Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N10B6A" level="3" n="4" type="chapter" type-free="capitulum">
              <pb chead="Prime partis" file="0011" n="11"/>
              <p xml:id="N10C08">
                <s xml:id="N10C09" xml:space="preserve">Quinta ſuppoſitio. </s>
                <s xml:id="N10C0C" xml:space="preserve">Si quantitatis
                  <lb/>
                moioris ad aliquã partē aliquota quãtitatis mi-
                  <lb/>
                noris ſit proportio rationalis: eiuſdē quãtitatis
                  <lb/>
                maioris ad totã quantitatē minorē erit ꝓportio-
                  <lb/>
                rationalis. </s>
                <s xml:id="N10C17" xml:space="preserve">Probatur. </s>
                <s xml:id="N10C1A" xml:space="preserve">q2 ſi quantitatis maioris
                  <lb/>
                ad partē aliquotã quantitatis minoris eſt ꝓpor-
                  <lb/>
                tio rationalis: iam quantitas maior: et pars ali-
                  <lb/>
                quota minoris quantitatis ſe habent vt duo nu-
                  <lb/>
                meri. </s>
                <s xml:id="N10C25" xml:space="preserve">et ꝑ cõſequens pars aliquota minoris quati­
                  <lb/>
                tatis ſe habet vt numerus. </s>
                <s xml:id="N10C2A" xml:space="preserve">et cū nõ ſit maior ratio
                  <lb/>
                de vna parte aliquota quã de qualibet tanta: ſe-
                  <lb/>
                quitur /  quelibet tanta: ſe habet vt numerꝰ. </s>
                <s xml:id="N10C31" xml:space="preserve">et per
                  <lb/>
                ↄ̨ñs aggregatū ex oībus partibꝰ aliquotꝪ ipſius
                  <lb/>
                mīoris: ſe habet vt nūerꝰ. </s>
                <s xml:id="N10C38" xml:space="preserve">vt ptꝫ ex ṗma ſuppoſiti­
                  <lb/>
                one: et illud aggregatū eſt ipſa mīor quãtitas: igr̄
                  <lb/>
                tp̄a mīor quãtitas ſe hꝫ vt numerꝰ: ad maiorē et ſic
                  <lb/>
                inter illas eſt ꝓportio rõnalis. </s>
                <s xml:id="N10C41" xml:space="preserve">et ſic ptꝫ ſuppoſitio</s>
              </p>
              <p xml:id="N10C44">
                <s xml:id="N10C45" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N10C48" xml:space="preserve">Si due quantita­
                  <lb/>
                tes inequales ſe habeant in ꝓportione rationali.
                  <lb/>
                </s>
                <s xml:id="N10C4E" xml:space="preserve">vtra illaꝝ ſe habet ad exceſſum quo maior exce-
                  <lb/>
                dit minorē in ꝓportione rationali: vĺ equalitatis
                  <lb/>
                </s>
                <s xml:id="N10C54" xml:space="preserve">Probatur hec ſuppoſitio. </s>
                <s xml:id="N10C57" xml:space="preserve">qm̄ ſi ille quantitates:
                  <lb/>
                ſe habent in ꝓportione rationali: ſe habēt vt duo
                  <lb/>
                numeri. </s>
                <s xml:id="N10C5E" xml:space="preserve">et vltra ſe habent vt duo numeri: ergo ex-
                  <lb/>
                ceſſus quo vna excedit alterã eſt numerꝰ. </s>
                <s xml:id="N10C63" xml:space="preserve">qm̄ ſemꝑ
                  <lb/>
                numerꝰ excedit numerū ꝑ numerū. </s>
                <s xml:id="N10C68" xml:space="preserve">et vltra exceſſus
                  <lb/>
                eſt numerꝰ: et quelibet aliarū ſe habet vt numerus
                  <lb/>
                reſpectu illiꝰ exceſſus. </s>
                <s xml:id="N10C6F" xml:space="preserve">igr̄ inter illū exceſſū et quãli­
                  <lb/>
                bet illarum quantitatem eſt proportio ratiõalis
                  <lb/>
                vel equalitatis: quod fuit probandum.</s>
              </p>
              <p xml:id="N10C76">
                <s xml:id="N10C77" xml:space="preserve">His ſuppoſitionibus poſitis: ſit pri-
                  <lb/>
                ma cõcluſio </s>
                <s xml:id="N10C7C" xml:space="preserve">Infinite ſunt ſpecies ꝓportionis irra­
                  <lb/>
                tionalis minores dupla: et illarū in īfinitū parua
                  <lb/>
                eſt aliqua. </s>
                <s xml:id="N10C83" xml:space="preserve">Probatur prima pars huiꝰ cõcluſiõis /
                  <lb/>
                et capio coſtã vniꝰ quadrati: et ſuã diametrū. </s>
                <s xml:id="N10C88" xml:space="preserve">et vo­
                  <lb/>
                lo /  vniformiter in hora diminuat̄̄ exceſſus quo
                  <lb/>
                diameter excedit coſtã ad nõ quantū. </s>
                <s xml:id="N10C8F" xml:space="preserve">ita  in fine
                  <lb/>
                diameter et coſta erūt equalia. </s>
                <s xml:id="N10C94" xml:space="preserve">quo poſito ſic argr̄
                  <lb/>
                </s>
                <s xml:id="N10C98" xml:space="preserve">Inter diametrū que ſic diminuitur et coſtaꝫ erunt
                  <lb/>
                infinite ꝓportiones irratiõales cõtinuo minores
                  <lb/>
                dupla: igitur infinite ſunt ſpecies ꝓportiõis irra-
                  <lb/>
                tionalis minores dupla. </s>
                <s xml:id="N10CA1" xml:space="preserve">Probatur ãtecedēs. </s>
                <s xml:id="N10CA4" xml:space="preserve">qm̄
                  <lb/>
                quãdo exceſſus: quo diameter excedit coſtã ꝑdide-
                  <lb/>
                rit medietatē ſui / tūc aggregatū ex alia medietate
                  <lb/>
                et coſta ſe habebit ad coſtã in ꝓportiõe irratiõali
                  <lb/>
                minori dupla. / et quãdo exceſſus diametri fuerit di­
                  <lb/>
                minutꝰ ad vnã quartã ſui: tūc aggregati ex coſta
                  <lb/>
                et illa quarta exceſſus diametri ad coſtã erit ꝓpor­
                  <lb/>
                tio irrationalis. </s>
                <s xml:id="N10CB5" xml:space="preserve">et ſic cõſequēter ſemꝑ aggregatū
                  <lb/>
                ex coſta: et aliqua parte aliquota exceſſus ſe habe­
                  <lb/>
                bit ad coſtã in ꝓportione irratiõali mīori dupla:
                  <lb/>
                et infinita ſunt talia aggregata ex coſta et aliqua
                  <lb/>
                parte aliquota exceſſus: igitur infinite erūt ꝓpor­
                  <lb/>
                tiones irrationales cõtinuo minores dupla. </s>
                <s xml:id="N10CC2" xml:space="preserve">Ptꝫ
                  <lb/>
                cõſequētia. </s>
                <s xml:id="N10CC7" xml:space="preserve">et arguit̄̄ maior videlicet /  aggregatū
                  <lb/>
                ex coſta et medietate exceſſus diametri: ſe habet in
                  <lb/>
                ꝓportione irrationali ad coſtã: q2 ſi nõ. </s>
                <s xml:id="N10CCE" xml:space="preserve">ſed ſe ba-
                  <lb/>
                bent in ꝓportione rationali. </s>
                <s xml:id="N10CD3" xml:space="preserve">ſequitur:  vtra il­
                  <lb/>
                laꝝ: ſe habet ad exceſſum quo maior excedit mino­
                  <lb/>
                rem in ꝓportione rationali vel equalitatis. </s>
                <s xml:id="N10CDA" xml:space="preserve">Ptꝫ
                  <lb/>
                ↄ̨ña ex ſexta ſuppoſitione. </s>
                <s xml:id="N10CDF" xml:space="preserve">et cõſequēs eſt falſū. </s>
                <s xml:id="N10CE2" xml:space="preserve">qm̄
                  <lb/>
                ſi vtra illarū ſe haberet ad exceſſum quo diame­
                  <lb/>
                ter excedit coſtã: in ꝓportione rationali .etc̈. cū al-
                  <lb/>
                tera illarum ſit coſta: et exceſſus quo maior excedit
                  <lb/>
                minorē ſit medietas exceſſus diametri: ſequitur / 
                  <cb chead="Capitulum quartū."/>
                coſte ad medietatē exceſſus diametri erit ꝓportio
                  <lb/>
                rationalis. </s>
                <s xml:id="N10CF2" xml:space="preserve">Patet hec cõſequētia ex ſe. </s>
                <s xml:id="N10CF5" xml:space="preserve">et vltra ſe-
                  <lb/>
                quitur /  coſte: ad exceſſum diametri erit ꝓportio
                  <lb/>
                rationalis. </s>
                <s xml:id="N10CFC" xml:space="preserve">Patet cõſequētia ex quīta ſuppoſitio­
                  <lb/>
                ne. </s>
                <s xml:id="N10D01" xml:space="preserve">hoc addito /  medietas exceſſus eſt pars aliq̊ta
                  <lb/>
                illius: cõſequēs eſt falſum: vt patet ex quarta igit̄̄
                  <lb/>
                et ãtecedēs. </s>
                <s xml:id="N10D08" xml:space="preserve">Et ſic ꝓbabis. </s>
                <s xml:id="N10D0B" xml:space="preserve"> aggregatū ex coſta et
                  <lb/>
                quarta parte exceſſus diametri ſe habet in ꝓpor-
                  <lb/>
                tione irratiõali ad coſtã: et ſimiliter  aggregatū
                  <lb/>
                ex coſta et octaua parte exceſſus / et ſic cõſequenter.
                  <lb/>
                </s>
                <s xml:id="N10D15" xml:space="preserve">Quod autē ille ꝓportiones cõtinuo ſint minores
                  <lb/>
                dupla: patet. </s>
                <s xml:id="N10D1A" xml:space="preserve">q2 a principio ꝓportio diametri ad
                  <lb/>
                coſtã erat minor dupla. cū eſſet medietas duple: et
                  <lb/>
                cõtinuo diminuet̄̄ vſ ad nõ gradū: vt ptꝫ ex ſcḋa
                  <lb/>
                parte. </s>
                <s xml:id="N10D23" xml:space="preserve">igr̄ cõtinuo erit minor dupla. </s>
                <s xml:id="N10D26" xml:space="preserve">Itē continuo
                  <lb/>
                exceſſus erit minor et minor reſpectu eiuſdē quãti-
                  <lb/>
                tatis: ergo cõtinuo ꝓportio erit minor et mīor. </s>
                <s xml:id="N10D2D" xml:space="preserve">Et
                  <lb/>
                ex hoc ptꝫ ſcḋa pars cõclſionis. </s>
                <s xml:id="N10D32" xml:space="preserve">q2 in infinitū mo-
                  <lb/>
                dicus erit exceſſus quãtitatis maioris ad quãtita­
                  <lb/>
                tē minorē: et ipſa quãtitas minor cõtinuo manebit
                  <lb/>
                equalis et īuariata. </s>
                <s xml:id="N10D3B" xml:space="preserve">igitur infinite modica erit ꝓ-
                  <lb/>
                portio maioris ad quantitatem minorem. </s>
                <s xml:id="N10D40" xml:space="preserve">Conſe­
                  <lb/>
                quentia patet ex ſecūda parte. </s>
                <s xml:id="N10D45" xml:space="preserve">Et ſic patet prima
                  <lb/>
                concluſio.
                  <note position="right" xlink:href="note-0011-01a" xlink:label="note-0011-01" xml:id="N10D67">
                    <s xml:id="N10D6B" xml:space="preserve">Correla-
                      <lb/>
                    rium.
                      <lb/>
                    </s>
                    <s xml:id="N10D71" xml:space="preserve">Gñatio
                      <lb/>
                    infinitoꝝ
                      <lb/>
                    ſpecierū
                      <lb/>
                    ꝓportio-
                      <lb/>
                    nis irra-
                      <lb/>
                    tionalis.</s>
                  </note>
                </s>
                <s xml:id="N10D4F" xml:space="preserve">¶ Ex hac concluſione ſequitur:  infini-
                  <lb/>
                tis modis poſſunt generari infinite ſpecies mino­
                  <lb/>
                res dupla irrationalis ꝓportiõis: vtpote ſi exceſ-
                  <lb/>
                ſus diametri diminuatur per partes ꝓportiona-
                  <lb/>
                les ꝓportione dupla: </s>
                <s xml:id="N10D5A" xml:space="preserve">Alio modo ꝓportiõe tripla
                  <lb/>
                alio quadrupla. alio ſexquialtera. / et ſic in infinitū
                  <lb/>
                </s>
                <s xml:id="N10D60" xml:space="preserve">Patet correlariū intelligēti ꝓbationē cõculſiõis</s>
              </p>
              <p xml:id="N10D7E">
                <s xml:id="N10D7F" xml:space="preserve">Secūda cõcluſio. </s>
                <s xml:id="N10D82" xml:space="preserve">Infinite ſunt ſpe-
                  <lb/>
                cies ꝓportionis irratiõalis maioris dupla: et illa­
                  <lb/>
                rū infinite magna eſt aliqua. </s>
                <s xml:id="N10D89" xml:space="preserve">Probatur hec con-
                  <lb/>
                cluſio: et pono /  exceſſus quo diameter excedit co-
                  <lb/>
                ſtam: diminuatur vniformiter in hora vſ ad nõ
                  <lb/>
                quantū. </s>
                <s xml:id="N10D92" xml:space="preserve">et capio ꝓportionē que eſt coſte ad exceſſū
                  <lb/>
                diametri: et arguo ſic. </s>
                <s xml:id="N10D97" xml:space="preserve">Illa ꝓportio eſt maior du-
                  <lb/>
                pla irrationalis. </s>
                <s xml:id="N10D9C" xml:space="preserve">et ꝓportio coſte ad medietatē il-
                  <lb/>
                lius exceſſus eſt etiã irratiõalis maior: et ꝓ-
                  <lb/>
                portio coſte ad quartã eſt etiã irrationalis maior
                  <lb/>
                dupla: et ſic in infinitū quelibet ꝓportio coſte ad
                  <lb/>
                aliquã partē aliquotã exceſſus eſt ꝓportio irrati-
                  <lb/>
                onalis et ſunt īfinite partes aliquote cõtinuo mi-
                  <lb/>
                nores et minores / ergo īfinite ſunt ꝓportiões irra­
                  <lb/>
                tiõales minores dupla. </s>
                <s xml:id="N10DAD" xml:space="preserve">Probat̄̄ maior. </s>
                <s xml:id="N10DB0" xml:space="preserve">qm̄ coſte
                  <lb/>
                ad exceſſū q̊ diameṫ excedit coſtã eſt ꝓportio irra-
                  <lb/>
                tionalis: ex q̈rta ſuppoſitiõe maior dupla: vt con-
                  <lb/>
                ſtat. </s>
                <s xml:id="N10DB9" xml:space="preserve">qm̄ ille exceſſus eſt minor quã medietas coſte.
                  <lb/>
                </s>
                <s xml:id="N10DBD" xml:space="preserve">qm̄ ſi eſſet medietas coſte aut moior: iam ibi eſſet
                  <lb/>
                ꝓportio ſexq̇altera īter diametrū et coſtã: vel ma-
                  <lb/>
                ior ſexquialtera: quod eſt falſum. </s>
                <s xml:id="N10DC4" xml:space="preserve">vt ptꝫ ex pcedēti
                  <lb/>
                capite. </s>
                <s xml:id="N10DC9" xml:space="preserve">ergo q̄libet ꝓportio coſte ad aliquã partē
                  <lb/>
                aliquotã exceſſus quo diameter excedit coſtam eſt
                  <lb/>
                ꝓportio irratiõalis maior dupla: qḋ fuit ꝓbãdū.
                  <lb/>
                </s>
                <s xml:id="N10DD1" xml:space="preserve">Patet cõſequētia ex quīta ſuppoſitiõe. </s>
                <s xml:id="N10DD4" xml:space="preserve">qm̄ ex illa
                  <lb/>
                ſuppoſitione. </s>
                <s xml:id="N10DD9" xml:space="preserve">ſi coſta ad aliquã partē aliquotã ex-
                  <lb/>
                ceſſus quo diameter excedit coſtã ſe habet in pro-
                  <lb/>
                portione ratiõali: ipſius coſte ad totū illū exceſſū
                  <lb/>
                erit ꝓportio rationalis: ſed nõ ipſiꝰ coſte ad totū
                  <lb/>
                illū exceſſū quo diameter excedit coſtã eſt ꝓportio
                  <lb/>
                rationalis. </s>
                <s xml:id="N10DE6" xml:space="preserve">vt ptꝫ ex quarta ſuppoſitiõe. </s>
                <s xml:id="N10DE9" xml:space="preserve">igitur nõ
                  <lb/>
                coſta ad aliquã partē aliquotã exceſſus quo dia-
                  <lb/>
                meter excedit coſtã: ſe habet in ꝓportiõe ratiõali.
                  <lb/>
                </s>
                <s xml:id="N10DF1" xml:space="preserve">Patet cõſequētia ꝑ ſyllogiſmū hypotheticum: a
                  <lb/>
                tota cõditionali cū deſtructiõe cõſequētis .etc̈. / et ſic
                  <lb/>
                patet prima pars. </s>
                <s xml:id="N10DF8" xml:space="preserve">Et ſcḋa ꝓbatur facile. </s>
                <s xml:id="N10DFB" xml:space="preserve">q2 in īfi- </s>
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