Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N13CDA" level="3" n="5" type="chapter" type-free="capitulum">
              <p xml:id="N14000">
                <s xml:id="N1408D" xml:space="preserve">
                  <pb chead="Secunde partis" file="0042" n="42"/>
                falſa et probatio nulla. </s>
                <s xml:id="N14095" xml:space="preserve">et ſecundumm baſanum ē
                  <lb/>
                quadrupla ad quadruplam: igitur dicta baſani
                  <lb/>
                et calculatoris non coherent. </s>
                <s xml:id="N1409C" xml:space="preserve">¶ Hoc idem ex mul-
                  <lb/>
                tis aliis locis calculatoris euidenter deprehēde-
                  <lb/>
                re potes. </s>
                <s xml:id="N140A3" xml:space="preserve">ſed hii loci ſufficiant. </s>
                <s xml:id="N140A6" xml:space="preserve">Et ſic relinquo po-
                  <lb/>
                ſitionem eius confutatam et exploſam: que tamē
                  <lb/>
                proterue defenſari poteſt: ſed nõ conſequenter ad
                  <lb/>
                mathemathica prīcipia vt dictū eſt.
                  <note position="left" xlink:href="note-0042-01a" xlink:label="note-0042-01" xml:id="N140DB" xml:space="preserve">correĺm.</note>
                </s>
                <s xml:id="N140B4" xml:space="preserve">¶ Ex his igit̄̄
                  <lb/>
                abunde apparet /  proportio proportionū nõ eſt
                  <lb/>
                ſicut proportio denominationum.</s>
              </p>
            </div>
            <div xml:id="N140E1" level="3" n="6" type="chapter" type-free="capitulum">
              <head xml:id="N140E6" xml:space="preserve">Capitulū ſextū / in quo agitur de pro-
                <lb/>
              portionū proportione: cõmenſurabilita­
                <lb/>
              te earūdem, et incõmenſurabilitate.</head>
              <p xml:id="N140ED">
                <s xml:id="N140EE" xml:space="preserve">PRo ſpecialiori noticia propor­
                  <lb/>
                tionis ꝓportionū habenda ſit.</s>
              </p>
              <p xml:id="N140F3">
                <s xml:id="N140F4" xml:space="preserve">Prima ſuppoſitio. </s>
                <s xml:id="N140F7" xml:space="preserve">Cõmenſurabilia
                  <lb/>
                ſiue in ꝓportione rationali ſe habentia ſunt illa
                  <lb/>
                quorū idem eſt pars aliquota vt .4. et .2. pedale et
                  <lb/>
                bipedale. </s>
                <s xml:id="N14100" xml:space="preserve">Unitas em̄ eſt pars aliquota et duorū et
                  <lb/>
                quatuor: et medietas pedalis eſt pars aliquota et
                  <lb/>
                pedalis et bipedalis.
                  <note position="left" xlink:href="note-0042-02a" xlink:label="note-0042-02" xml:id="N14115" xml:space="preserve">eu. 10. ele.</note>
                </s>
                <s xml:id="N1410C" xml:space="preserve">Hec eſt diffinitio cõmenſura­
                  <lb/>
                biliū in principio decimi elementoꝝ euclidis.</s>
              </p>
              <p xml:id="N1411B">
                <s xml:id="N1411C" xml:space="preserve">Secunda ſuppoſitio. </s>
                <s xml:id="N1411F" xml:space="preserve">Ille proportio­
                  <lb/>
                nes dicūtur cõmenſurabiles quarum eadem pro-
                  <lb/>
                portio eſt pars aliquota. </s>
                <s xml:id="N14126" xml:space="preserve">Patet ex priori.</s>
              </p>
              <p xml:id="N14129">
                <s xml:id="N1412A" xml:space="preserve">Tertia ſuppoſitio. </s>
                <s xml:id="N1412D" xml:space="preserve">Quando aliqua
                  <lb/>
                ꝓportio cõponitur ex aliquot ꝓportionibus ade-
                  <lb/>
                quate ſemꝑ altera illarū eſt ꝓportio que eſt alicu-
                  <lb/>
                ius termini intermedii ad minimū extremū: vt ꝓ-
                  <lb/>
                portio quatuor ad duo componitur ex proportio­
                  <lb/>
                ne .4. ad .3. et trium ad duo que eſt alicuius termi-
                  <lb/>
                ni intermedii ad minimum extremum. </s>
                <s xml:id="N1413C" xml:space="preserve">Patet hec
                  <lb/>
                ſatis ex his que dicta ſunt in quarto capite huius
                  <lb/>
                partis.</s>
              </p>
              <p xml:id="N14143">
                <s xml:id="N14144" xml:space="preserve">Quarta ſuppoſitio </s>
                <s xml:id="N14147" xml:space="preserve">Quilibet nume-
                  <lb/>
                rus eſt multiplex ad vnitatem </s>
                <s xml:id="N1414C" xml:space="preserve">Patet ex his que
                  <lb/>
                dicta ſunt in quarto capite: </s>
                <s xml:id="N14151" xml:space="preserve">Et rurſns quia omīs
                  <lb/>
                numerus aut componitur ex duabus vnitatibus:
                  <lb/>
                et ſic eſt duplus ad vnitatem. </s>
                <s xml:id="N14158" xml:space="preserve">vel ex tribus / et ſic eſt
                  <lb/>
                triplus, vel ex quatuor / et ſic eſt quadruplus: et ſic
                  <lb/>
                in infinitum. </s>
                <s xml:id="N1415F" xml:space="preserve">¶ Ex hac ſequitur.</s>
              </p>
              <p xml:id="N14162">
                <s xml:id="N14163" xml:space="preserve">Quinta ſuppoſitio </s>
                <s xml:id="N14166" xml:space="preserve">Cuiuſlibet pro-
                  <lb/>
                portionis multiplicis vnitas eſt minimum extre-
                  <lb/>
                mum.</s>
              </p>
              <p xml:id="N1416D">
                <s xml:id="N1416E" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N14171" xml:space="preserve">Nullus numerus
                  <lb/>
                eſt ſuprapartiēs, aut ſuperparticularis: aut mul­
                  <lb/>
                tiplex ſuprapartiens, aut multiplex ſuperparti-
                  <lb/>
                cularis ad vnitatem. </s>
                <s xml:id="N1417A" xml:space="preserve">Probatur / quoniã quilibet
                  <lb/>
                numerus adequate eſt multiplex ad vnitatem / vt
                  <lb/>
                patet ex quarta: igitur nullꝰ eſt ſuprapartiēs aut
                  <lb/>
                ſuperparticularis: aut multiplex etc. ad vnitatem</s>
              </p>
              <p xml:id="N14183">
                <s xml:id="N14184" xml:space="preserve">His ſuppoſitis ſit </s>
                <s xml:id="N14187" xml:space="preserve">Prima concluſio
                  <lb/>
                </s>
                <s xml:id="N1418B" xml:space="preserve">Nulla proportio multiplex eſt pars aliquota ali­
                  <lb/>
                cuius proportionis non multiplicis. </s>
                <s xml:id="N14190" xml:space="preserve">Probatur /
                  <lb/>
                quoniaꝫ multiplex nullius proportionis ſuperꝑ-
                  <lb/>
                ticularis aut ſuprapartientis eſt pars: cum quali­
                  <lb/>
                bet tali ſit maior: nec etiam alicuius non multipli­
                  <lb/>
                cis alterius: quia ſi ſic detur illa proportio et ſit a. /
                  <lb/>
                et multiplex pars aliquota eius ſit b. inter d. et e.
                  <lb/>
                terminos primos / et arguitur ſic b. proportio mul­
                  <lb/>
                tiplex eſt pars aliquota ipſius a. / igitur a. eſt pro-
                  <lb/>
                portio multiplex / quod eſt oppoſitum dati. </s>
                <s xml:id="N141A3" xml:space="preserve">Pro-
                  <lb/>
                batur conſequentia / quia ſi b. eſt pars aliquota ip­
                  <lb/>
                ſius a. / ſequitur /  ipſa b. proportio multiplex ali-
                  <cb chead="Capitulum ſextum"/>
                quoties ſumpta reddit et componit ipſam a. pro-
                  <lb/>
                portionem: cõponat igitur c. vicibus ſumpta ade­
                  <lb/>
                quate: et tūc capio proportionem b. inter primos
                  <lb/>
                numeros eius ſiue terminos d. videlicet maiorem
                  <lb/>
                et e. minorem: et manifeſtum eſt /  e. eſt vnitas vt
                  <lb/>
                patet ex quinta ſuppoſitione: capio igitur / tūc vnū
                  <lb/>
                alium numerum que ſe habeat in proportione b.
                  <lb/>
                ad ipſum d. qui ſit f. et iterum vnum alterum qui
                  <lb/>
                ſe habeat in proportione b. ad f: et ſic c. vicibus: et
                  <lb/>
                ſit vltimus numerus ſic ſumptus g. / et manifeſtum
                  <lb/>
                eſt /  g. ad e. erit proportio compoſita ex b. ꝓpro-
                  <lb/>
                tione c. vicibus adequate: et illa proportio g. ad e.
                  <lb/>
                eſt multiplex quia eſt inter g. numerum et e. vnita-
                  <lb/>
                tem. </s>
                <s xml:id="N141C7" xml:space="preserve">Conſequentia patet ex quarta ſuppoſitione
                  <lb/>
                et ſexta: et illa eſt a. proportio per te / ergo a. ē mul
                  <lb/>
                multiplex / quod fuit probandum. </s>
                <s xml:id="N141CE" xml:space="preserve">Et ſic patet con-
                  <lb/>
                cluſio. </s>
                <s xml:id="N141D3" xml:space="preserve">¶ Ex qua ſequitur /  nulla proportio non
                  <lb/>
                multiplex eſt dupla, quadrupla, aut aliqua alia
                  <lb/>
                de genere multiplici, ad aliquam multiplicem.</s>
              </p>
              <p xml:id="N141DA">
                <s xml:id="N141DB" xml:space="preserve">Probatur facile ex concluſione: quia ſi ſic: iã mul­
                  <lb/>
                tiplex eſſet pars aliquota illius nõ multiplicis / vt
                  <lb/>
                conſtat / quod eſt contra concluſionem.</s>
              </p>
              <p xml:id="N141E2">
                <s xml:id="N141E3" xml:space="preserve">Secunda concluſio </s>
                <s xml:id="N141E6" xml:space="preserve">Nulla propor-
                  <lb/>
                tio multiplex eſt cõmenſurabilis alicui proportio­
                  <lb/>
                ni ſuperparticulari aut ſuprapartienti. </s>
                <s xml:id="N141ED" xml:space="preserve">Proba-
                  <lb/>
                tur / quoniam cuiuſlibet proportionis multiplicis
                  <lb/>
                vnitas eſt minimum extremum: igitur nulla ꝓpor­
                  <lb/>
                tio multiplex eſt cõmenſurabilis alicui proportio­
                  <lb/>
                ni ſuperparticulari aut ſuprapartienti. </s>
                <s xml:id="N141F8" xml:space="preserve">Antece-
                  <lb/>
                dens patet ex quinta ſuppoſitione: et conſequen-
                  <lb/>
                tia probatur / quia detur oppoſitum conſequētis:
                  <lb/>
                et ſit illa proportio ſuperparticularis aut ſuper-
                  <lb/>
                partiens b. et multiplex et commenſurabilis a. / et
                  <lb/>
                ſequitur /  aliqua proportio eſt pars aliquota ip­
                  <lb/>
                ſius b. et ipſius a. / vt patet ex ſecunda ſuppoſitio-
                  <lb/>
                ne: ſit igitur illa proportio que eſt pars aliquota
                  <lb/>
                c. / et arguit̄̄ ſic / c. ē pars aliq̊ta ipſius a. / igr̄ a. ex ali­
                  <lb/>
                quot c. proportionibus adequate componitur.</s>
              </p>
              <p xml:id="N1420D">
                <s xml:id="N1420E" xml:space="preserve">Patet hec conſequentia ex definitione partis ali­
                  <lb/>
                quote: et vltra ex aliquot proportionibus c. ade-
                  <lb/>
                quate componitur: ergo altera illarum c. propor­
                  <lb/>
                tionum eſt alicuius termini ītermedii ad minimū
                  <lb/>
                extremum ipſius proportionis a. </s>
                <s xml:id="N14219" xml:space="preserve">Patet hec con­
                  <lb/>
                ſequentia ex tertia ſuppoſitione. </s>
                <s xml:id="N1421E" xml:space="preserve">et c. non eſt ꝓpor­
                  <lb/>
                tio multiplex / vt conſtat: cum ſit pars aliquota ꝓ-
                  <lb/>
                portionis qualibet multiplice minoris. </s>
                <s xml:id="N14225" xml:space="preserve">ergo ſeq̇-
                  <lb/>
                tur /  minimum extremum talis ꝓportionis c. nõ
                  <lb/>
                eſt vnitas: et illud minimum extremum proportio­
                  <lb/>
                nis .c. eſt minimum extremum proportionis a. / igi­
                  <lb/>
                tur illud minimum extremum proportionis a. nõ
                  <lb/>
                eſt vnitas: et a. eſt multiplex per te: ergo non cuiuſ­
                  <lb/>
                libet multiplicis vnitas eſt minimum extremum /
                  <lb/>
                quod eſt oppoſitum antecedentis conſequentie ꝓ­
                  <lb/>
                bande et quinte ſuppoſitionis.</s>
              </p>
              <p xml:id="N14238">
                <s xml:id="N14239" xml:space="preserve">Tertia concluſio. </s>
                <s xml:id="N1423C" xml:space="preserve">Nulla proportio
                  <lb/>
                multiplex eſt commenſurabilis alicui multiplici
                  <lb/>
                ſuperparticulari aut multiplici ſuprapartienti.</s>
              </p>
              <p xml:id="N14243">
                <s xml:id="N14244" xml:space="preserve">Probatur: quia ſi aliqua proportio multiplex
                  <lb/>
                ſit commenſurabilis alicui proportioni multipli­
                  <lb/>
                ci ſuperparticulari: aut ſuprapartienti: aliqua ꝓ­
                  <lb/>
                portio eſſet pars aliquota vtriuſ puta multipli­
                  <lb/>
                cis, et multiplicis ſuperparticularis, vel multipli­
                  <lb/>
                cis ſuprapartientis que ſit c. / et arguo ſic / c. non eſt
                  <lb/>
                proportio multiplex / vt patet ex prima concluſio-
                  <lb/>
                ne huius: nec eſt ſuperparticularis: aut ſuprapar­
                  <lb/>
                tiens vt patet ex ſecunda: igitur erit multiplex ſu­
                  <lb/>
                perparticularis, aut multiplex ſuprapartiens: ſꝫ
                  <lb/>
                hoc eſt falſum / igitur c. non eſt pars aliquota pro­ </s>
              </p>
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