Alvarus, Thomas, Liber de triplici motu, 1509

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              <p xml:id="N113C9">
                <s xml:id="N113CD" xml:space="preserve">
                  <pb chead="Prime partis" file="0016" n="16"/>
                et ſic cõſequenter / vt capite quarto oſtenſum eſt: de­
                  <lb/>
                bet ꝓ prima parte capi exceſſus quo maior quan-
                  <lb/>
                titas excedit minorem ita  reſiduum a prima ſit
                  <lb/>
                minor quantitas et totum corpus ſit maior quan-
                  <lb/>
                titas talis proportionis. </s>
                <s xml:id="N113E5" xml:space="preserve">Probatur hec cõcluſio
                  <lb/>
                ex precedenti / quoniam totū corpus diuiſum pro-
                  <lb/>
                portiõe aliqua irrationali ſe debet habere ad ag­
                  <lb/>
                gregatum ex omnibus ſequentibus primam tali
                  <lb/>
                diuiſione: in ea proportione qua ipſum corpus di­
                  <lb/>
                uiditur: igitur oportet /  totum corpus ſe habeat
                  <lb/>
                vt maior quantitas talis proportionis: et aggre-
                  <lb/>
                gatum ex omnibus ſequentibus primam vt minor
                  <lb/>
                quantitas: et per conſequens exceſſus / quo totum
                  <lb/>
                corpus excedit aggregatum ex omnibus ſequen-
                  <lb/>
                tibus primã erit prima pars proportionalis tali
                  <lb/>
                proportione. </s>
                <s xml:id="N113FE" xml:space="preserve">Patet conſequentia / quia reſiduum
                  <lb/>
                eſt aggregatū ex omnibus aliis a prima: ille igit̄̄
                  <lb/>
                exceſſus erit prima / quod fuit probandū.
                  <note position="left" xlink:href="note-0016-01a" xlink:label="note-0016-01" xml:id="N114DE" xml:space="preserve">Primuꝫ
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N1140A" xml:space="preserve">¶ Ex hac
                  <lb/>
                concluſione ſequitur primo /  ad diuidendum cor­
                  <lb/>
                pus proportione irrationali diametri ad coſtam
                  <lb/>
                oportet / pro prima parte proportionali capere ex­
                  <lb/>
                ceſſum quo diameter excedit coſtam: et pro ſecūda
                  <lb/>
                capere etiam exceſſum / quo illa coſta cum eſt dia-
                  <lb/>
                meter quadrati excedit coſtam illius quadrati / et
                  <lb/>
                ſic conſequenter: et addandam primã partem pro­
                  <lb/>
                portionale proportionis irrationalis / que eſt ag-
                  <lb/>
                gregati ex coſta et medietate exceſſus diametri ad
                  <lb/>
                ipſam coſtam capiatur pro prima parte propor-
                  <lb/>
                tionali illa medietas exceſſus: et pro ſecūda parte
                  <lb/>
                proportiõali capiatur tanta pars reſidui ad quã
                  <lb/>
                prima habeat illam proportionem / que eſt totius
                  <lb/>
                corporis ad aggregatum ex omnibus ſequen-
                  <lb/>
                tibus primam: et iterum in reſiduo a prima parte
                  <lb/>
                et ſecunda, pro tertia parte capiatur tanta pars
                  <lb/>
                ad quam ſecunda habeat illam proportionē quã
                  <lb/>
                prima habet ad ipſam: et ſic cõſequenter. </s>
                <s xml:id="N11431" xml:space="preserve">Et ſimili
                  <lb/>
                modo operandum eſſet / ſi diuideretur corpus pro­
                  <lb/>
                portione irrationali / que eſt aggregati ex coſta et
                  <lb/>
                q̈rta parte, vel octaua, vel decimaſexta exceſſus / q̇
                  <lb/>
                diameter excedit coſtã ad ipſã coſtã. </s>
                <s xml:id="N1143C" xml:space="preserve">Ptꝫ correla-
                  <lb/>
                riū ex cõcluſione addita ſuppoſitiõe ſecunda pre­
                  <lb/>
                cedētis capitis: ille enim partes infinite continue
                  <lb/>
                ſe habent in proportione diuiſionis et totum ab-
                  <lb/>
                ſoluūt.
                  <note position="left" xlink:href="note-0016-02a" xlink:label="note-0016-02" xml:id="N114E8" xml:space="preserve">Secūduꝫ
                    <lb/>
                  correlar̄.</note>
                </s>
                <s xml:id="N1144C" xml:space="preserve">¶ Sequitur ſecundo /  diuiſo corpore per
                  <lb/>
                partes proportionales proportione irrationali /
                  <lb/>
                que eſt diametri ad coſtam: omnes partes impa-
                  <lb/>
                res continuo ſe habent in proportione dupla: et
                  <lb/>
                omnes pares ſimiliter: et oēs due inter quas me-
                  <lb/>
                diant due ſe habent continuo in proportione ſex-
                  <lb/>
                quialtera ad duplam: et omnes inter quas mediãt
                  <lb/>
                tres ſe habent in proportione quadrupla: et ſic cõ­
                  <lb/>
                ſequenter. </s>
                <s xml:id="N1145F" xml:space="preserve">Probatur / quia proportio que eſt pri-
                  <lb/>
                me partis proportionalis ad tertiam componi-
                  <lb/>
                tur ex duabus proportionibus equalibus quarū
                  <lb/>
                vtra eſt medietas duple: ergo ſequitur /  illa eſt
                  <lb/>
                dupla. </s>
                <s xml:id="N1146A" xml:space="preserve">Patet conſequentia: et probatur antece-
                  <lb/>
                dens: quia componitur illa proportio ex propor-
                  <lb/>
                tione prime partis ad ſecundam que eſt medietas
                  <lb/>
                duple: et ex proportione ſecunde ad tertiã que etiã
                  <lb/>
                eſt medietas duple: quoniam proportio diametri
                  <lb/>
                ad coſtã eſt medietas duple: vt patet ex tertia ſup­
                  <lb/>
                poſitione tertii capitꝪ. </s>
                <s xml:id="N11479" xml:space="preserve">Et ſic probabis de quibuſ-
                  <lb/>
                cun duabus partibus paribus īmediatis: et etiã
                  <lb/>
                īparibus. </s>
                <s xml:id="N11480" xml:space="preserve">Sed iam probo partes inter quas me-
                  <lb/>
                diant due ſe habere in proportione ſexquialtera
                  <lb/>
                ad duplam quia proportio inter tales partes cõ-
                  <cb chead="Capitulum ſextū."/>
                ponitur ex proportione prime ad ſecundam: et ſe-
                  <lb/>
                cunde ad tertiam: et tertie ad quartam: ſed pro-
                  <lb/>
                portio prime ad tertiam eſt dupla: vt patet ex pro­
                  <lb/>
                batione precedentis partis: et proportio tertie ad
                  <lb/>
                quartam eſt proportio que eſt medietas duple: vt
                  <lb/>
                conſtat: ergo proportio prime ad quartam con-
                  <lb/>
                tinet duplam et medietateꝫ duple adequate: et per
                  <lb/>
                conſequēs talis proportio que eſt prime ad quar-
                  <lb/>
                tam eſt ſexquialtera ad duplam. </s>
                <s xml:id="N1149A" xml:space="preserve">Patet hec conſe­
                  <lb/>
                quentia ex diffinitione ſexquialtere. </s>
                <s xml:id="N1149F" xml:space="preserve">Et ſic proba-
                  <lb/>
                bis de aliis huiuſcemodi partibus. </s>
                <s xml:id="N114A4" xml:space="preserve">Sed iam ꝓbo
                  <lb/>
                tertiam parteꝫ / quia proportio partiū inter quas
                  <lb/>
                manent tres cuiuſmodi eſt proportio prime par-
                  <lb/>
                tis ad quintaꝫ cõponitur ex duabus duplis: puta
                  <lb/>
                ex proportione que eſt prime ad tertiaꝫ et tertie ad
                  <lb/>
                quintam que ſunt duple: vt patet ex prima parte
                  <lb/>
                huius correlarii: et per conſequens talis propor-
                  <lb/>
                tio prime ad quintam eſt dupla ad duplam cū con­
                  <lb/>
                tineat ipſam duplam bis: et per conſequens qua-
                  <lb/>
                drupla. </s>
                <s xml:id="N114B9" xml:space="preserve">Patet conſequētia ex diffinitione duple
                  <lb/>
                et ſecunda parte. </s>
                <s xml:id="N114BE" xml:space="preserve">Et hoc modo probabis de omni­
                  <lb/>
                bus ſimilibus. </s>
                <s xml:id="N114C3" xml:space="preserve">Patet hoc correlarium ſenſui in fi­
                  <lb/>
                gura ſequēti / in qua prima pars eſt diameter qua­
                  <lb/>
                drati maioris ibidem poſiti: et ſecunda eſt coſta
                  <lb/>
                eiuſdem quadrati: et tertia eſt coſta quadrati ſe-
                  <lb/>
                quentis: et tertia eſt coſta tertii quadrati: et diame­
                  <lb/>
                ter quarti: et quarta eſt coſta quarti quadrati: et
                  <lb/>
                diametri quinti: et quinta eſt coſta ipſius quinti
                  <lb/>
                quadrati: et ſic in infinitum poteris procedere ibi
                  <lb/>
                n. conſpicies /  prime ad tertiã eſt proportio du-
                  <lb/>
                pla et ſecunde ad quartam etiam dupla: et prime
                  <lb/>
                ad quintam eſt quadrupla.</s>
              </p>
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                <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YHKVZ7B4/figures/0016-01"/>
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              <note position="right" xml:id="N114F4" xml:space="preserve">Tertium
                <lb/>
              correlar̄.</note>
              <p xml:id="N114FA">
                <s xml:id="N114FB" xml:space="preserve">¶ Ex quo ſequitur tertio /  in tali diuiſiõe aggre-
                  <lb/>
                gatuꝫ ex oībus īparibus a prima īpari eſt equale
                  <lb/>
                ṗme: et aggregatū ex oībus paribꝰ a ſecunda q̄ eſt
                  <lb/>
                prima par eſt equale ſecunde: et aggregatum ex
                  <lb/>
                oībus imparibus ſe habet ad aggregatum ex om­
                  <lb/>
                nibus paribus in proportione que eſt medietas
                  <lb/>
                duple. </s>
                <s xml:id="N1150A" xml:space="preserve">Probatur prima pars huius correlarii /
                  <lb/>
                quia partes impares continuo ſe habent in pro-
                  <lb/>
                portione dupla / vt patet ex proximo correlario:
                  <lb/>
                igitur reſiduum ex omnibus īparibus ſequētibus
                  <lb/>
                primã imparem eſt equale prime impari. </s>
                <s xml:id="N11515" xml:space="preserve">Patet
                  <lb/>
                conſequentia ex ſecundo correlario tertie conclu-
                  <lb/>
                ſionis quinti capitis. </s>
                <s xml:id="N1151C" xml:space="preserve">Et eodem modo probabis
                  <lb/>
                ſecundam partem. </s>
                <s xml:id="N11521" xml:space="preserve">Sed iam probatur tertia / quo-
                  <lb/>
                niam medietas aggregati ex omnibus impari-
                  <lb/>
                bus ſe habet ad medietatem aggregati ex omni-
                  <lb/>
                bus paribus in proportione que eſt medietas du-
                  <lb/>
                ple: ergo totum aggregatum imparium ſe habet
                  <lb/>
                ad totum aggregatuꝫ parium in proportione du-
                  <lb/>
                pla. </s>
                <s xml:id="N11530" xml:space="preserve">Patet conſequentia / per hanc regulam in
                  <lb/>
                quacun proportione ſe habent partes aliquote
                  <lb/>
                aliquarum quantitatum eiuſdem denominatio-
                  <lb/>
                nis in eadem ſe habent et ille quantitates totales /
                  <lb/>
                et per conſequens in proportione qua ſe habent
                  <lb/>
                due medietates aliquoꝝ in eadē ſe hñt tota illarū
                  <lb/>
                medietatū. </s>
                <s xml:id="N1153F" xml:space="preserve">Sed ꝓbat̄̄ añs / q2 prima pars ꝓporti-
                  <lb/>
                onalis īpar ſe habet ad ṗmã parē: que eſt ſecūda.
                  <lb/>
                </s>
                <s xml:id="N11545" xml:space="preserve"/>
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