Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N1F761">
                  <s xml:id="N1F78A" xml:space="preserve">
                    <pb chead="Secundi tractatus" file="0159" n="159"/>
                  a. mobile moueatur aliqua velocitate, et in ſecunda
                    <lb/>
                  in ſexquialtero velocius ꝙ̄ in prima, et in tertia in
                    <lb/>
                  ſexquialtero velocius ꝙ̄ in ſecunda, et in quarta in
                    <lb/>
                  ſexquialtero velocius ꝙ̄ in tertia, et ſic conſeque-
                    <lb/>
                  ter: ſpacium pertranſitum in tota hora erit infi-
                    <lb/>
                  nitum. </s>
                  <s xml:id="N1F7A0" xml:space="preserve">Probatio: quia in qualibet parte ſequen-
                    <lb/>
                  ti primam a. mobile maius ſpacium abſoluet ̄
                    <lb/>
                  in prima: qm̄ contiuo maior eſt proportio velocita­
                    <lb/>
                  tis minoris ad velocitatē maioris ꝙ̄ ſit temporis
                    <lb/>
                  maiors ad tempus minus: igitur per quintã pro-
                    <lb/>
                  poſitionem ſecundi notabilis in qualibet ſequenti
                    <lb/>
                  primã maius ſpacium ꝑtranſibit ꝙ̄ in prima: et ꝑ
                    <lb/>
                  conſequens in tota hora infinitum ſpacium tranſ-
                    <lb/>
                  curret: quod fuit probandum.
                    <note position="left" xlink:href="note-0159-01a" xlink:label="note-0159-01" xml:id="N1F7E1" xml:space="preserve">3. correĺ.</note>
                  </s>
                  <s xml:id="N1F7B8" xml:space="preserve">¶ Tertio ſequitur:
                    <lb/>
                   ſi hora fuerit diuiſa per partes proportionales
                    <lb/>
                  proportione aliqua ſuprapartienti: et continuo ve­
                    <lb/>
                  locitates partium proportionaliū immediataruꝫ
                    <lb/>
                  puta velocitas minoris partis ad velocitatem ma­
                    <lb/>
                  ioris ſe habuerit in aliqua proportione multiplici
                    <lb/>
                  vel multiplici ſuperparticulari, vel multiplici ſu-
                    <lb/>
                  perpartienti: ſpaciū ꝑtranſitū in tota hora erit in-
                    <lb/>
                  finitum. </s>
                  <s xml:id="N1F7CB" xml:space="preserve">Patet hoc correlarium / quia continuo ma­
                    <lb/>
                  ior erit ibi proportio velocitatum temporum ma-
                    <lb/>
                  iorum et minorum ꝙ̄ proportio maioris temporis
                    <lb/>
                  ad minꝰ tēpus igitur. </s>
                  <s xml:id="N1F7D4" xml:space="preserve">Interas ad libitū correlaria</s>
                </p>
                <p xml:id="N1F7E7">
                  <s xml:id="N1F7E8" xml:space="preserve">Septima cõcluſio. </s>
                  <s xml:id="N1F7EB" xml:space="preserve">Partita hora per
                    <lb/>
                  partes proportionales qua libuerit proportione
                    <lb/>
                  mobile continuo mouente velocius in parte ſequē­
                    <lb/>
                  ti quam in parte p̄cepenti: velocius nihilominus in
                    <lb/>
                  proportiõe minori ꝙ̄ ſit proportio diuiſionis) ſpa­
                    <lb/>
                  cium ꝑtranſitum in tota hora ſe habebit ad ſpaci-
                    <lb/>
                  um ꝑtranſitum in prima parte ꝓportionali in pro­
                    <lb/>
                  portione qua aliquod totum diuiſum proportione
                    <lb/>
                  qua maior proportio temporis excedit proportio-
                    <lb/>
                  nem velocitatum ſe habet in ordine ad primã par-
                    <lb/>
                  tem ꝓportionalem. </s>
                  <s xml:id="N1F802" xml:space="preserve">Hoc theorema multiplicibus
                    <lb/>
                  verbis implicitum et intricatū familiarem et exem-
                    <lb/>
                  plarem enucleationem efflagitat. </s>
                  <s xml:id="N1F809" xml:space="preserve">Exemplo / igitur
                    <lb/>
                  vtens volo dicere:  ſi hora fuerit diuiſa ꝑ partes
                    <lb/>
                  ꝓportionales ꝓportione quadrupla exempli gra­
                    <lb/>
                  tia: et a. mobile moueatur in prima parte ꝓportio-
                    <lb/>
                  nali aliquanta velocitate, et in ſecūda in duplo ma­
                    <lb/>
                  iori velocitate, et in tertia in duplo maiori ꝙ̄ in ſe-
                    <lb/>
                  cunda, et ſic in qualibet ſequenti in duplo maiori
                    <lb/>
                  velocitate quam in immediate ꝑcedenti (quoniam
                    <lb/>
                  ꝓportio illarum velocitatum que eſt dupla excedi-
                    <lb/>
                  tur a ꝓportione tempoꝝ que eſt quadrupla ꝑ ꝓpor­
                    <lb/>
                  tionem duplam) / dico /  totale ſpacium ꝑtranſituꝫ
                    <lb/>
                  in illa totali hora ſe habet ad ſpaciū ꝑtranſitū in
                    <lb/>
                  prima parte proportionali: ſicut ſe habet aliquod
                    <lb/>
                  corpus diuiſum ꝓportione dupla in ordine ad ſuã
                    <lb/>
                  primam partem / vt poſt modum correlaria fami-
                    <lb/>
                  liariter oſtendent. </s>
                  <s xml:id="N1F82A" xml:space="preserve">Probatur tamen concluſio ge-
                    <lb/>
                  neraliter / et ſit hora diuiſa ꝑ partes ꝓportionales
                    <lb/>
                  ꝓportione g. maiore: ſit continuo velocitatis par­
                    <lb/>
                  tis minoris ad velocitatē partis maioris īmedia-
                    <lb/>
                  te p̄cedentis ꝓportio f. minor quã ſit ꝓportio g. ex-
                    <lb/>
                  cedat ꝓportio g. ꝓportionem f. mediante ꝓpor-
                    <lb/>
                  tione h. </s>
                  <s xml:id="N1F839" xml:space="preserve">Tunc dicit / theorema ſpacium ꝑtranſitum
                    <lb/>
                  in totali hora ſe habere ad ſpacium ꝑtranſitum in
                    <lb/>
                  prima parte ꝓportionali illius hore, in ea ꝓporti­
                    <lb/>
                  one in qua ſe habet aliquod diuiſum ꝓportione h.
                    <lb/>
                  ad primam partem ꝓportionalem eiuſdem ꝓpor-
                    <lb/>
                  tionis h. </s>
                  <s xml:id="N1F846" xml:space="preserve">Quod ſic ꝓbatur / quia prime partis pro-
                    <lb/>
                  portionalis hore ad ſecundã partem ꝓportialē
                    <lb/>
                  eiuſdem eſt ꝓportio g. maior: et velocitatis ſecunde
                    <lb/>
                  partis proportionalis ad velocitatē prime partis
                    <lb/>
                  ꝓportionalis eſt ꝓportio f. minor / vt ponit caſus: et
                    <cb chead="Capitulū tertiū"/>
                  g. ꝓportio temporis maioris ad tempus minus ex­
                    <lb/>
                  cedit f. ꝓportionem velocitatis temporis minoris
                    <lb/>
                  ad velocitatem temporis maioris (quod tēpus ma­
                    <lb/>
                  ius eſt prima pars proportionalis et minus ſecun-
                    <lb/>
                  da) per h. ꝓportionem / vt ponitur in caſu: igitur in
                    <lb/>
                  h. ꝓportione maius ſpacium ꝑtranſitur a mobili
                    <lb/>
                  in prima parte proportionali quã in ſecunda. </s>
                  <s xml:id="N1F860" xml:space="preserve">Ptꝫ
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                  hec conſequentia ex ſexta ꝓpoſitione ſecundi nota­
                    <lb/>
                  bilis huius queſtionis. </s>
                  <s xml:id="N1F867" xml:space="preserve">Et ſic argumentaberis de
                    <lb/>
                  ſecunda et tertia /  in h. proportione maius ſpaci-
                    <lb/>
                  um pertranſitur in ſecunda quam in tertia: et ſic de
                    <lb/>
                  quibuſcun duabus partibus immediatis argu-
                    <lb/>
                  mentatione exordiri licebit: igitur illa ſpacia per-
                    <lb/>
                  tranſita ſe habent continuo in h. proportiõe ita 
                    <lb/>
                  primi ad ſecundum ſit h. proportio et ſecundi ad ter­
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                  tium / et ſic conſequenter: igitur aggregatum ex om­
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                  nibus illis ſpaciis ſe habebit ad ſpacium pertran­
                    <lb/>
                  ſitum in prima parte proportionali in ꝓportione in
                    <lb/>
                  qua ſe habet totum diuiſum in ꝓportione h. ad pri­
                    <lb/>
                  mam partem ꝓportionaleꝫ eiuſdem ꝓportionis
                    <lb/>
                  h. / quod fuit probandū.
                    <note position="right" xlink:href="note-0159-02a" xlink:label="note-0159-02" xml:id="N1F8FD" xml:space="preserve">1. correĺ.</note>
                  </s>
                  <s xml:id="N1F887" xml:space="preserve">¶ Ex hac concluſione ſequi­
                    <lb/>
                  tur primo:  partitione hore facta per partes pro­
                    <lb/>
                  portionales ꝓportione quadrupla: velocitatibus
                    <lb/>
                  continuo ſe habentibus in ꝓportione dupla: ita 
                    <lb/>
                  velocitatis ſecunde partis ꝓportionalis ad velo-
                    <lb/>
                  citatem prime ſit proportio dupla, et velocitatis
                    <lb/>
                  tertie ad velocitatem ſecunde ſit etiam proportio
                    <lb/>
                  dupla .etc̈. ſpacium pertranſitum in tota hora eſt
                    <lb/>
                  duplum ad ſpacium pertranſitum in prima parte
                    <lb/>
                  proportionali. </s>
                  <s xml:id="N1F89C" xml:space="preserve">Probatur / quia proportio illoruꝫ
                    <lb/>
                  temporum quadrupla excedit ꝓportionem duplã
                    <lb/>
                  velocitatum per proportionem duplam / vt patet ex
                    <lb/>
                  quarta concluſioue quarti capitis ſecunde partis:
                    <lb/>
                  igitur totale ſpacium pertranſitum in illa hora eſt
                    <lb/>
                  duplum ad ſpacium pertranſitum in prima parte
                    <lb/>
                  proportionali hore. </s>
                  <s xml:id="N1F8AB" xml:space="preserve">Patet conſequentia ex prece-
                    <lb/>
                  denti concluſione: hoc addito /  quodlibet diuiſuꝫ
                    <lb/>
                  per partes proportionales proportione dupla ſe
                    <lb/>
                  habet ad primam partem proportionalem in pro-
                    <lb/>
                  portiõe dupla. </s>
                  <s xml:id="N1F8B6" xml:space="preserve">Arguitur tamen et familiarius pro­
                    <lb/>
                  batur correlarium: et volo /  ſpacium pertranſituꝫ
                    <lb/>
                  in prima parte proportionali proportione dupla
                    <lb/>
                  ſit pedale: et arguo ſic / ſpacium pertranſitum in ſe-
                    <lb/>
                  cunda parte proportionali eſt ſubduplum ad ſpa-
                    <lb/>
                  tium pertranſitum in prima, et ſpacium pertranſi-
                    <lb/>
                  tum in tertia ad ſpacium pertranſitum in ſecunda /
                    <lb/>
                  et ſic conſequenter ſe habent illa ſpacia in propor-
                    <lb/>
                  tione ſubdupla: et primuꝫ illorum eſt pedale: igitur
                    <lb/>
                  totum aggregatum ex omnibus ſequentibus pri-
                    <lb/>
                  mum eſt pedale: et per conſequens totum ſpacium
                    <lb/>
                  eſt bipedale: et ſic duplum ad ſpacium pertranſituꝫ
                    <lb/>
                  in prima parte proportiõali quod eſt pedale: quod
                    <lb/>
                  fuit inferendū. </s>
                  <s xml:id="N1F8D3" xml:space="preserve">Probatur tamen maior /  illa ſpa­
                    <lb/>
                  cia pertranſita in partibus proportionalibus ſe
                    <lb/>
                  habent in proportione ſubdupla quoniam prime
                    <lb/>
                  partis ad ſecundam eſt proportio quadrupla per
                    <lb/>
                  caſum: et velocitatis ſecunde ad velocitatem prime
                    <lb/>
                  eſt proportio dupla per caſum: igitur ſpacium per­
                    <lb/>
                  tranſitum in ſecunda eſt ſubduplum ad ſpaciū per­
                    <lb/>
                  tranſitū in prima: et ſic argues de ſpacio pertran-
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                  ſito in tertia ad ſpacium pertranſitum in ſecunda:
                    <lb/>
                  et de quibuſcun ſpaciis pertranſitis in duabus
                    <lb/>
                  partibus īmediatis proportionalibus: igitur illa
                    <lb/>
                  ſpacia continuo ſe habent in proportione ſubdu-
                    <lb/>
                  pla: quod fuit probandum. </s>
                  <s xml:id="N1F8EE" xml:space="preserve">Patet conſequentia
                    <lb/>
                  ex ſexta propoſitione ſecundi notabilis: hoc addi-
                    <lb/>
                  to /  proportio quadrupla excedit proportionem
                    <lb/>
                  duplam per ipſammet duplam: vt ſecunda pars
                    <lb/>
                  loco preallegato docet.</s>
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