preserve">Probatur aliter correlariū tam in nu­
                  <lb/>
                meris quã in quãtitatibus cõtinuis: et retēta eadē
                  <lb/>
                hypotheſi: manifeſtū eſt /  ipſiꝰ a. ad d. et ipſiꝰ b.
                  <lb/>
                ad c. et ipſius c. ad f. eſt eadē ꝓportio: que ſit l. / qm̄
                  <lb/>
                ex hypotheſi ſicut ſe habet a. ad b. ita ſe habet d.
                  <lb/>
                ad e. / ergo per locū a. permutata proportiõe ſicut
                  <lb/>
                ſe habet a. ad d. ita b. ad e. et vltra ſicut ſe habet b
                  <lb/>
                ad c. ita e. ad f. ex hypotheſi: ergo ꝑmutatim: ſicut
                  <lb/>
                ſe habet b. ad e. ita c. ad f. et a. ad d. eſt etiã ꝓportio
                  <lb/>
                illa que eſt b. ad c. / igit̄̄ eadē proportio eſt a. ad d. et
                  <lb/>
                b. ad e. et c. ad f. puta l. </s>
                <s xml:id="N13648" xml:space="preserve">Quo ſuppoſito: probatur
                  <lb/>
                correlariū: q2 i. et k. ſūt equales: igit̄̄ .d.e.f. ſunt ter­
                  <lb/>
                mini cõtinuo proportionabiles arithmetice. </s>
                <s xml:id="N1364F" xml:space="preserve">Ptꝫ
                  <lb/>
                cõſequentia ex hypotheſi: iūcta diffinitione ꝓpor­
                  <lb/>
                tionalitatis arithmetice. </s>
                <s xml:id="N13656" xml:space="preserve">Probat̄̄ antecedens: q2
                  <lb/>
                ſicut ſe habet g. ad h. ita ſe habet i. ad k. ſed g et h.
                  <lb/>
                ſe habent in proportiõe equalitatis / vt ptꝫ ex hy-
                  <lb/>
                potheſi: igit̄̄ i. et k. ſe habent in proportione equa-
                  <lb/>
                litatis: et ſic ſunt equalia igit̄̄. </s>
                <s xml:id="N13661" xml:space="preserve">Probat̄̄ antecedēs /
                  <lb/>
                q2 ſicut ſe habet g. ad i. ita h. ad k. / ergo ꝑmutatim
                  <lb/>
                ſicut ſe habet g. ad h. ita i. ad k. / qḋ fuit probandū.
                  <lb/>
                </s>
                <s xml:id="N13669" xml:space="preserve">Probatur antecedens: q2 g. ſe habet ad i. in l. ꝓ-
                  <lb/>
                portione: et h. ſe habet ad k. in eadē l. proportione /
                  <lb/>
                igitur intentū. </s>
                <s xml:id="N13670" xml:space="preserve">Probat̄̄ maior / q2 g. ſe hꝫ ad i. ſicut
                  <lb/>
                a. ſe hꝫ ad d. / igitur ſe hꝫ in l. ꝓportione. </s>
                <s xml:id="N13675" xml:space="preserve">Patꝫ ↄ̨ña
                  <lb/>
                ex hypotheſi. </s>
                <s xml:id="N1367A" xml:space="preserve">Probat̄̄ antecedēs: et volo /  a. dimi­
                  <lb/>
                nuatur ad equalitatē b. ꝑdendo g. differentiã per
                  <lb/>
                quã excedit ipſum b. ex hypotheſi: et d. diminuatur
                  <lb/>
                ad equalitatē c. ꝑdendo i. differentiã ꝑ quã excedit
                  <lb/>
                e. ex hypotheſi: et manifeſtū eſt /  reſidui ex ipſo a. /
                  <lb/>
                qḋ eſt b. ad reſiduū ex ipſo d. / qḋ eſt e. adhuc eſt l. ꝓ­
                  <lb/>
                portio: vt ptꝫ ex hypotheſi: g̊ inṫ deꝑditū ab ip̄o a
                  <lb/>
                et deꝑditū ab ip̄o d. eſt etiã l. ꝓportio: et deꝑditū ab
                  <lb/>
                ip̄o a eſt g. et deꝑditū ab ipſo d. eſt i. / g̊ g. ſe hꝫ ad i.
                  <lb/>
                ſicut a. ad d. puta in l. ꝓportione. </s>
                <s xml:id="N1368F" xml:space="preserve">Ptꝫ tamen ↄ̨ña
                  <lb/>
                ex primo correlario quinte cõcluſionis ſecūdi ca-
                  <lb/>
                pitis huiꝰ partis. </s>
                <s xml:id="N13696" xml:space="preserve">Et ſic ptꝫ maior. </s>
                <s xml:id="N13699" xml:space="preserve">Iam ꝓbo mi-
                  <lb/>
                norē / q2 h. ſe hꝫ ad k. ſicut b. ſi ſe hꝫ ad e. / igr̄ ꝓpoſitū
                  <lb/>
                </s>
                <s xml:id="N1369F" xml:space="preserve">Probat̄̄ ãtecedēs: et volo /  b. diminuat̄̄ ad equa-
                  <lb/>
                litatē c: perdendo h. differentiã: et e. diminuat̄̄ ad
                  <lb/>
                equalitatē f. perdendo k. differentiã: et manifeſtuꝫ
                  <lb/>
                eſt /  reſidui ex ipſo b. / qḋ eſt c. ad reſiduū ex ipſo e.
                  <lb/>
                qḋ eſt f. eſt adhuc l. ꝓportio: vt patet ex hypotheſi:
                  <lb/>
                igitur inter h. deperditū a b. termino maiori, et
                  <lb/>
                k. deꝑditū ab c. ṫmīo minori eſt ēt l ꝓportio: vt ſu-
                  <lb/>
                pra argutū eſt / igr̄ h. ſe hꝫ ad k. ſicut b. ad e. puta in
                  <lb/>
                l. ꝓportione: qḋ fuit probandū. </s>
                <s xml:id="N136B2" xml:space="preserve">Et ſic ptꝫ correla-
                  <lb/>
                riū.
                  <note position="right" xlink:href="note-0035-03a" xlink:label="note-0035-03" xml:id="N13713" xml:space="preserve">Calcu. de
                    <lb/>
                  īduc. gra­
                    <lb/>
                  dus ſūmi</note>
                </s>
                <s xml:id="N136BC" xml:space="preserve">Et hec ē ſuppoſitio quã calculator ponit ī ca-
                  <lb/>
                pitulo de inductione gradus ſummi circa princi-
                  <lb/>
                piū ſub iſta forma. </s>
                <s xml:id="N136C3" xml:space="preserve">Si ſint tria cõtinuo ꝓportio-
                  <lb/>
                nabilia ꝓportione arithmetica: et ſint alia tria cõ­
                  <lb/>
                ſimiliter ꝓportionabilia proportiõe geometrica
                  <lb/>
                ſicut prima tria: illa etiã ſunt cãtinuo ꝓportiõabi­
                  <lb/>
                lia proportiõe arithmetica.
                  <note position="right" xlink:href="note-0035-04a" xlink:label="note-0035-04" xml:id="N1371D" xml:space="preserve">Tertium
                    <lb/>
                  correlar̄.</note>
                </s>
                <s xml:id="N136D3" xml:space="preserve">¶ Sequit̄̄ ex hoc ter-
                  <lb/>
                tio /  ſi ſint tres termini arithmetice proportiõa-
                  <lb/>
                biles: et quilibet illoꝝ dupletur, aut tripletur, aut
                  <lb/>
                ſexquialteretur .etc̈. ſemꝑ ꝓportio extremi ad ex-
                  <lb/>
                tremū manet equalis: et cõtinuo manebūt illi tres
                  <lb/>
                termini arithmetice ꝓportiõabiles: et in ea ꝓpor-
                  <lb/>
                tiõe in qua termini augmētant̄̄ exceſſus augmētat̄̄
                  <lb/>
                </s>
                <s xml:id="N136E3" xml:space="preserve"/>
              </p>
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