Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N1C8AF" level="3" n="2" type="other" type-free="tractatus">
              <div xml:id="N1DD6A" level="4" n="3" type="chapter" type-free="capitulum">
                <p xml:id="N1E4F5">
                  <s xml:id="N1E4F9" xml:space="preserve">
                    <pb chead="Secundi tractatus" file="0147" n="147"/>
                  plex eſt: </s>
                  <s xml:id="N1E503" xml:space="preserve">Nam quidam eſt vniformiter difformis ter­
                    <lb/>
                  minatus ad non gradum in altero extremo. </s>
                  <s xml:id="N1E508" xml:space="preserve">Alter
                    <lb/>
                  vero eſt vniformiter difformis vtrobi ad graduꝫ
                    <lb/>
                  terminatus. </s>
                  <s xml:id="N1E50F" xml:space="preserve">Et de vtro iſtorum dicitur /  gradui
                    <lb/>
                  ſuo medio correſpondet: id eſt gradui motus quem
                    <lb/>
                  habet in medio temporis. </s>
                  <s xml:id="N1E516" xml:space="preserve">Nam quanto velociꝰ mo­
                    <lb/>
                  uetur mobile motum vniformiter difformiter medi­
                    <lb/>
                  ante medietate talis motus intenſiori tanto tardiꝰ
                    <lb/>
                  mouetur mediãte medietate remiſſiori, et ſic eque ve­
                    <lb/>
                  lociter mouetur ac ſi moueretur gradu medio. </s>
                  <s xml:id="N1E521" xml:space="preserve">Et
                    <lb/>
                  ad cognitionem talis gradus medii pono aliq̈s ꝓ­
                    <lb/>
                  poſitiones.</s>
                </p>
                <p xml:id="N1E528">
                  <s xml:id="N1E529" xml:space="preserve">Prima propoſitio </s>
                  <s xml:id="N1E52C" xml:space="preserve">In omni latitudīe
                    <lb/>
                  vniformiter difformi incipiente a gradu a termina­
                    <lb/>
                  ta ad non gradum: gradus medius eſt ſubduplus
                    <lb/>
                  ad extremuꝫ intenſius: ita  ſi latitudo incipiat ad
                    <lb/>
                  octauo et terminatur ad nõ gradū: gradus medius
                    <lb/>
                  eſt gradus quartus q2 quartus gradꝰ eſt ſnbduplꝰ
                    <lb/>
                  ad octauum. </s>
                  <s xml:id="N1E53B" xml:space="preserve">Ad quam ꝓpoſitionem oſtendendam
                    <lb/>
                  ſupponendum eſt /  quandocun ſunt iufiniti ter­
                    <lb/>
                  mini cõtinuo ꝓportionales ꝓportione dupla / tūc to­
                    <lb/>
                  tum aggregatum ex eis eſt duplum ad totuꝫ aggre­
                    <lb/>
                  gatū ex oībus ſequētibus primū. </s>
                  <s xml:id="N1E546" xml:space="preserve">Secūdo ſupponē­
                    <lb/>
                  dum eſt /  medium eſt illḋ quod equaliter dlſtat ab
                    <lb/>
                  extremis </s>
                  <s xml:id="N1E54D" xml:space="preserve">Hee ſuppoſitiones ſatis aperte ſunt ex ṗ­
                    <lb/>
                  ma et ſecunda partibus. </s>
                  <s xml:id="N1E552" xml:space="preserve">His ſuppoſitis arguitur ꝓ­
                    <lb/>
                  poſitio: et volo /  diuidatur latitudo vniformiter
                    <lb/>
                  difformis a nõ gradu vſ ad certum gradum ī par­
                    <lb/>
                  tes ꝓportionales continuo ſe habentes ī ꝓportio­
                    <lb/>
                  ne dupla: et arguo ſic / gradus initians aggregatuꝫ
                    <lb/>
                  ex omnibus latitudinibus ſequentibus primam eſt
                    <lb/>
                  medius: et talis eſt ſubduplus ad gradum intenſio­
                    <lb/>
                  rem illius latitudinis / igitur talis latitudinis vni-
                    <lb/>
                  formiter difformis terminate ad non gtadum: gra­
                    <lb/>
                  dus medius eſt ſubduplus ad extremum intenſius
                    <lb/>
                  eiuſdem latitudinis: et ſic ꝓbabis de qualibet alia
                    <lb/>
                  </s>
                  <s xml:id="N1E56A" xml:space="preserve">Conſequentia patet, et arguitur maior / q2 talis gra­
                    <lb/>
                  dus equaliter diſtat ab extremis illius latitudinis /
                    <lb/>
                  vt patet ex prima ſuppoſitione </s>
                  <s xml:id="N1E571" xml:space="preserve">Nam initiat ſecun­
                    <lb/>
                  dam medietatem latitudinis: et terminat primam:
                    <lb/>
                  igitur eſt medius gradus: </s>
                  <s xml:id="N1E578" xml:space="preserve">Patet conſequentia ex
                    <lb/>
                  ſecunda ſuppoſitione. </s>
                  <s xml:id="N1E57D" xml:space="preserve">Sed  iſte ſit ſubduplus ad
                    <lb/>
                  extremum intenſius probatur: quia ipſe bis ſūptꝰ
                    <lb/>
                  conſtituit extremum intenſius adequate: igitur.</s>
                </p>
                <p xml:id="N1E584">
                  <s xml:id="N1E585" xml:space="preserve">Alio modo Hentiſber deducit hanc concluſionem
                    <lb/>
                  in ſuo tractatu de motu locali capite primo.</s>
                </p>
                <p xml:id="N1E58A">
                  <s xml:id="N1E58B" xml:space="preserve">Secunda propoſitio </s>
                  <s xml:id="N1E58E" xml:space="preserve">Gradus mediꝰ
                    <lb/>
                  motus vniformiter difformis vtrobi ad gradum
                    <lb/>
                  terminati eſt intenſior quaꝫ ſubduplus ad extremū
                    <lb/>
                  intenſius. </s>
                  <s xml:id="N1E597" xml:space="preserve">Probatur hec ꝓpoſitio / quia omnis gra­
                    <lb/>
                  dus ſubduplus ad extremum intenſius tantum di-
                    <lb/>
                  ſtat ab extremo intenſiori quantum a nõ gradu: ſꝫ
                    <lb/>
                  uullus gradus medius latitudinis vtrobi ad gra­
                    <lb/>
                  dum terminate tantum diſtat ab extremo intenſio-
                    <lb/>
                  ri eius quantum a non gradu: igitur nullus gradꝰ
                    <lb/>
                  medius latitudinis vtrobi ad gradum terminate
                    <lb/>
                  eſt ſubduplus ad extremum intenſius eiuſdem lati-
                    <lb/>
                  tudinis: nec remiſſior / vt ꝓbabītur: ergo intenſior.</s>
                </p>
                <p xml:id="N1E5AA">
                  <s xml:id="N1E5AB" xml:space="preserve">Conſequentia patet in ſecundo ſecunde. </s>
                  <s xml:id="N1E5AE" xml:space="preserve">Et maior
                    <lb/>
                  patet ex precedēti propoſitione: et minor probatur /
                    <lb/>
                  quia tantum talis gradus diſtat ab extremo inten­
                    <lb/>
                  ſiori quantū diſtet adequate ab extremo remiſſiori
                    <lb/>
                  ſed non tantum talis gradus medius diſtat ab ex-
                    <lb/>
                  tremo intenſiori quantum diſtat a non gradu / vt ſa­
                    <lb/>
                  tis patet de ſe: igitur non tantuꝫ diſtat ab extremo
                    <lb/>
                  intenſiori quãtum a non gradu </s>
                  <s xml:id="N1E5BF" xml:space="preserve">Patet conſequētia
                    <lb/>
                  per hanc maximam </s>
                  <s xml:id="N1E5C4" xml:space="preserve">Quando aliqua duo ſunt eq̈-
                    <cb chead="Capitulum tertium"/>
                  lia q̇cq̇d eſt maius vno eſt maius altero. </s>
                  <s xml:id="N1E5CA" xml:space="preserve">Et per hoc
                    <lb/>
                  patet facile /  talis gradꝰ ē intenſior gradu ſudu-
                    <lb/>
                  plo ad extremum intenſius. </s>
                  <s xml:id="N1E5D1" xml:space="preserve">q2 magis diſtat a non
                    <lb/>
                  gradu quam gradus ſubduplus ad extremum in-
                    <lb/>
                  tenſius / et ſic patet propoſitio.</s>
                </p>
                <p xml:id="N1E5D8">
                  <s xml:id="N1E5D9" xml:space="preserve">Tertia proportio </s>
                  <s xml:id="N1E5DC" xml:space="preserve">Cuiuſlibet latitudi­
                    <lb/>
                  nis motus vniformiter difformis terminati ad nõ
                    <lb/>
                  gradum: medietas intenſior eſt in triplo intenſior
                    <lb/>
                  medietate remiſſiori. </s>
                  <s xml:id="N1E5E5" xml:space="preserve">Probatur hec ꝓpoſitio ſup-
                    <lb/>
                  ponendo /  quando ſunt tres termini continuo ꝓ-
                    <lb/>
                  portionabiles ꝓportione dupla / tūc extremi ad ex-
                    <lb/>
                  tremū eſt proportio duplicata / et per conſequens q̈­
                    <lb/>
                  drupla. </s>
                  <s xml:id="N1E5F0" xml:space="preserve">Hoc ſuperius oſtenſum eſt in ſecunda par-
                    <lb/>
                  te ſexti capitis octaua concluſione. </s>
                  <s xml:id="N1E5F5" xml:space="preserve">Secundo ſup-
                    <lb/>
                  ponendum eſt /  in qualibet tali latitudine motus
                    <lb/>
                  vniformiter difformis terminati ad non gradum
                    <lb/>
                  gradus initians ſecundam partem proportionalē
                    <lb/>
                  ꝓportione dupla eſt ſubduplus ad extremum inten­
                    <lb/>
                  ſius: et gradus initians tertia tem proportio­
                    <lb/>
                  nalem eſt ſubduplus ad gradum initiantē ſecundã:
                    <lb/>
                  et ſic conſequenter (loquor de partibus proportiõa­
                    <lb/>
                  libus quantitatiuis) </s>
                  <s xml:id="N1E608" xml:space="preserve">Suppono vlterius /  ſubſexq̇­
                    <lb/>
                  tertium ad quadruplum alicuius eſt triplum ad il-
                    <lb/>
                  lud ſubquadruplum. </s>
                  <s xml:id="N1E60F" xml:space="preserve">Quod probatur facile / quia ſi
                    <lb/>
                  eſt ſubſexquitertium ad illud eſt tres quarte eius: et
                    <lb/>
                  ſubquadruplum ad illud quadruplum eſt vna quar­
                    <lb/>
                  ta: igitur illud ſubſexquitertium erit triplum ad il­
                    <lb/>
                  lud ſubquadruplum. </s>
                  <s xml:id="N1E61A" xml:space="preserve">Patet conſequentia / q2 triuꝫ
                    <lb/>
                  quartarum ad vnam quartam eſt ꝓportio tripla.
                    <lb/>
                  </s>
                  <s xml:id="N1E620" xml:space="preserve">His ſuppoſitis probatur ꝓpoſitio: et diuido vnam
                    <lb/>
                  talem latitudinem per partes ꝓportionales ꝓpor­
                    <lb/>
                  tione dupla: quo poſito arguitur ſic / gradus mediꝰ
                    <lb/>
                  medietatis intenſioris eſt triplus ad graduꝫ medi­
                    <lb/>
                  um medietatis remiſſioris et penes tales gradꝰ me­
                    <lb/>
                  tri habent velocitates illarum medietatū / vt dictū
                    <lb/>
                  eſt. </s>
                  <s xml:id="N1E62F" xml:space="preserve">igitur medietas intenſior eſt triple intenſionis
                    <lb/>
                  ad medietatem remiſſiorem / quod fuit probandum
                    <lb/>
                  </s>
                  <s xml:id="N1E635" xml:space="preserve">Patet conſequentia cuꝫ minore / et arguitur maior /
                    <lb/>
                  quia vt patet ex ſecunda ſuppoſitione gradus ini-
                    <lb/>
                  tians tertiã partem proportionalem eſt ſubduplꝰ
                    <lb/>
                  ad initiantem ſecundam: et intians ſecundam ad in­
                    <lb/>
                  itiantiantem primam: igitur initians primaꝫ eſt q̈­
                    <lb/>
                  druplus ad initiantem tertiam / vt patet ex prīa ſup­
                    <lb/>
                  poſitione: et ille eſt gradus medius ſecunde medie-
                    <lb/>
                  tatis puta remiſſioris: igitur gradus medius me-
                    <lb/>
                  dietatis remiſſioris ē ſubquadruplus ad extremuꝫ
                    <lb/>
                  intenſius medietatis intenſioris: et gradus mediꝰ
                    <lb/>
                  medietatis intenſioris eſt ſubſexquitertius ad ex-
                    <lb/>
                  tremum intenſius: ergo eſt triplus ad gradum me­
                    <lb/>
                  dium medietatis remiſſioris qui eſt ſubquadruplꝰ
                    <lb/>
                  ad extremum intenſius latitudinis. </s>
                  <s xml:id="N1E652" xml:space="preserve">Patet conſe-
                    <lb/>
                  quentia ex tertia ſuppoſitione. </s>
                  <s xml:id="N1E657" xml:space="preserve">Sed reſtat ꝓbare /
                    <lb/>
                   gradus medius medietatis ītenſioris eſt ſubſex­
                    <lb/>
                  quitertius ad extremum intenſius eiuſdcm medie­
                    <lb/>
                  tatis: </s>
                  <s xml:id="N1E660" xml:space="preserve">Quod probatur ſic / quia talis gradus ē me-
                    <lb/>
                  dius inter duplum et ſubduplum puta inter extre-
                    <lb/>
                  mum intenſius illius medietatis et extremuꝫ remiſ­
                    <lb/>
                  ſius eiuſdem qui eſt ſubduplus ad illum: igitur ta-
                    <lb/>
                  lis gradus medius eſt ſubſexquitertius ad illū du­
                    <lb/>
                  plum puta ad illud extremum intenſius / quod fuit
                    <lb/>
                  probandum. </s>
                  <s xml:id="N1E66F" xml:space="preserve">Patet conſequētia per hanc maximã
                    <lb/>
                  </s>
                  <s xml:id="N1E673" xml:space="preserve">Omnis gradus medius inter duplum et ſubduplū
                    <lb/>
                  eſt ſexquialterꝰ ad ſubduplum et ſexquitertius ad
                    <lb/>
                  duplum / vt patet de ſenario mediãte inter .4. et .8.
                    <lb/>
                  de ternario mediante inter binarium et quarterna­
                    <lb/>
                  rium et de nouenario mediante inter ſenariū et duo­
                    <lb/>
                  denarium: et vniuerſaliter in omnibus.</s>
                </p>
                <p xml:id="N1E680">
                  <s xml:id="N1E681" xml:space="preserve">Quarta ꝓpoſitio / que ſequit̄̄ ex priori
                    <lb/>
                  </s>
                  <s xml:id="N1E685" xml:space="preserve"/>
                </p>
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