Alvarus, Thomas, Liber de triplici motu, 1509

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              <div xml:id="N1DD6A" level="4" n="3" type="chapter" type-free="capitulum">
                <p xml:id="N1F5F5">
                  <s xml:id="N1F5F6" xml:space="preserve">
                    <pb chead="De motu locali quo ad effectū ſcḋm tempus difformi." file="0158" n="158"/>
                  ad primam partcm in proportione tripla: et totuꝫ
                    <lb/>
                  diuiſum proportione ſexquitertia in proportione
                    <lb/>
                  quadrupla: et ſic conſequenter / vt prima pars quin-
                    <lb/>
                  to ſuo capitulo oſtendit. </s>
                  <s xml:id="N1F60C" xml:space="preserve">Et ſic patet prima pars.
                    <lb/>
                  </s>
                  <s xml:id="N1F610" xml:space="preserve">Secunda vero patet ex quarta concluſione hoc ad­
                    <lb/>
                  dito /  in caſu concluſionis proportio ſpacii per-
                    <lb/>
                  tranſiti in tota hora ad ſpacium pertranſitum in
                    <lb/>
                  prima parte eſt dupla ad proportionem totius ve­
                    <lb/>
                  locitatis ad velocitatem prime partis proportio-
                    <lb/>
                  nalis temporis.</s>
                </p>
                <note position="left" xml:id="N1F61D" xml:space="preserve">1. correĺ.</note>
                <p xml:id="N1F621">
                  <s xml:id="N1F622" xml:space="preserve">¶ Ex hac ↄ̨cluſione ſequitur primo /  diuiſa hora
                    <lb/>
                  per partes proportionales proportione ſuperpar­
                    <lb/>
                  ticulari quauis libuerit: diſtributa velocitate vt
                    <lb/>
                  in caſu ſecunde concluſionis ponitur, ita videlicet /
                    <lb/>
                   mobile in prima parte proportiõali moueatur
                    <lb/>
                  aliquantulum, et in ſecunda in duplo velocius, et in
                    <lb/>
                  tertio in triplo velocius ꝙ̄ in prima, et in quarta in
                    <lb/>
                  quadruplo velocius ꝙ̄ in prima, et ſic conſequenter /
                    <lb/>
                  tūc tota velocitas erit equalis velocitati tertie par­
                    <lb/>
                  tis proportiõalis ſi fuerit facta diuiſio ꝓportione
                    <lb/>
                  ſexquialtera: et ſi fuerit diuiſio facta ſexquitertia
                    <lb/>
                  tota velocitas erit equalis velocitati quarta par-
                    <lb/>
                  tis proportionalis: et ſi fuerit facta diuiſio propor­
                    <lb/>
                  tione ſexquiquarta erit equalis velocitati quinte
                    <lb/>
                  partis proportionalis: et ſic conſequenter aſcendē­
                    <lb/>
                  do per ſpecies proportionis ſuperparticularis et
                    <lb/>
                  per partes proportionales. </s>
                  <s xml:id="N1F645" xml:space="preserve">Probatur correlariū
                    <lb/>
                  facile ex ſecunda concluſione: quoniã facta diuiſio-
                    <lb/>
                  ne hore proportione ſexquialtera: tota hora ſe ha-
                    <lb/>
                  bet ad primam partē in proportione tripla / vt con­
                    <lb/>
                  ſtat: ergo tota velocitas vt dicit concluſio ſe habet
                    <lb/>
                  ad velocitatē prime partis proportionalis in pro-
                    <lb/>
                  portione tripla et in tali proportione ſe habet ve-
                    <lb/>
                  locitas tertie partis proportionalis ad velocitatē
                    <lb/>
                  prime / vt dicit caſus igr̄. </s>
                  <s xml:id="N1F658" xml:space="preserve">Itē diuiſiõe facta ꝑ partes
                    <lb/>
                  ꝓportiõales ꝓportiõe ſexquitertia: totū ſic diuiſuꝫ
                    <lb/>
                  ſe habet ad primam partem proportionalem in ꝓ­
                    <lb/>
                  portione quadrupla: ergo totalis velocitas ſe ha-
                    <lb/>
                  bet ad velocitatem prime partis proportiõalis in
                    <lb/>
                  proportione quadrupla / vt patet ex ſecūda conclu-
                    <lb/>
                  ſione: et tanta eſt velocitas quarte partis / igitur. </s>
                  <s xml:id="N1F667" xml:space="preserve">Et
                    <lb/>
                  ſic probabis reſiduas partes in infinitum.</s>
                </p>
                <note position="left" xml:id="N1F66C" xml:space="preserve">2. correĺ.</note>
                <p xml:id="N1F670">
                  <s xml:id="N1F671" xml:space="preserve">¶ Sequitur ſecūdo /  hora diuiſa per partes pro-
                    <lb/>
                  portionales proportione ſexquialtera et mobile a.
                    <lb/>
                  in prima parte moueatur aliquantulū, et in ſecun-
                    <lb/>
                  da parte in duplo velocius, et in tertia in triplo ve-
                    <lb/>
                  locius q̈ in prima, et ſic conſequenter: vt in prima
                    <lb/>
                  parte proportionali pertranſit vnum pedale: in to­
                    <lb/>
                  ta hora pcriranſit nouē. </s>
                  <s xml:id="N1F680" xml:space="preserve">Probatur / quia illo caſu
                    <lb/>
                  poſito velocitatis totius ad velocitatē prime par-
                    <lb/>
                  tis eſt proportio tripla: vt patet ex precedenti: igi-
                    <lb/>
                  tur ſpacii pertraſiti in tota hora ad ſpaciū ꝑtran-
                    <lb/>
                  ſitum in prima parte eſt ꝓportio dupla ad triplam /
                    <lb/>
                  vt ptꝫ ex quarta huius: ſed noncupla eſt dupla ad
                    <lb/>
                  triplam ex ſecūda parte huius operis capite ſexto /
                    <lb/>
                  igitur totius ſpacii pertrãſiti in tota hora ad ſpa­
                    <lb/>
                  cium pertranſitum in prima parte eſt ꝓportio non­
                    <lb/>
                  cupla / quod fuit probandum.
                    <note position="left" xlink:href="note-0158-01a" xlink:label="note-0158-01" xml:id="N1F6F7" xml:space="preserve">3. correĺ.</note>
                  </s>
                  <s xml:id="N1F69A" xml:space="preserve">¶ Sequitur tertio / 
                    <lb/>
                  diuiſa hora vel tempore aliquo proportione qua-
                    <lb/>
                  uis ſuperparticulari / vt poſitum eſt in primo corre­
                    <lb/>
                  lario: ſpacii pertrãſiti in tota hora ad ſpaciū per-
                    <lb/>
                  tranſitum in prima parte eſt proportio dupla ad
                    <lb/>
                  proportionem quam habet velocitas tertie partis
                    <lb/>
                  ad velocitatem prime partis ſi fuerit diuiſio facta
                    <lb/>
                  proportione ſexquialtera: ſi vero fiat proportione
                    <lb/>
                  ſexquitertia in proportione dupla ad proportio-
                    <lb/>
                  nem velocitatis quarte partis ad velocitatem pri-
                    <lb/>
                  me. </s>
                  <s xml:id="N1F6B1" xml:space="preserve">ſi ſexquiquarta in proportione dupla ad pro-
                    <lb/>
                  portionem velocitatis quinte partis ad velocita-
                    <cb chead="De motu locali quo ad effectū ſcḋm tempus difformi."/>
                  tem prime / et ſic conſequenter. </s>
                  <s xml:id="N1F6B9" xml:space="preserve">Et quia hoc correla-
                    <lb/>
                  rium manifeſte ſequitur ex predictes ꝓbatione non
                    <lb/>
                  indiget.
                    <note position="right" xlink:href="note-0158-02a" xlink:label="note-0158-02" xml:id="N1F6FD" xml:space="preserve">4. correĺ</note>
                  </s>
                  <s xml:id="N1F6C5" xml:space="preserve">¶ Ex quo ſequitur quarto /  hora diuiſa
                    <lb/>
                  per partes proportionales proportione aliqua
                    <lb/>
                  ſuperparticulari quauis volueris: et aliquod mo-
                    <lb/>
                  bile moueatur in prima etc̈. / vt poſitū eſt: ſpacii per­
                    <lb/>
                  tranſiti eſt tota hora eſt noncuplum ad ſpaciū per-
                    <lb/>
                  tranſitum in prima parte proportionali ſi fuerit
                    <lb/>
                  diuiſio facta proportione ſexquialtera: ſi vero pro­
                    <lb/>
                  portio eſt ſequitertia: eſt ſexdecuplum: ſi autē pro­
                    <lb/>
                  portione ſexquiquarta: eſt vicecuplum quintuplū.
                    <lb/>
                  </s>
                  <s xml:id="N1F6D9" xml:space="preserve">ita  in prima parte pertranſit vnum pedale in to­
                    <lb/>
                  ta hora viginti quin pedalia: et ſic conſequenter.
                    <lb/>
                  </s>
                  <s xml:id="N1F6DF" xml:space="preserve">Patet hoc correlarium ex predictis. </s>
                  <s xml:id="N1F6E2" xml:space="preserve">¶ Innumera
                    <lb/>
                  alia correlaria inferre poteris ſi virtutē et robur,
                    <lb/>
                  ſecunde et quarte concluſionis diligenter inſpexe-
                    <lb/>
                  ris: non ſolum in generibus proportionum multi-
                    <lb/>
                  plicis at ſuperparticularis: verū etiam pari faci­
                    <lb/>
                  litate in omnibus aliis generibus puta ſuprapar-
                    <lb/>
                  tiente multiplici ſuperparticulari multiplici ſu-
                    <lb/>
                  perpartiente.</s>
                </p>
                <p xml:id="N1F703">
                  <s xml:id="N1F704" xml:space="preserve">Sexta cõcluſio. </s>
                  <s xml:id="N1F707" xml:space="preserve">Diuiſa hora quauis
                    <lb/>
                  proportione libuerit et in quacun proportione ſe
                    <lb/>
                  habuerint due partes immediate in eadem propor­
                    <lb/>
                  tione vel maiori ſe habuerit velocitas minoris par­
                    <lb/>
                  tis ad velocitatem maioris: tota illa velocitas eſt
                    <lb/>
                  infinita: ſpacium pertranſitum pari ratione infi­
                    <lb/>
                  nitum erit. </s>
                  <s xml:id="N1F716" xml:space="preserve">Probatur ſecnnda pars concluſionis
                    <lb/>
                  quoniam in illo caſu mobile / quod ſic mouetur tan­
                    <lb/>
                  tum ſpacium pertranſit in ſequenti parte ſicut in
                    <lb/>
                  priori vel maius et ſunt infinite partes proportio-
                    <lb/>
                  nales: ergo in totali hora infinitum pertranſibit.
                    <lb/>
                  </s>
                  <s xml:id="N1F722" xml:space="preserve">Patet cõſequentia cum minore: et arguitur maior /
                    <lb/>
                  qm̄ qualis eſt proportio prime partis ad ſecundaꝫ
                    <lb/>
                  partē proportionalē talis eſt ꝓportio velocitatis
                    <lb/>
                  ſecunde partis proportionalis ad velocitatem pri­
                    <lb/>
                  me partis vel maior: igitur tantum ſpaciū pertran­
                    <lb/>
                  ſit in ſecūda ſicut in prima vel maius. </s>
                  <s xml:id="N1F72F" xml:space="preserve">Item qualis
                    <lb/>
                  eſt proportio ſecunde partis ad tertiam partem ta­
                    <lb/>
                  lis eſt proportio velocitatis tertie partis ad ſecun­
                    <lb/>
                  de / et ſic conſequenter de quibuſcun duabus par-
                    <lb/>
                  tibus proportionalibus immediatis / vt ptꝫ ex caſu
                    <lb/>
                  concluſionis: igitur in qualibet pate immediate ſe­
                    <lb/>
                  quente alteram maiorem, mobile motum tali velo­
                    <lb/>
                  citate pertranſit tantum ſpacium ſicut in īmediate
                    <lb/>
                  precedenti vel maius / quod fuit probãdum. </s>
                  <s xml:id="N1F742" xml:space="preserve">Patet
                    <lb/>
                  tamen conſequentia ex quarta et quīta ꝓpoſitioni­
                    <lb/>
                  bus ſecundi notabilis. </s>
                  <s xml:id="N1F749" xml:space="preserve">Et ſic patet ſecunda pars et
                    <lb/>
                  per conſequens prima. </s>
                  <s xml:id="N1F74E" xml:space="preserve">Si enim mediante illa velo­
                    <lb/>
                  citate mobile pertranſit infinitum ſpacium: conſe­
                    <lb/>
                  quens eſt illam velocitatem infinitam eſſe. </s>
                  <s xml:id="N1F755" xml:space="preserve">(Eſt em̄
                    <lb/>
                  in tempore fiuito) </s>
                  <s xml:id="N1F75A" xml:space="preserve">Patet igitur concluſio.</s>
                </p>
                <note position="right" xml:id="N1F75D" xml:space="preserve">1. correĺ.</note>
                <p xml:id="N1F761">
                  <s xml:id="N1F762" xml:space="preserve">¶ Ex quo ſequitur primo /  ſi hora diuidatur per
                    <lb/>
                  partes proportionales proportione dupla: vt mo­
                    <lb/>
                  bile moueatur in prima parte aliquantulum, et in
                    <lb/>
                  ſecunda in duplo velocius ꝙ̄ in prima, et in tertia
                    <lb/>
                  in duplo velocius ꝙ̄ in ſecunda, et in quarta in du-
                    <lb/>
                  plo velocius ꝙ̄ in tertia, ſpacium pertranſitum erit
                    <lb/>
                  infinitum. </s>
                  <s xml:id="N1F771" xml:space="preserve">Patet correlarium ex concluſione quo-
                    <lb/>
                  niam in quacun proportione ſe habent partes ꝓ­
                    <lb/>
                  portionales īmediate continuo: in eadem propor-
                    <lb/>
                  tione ſe habet velocitas partis minoris ad veloci­
                    <lb/>
                  tatem partis maioris: et per conſequens totum il-
                    <lb/>
                  lud mobile pertranſit in qualibet ſequenti primaꝫ
                    <lb/>
                  tantum quantum iu prima. </s>
                  <s xml:id="N1F780" xml:space="preserve">Infinitum igitur ſpa-
                    <lb/>
                  cium tranſcurret / quod fuit probandū.
                    <note position="right" xlink:href="note-0158-03a" xlink:label="note-0158-03" xml:id="N1F7DB" xml:space="preserve">2. correĺ.</note>
                  </s>
                  <s xml:id="N1F78A" xml:space="preserve">¶ Sequitur
                    <lb/>
                  ſcḋo /  partita hora per partes ꝓportiõales ꝓpor­
                    <lb/>
                  tione ſexquitertia: et in prima parte proportionali </s>
                </p>
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