Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N1E7EE">
                  <s xml:id="N1E811" xml:space="preserve">
                    <pb chead="Secundi tractatus" file="0149" n="149"/>
                  per mediam horam, et velocitate vt duo per horam
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                  </s>
                  <s xml:id="N1E81C" xml:space="preserve">Item ſi maior velocitas coextendatur tēpori mīori
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                  et minor maiori. </s>
                  <s xml:id="N1E821" xml:space="preserve">hoc coutingit tripliciter / quia aut
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                  ꝓportio tempoꝝ excedit ꝓportionē velocitatū aut
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                  ꝓportio velocitatū excedit ꝓportionē tempoꝝ aut
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                  ꝓportiones tempoꝝ et velocitatū ſunt equales. </s>
                  <s xml:id="N1E82A" xml:space="preserve">Exē-
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                  plum primi / vt ſi aliquod mobile in hora moueatur
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                  vt duo, et in quarta hore vt quatuor: tunc ꝓportio
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                  tempoꝝ excedit proportionē velocitatū. </s>
                  <s xml:id="N1E833" xml:space="preserve">Nam ipſa
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                  tēpoꝝ proportio quadrupla eſt: velocitatū vero du­
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                  pla / vt patet aſpicienti. </s>
                  <s xml:id="N1E83A" xml:space="preserve">Exemplū ſecundi / vt ſi mo-
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                  bile moueatur vt vnū per horã, et in media vt .3. / tūc
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                  proportio tempoꝝ eſt dupla, velocitatū o tripla:
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                  exuperat igitur velocitatū proportio tempoꝝ pro-
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                  portionē. </s>
                  <s xml:id="N1E845" xml:space="preserve">Exemplū tertii / vt ſi aliquod mobile mo-
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                  ueatur in hora vt vnū, et aliud in media vt duo: con­
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                  ſtat ꝓportionē tempoꝝ ꝓportioni velocitatū equa­
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                  ri: vtra em̄ dupla eſt: et velocitatū, et tempoꝝ. </s>
                  <s xml:id="N1E84E" xml:space="preserve">Hac
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                  longa diuiſione velocitatū exacta: ipſa velocita­
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                  te fruſtrat ī conciſa: opere preciū eſt cuilibet huiꝰ di-
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                  uiſionis fruſto et membro peculiarē propoſitioneꝫ
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                  aſſcriberet. </s>
                  <s xml:id="N1E859" xml:space="preserve">Sit igitur.</s>
                </p>
                <p xml:id="N1E85C">
                  <s xml:id="N1E85D" xml:space="preserve">Capitalis propoſitio. </s>
                  <s xml:id="N1E860" xml:space="preserve">Si velocitates
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                  ſint equales equalibus coextenſe temporibus: mo-
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                  bilia in eiſdem mota equalia ſpacia in eiſdē tēpo-
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                  ribus abſoluūt (ceteris aliis deductis) vt puta ra-
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                  refactione condenſatione ſpacii et prepoſtera mo-
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                  tione / vt concluſiones ſexto phiſicoꝝ oſtendunt. </s>
                  <s xml:id="N1E86D" xml:space="preserve">Si
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                  vero velocitates equales per īequalia labantur tē­
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                  pora: tunc in ea ꝓportione mobile in maiori tēpo-
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                  re maius ſpaciū pertranſit quam in minori: in qua
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                  ipſū maius tēpus ſe habet ad minus. </s>
                  <s xml:id="N1E878" xml:space="preserve">Prima pars
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                  huiꝰ ꝓpoſitionis patet ex ſe: et ſecunda ꝓbatur: ſup­
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                  poſito /  quando aliquid mobile mouetur vnifor-
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                  miter per aliquod tēpus in quacū ꝓportione ſe
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                  habent partes tēporis ad totū: in ea ꝓportione ſe
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                  habent ſpacia pertranſita in illis temporibus ad
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                  ad ſpaciū pertranſitū in toto tēpore: quo ſuppoſi-
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                  to arguitur ſic mobile quod mouetur in maiori tē-
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                  pore et mobile motū in minori tēpore mouētur vni-
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                  formiter et eque velociter. </s>
                  <s xml:id="N1E88D" xml:space="preserve">ergo in eq̈libꝰ tēporibus
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                  equalia ſpacia ꝑtranſeunt / vt patet ex priori parte:
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                  ergo quantū ſpaciū mobile motū in minori tēpore
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                  ꝑtranſit in totali ſuo tēpore: tantū adequate per-
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                  trãſit mobile motū in maiori tēpore in tēpore ſibi
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                  equali: ergo qualis eſt ꝓportio illius tēporis ma-
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                  ioris ad tēpus minus talis eſt ꝓportio ſpacii ꝑtrã­
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                  ſiti in tēpore maiori ad ſpaciū ꝑtranſitū in tēpore
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                  minori / quod fuit ꝓbandū: et cõſequentia patet ex
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                  ſuppoſito hoc adiecto /  qualis eſt ꝓportio totius
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                  tēporis ad illam ſuã partem equalē tēpori minori
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                  talis eſt proportio ipſius maioris temporis ad il-
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                  lud minus tempus / vt patet de ſe.</s>
                </p>
                <p xml:id="N1E8A8">
                  <s xml:id="N1E8A9" xml:space="preserve">Secūda ꝓpoſitio. </s>
                  <s xml:id="N1E8AC" xml:space="preserve">Quãdo inequales
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                  velocitates equalibus tēporibus coextendūtur: tūc
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                  mobile quod maiore velocitate mouetur in ea pro­
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                  portione maius ſpaciū ꝑtranſit ꝙ̄ alterum / mobile
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                  in qua ſe habet velocitas maior ad minorē. </s>
                  <s xml:id="N1E8B7" xml:space="preserve">Pro-
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                  batur hec ꝓpoſitio (̄uis facilis ſit) / quia ſi mobile
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                  motū velocitate maiori in tēpore a. moueretur ade­
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                  quate equali velocitate ſicut mouetur aliud mobile
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                  motū velocitate minori in eodē a. tempore / tūc illa
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                  duo mobilia equalia ſpacia ꝑtranſirent in a. tēpo­
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                  re / vt ptꝫ ex priori parte p̄cedentis ꝓpoſitionis: ſed
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                  modo illud mobile mouetur in aliqua proportione
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                  puta in f. velocius quã tunc: ergo in f. proportione
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                  maius ſpaciū pertranſit quã tunc: et per conſequēs
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                  maius ſpaciū ꝑtranſit in eodem tēpori in f. ꝓpor-
                    <cb chead="Capitulū tertiū."/>
                  tione quã alterū mobile motum in eodem tempore
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                  velocitate in f. proportione minori.</s>
                </p>
                <p xml:id="N1E8D3">
                  <s xml:id="N1E8D4" xml:space="preserve">Tertia ꝓpoſitio. </s>
                  <s xml:id="N1E8D7" xml:space="preserve">Si inequales velo-
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                  citates in equalibus tēporibus coextendunt̄̄, et ma­
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                  ior velocitas maiori tempori coextendatur: et mīor
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                  minori: tunc mobile quod mouetur in maiori tem-
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                  pore maius ſpaciū ꝑtranſit in ꝓportione cõpoſita
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                  temporis maioris ad tempus minus: et velocitatis
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                  maioris ad velocitatē minorē. </s>
                  <s xml:id="N1E8E6" xml:space="preserve">Exemplū / vt ſi mobi­
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                  le a. moueatur per horã vt quatuor, et b. per mediã
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                  horã, vt .2. / tunc dico /  a ꝑtranſit maius ſpacium
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                  quã b. in ꝓportione cõpoſita ex ꝓportione hore ad
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                  mediã horam: et velocitatis vt .4. ad velocitatem vt
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                  duo. / et cū vtra illaꝝ ꝓportionū ſit dupla: conſe-
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                  quens eſt /  cõpoſita ex eis ſit quadrupla / vt patet
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                  ex ſecunda parte: et per conſequens in quadruplo
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                  maius ſpaciū ꝑtranſit a. in hora quam b. in media
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                  hora. </s>
                  <s xml:id="N1E8FB" xml:space="preserve">Probatur hec concluſio / quia ſi a. et b. moue-
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                  rentur equaliter in illis duobus temporibus ine-
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                  qualibus: tunc a. ꝑtranſit maius ſpaciū quam b. in
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                  illa ꝓportione in qua ſe habent tēpora / vt patet ex
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                  ſecunda parte prime ꝓpoſitionis: et modo a. in ali-
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                  qua proportione que ſit f. maiori velocitate mouet̄̄
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                  quã tunc: ergo in f. proportione maius ſpaciū per-
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                  tranſit quã tunc. </s>
                  <s xml:id="N1E90C" xml:space="preserve">Patet conſequētia / quia quanto
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                  in eodem tempore velocitas eſt maior: tanto in eo-
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                  dem tempore per eandeꝫ maius ſpaciū ꝑtranſitur.
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                  </s>
                  <s xml:id="N1E914" xml:space="preserve">Ergo proportio ſpacii ꝑtranſiti a mobili quod ve-
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                  lociꝰ mouetur ad ſpaciū ꝑtranſitū a mobili / quod
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                  tardius mouetur componitur adequate ex propor­
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                  tione tēporū: et ex proportione velocitatū que eſt f. /
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                  quod fuit probandū. </s>
                  <s xml:id="N1E91F" xml:space="preserve">Patet / quia inter terminos il­
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                  lius proportionis reperiūtur iſti termini puta ſpa­
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                  ciū ꝑtranſitum ab illa velocitate maiori in maiori
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                  tempore et ſpaciū pertranſitū in eodem maiori tē-
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                  pore a velocitate equali velocitate minoris tempo­
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                  ris: et ſpacium ꝑtranſitum a velocitate minoris tē-
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                  poris in minori tempore: ſed primi termini ad ſe-
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                  cundū eſt proportio f. que eſt proportio velocitatū
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                  et ſecundi ad tertiū eſt proportio tempoꝝ: et totalis
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                  illa ꝓportio q̄ cõponit̄̄ ex illis duabꝰ eſt proportio
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                  ſpacii ad ſpaciū: g̊ ꝓportio ſpacii ꝑtrãſiti a mobi­
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                  li velociori ad ſpaciū ꝑtrãſitū a mobili tardiori cõ-
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                  ponitur ex ꝓportiꝰe velocitatis ad velocitatē: et tꝑs
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                  ad tēpus / quod fuit probandū: et ſic ptꝫ propoſitio
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                    <note position="right" xlink:href="note-0149-01a" xlink:label="note-0149-01" xml:id="N1E95D" xml:space="preserve">Correĺ.</note>
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                  <s xml:id="N1E943" xml:space="preserve">¶ Ex hac propoſitione ſequitur primo /  ſi a. mo-
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                  ueatur per vnã horã velocitate vt .6. et b. ꝑ mediam
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                  horã velocitate vt .4. /  ſpaciū ꝑtranſitū ab a. erit
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                  triplū ad ſpaciū ꝑtranſitū a.b. </s>
                  <s xml:id="N1E94C" xml:space="preserve">Ptꝫ / qm̄ ex propor­
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                  tione tēporis ad tēpus, et velocitatis ad velocitatē
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                  quaꝝ prima eſt dupla: et ſecūda ſexq̇altera cõponit̄̄
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                  tripla proportio / vt ptꝫ in his terminis .6. ad .4. et
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                  4. ad .2. et in illa proportione a. mouet̄̄ velociꝰ b. / vt
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                  ptꝫ ex precedenti propoſitione: igitur propoſitum.</s>
                </p>
                <p xml:id="N1E963">
                  <s xml:id="N1E964" xml:space="preserve">Sequit̄̄ ſcḋo /  ſi a. mobile moueat̄̄ ꝑ
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                  horã velocitate vt .6. et b. ꝑ duas tertias hore velo-
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                  citate vt .4. /  in minori proportione maiꝰ ſpacium
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                  ꝑtrãſit a. ꝙ̄ b. ꝙ̄ in priori caſu. </s>
                  <s xml:id="N1E96D" xml:space="preserve">Ptꝫ / q2 tūc ſpaciū ꝑ-
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                  trãſitū ab a. erit duplū ſexquiq̈rtū ad ſpaciū ꝑtrã-
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                  ſitū a.b. et in priori caſu erat triplū: g̊ in minori pro­
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                  portione maiꝰ ſpaciū ꝑtraſit a. quã b. in iſto caſu ̄
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                  in priori. </s>
                  <s xml:id="N1E978" xml:space="preserve">Ptꝫ ↄ̨ña / q2 tripla eſt maior ꝙ̄ dupla ſex-
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                  quiquarta proportio. </s>
                  <s xml:id="N1E97D" xml:space="preserve">Probo tamen maiorē / quia
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                  proportio tēporis ad tēpus eſt ſexq̇altera: et ſimili­
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                  ter velocitatis ad velocitatē: ergo ſpaciū ꝑtrãſitū
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                  ab a. eſt maiꝰ ſpacio ꝑtrãſito a b. in proportione cõ­
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                  poſita ex duabꝰ ſexq̇alteris, q̄ eſt dupla ſexquiq̈rta /
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                  vt patet in his terminis: 9.6.4. auxiliantibus his </s>
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