Alvarus, Thomas, Liber de triplici motu, 1509

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                <pb chead="Secūdi. De motu locali quo ad effectū ſcḋm tempus difformi." file="0160" n="160"/>
                <p xml:id="N1F907">
                  <s xml:id="N1F908" xml:space="preserve">¶ Sequitur ſecūdo /  diuiſa hora ꝑ partes ꝓpor-
                    <lb/>
                  tionales ꝓportiõe ſuꝑtripartienti quartas cuiuſli­
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                  bet partis velocitate ſe habente ad velocitatē par-
                    <lb/>
                  tis maioris īmediate precedentis in ꝓportione ſex­
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                  quialtera ſpaciū pertranſitū in tota hora ſe habet
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                  ad ſpaciū pertranſitū in prima parte proportionali
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                  in ꝓportione ſeptupla: abſoluto pedali in prima
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                  parte: ſeptē pedalia in tota hora abſoluētur. </s>
                  <s xml:id="N1F919" xml:space="preserve">Pro­
                    <lb/>
                  batur hoc correlariū ex cõcluſione īmediate precedē­
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                  ti: quia partes ꝓportionales tēporis ſe habent cõ-
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                  tinuo in ꝓportione ſuꝑtripartienti quartas: et ve-
                    <lb/>
                  locitates partiū īmediatarū ſe habent in ꝓportiõe
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                  ſexq̇altera / vt ponit caſus: et ꝓportio ſuꝑtripartiēs
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                  q̈rtas excedit ꝓportionē ſexquialteram ꝑ .4. ꝓpor-
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                  tionē ſexquiſextaꝫ / vt ptꝫ in his terminis .7.6.4. / igr̄
                    <lb/>
                  ſpaciū ꝑtranſitū in toto tēpore ſe habebit ad ſpa-
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                  cium pertranſitū in prima parte proportionali in
                    <lb/>
                  ꝓportione ſeptupla / quod fuit ꝓbandū. </s>
                  <s xml:id="N1F930" xml:space="preserve">Patet cõ-
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                  ſequentia ex cõcluſione ſeptima: hoc adiecto /  cor­
                    <lb/>
                  pus diuiſum proportione ſexquiſexta ſe habet ad
                    <lb/>
                  primã ſui partē in ꝓportione ſeptupla: vt patet ex
                    <lb/>
                  prima parte huiꝰ operis. </s>
                  <s xml:id="N1F93B" xml:space="preserve">Familiarius tamen ꝓba­
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                  tur ſic, et ſuppono /  mobile ꝑtranſit in prima par­
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                  te proportionali vnum pedale, et arguo ſic / mobile
                    <lb/>
                  pertranſit in prima parte proportionali vnum pe­
                    <lb/>
                  dale: et in ſecunda in ſexquiſexto minus, et in tertia
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                  in ſexquiſexto minꝰ ꝙ̄ in ſecunda: et ſic conſequēter
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                  ꝓcedendo per ꝓportiones ſexquiſextas: igitur to-
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                  tale ſpaciū componitur ex illis infinitis continuo
                    <lb/>
                  ſe habentibus in ꝓportione ſexquiſexta: ergo ag-
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                  gregatū ex omnibꝰ ſequētibus primã eſt ſextupluꝫ
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                  ad primū / vt ptꝫ ex prima parte huiꝰ operis capite
                    <lb/>
                  quinto: et primū eſt vnū pedale: ergo totū reſiduum
                    <lb/>
                  eſt ſextupedale, et ꝑ cõſequens totū ſpaciū eſt ſeptē
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                  pedū / quod ſe habet in proportiõe ſeptupla ad vnū
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                  pedale ꝑtranſitū in prima parte ꝓportiõali / quod
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                  fuit ꝓbandum. </s>
                  <s xml:id="N1F95C" xml:space="preserve">Probatur tamen antecedens vide­
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                  licet /  illud mobile in qualibet parte ſequenti per­
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                  tranſit ſubſexquiſextū ſpaciū ad ſpaciū ꝑtranſitū
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                  in īmediate p̄cedenti, quia prime partis ꝓportio-
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                  nalis ad ſecundã eſt ꝓportio ſuꝑtripartiens quar-
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                  tas, et velocitatis ſecunde partis ꝓportionalis ad
                    <lb/>
                  velocitatē prime eſt ꝓportio ſexquialtera: ſed ꝓpor­
                    <lb/>
                  tio ſuꝑtripartiens quartas tempoꝝ excedit ꝓpor-
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                  tionē velocitatū ſexquialterã per ꝓportionē ſexqni­
                    <lb/>
                  ſextam / vt notū eſt: igitur ſpaciū pertranſitū in ſe-
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                  cunda parte ꝓportionali eſt ſubſexquiſextū ad ſpa­
                    <lb/>
                  cium pertranſitū in prima. </s>
                  <s xml:id="N1F975" xml:space="preserve">Patet conſequentia / ex
                    <lb/>
                  ſexta ꝓpoſitione ſecundi notabilis ſepius allega-
                    <lb/>
                  ta. </s>
                  <s xml:id="N1F97C" xml:space="preserve">Et ſic ꝓbabis de ſpacio pertranſito in tertia ad
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                  ſpacium ꝑtranſitum in ſecunda, et de ſpaciis ꝑtrã-
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                  ſitis in duabus partibus īmmediatis quibuſcun
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                  ſignatis: ergo cõtinuo ſpaciū pertranſitū in aliqua
                    <lb/>
                  parte ꝓportionali ſequente eſt ſubſexquiſextū ad
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                  ſpacium ꝑtranſiium in parte īmediate precedente:
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                  quod fuit ꝓbanduꝫ. </s>
                  <s xml:id="N1F98B" xml:space="preserve">Inferas tuo ingenio et labore
                    <lb/>
                  ſimilia infinita correlaria. </s>
                  <s xml:id="N1F990" xml:space="preserve">Iſta enim ſufficiūt pro
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                  praxi concluſionis.</s>
                </p>
                <p xml:id="N1F995">
                  <s xml:id="N1F996" xml:space="preserve">Octaua cõcluſio. </s>
                  <s xml:id="N1F999" xml:space="preserve">Partita hora ꝑ part­
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                  tes ꝓportionales quauis ꝓportione volueris, et in
                    <lb/>
                  certa ꝓportione continuo velocius mobile moueat̄̄
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                  in parte p̄cedente maiore quã in īmediate ſequenti
                    <lb/>
                  minori: ſpaciū ꝑtranſitum in totali hora ſe habe-
                    <lb/>
                  bit ad ſpaciū ꝑtranſitum in prima parte ꝓportio-
                    <lb/>
                  nali in ꝓportione qua ſe habet aliquod totū diui-
                    <lb/>
                  ſum in partes ꝓportionales ꝓportione compoſi-
                    <lb/>
                  ta ex proportione temporis puta partis propor-
                    <lb/>
                  tionalis maioris ad partem immediate ſequenteꝫ
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                  minorem, et velocitatis partis maioris ad veloci-
                    <cb chead="Secūdi. De motu locali quo ad effectū ſcḋm tempus difformi."/>
                  tatem partis minoris ad primam partem prpor-
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                  tionalem talis diuiſionis. </s>
                  <s xml:id="N1F9B5" xml:space="preserve">Hoc inuolutum theo-
                    <lb/>
                  rema exemplari declaratione reſoluatur: volo em̄
                    <lb/>
                  dicere /  conſciſa hora per partes proportionales
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                  proportione dupla, et in prima parte proportiõa-
                    <lb/>
                  li aliquod mobile moueatur aliquanta velocitate
                    <lb/>
                  q̄ in ſecunda parte proportionali in ſexquialtero
                    <lb/>
                  minori velocitate, et in tertia in ſexquialtero mīor
                    <lb/>
                  velocitate quã in ſecūda, et ſic cõſequēter </s>
                  <s xml:id="N1F9C6" xml:space="preserve">ita  cu-
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                  iuſlꝫ ꝑtis p̄cedētis maioris velocitas ad velocitatē
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                  mīoris īmediate ſequētꝪ ſexq̇alterã ꝓportionē ha-
                    <lb/>
                  beat: tūc dicit theorema poſitū. </s>
                  <s xml:id="N1F9CF" xml:space="preserve">ſpaciū ꝑtranſitū in
                    <lb/>
                  totali hora ſe habere ad ſpaciū ꝑtranſitū in prima
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                  parte proportionali in proportione ſequialtera:
                    <lb/>
                  qm̄ proportio compoſita ex proportione dupla tē­
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                  porum et ſexquialtera velocitatū eſt tripla: et quod­
                    <lb/>
                  libet totū diuiſum per partes proportione tripla
                    <lb/>
                  ſe habet ad primã proportionalem partem eius in
                    <lb/>
                  proportione ſexquialtera. </s>
                  <s xml:id="N1F9E0" xml:space="preserve">Probatur tamen vni-
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                  uerſaliter cõcluſio: ſit hora diuiſa per partes pro-
                    <lb/>
                  portionales portiõe g. et moueatur mobile in ali-
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                  qua certa proportione velocius continuo in parte
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                  p̄cedenti maiore quam in minore ſequente ita  cõ­
                    <lb/>
                  tinuo maior velocitas ſit in parte maiori quam in
                    <lb/>
                  minore īmediate ſequente, ſit proportio cõtinuo
                    <lb/>
                  velocitatis partis maioris ad velocitatem partis
                    <lb/>
                  minoris f. compoſita proportio ex g. et f. ſit h. / tūc
                    <lb/>
                  ſpaciū ꝑtranſitum in totali hora ſe pabet ad ſpa-
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                  cium ꝑtranſitum in prima parte proportionali in
                    <lb/>
                  ꝓportione in qua ſe habet aliquod totum diuiſum
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                  in partes proportionales ꝓportione h. ad primaꝫ
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                  partem ꝓportionalē eiuſdem diuiſionis videlicet
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                  proportione h. </s>
                  <s xml:id="N1F9FF" xml:space="preserve">Quod probatur ſic / quia ſpacii ꝑ-
                    <lb/>
                  tranſiti in prima parte proportionali ad ſpacium
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                  ꝑtranſitū in ſecunda parte ꝓportionali eſt propor­
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                  tio h. et ſpacii ꝑtranſiti in ſecunda ad ſpaciū ꝑtran­
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                  ſiti in tertia eſt etiam proportio h. / et ſic conſequen-
                    <lb/>
                  ter de ſpaciis ꝑtranſitis in duabus partibus pro-
                    <lb/>
                  portionalibus īmediatis quibuſuis demonſtratis /
                    <lb/>
                  ergo totale ſpaciū ꝑtrãſitū in tota hora componit̄̄
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                  ex infinitis continuo ſe habentibus in proportione
                    <lb/>
                  h. / igitur totale ſpaciū ſe habet ad primū illoꝝ ſpa­
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                  ciorum / quod eſt ꝑtranſitū in prima parte propor-
                    <lb/>
                  tionali in proportiõe in qua ſe habet aliquod totū
                    <lb/>
                  diuiſum ꝑ partes ꝓportionales proportione h. ad
                    <lb/>
                  primã eius partē / quod fuit probandū. </s>
                  <s xml:id="N1FA1C" xml:space="preserve">Patet con-
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                  ſequentia / quia eodem modo ſe habent illa ſpacia
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                  continuo ſe habentia in proportione h. ſicut ſe ha-
                    <lb/>
                  bent partes proportionales alicuiꝰ continui pro-
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                  portiõe h. </s>
                  <s xml:id="N1FA27" xml:space="preserve">Probatur tamen añs videlicet /  ſpacii
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                  ꝑtranſiti in prima parte ꝓportionali ad ſpaciū ꝑ-
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                  tranſitū in ſecūda eſt ꝓpertio h. et ſpacii ꝑtrãſiti in
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                  ſcḋa ad ſpaciū ꝑtranſitum in tertia etc̈. quia prima
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                  pars proportionalis eſt maius tempus quã ſecun-
                    <lb/>
                  da in g. proportione, et ei coextenditur velocitas in­
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                  tenſior quam ſeēunde in f. proportione / vt dici hipo­
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                  teſis: et h. proportio eſt proportio cõpoſita ex g. et f.
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                  proportionibus ex hypoteſi: igitur ſpaciū ꝑtranſi­
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                  tum in prima parte ꝓportiõali ſe habet ad ſpaciū
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                  ꝑtrãſitū in ſecūda in h. ꝓportiõe. </s>
                  <s xml:id="N1FA3E" xml:space="preserve">Cõſimili argumē­
                    <lb/>
                  to ꝓbabis de quibuſcū ſpaciis ꝑtranſitis in qui­
                    <lb/>
                  buſcū duabus partibus immediatis: quod erat
                    <lb/>
                  inferendum. </s>
                  <s xml:id="N1FA47" xml:space="preserve">Patet tamen conſequentia ꝑ tertiam
                    <lb/>
                  propoſitionem ſecundi notabilis huius queſtiõis.
                    <lb/>
                    <note position="right" xlink:href="note-0160-01a" xlink:label="note-0160-01" xml:id="N1FB10" xml:space="preserve">Correĺ.</note>
                  </s>
                  <s xml:id="N1FA53" xml:space="preserve">¶ Ex hac ſolutione ſequitur primo /  partitiõe ho­
                    <lb/>
                  re facta ꝑ partes proportionales proportione ſu-
                    <lb/>
                  prabipartiēte tertias, et in prima parte ꝓporõali
                    <lb/>
                  moueatur aliquod mobile aliquãta velocitate, et ī
                    <lb/>
                  ſecunda in ſuprabipartiente quintas minore et in
                    <lb/>
                  tertia in eadē proportiõe ſuprabipartiēte quintas </s>
                </p>
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