Alvarus, Thomas, Liber de triplici motu, 1509

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          <div xml:id="N1194D" level="2" n="2" type="other" type-free="pars">
            <div xml:id="N140E1" level="3" n="6" type="chapter" type-free="capitulum">
              <p xml:id="N14243">
                <s xml:id="N14244" xml:space="preserve">
                  <pb chead="Secūde partis" file="0043" n="43"/>
                portionis multiplicis vel multiplicis ſuperparti­
                  <lb/>
                cularis, vel multiplicis ſuprapartientis. </s>
                <s xml:id="N14262" xml:space="preserve">Falſitas
                  <lb/>
                conſequentis probatur: quoniam ſi c. eſt pars ali­
                  <lb/>
                quota multiplicis ꝓportionis: capio talem ꝓpor­
                  <lb/>
                tionem multiplicem inter primos terminos eius:
                  <lb/>
                et arguo ſic: c: proportio multiplex ſuperparticu-
                  <lb/>
                laris, aut multiplex ſnpraꝑtiens, eſt pars aliquo­
                  <lb/>
                ta alicuius ꝓportionis multiplicis: igitur ex ali-
                  <lb/>
                quot c. illa proportio multiplex componitur. </s>
                <s xml:id="N14273" xml:space="preserve">igi-
                  <lb/>
                tur ex conſequenti ſequitur /  alicuius termini in-
                  <lb/>
                termedii ad minimum extremū ipſius proportio-
                  <lb/>
                nis mĺtiplicis / qḋ minimū externū ē vnitas ē ꝓpor­
                  <lb/>
                tio c. / vt patet ex tertia ſuppoſitione: et illa ꝓpor-
                  <lb/>
                tio c. eſt multiplex ſuꝑparticularis, aut multiplex
                  <lb/>
                ſuꝑperpartiens: igitur alicuius numeri ad vnita-
                  <lb/>
                tem eſt ꝓportio multiplex ſuprapartiens aut mul­
                  <lb/>
                tiplex ſuperparticularis quod eſt oppoſitum ſex-
                  <lb/>
                te ſuppoſitionis: et per conſequens falſum: et ex
                  <lb/>
                conſequenti illud ex quo ſequitur videlicet /  c. eſt
                  <lb/>
                ꝓportio multiplex ſuperparticularis, aut multi-
                  <lb/>
                plex ſuprapartiens. </s>
                <s xml:id="N1428E" xml:space="preserve">Et ſic patet concluſio.</s>
              </p>
              <p xml:id="N14291">
                <s xml:id="N14292" xml:space="preserve">Quarta concluſio. </s>
                <s xml:id="N14295" xml:space="preserve">Nulla proportio
                  <lb/>
                multiplex eſt commenſurabilis alicui proportio-
                  <lb/>
                ni rationali non multiplici. </s>
                <s xml:id="N1429C" xml:space="preserve">Probatur: quia nul­
                  <lb/>
                la ꝓportio multiplex eſt commenſurabilis alicui
                  <lb/>
                ſuperparticulari, aut ſuprapartienti / vt patet ex
                  <lb/>
                ſecunda, nec alicui multiplici ſuꝑparticulari, aut
                  <lb/>
                multiplici ſuprapartienti / vt ptꝫ ex tertia, igit̄̄ nul­
                  <lb/>
                la ꝓportio multiplex ↄ̨menſurabilis eſt alicui ꝓ-
                  <lb/>
                portioni rationali non multiplici. </s>
                <s xml:id="N142AB" xml:space="preserve">Et ſic patet cõ­
                  <lb/>
                cluſio.</s>
              </p>
              <p xml:id="N142B0">
                <s xml:id="N142B1" xml:space="preserve">Quinta concluſio </s>
                <s xml:id="N142B4" xml:space="preserve">Nulla proportio
                  <lb/>
                ſuperparticularis eſt commenſurabilis alicui ꝓ-
                  <lb/>
                portioni ſuperparticulari. </s>
                <s xml:id="N142BB" xml:space="preserve">Probatur ſupponen­
                  <lb/>
                do /  inter cuiuſlibet ꝓportionis ſuperparticula­
                  <lb/>
                ris primos numeros nullus numerus mediat vt
                  <lb/>
                viſum eſt in prima parte vbi agebatur de genera­
                  <lb/>
                tione ꝓportionum ſuperparticularium. </s>
                <s xml:id="N142C6" xml:space="preserve">quo ſup­
                  <lb/>
                poſito arguitur ſic: inter cuiuſlibet ꝓportionis ſu­
                  <lb/>
                perparticularis primos numeros nullus mediat
                  <lb/>
                numerus: igitur nulla talis ex aliquot ītermediis
                  <lb/>
                ꝓportionibus adequate componitur. </s>
                <s xml:id="N142D1" xml:space="preserve">Patet con­
                  <lb/>
                ſequentia / quia nulla eſt ꝓportio intermedia niſi
                  <lb/>
                ſit numerus intermedius: et vltra ex nullis ꝓpor-
                  <lb/>
                tionibus componitur. </s>
                <s xml:id="N142DA" xml:space="preserve">igitur nulla ꝓportio ē pars
                  <lb/>
                aliquota eius: et per conſequens ipſa non eſt com­
                  <lb/>
                menſurabilis alicui proportioni ſuperparticula­
                  <lb/>
                ri. </s>
                <s xml:id="N142E3" xml:space="preserve">Patet conſequentia / quia alias aliquid eſſet
                  <lb/>
                pars aliquota vtriuſ. </s>
                <s xml:id="N142E8" xml:space="preserve">Et ſic patet concluſio.</s>
              </p>
              <note position="left" xml:id="N142EB" xml:space="preserve">obiectio.</note>
              <p xml:id="N142EF">
                <s xml:id="N142F0" xml:space="preserve">¶ Sed tu dices /  hec ꝓbatio eſt inefficax: quoniã
                  <lb/>
                concedit /  aliqua proportio ex nullis ꝓportioni­
                  <lb/>
                bus componitur quod eſt contra ea que dicta ſūt
                  <lb/>
                capite quarto huius partis. </s>
                <s xml:id="N142F9" xml:space="preserve">imo ꝓbatio nihil ali­
                  <lb/>
                ud probat niſi /  ex nullis ꝓportionibus equalibꝰ
                  <lb/>
                rationalibus componitur que ſint partes aliquo­
                  <lb/>
                te illius: cum hoc tamen ſtat /  aliqua ꝓportio ir-
                  <lb/>
                rationalis eſt pars aliquota duarum ꝓportionū
                  <lb/>
                ſuperparticularium: et ſic erunt commenſurabi-
                  <lb/>
                les.
                  <note position="left" xlink:href="note-0043-01a" xlink:label="note-0043-01" xml:id="N14340" xml:space="preserve">reiicitur
                    <lb/>
                  obiectio.</note>
                </s>
                <s xml:id="N1430D" xml:space="preserve">¶ Sed hoc non obſtat / quia nulla ꝓportio ſuꝑ­
                  <lb/>
                particularis componitur ex alia ſuperparticula­
                  <lb/>
                ri et vna irrationali: ſicut nec aliq̄ rationalis cõ-
                  <lb/>
                ponitur ex vna rationali et altera irrationali a-
                  <lb/>
                dequate / vt probãt mathemathici. </s>
                <s xml:id="N14318" xml:space="preserve">igitur nulla ſu­
                  <lb/>
                ꝑparticularis continet alteram ſuꝑparticularem
                  <lb/>
                ſemel aut aliquoties et vnam partem aliquotam
                  <lb/>
                eius que ſit ꝓportio irrationalis: quia tunc com-
                  <lb/>
                poneretur ex rationali et irrationali adequate:
                  <lb/>
                nec aliqua ſuꝑparticularis continet alteram ſe-
                  <cb chead="Capitulum ſextum"/>
                mel vel aliquoties et aliquot partes eius aliquo-
                  <lb/>
                tas que ſint proportiones irrationales: quia tunc
                  <lb/>
                iam ille proportiones irrationales componerent
                  <lb/>
                vnam rationalem: quia alias componeretur illa
                  <lb/>
                ſuperparticularis ex rationali et irrationali: et ſi
                  <lb/>
                ille partes aliquote faciant vnam rationalem iaꝫ
                  <lb/>
                inter terminos illius ꝓportionis ſuꝑparticularis
                  <lb/>
                reperientur aliquot ꝓportiones rationales equa­
                  <lb/>
                les / vt patet intuenti: quod tamen eſt falſum cum
                  <lb/>
                non reperiantur inter primos numeros alicuius
                  <lb/>
                ꝓportionis ſuꝑparticularis.</s>
              </p>
              <p xml:id="N14348">
                <s xml:id="N14349" xml:space="preserve">Sexta concluſio </s>
                <s xml:id="N1434C" xml:space="preserve">Inter rationales.
                  <lb/>
                </s>
                <s xml:id="N14350" xml:space="preserve">tantum ꝓportio multiplex commenſuratur ꝓpor­
                  <lb/>
                tioni multiplici. </s>
                <s xml:id="N14355" xml:space="preserve">Probatur / quia proportio multi­
                  <lb/>
                plex eſt commenſurabilis ꝓportioni multiplici / vt
                  <lb/>
                patet de quadrupla reſpectu duple: et inter ratio-
                  <lb/>
                nales nulla non multiplex eſt cõmenſurabilis ali­
                  <lb/>
                cui ꝓportioni multiplici / vt patet ex quarta cõclu­
                  <lb/>
                ſione / igitur propoſitum. </s>
                <s xml:id="N14362" xml:space="preserve">Conſequentia patet ex
                  <lb/>
                dialectica.</s>
              </p>
              <p xml:id="N14367">
                <s xml:id="N14368" xml:space="preserve">Septima concluſio </s>
                <s xml:id="N1436B" xml:space="preserve">Omēs propor-
                  <lb/>
                tiones multiplices quarum denominationes ſunt
                  <lb/>
                de numero numerorum ſunt inter ſe cõmenſurabi­
                  <lb/>
                les.
                  <note position="right" xlink:href="note-0043-02a" xlink:label="note-0043-02" xml:id="N143C0" xml:space="preserve">nicholaꝰ
                    <lb/>
                  horen.</note>
                </s>
                <s xml:id="N14379" xml:space="preserve">Hanc concluſionem ponit Nicholaus horen
                  <lb/>
                ſub forma dicta: ſed pono eam ſub alia forma cla­
                  <lb/>
                riori. </s>
                <s xml:id="N14380" xml:space="preserve">Omnes ꝓportiones multiplices ꝓcedentes
                  <lb/>
                ſemper ſecundum dendminationem prime illarū
                  <lb/>
                ſunt cõmenſurabiles: ita  ſi prima illarum ſit du­
                  <lb/>
                pla. </s>
                <s xml:id="N14389" xml:space="preserve">ſecunda immediate ſequens ſit etiam dupla:
                  <lb/>
                et ſic conſequenter tales ſunt cõmenſurabiles. </s>
                <s xml:id="N1438E" xml:space="preserve">Et
                  <lb/>
                vt paucis abſoluam omnes ꝓportiones quarum
                  <lb/>
                quelibet īmediate ſequētes ſunt eiuſdem denomi­
                  <lb/>
                nationis cum prima ſunt commenſurabiles </s>
                <s xml:id="N14397" xml:space="preserve">Pa-
                  <lb/>
                tet hec concluſio / quoniam omnes tales ita ſe ha-
                  <lb/>
                bent  aliquid eſt pars aliquota vtriuſ / igitur.
                  <lb/>
                </s>
                <s xml:id="N1439F" xml:space="preserve">Et ad hoc videndum diſponatur vna ſeries nūe-
                  <lb/>
                rorum incipiendo ab vnitate ſemper duplando et
                  <lb/>
                vna alia ſemper triplando, et alia quadruplan-
                  <lb/>
                do, et alia quintuplando, et ſic in infinitum. </s>
                <s xml:id="N143A8" xml:space="preserve">et tunc
                  <lb/>
                dico /  omnes ꝓportiones primi ordinis ſunt cõ-
                  <lb/>
                menſurabiles inter ſe. </s>
                <s xml:id="N143AF" xml:space="preserve">et quelibet cuilibet alteri il­
                  <lb/>
                lius ordines: </s>
                <s xml:id="N143B4" xml:space="preserve">Et ſic etiam dicendum eſt de ꝓportio­
                  <lb/>
                nibus alioruꝫ ordinum. </s>
                <s xml:id="N143B9" xml:space="preserve">Patet hoc in his figuris</s>
              </p>
              <xhtml:table xml:id="N143C8">
                <xhtml:tr xml:id="N143C9">
                  <xhtml:td xml:id="N143CA" xml:space="preserve"/>
                </xhtml:tr>
              </xhtml:table>
              <p xml:id="N143CC">
                <s xml:id="N143CD" xml:space="preserve">Et ſic etiam conſtitues ordines multarum ſuper-
                  <lb/>
                particularium et ſuprapartientium etc. </s>
                <s xml:id="N143D2" xml:space="preserve">Quod au­
                  <lb/>
                tem iſte ſunt commenſurabiles probatur / quoniã
                  <lb/>
                quelibet illius ordinis eſt equalis prime aut com­
                  <lb/>
                ponitur ex aliquot equalibus illi: igitur. </s>
                <s xml:id="N143DB" xml:space="preserve">¶ Iſte
                  <lb/>
                concluſiones dempta prima et ſexta ſunt Nicho-
                  <lb/>
                lai horen cum ſuis probationibus ſaltem virtu-
                  <lb/>
                tes probationum et fundamenta ſunt ex ipſo.</s>
              </p>
              <note position="right" xml:id="N143E4" xml:space="preserve">cõtra ni-
                <lb/>
              cholauꝫ
                <lb/>
              horen.</note>
              <p xml:id="N143EC">
                <s xml:id="N143ED" xml:space="preserve">¶ Sed videntur mihi ille probationes inefficaces
                  <lb/>
                </s>
                <s xml:id="N143F1" xml:space="preserve">Fundatur enim principaliter probatio ſecūde ter­
                  <lb/>
                tie et quarte in hac ſuppoſitione cuiuſlibet ꝓpor­
                  <lb/>
                tionis multiplicis vnitas eſt minimum extremum
                  <lb/>
                </s>
                <s xml:id="N143F9" xml:space="preserve">Modo illa ſuppoſitio falſa eſt / quoniam octo ad
                  <lb/>
                quatuor eſt proportio multiplex: tamen neutrum
                  <lb/>
                extremorum eius eſt vnitas: </s>
                <s xml:id="N14400" xml:space="preserve">Sed diceret Nicho-
                  <lb/>
                laus horen et bene /  illa ſuppoſitio et ſi nõ ſit ve-
                  <lb/>
                ra diſtribuendo pro ſingulis generum. </s>
                <s xml:id="N14407" xml:space="preserve">eſt tamen
                  <lb/>
                vera diſtribuendo pro generibus ſingulorum: et ī </s>
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