Alvarus, Thomas, Liber de triplici motu, 1509

Page concordance

< >
Scan Original
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
< >
page |< < of 290 > >|
    <echo version="1.0">
      <text xml:lang="la">
        <div xml:id="N10132" level="1" n="1" type="body">
          <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="N1484D">
                <s xml:id="N148E4" xml:space="preserve">
                  <pb chead="Secūde partis" file="0047" n="47"/>
                nominatnr data ꝓportio multiplex: et ſi ſic iã in-
                  <lb/>
                ter terminos eius computatis extremis reperiren­
                  <lb/>
                tur tot numeri continuo ꝓportionabiles quotus
                  <lb/>
                eſt numerus a quo denominatur dicta proportio
                  <lb/>
                multiplex: puta quoties a. cõtinet b. vno plus. </s>
                <s xml:id="N148FE" xml:space="preserve">igi­
                  <lb/>
                tur ex oppoſito: ſi non reperiantur tot numeri cõ-
                  <lb/>
                putatis extremis iam a. non ſe habet in tali ꝓpor­
                  <lb/>
                tione multiplici ad b. ꝓportionem rationalem.</s>
              </p>
              <note position="left" xml:id="N14907" xml:space="preserve">nota.</note>
              <p xml:id="N1490B">
                <s xml:id="N1490C" xml:space="preserve">¶ Utrum autē inter aliquos numeros date ꝓpor­
                  <lb/>
                tionis a. reperiantur tot numeri continuo ꝓpor-
                  <lb/>
                tionabiles computatis extremis vno plus quotꝰ
                  <lb/>
                eſt numerus a quo denominatur proportio multi­
                  <lb/>
                plex in qua ponitur a. ſe habere ad b. videndū eſt
                  <lb/>
                vtrum inter primos numeros eius inueniant̄̄ tot
                  <lb/>
                numeri continuo proportionabiles: et ſi ſic conclu­
                  <lb/>
                das /  inter numeros ipſius a. reperiuntur tot nu­
                  <lb/>
                meri continuo ꝓportionabiles: et ſi non inuenian­
                  <lb/>
                tur tot inter primos numeros date ꝓportionis:
                  <lb/>
                dicas /  inter nullos numeros eius reperiunt̄̄ tot
                  <lb/>
                numeri continuo ꝓportinoabiles computatis ex­
                  <lb/>
                tremis. </s>
                <s xml:id="N14927" xml:space="preserve">Patet hec conſequentia / et deductio tota
                  <lb/>
                ex octaua ꝓpoſitione octaui elementorum eucli-
                  <lb/>
                dis in qua habetur /  ſi inter duos numeros ceci-
                  <lb/>
                derint aliqui numeri continuo ꝓportionabiles:
                  <lb/>
                inter quoſcun duos in eadem ꝓportione ſe ha-
                  <lb/>
                bentes cadent tot numeri continuo ꝓportionabi­
                  <lb/>
                les eadem ꝓportione qua ꝓportionautur alii. </s>
                <s xml:id="N14936" xml:space="preserve">ex
                  <lb/>
                qua immediate infertur /  ſi inter duos numeros
                  <lb/>
                ſe habentes in ꝓportio a. ceciderint aliqui nume-
                  <lb/>
                ri continuo ꝓportionabiles ꝓportiõe que eſt vna
                  <lb/>
                tertia: aut vna quarta: aut vna quinta: ipſius a. in­
                  <lb/>
                ter primos numeros ipſius a. tot numeri cadēt ꝓ­
                  <lb/>
                portionabiles eadeꝫ ꝓportione que ſit tertia aut
                  <lb/>
                quarta: aut quinta ipſius a. / igitur ex oppoſito cõ­
                  <lb/>
                ſequentis ſi inter primos numeros a. proportio-
                  <lb/>
                nis non reperiantur aliqui numeri continuo pro­
                  <lb/>
                portionabiles ꝓportione que eſt vna tertia: vna
                  <lb/>
                quarta: quinta: ipſius a. et c. nec inter aliquos nūe­
                  <lb/>
                ros ipſius a. reperientur: quod fuit oſtendendum:
                  <lb/>
                </s>
                <s xml:id="N14952" xml:space="preserve">Et ſic patet concluſio.
                  <note position="left" xlink:href="note-0047-01a" xlink:label="note-0047-01" xml:id="N14B1B" xml:space="preserve">1. correl.</note>
                </s>
                <s xml:id="N1495A" xml:space="preserve">¶ Ex quo ſequitur primo. / 
                  <lb/>
                ꝓportio dupla ad nullam ꝓportionem rationa-
                  <lb/>
                lem ſe habet in ꝓportione dupla: aut tripla. aut
                  <lb/>
                quadrupla: aut in aliqua alia multiplici: nec quin­
                  <lb/>
                tupla, nec ſextupla etc. </s>
                <s xml:id="N14965" xml:space="preserve">Probatur / quia inter pri-
                  <lb/>
                mos numeros ꝓportionis duple nullus numerus
                  <lb/>
                reperitur (computamus enim vnitatem pro nume­
                  <lb/>
                ro). </s>
                <s xml:id="N1496E" xml:space="preserve">Item inter primos numeros proportionis
                  <lb/>
                quintuple qui ſunt .5. et .1. non reperiuntur aliqui
                  <lb/>
                numeri continuo ꝓportionabiles adequate com­
                  <lb/>
                putatis extremis / vt conſtat. </s>
                <s xml:id="N14977" xml:space="preserve">Et ſic patet etiam de
                  <lb/>
                ſextupla. </s>
                <s xml:id="N1497C" xml:space="preserve">Patet igitur correlarium.
                  <note position="left" xlink:href="note-0047-02a" xlink:label="note-0047-02" xml:id="N14B21" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N14984" xml:space="preserve">¶ Sequitur
                  <lb/>
                ſecundo /  nulla ꝓportio ſuperparticularis ſe ha­
                  <lb/>
                bet in aliqua ꝓportione multiplici ad aliquam ꝓ­
                  <lb/>
                portionem rationalem. </s>
                <s xml:id="N1498D" xml:space="preserve">Patet / quia inter cuiuſli­
                  <lb/>
                bet ſuperparticularis primos terminos nullꝰ re-
                  <lb/>
                peritur numerus: igitur.
                  <note position="left" xlink:href="note-0047-03a" xlink:label="note-0047-03" xml:id="N14B27" xml:space="preserve">3. correl.</note>
                </s>
                <s xml:id="N14999" xml:space="preserve">¶ Sequitur tertio /  pro­
                  <lb/>
                poſita quauis proportione rationali inueſtigare
                  <lb/>
                poſſumus an habeat aliquam ꝓportionem ratio­
                  <lb/>
                nalem que ſe habeat ad ipſam in ꝓportione ſexq̇-
                  <lb/>
                altera: ſexquitertia: ſexquiquarta etc. / vt ꝓpoſita
                  <lb/>
                ꝓportione dupla: videre an ſit aliqua ꝓportio ra­
                  <lb/>
                tionalis que ſe habeat ad ipſam duplam in pro-
                  <lb/>
                portione ſexquialtera, ſexquitertia, aut in aliqua
                  <lb/>
                alia ſuperparticulari. </s>
                <s xml:id="N149AC" xml:space="preserve">Ad quod inueſtiganduꝫ et
                  <lb/>
                ſciendum videndum eſt an inter primos numeros
                  <lb/>
                ꝓportiouis duple aut cuiuſuis alterius rationa-
                  <lb/>
                lis ſint tres numeri continuo ꝓportionabiles cõ-
                  <lb/>
                putatis extremis: et ſi ſic: talis ꝓportio habet me­
                  <lb/>
                dietatem rationalem: et per conſequens ſexquial­
                  <cb chead="Capitulum ſextum"/>
                teram rationalem ad ipſam. </s>
                <s xml:id="N149BC" xml:space="preserve">Addendo enī et me-
                  <lb/>
                dietatem ſui conſtituetur ſexquialtera rationalis
                  <lb/>
                ad ipſaꝫ. </s>
                <s xml:id="N149C3" xml:space="preserve">Et ſi inter primos numeros eius compu­
                  <lb/>
                tatis extremis inueniantur quatuor numeri conti­
                  <lb/>
                nuo ꝓportionabiles: ipſa habebit tertiam ratio­
                  <lb/>
                nalem et per conſequens ſexquitertiam rationa-
                  <lb/>
                lem ad ſeipſam: et ſi reperiuntur .5. numeri conti-
                  <lb/>
                nuo ꝓportionabiles computatis extremis ip̄a ha­
                  <lb/>
                bebit quartam rationalem: et per conſequens ſex­
                  <lb/>
                quiquartam rationalem / et ſic conſequenter. </s>
                <s xml:id="N149D4" xml:space="preserve">Et
                  <lb/>
                ſic patet correlarium.
                  <note position="right" xlink:href="note-0047-04a" xlink:label="note-0047-04" xml:id="N14B2D" xml:space="preserve">4. correl.</note>
                </s>
                <s xml:id="N149DE" xml:space="preserve">¶ Sequitur quarto /  ꝓpo­
                  <lb/>
                ſita quauis ꝓportione rationali: inquirere et ſci-
                  <lb/>
                re poterimus an habeat aliquam ſuprapartien-
                  <lb/>
                tem, multiplicem ſuperparticulareꝫ, vel multipli­
                  <lb/>
                cem ſuprapartientem, rationales. </s>
                <s xml:id="N149E9" xml:space="preserve">vt ꝓpoſita pro­
                  <lb/>
                portione octupla īueſtigare poterimus et ſcire ex
                  <lb/>
                dictis an habeat ſuprabipartientem tertias ſu-
                  <lb/>
                prapartientem quartas rationales etc. </s>
                <s xml:id="N149F2" xml:space="preserve">Ad quod
                  <lb/>
                ſciendum et inueſtigandum: conſiderandum ē an
                  <lb/>
                data proportio rationalis habeat illam partem
                  <lb/>
                aliquotam rationalem: hoc eſt an aliqua propor­
                  <lb/>
                tio rationalis ſit tota pars aliquota eius quota
                  <lb/>
                eſt illa a qua denominatur dicta proportio ſupra­
                  <lb/>
                partiens, ant multiplex ſuperparticularis, aut
                  <lb/>
                multiplex ſuprapartiens: quod inueſtigari et ſciri
                  <lb/>
                debet ex vndecima concluſione: et ſi repperias / 
                  <lb/>
                habet proportionem aliquam rationalem que ſit
                  <lb/>
                talis pars aliquota eius: tunc manifeſtum ē /  ha­
                  <lb/>
                bet proportionem rationalem que denominatur
                  <lb/>
                a tali parte aliquota vel talibus partibus aliquo­
                  <lb/>
                tis (quod dico ꝓpter ſuprapartientes) ſi vero nõ:
                  <lb/>
                tunc manifeſtum eſt illam proportionem rationa­
                  <lb/>
                lem propoſitam non habere proportionem ratio­
                  <lb/>
                nalem denominatam a tali parte aliquota vel ta­
                  <lb/>
                libus partibus. </s>
                <s xml:id="N14A17" xml:space="preserve">Probatur hoc demonſtratione
                  <lb/>
                particulari que equiualebit vniuerſali. </s>
                <s xml:id="N14A1C" xml:space="preserve">Data em̄
                  <lb/>
                ꝓportione ſexdecupla volo inueſtigare et ſcire an
                  <lb/>
                habeat proportionem ſupratripartientem quar-
                  <lb/>
                tas ad quod inueſtigandum conſiderabo ex doc-
                  <lb/>
                trina vndecime concluſionis an talis ꝓportio ſex­
                  <lb/>
                decupla habeat ſubquadruplam rationaleꝫ que
                  <lb/>
                ſit vna quarta eius: et inuento  ſic eo /  inter ter­
                  <lb/>
                minos eius computatis extremis inueniuntur
                  <lb/>
                quin numeri continuo ꝓportionabiles ꝓportio­
                  <lb/>
                ne dupla: aſſeuerabo conſtanter illam proportio­
                  <lb/>
                nem habere proportionem rationalem ſupertri-
                  <lb/>
                partientem quartas: et multiplicem ſexquiquar-
                  <lb/>
                tam et multiplicem ſupratripartientem quartas
                  <lb/>
                rationales. </s>
                <s xml:id="N14A39" xml:space="preserve">Quod ſic monſtratur </s>
                <s xml:id="N14A3C" xml:space="preserve">Nam ſi ſupra il­
                  <lb/>
                lam proportionem ſexdecuplam que eſt .16. ad .1.
                  <lb/>
                addantur tres proportiones duple: tunc aggre-
                  <lb/>
                gatum ex ſexdecupla et illis tribus duplis ſuꝑ ad­
                  <lb/>
                ditis qualis eſt proportio .128. ad .1. ſe habebit ad
                  <lb/>
                proportionem ſexdecuplam in proportiõe ſupra-
                  <lb/>
                tripartiente quartas. </s>
                <s xml:id="N14A4B" xml:space="preserve">Continet enim ſexdecu-
                  <lb/>
                plam et tres quartas eius. </s>
                <s xml:id="N14A50" xml:space="preserve">Item triplando illam
                  <lb/>
                proportionem ſexdecuplam / et addendo vnam ſui
                  <lb/>
                quartam habebis ꝓportionem triplam ſexquiq̈r­
                  <lb/>
                tam ad ſexdecuplam: et addendo ei duas quartas
                  <lb/>
                habebis triplam ſexquialteram: et addendo ſuꝑ
                  <lb/>
                illam triplatam .3. quartas habebis triplam ſu-
                  <lb/>
                pratripartientem quartas rationalem ad ſexde-
                  <lb/>
                cuplam. </s>
                <s xml:id="N14A61" xml:space="preserve">Omnia iſta patet ex diffinitionibus ſu-
                  <lb/>
                prapartiētis, multiplicis ſuperparticularis. </s>
                <s xml:id="N14A66" xml:space="preserve">aut
                  <lb/>
                multiplicis ſuprapartientis. </s>
                <s xml:id="N14A6B" xml:space="preserve">hoc addito /  cuili-
                  <lb/>
                bet proportioni rationali addi poteſt queuis alia
                  <lb/>
                rationalis: aggregato ex ipſis manente rationa­
                  <lb/>
                li proportione. </s>
                <s xml:id="N14A74" xml:space="preserve">Ex quibuſcnn enim rationalibꝰ
                  <lb/>
                et quotcun: rationalis componitur: q2 alias in </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum