Alvarus, Thomas, Liber de triplici motu, 1509

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                  <s xml:id="N216E8" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0179" n="179"/>
                  inuicem approximari: et tūc tale condenſaret̄̄: igi-
                    <lb/>
                  tur non eſſet ante illam approximationem puncto­
                    <lb/>
                  rum infinite denſum. </s>
                  <s xml:id="N216F4" xml:space="preserve">Conſequentia patet et mi-
                    <lb/>
                  nor ꝓbatur. </s>
                  <s xml:id="N216F9" xml:space="preserve">q2 condenſari nihil aliud eſt ꝙ̄ puncta
                    <lb/>
                  approximari / vt patet ex deſcriptione cõdēſatiõis
                    <lb/>
                  </s>
                  <s xml:id="N216FF" xml:space="preserve">¶ Dices et bñ cõcedēdo ſeq̄lã et negãdo falſitatē cõ­
                    <lb/>
                  ſequētis: et ad ꝓbatiouē concedo /  pūcta illiꝰ cor-
                    <lb/>
                  poris poſſūt ad inuicē aproximari: et nego  tunc
                    <lb/>
                  condenſaretur tale corpus: et cū ꝓbat̄̄ /  ſic per dif­
                    <lb/>
                  finitionem condenſationis: dico /  non ſic deſcribi­
                    <lb/>
                  tur condēſatio. </s>
                  <s xml:id="N2170C" xml:space="preserve">Sed de hoc videbit̄̄ poſtea. </s>
                  <s xml:id="N2170F" xml:space="preserve">Si enim
                    <lb/>
                  alicuius pedalis prīa pars ꝓportionalis propor-
                    <lb/>
                  tione dupla aliq̇d cõtineat de materia: et ſecūda tm̄
                    <lb/>
                  de materia: et tertia tm̄: et ſic ↄ̨ñter. </s>
                  <s xml:id="N21718" xml:space="preserve">Ita  prima ſit
                    <lb/>
                  aliquãtulū denſa: ſecūda ī duplo dēſior: et tertia ī q̈­
                    <lb/>
                  druplo: et ſic cõſequēter: tūc cõſtat  tale corpꝰ ē īfi-
                    <lb/>
                  nite dēſū: et ſub pedali quantitate infinitam mate-
                    <lb/>
                  riam continet.</s>
                </p>
                <p xml:id="N2172D">
                  <s xml:id="N2172E" xml:space="preserve">Sꝫ ↄ̨̨tra / q2 ſi ſolutio eſſet a ſeq̄ret̄̄ / 
                    <lb/>
                  poſſet dari finitū īfinite dēſū vniformiter: ſꝫ ↄ̨ñs eſt
                    <lb/>
                  falſū: igr̄ ſolutio nulla. </s>
                  <s xml:id="N21735" xml:space="preserve">Seq̄la ꝓbat̄̄ / q2 tale corpus
                    <lb/>
                  de quo fit mētio in ſolntiõe eſt finitū īfinite dēſū dif­
                    <lb/>
                  formiter / vt dictis: igr̄ illud corpꝰ finitū p̄t reduci ad
                    <lb/>
                  vniformitatē: q̊ facto tale corpꝰ finitū eſſet īfinite dē­
                    <lb/>
                  ſū vniformiter: igit̄̄. </s>
                  <s xml:id="N21740" xml:space="preserve">Sꝫ iã ꝓbat̄̄ falſitas ↄ̨ñtis: q2 ſi
                    <lb/>
                  aliq̇d eſt finitum infinite dēſū vniformiter ſeq̇tur / 
                    <lb/>
                  prīa pars ꝓportionalis eſt ita denſa ſicut ſcḋa ade­
                    <lb/>
                  quate: et ſecunda ſicut tertia et tertia ſicut quarta / et
                    <lb/>
                  ſic ↄ̨ñter: et vltra prīa pars ꝓportiõalis eius eſt ita
                    <lb/>
                  dēſa ſicut ſcḋa adequate etc. / igit̄̄ ſecūda ī duplo mi­
                    <lb/>
                  nus continet de materia ꝙ̄ tertia: et ſic ↄ̨ñter: g̊ reſi­
                    <lb/>
                  duū ex oībus dēpta prīa habet tm̄ de materia ſicut
                    <lb/>
                  prima: ſꝫ materia prime eſt finita: igit̄̄ materia to-
                    <lb/>
                  tius corporis ē finita: et quãtitas ſimiliter finita: igr̄
                    <lb/>
                  totū corpꝰ ē finite denſū. </s>
                  <s xml:id="N21757" xml:space="preserve">et ſic nõ eſt vniformiter īfini­
                    <lb/>
                  te dēſū / qḋ fuit ꝓbandū. </s>
                  <s xml:id="N2175C" xml:space="preserve">Et ſi dicas /  ſecūda ꝑs pro­
                    <lb/>
                  portionalis continet tãtã materiã ſicut prīa et q̄lib3
                    <lb/>
                  ſequens ſimiliter quia īfinitã: iã ſeq̇t̄̄ /  ad quodlib3
                    <lb/>
                  pūctū talis corporis ē materia īfinita: et  ē penetra­
                    <lb/>
                  tio dimenſionū vel  materia ṗme ꝑtis ꝓportiona­
                    <lb/>
                  lis ē reducta ad nõ quãtū: et ſiĺr materia ſcḋe. </s>
                  <s xml:id="N21769" xml:space="preserve">et ter-
                    <lb/>
                  tie / et ſic ↄ̨ñter: et ꝑ ↄ̨ñs totū illud corpꝰ erit reductum
                    <lb/>
                  ad nõ quãtū et ſic nõ erit finitū īfinite dēſū vniformi­
                    <lb/>
                  ter / qḋ fuerat demonſtrãdū. </s>
                  <s xml:id="N21772" xml:space="preserve">¶ Cõfirmat̄̄ ſcḋo </s>
                  <s xml:id="N21775" xml:space="preserve">Q2 ſi ra­
                    <lb/>
                  ritas eēt poſſibilis: ēt poſſibilis eēt raritas īfinita
                    <lb/>
                  ī ſubiecto finito: ſꝫ ↄ̨ñs eſt falſū. </s>
                  <s xml:id="N2177C" xml:space="preserve">igr̄ illud ex quo ſeq̇­
                    <lb/>
                  tur. </s>
                  <s xml:id="N21781" xml:space="preserve">Seq̄la apparet et falſitas ↄ̨ñtis deducir̄: q2 vel
                    <lb/>
                  tale ſubiectū finitū cõtinet infinitã materiã vel fini-
                    <lb/>
                  tã ſi infinitã iã illud nõ ē rarū: et ꝑ ↄ̨ñs nõ ē īfinite ra­
                    <lb/>
                  rū. </s>
                  <s xml:id="N2178A" xml:space="preserve">Si finitã vel igr̄ cõtinet tãtã quantã vnū aliḋ ſub­
                    <lb/>
                  ieetū eq̈le illi finite rarū vel maiorē vel minorē. </s>
                  <s xml:id="N2178F" xml:space="preserve">Si
                    <lb/>
                  tantã ſeq̇t̄̄ /  illa ſubiecta ſūt eq̄ rara: et vnū ē finite
                    <lb/>
                  raꝝ. </s>
                  <s xml:id="N21796" xml:space="preserve">ir̄ et aliud. </s>
                  <s xml:id="N21799" xml:space="preserve">Si maiorē iã ſeq̇t̄̄ /  hoc nõ eſt ita ra­
                    <lb/>
                  rū. </s>
                  <s xml:id="N2179E" xml:space="preserve">Si minorē cū nõ ſit poſſibile  aliq̈ materia ſit ī­
                    <lb/>
                  finite modica ſeq̇t̄̄ /  ī aliq̈ ꝓportiõe materiã mino-
                    <lb/>
                  rē cõtinebit et ſic in eadē ꝓportiõe erit magꝪ rarū et
                    <lb/>
                  ꝑ ↄ̨ñs nõ erit īfinite rarū / quod fuit ꝓbandum.</s>
                </p>
                <p xml:id="N217A7">
                  <s xml:id="N217A8" xml:space="preserve">Septīo prīcipaliṫ argr̄ ſic īq̇rēdo ma­
                    <lb/>
                  teriam de raritate et dēſitate difformi. </s>
                  <s xml:id="N217AD" xml:space="preserve">q2 ſi raritas
                    <lb/>
                  et dēſitas eſſent poſſibiles ſeq̄ret̄̄ /  pedale cuius pri­
                    <lb/>
                  ma ꝑs ꝓportionalis ꝓportione dupla eſſet aliquã­
                    <lb/>
                  tulū rara et ſecunda in duplo rarior ꝙ̄ prīa: et tertia
                    <lb/>
                  ī duplo rarior ꝙ̄ ſcḋa et q̈rta in duplo rarior ꝙ̄ ter­
                    <lb/>
                  tia: et ſic ↄ̨ñter eſſet infinite rarū: ſed ↄ̨ñs eſt flm̄: igit̄̄
                    <lb/>
                  illud ex q̊ ſeq̇tur </s>
                  <s xml:id="N217BC" xml:space="preserve">Seq̄lã ꝓbat̄̄ / q2 raritas prīe ꝑtis ꝓ­
                    <lb/>
                  portiõalis illiꝰ corꝑis denoīat totale corpꝰ aliquã­
                    <lb/>
                  tū rarū et raritas ſcḋe ꝑtis ꝓportionalis tm̄ deno-
                    <lb/>
                  minat et raritas tertie ꝑtis: ſiĺr / et ſic ↄ̨ñter: igit̄̄ ibi
                    <cb chead="Capitulum tertium"/>
                  ſūt īfinite denoīatiões eq̈les nõ cõicãtes illud corpꝰ
                    <lb/>
                  denoīantes: igit̄̄ illud corpꝰ ē īfinite raꝝ. </s>
                  <s xml:id="N217CA" xml:space="preserve">Añs pꝫ / q2
                    <lb/>
                  raritas ſcḋe ꝑtis eſt in ſubduplo ſubiecto: et ī duplo
                    <lb/>
                  maior ꝙ̄ prime ꝑtis raritas: igr̄ tm̄ denoīat totale
                    <lb/>
                  corpꝰ ſicut raritas prīe partis et eadē rõne raritas
                    <lb/>
                  tertie tm̄ ſicut raritas ſcḋe / et ſic ↄ̨ñter: igt̄̄ intētū </s>
                  <s xml:id="N217D5" xml:space="preserve">Sꝫ
                    <lb/>
                  falſitas ↄ̨ñtis ꝓbat̄̄: q2 illud corpꝰ pedale ſub finita
                    <lb/>
                  quãtitate cõtinet aliquãtã materiã: igr̄ nõ ē īfinite
                    <lb/>
                  rarū. </s>
                  <s xml:id="N217DE" xml:space="preserve">itē illud pedale ē aliq̈liṫ denſū: igr̄ nõ ē īfinite
                    <lb/>
                  raꝝ. </s>
                  <s xml:id="N217E3" xml:space="preserve">Coña pꝫ et arguit̄̄ añs / q2 prīa ꝑs ꝓportiõalis
                    <lb/>
                  illiꝰ pedalis eſt aliq̈liṫ denſa: et ſcḋa in duplo minꝰ
                    <lb/>
                  et tertia ī duplo minꝰ ꝙ̄ ſcḋa: et ſic ↄ̨ñter: igr̄ prima
                    <lb/>
                  ꝑs ꝓportionalis cõtinet aliquãtã materiã et ſcḋa in
                    <lb/>
                  q̈druplo minorē: et tertia in q̈druplo minorē ꝙ̄ ſcḋa /
                    <lb/>
                  et ſic ↄ̨ñter: igit̄̄ aggregatū ex illis oībꝰ materiebꝰ
                    <lb/>
                  dēpta mã prīe ꝑtis eſt ſubtriplū ad materiaꝫ prīe
                    <lb/>
                  ꝑtis ſed materia prime ꝑtis eſt vt tria (vt ſuppono) /
                    <lb/>
                  igit̄̄ tota materia illiꝰ corꝑis pedalis eſt vt q̈tuor: et
                    <lb/>
                  ꝑ ↄ̨ñs illud corpus eſt ita dēſū adeq̈te ſicut vnū aliḋ
                    <lb/>
                  pedale vniformite qḋ hꝫ q̈tuor gradꝰ materie / qḋ fuit
                    <lb/>
                  ꝓbãdū.
                    <note position="right" xlink:href="note-0179-01a" xlink:label="note-0179-01" xml:id="N218C4" xml:space="preserve">.1. confir.</note>
                  </s>
                  <s xml:id="N21801" xml:space="preserve">Et ↄ̨firmat̄̄ </s>
                  <s xml:id="N21804" xml:space="preserve">Et capio vnū corpꝰ cuiꝰ prīa ꝑs
                    <lb/>
                  ꝓportiõalis ꝓportiõe dupla ſit aliquãtulum rara
                    <lb/>
                  vniformitet puta vt duo: et ſecūda in duplo minus
                    <lb/>
                  et tertia in duplo minus ꝙ̄ ſcḋa / et ſic ↄ̨ñter ſequitur /
                    <lb/>
                   illud corpus eſſet rarum et nõ eſſet rarum: ſed cõ-
                    <lb/>
                  ſequens implicat: igit̄̄ et q̄ſtio </s>
                  <s xml:id="N21811" xml:space="preserve">Sequela ꝓbatur / q2
                    <lb/>
                  illud eſt rarū vt vnū cuꝫ vna tertia: igr̄ illud eſt raꝝ
                    <lb/>
                  </s>
                  <s xml:id="N21817" xml:space="preserve">Añs ꝓbatur / q2 ſi eſſet vnum corpus cuius prīa pro­
                    <lb/>
                  portionalis ꝓportione dupla eēt intenſa vt duo: et
                    <lb/>
                  ſecunda in duplo minus. </s>
                  <s xml:id="N2181E" xml:space="preserve">et tertia in duplo minus ̄
                    <lb/>
                  ſecunda / et ſic couſequenter. </s>
                  <s xml:id="N21823" xml:space="preserve">totū eēt intenſū vt vnuꝫ
                    <lb/>
                  cū vna tertia / vt ꝓbabitur infra. de intenſione: igit̄̄
                    <lb/>
                  pari ratione illud corpꝰ cuiꝰ vna ꝑs ꝓportionalis
                    <lb/>
                  ꝓportione dupla eſt rara vt duo: et ſcḋa in duplo
                    <lb/>
                  minus et tertia in duplo minus ꝙ̄ ſcḋa / et ſic cõſequē­
                    <lb/>
                  ter eſt rarū vt vnū cū vna tertia / quod fuit ꝓbanduꝫ
                    <lb/>
                  </s>
                  <s xml:id="N21831" xml:space="preserve">Sed  nõ ſit rarū ꝓbat̄̄ / q2 eſt infinite denſū: g̊ nõ eſt
                    <lb/>
                  rarum antecedens ꝓbatur / q2 ſub finita quantitate
                    <lb/>
                  infinitam materiam continet / quod probatur / q2 q̄-
                    <lb/>
                  libet pars proportionalis continet tantum de ma­
                    <lb/>
                  teria ſicut prima: ergo tota materia illius totiꝰ eſt
                    <lb/>
                  infinita añs ꝓbatur / q2 cū ſecunda pars ꝓportiõa-
                    <lb/>
                  lis eſt in duplo minus rara ꝙ̄ prīa ipſa eſt in duplo
                    <lb/>
                  denſior ꝙ̄ prīa et eſt in duplo minor: g̊ tm̄ cõtinet de
                    <lb/>
                  materia adeq̈te quãtã cõtinet prīa. </s>
                  <s xml:id="N21844" xml:space="preserve">Coña ptꝫ / q2 ſi ſe­
                    <lb/>
                  cūda eēt eq̄ dēſa cū prīa in duplo minorē materiaꝫ
                    <lb/>
                  cõtiueret ꝙ̄ prīa / vt patet: ergo cū modo ſit ī duplo
                    <lb/>
                  denſior ꝙ̄ tunc eſſet mõ ſub eadē quãtitate in duplo
                    <lb/>
                  maiorē materiã cõtinet ꝙ̄ tunc contineret. </s>
                  <s xml:id="N2184F" xml:space="preserve">Et eodē°
                    <lb/>
                  ꝓbabis /  tertia tãtã materiã cõtinet ſicut ſecūda et
                    <lb/>
                  q̈rta ſicut tertia et ſic ī iufinitū: et ſic pꝫ /  iliud conti­
                    <lb/>
                  net infinitã materiã ſub finita quãtitate / qḋ fuit pro­
                    <lb/>
                  bãdū.
                    <note position="right" xlink:href="note-0179-02a" xlink:label="note-0179-02" xml:id="N218CA" xml:space="preserve">2. confir.</note>
                  </s>
                  <s xml:id="N2185F" xml:space="preserve">¶ Cõfirmaṫ ſcḋo </s>
                  <s xml:id="N21862" xml:space="preserve">Et capio vnū pedale cuiꝰ pri­
                    <lb/>
                  ma ꝑs ꝓportiõalis ꝓportione decupla ſit dēſa ali­
                    <lb/>
                  qualiter et ſcḋa ī duplo magis: et tertia ī duplo ma­
                    <lb/>
                  gis ꝙ̄ ſcḋa et quarta in duplo magis ꝙ̄ tertia: et ſic
                    <lb/>
                  couſequenter: et ſic arguo ſequeretur ex queſtiõe 
                    <lb/>
                  illud corpus eſſet infinite denſum: ſed conſequēs eſt
                    <lb/>
                  falſum: igitur illud ex quo ſequitur. </s>
                  <s xml:id="N21871" xml:space="preserve">Sequela pro-
                    <lb/>
                  batur / quia ſi alicuius corporis diuiſi per partes ꝓ-
                    <lb/>
                  portionales propoſitione dupla prima pars ꝓpor­
                    <lb/>
                  tionalis ſit aliquantulum denſa: et ſecunda in du-
                    <lb/>
                  plo denſior: et tertia in duplo denſior ꝙ̄ ſecun-
                    <lb/>
                  da: et quarta in duplo denſior ꝙ̄ tertia: et ſic conſe-
                    <lb/>
                  quenter: totum illud corpus eſt infinite denſum cuꝫ
                    <lb/>
                  contineat ſub finita quantitate infinitam materi-
                    <lb/>
                  am / vt probatum eſt in confirmatione ſuperiori:
                    <lb/>
                  igitur pari ratione etiam corpus diuiſum per par­
                    <lb/>
                  tes ꝓportionales ꝓportione decupla cuius prima </s>
                </p>
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