Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N20F21" level="3" n="3" type="other" type-free="tractatus">
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                <p xml:id="N225D0">
                  <s xml:id="N225E8" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0188" n="188"/>
                  quoniã ſignetur illud et ſit vnū pedale / et arguo ſic /
                    <lb/>
                  illud pedale eſt infinite rarum: igitur in eo eſt infi-
                    <lb/>
                  nita ꝓportio quantitatis ad materiam: ſed quã-
                    <lb/>
                  titas eſt finita: ergo materia eſt infinite modica:
                    <lb/>
                  ſed non eſt dabilis materia infinite modica: igitur
                    <lb/>
                  eo nulla eſt materia vel ipſum nõ eſt infinite rarum
                    <lb/>
                  ſed non eſt dicendum  in eo nulla eſt materia: er-
                    <lb/>
                  go eſt dicendum  non eſt infinite rarum / quod
                    <lb/>
                  fuit probandum.</s>
                </p>
                <p xml:id="N22600">
                  <s xml:id="N22601" xml:space="preserve">In oppoſitū tamen arguitur ſic / quia
                    <lb/>
                  hec apinio eſt adeo ſuſtentabilis et rationabilis ſi­
                    <lb/>
                  cut ſecunda: ergo eo modo poteſt deffenſari vera
                    <lb/>
                  ſicut ſecunda. </s>
                  <s xml:id="N2260A" xml:space="preserve">Antecedens patebit ſoluendo. </s>
                  <s xml:id="N2260D" xml:space="preserve">ea que
                    <lb/>
                  hanc poſitionem opugnant.</s>
                </p>
                <p xml:id="N22612">
                  <s xml:id="N22613" xml:space="preserve">Pro ſolutione huius dubitationis:
                    <lb/>
                  et exacta huius opinionis inquiſitione. </s>
                  <s xml:id="N22618" xml:space="preserve">Cõſideran­
                    <lb/>
                  dum eſt /  in hac opinioue ſicut et in aliis, peculia-
                    <lb/>
                  ribus definitionibus raritatis et denſitatis ſiue ra­
                    <lb/>
                  ri et denſi vtendum eſt. </s>
                  <s xml:id="N22621" xml:space="preserve">Cum enim hec opinio dicat
                    <lb/>
                  ad raritatem requiri proportionem maioris ine-
                    <lb/>
                  qualitatis quantitatis ad materiam: et ad denſita­
                    <lb/>
                  tem ecõtra requiri proportionem maioris inequa-
                    <lb/>
                  litatis materie ad quantitatem id ſignum nobis
                    <lb/>
                  erit, et fidem faciet rarum hoc pacto diffiniri debe-
                    <lb/>
                  re.
                    <note position="left" xlink:href="note-0188-01a" xlink:label="note-0188-01" xml:id="N226D0" xml:space="preserve">q̇d rarū</note>
                  </s>
                  <s xml:id="N22635" xml:space="preserve">Rarum eſt illud in quo eſt proportio maioris in­
                    <lb/>
                  equalitatis quantitatis ad materiam.
                    <note position="left" xlink:href="note-0188-02a" xlink:label="note-0188-02" xml:id="N226D6" xml:space="preserve">q̇d dēſuꝫ.</note>
                  </s>
                  <s xml:id="N2263F" xml:space="preserve">Denſum ve-
                    <lb/>
                  ro ita deſcribi debet. </s>
                  <s xml:id="N22644" xml:space="preserve">denſum eſt illud in quo eſt ꝓ-
                    <lb/>
                  portio maioris inequalitatis materie ad quanti-
                    <lb/>
                  tatem. </s>
                  <s xml:id="N2264B" xml:space="preserve">Aliter tamen poſſunt iſti termini ſic deſcri-
                    <lb/>
                  bi manente eadem ſententia paululum verbis va-
                    <lb/>
                  riatis. </s>
                  <s xml:id="N22652" xml:space="preserve">Rarum eſt cuius quãtitas eiuſdem materi-
                    <lb/>
                  am exuperat. </s>
                  <s xml:id="N22657" xml:space="preserve">Denſum vero eſt cuius materia ſuam
                    <lb/>
                  excedit quantitatem. </s>
                  <s xml:id="N2265C" xml:space="preserve">Quo in loco intelligendum
                    <lb/>
                  eſt hanc opinionem, et materie, et quantitati gra-
                    <lb/>
                  dus aſcribere: nõ quidem intenſionales: ita  ipſa
                    <lb/>
                  quantitas ſit intenſa, aut ipſa materia, velut albe­
                    <lb/>
                  do ſiue nigredo: ſed habet certas partes ſue ſubſtã­
                    <lb/>
                  tie ſiue entitatis ipſa materia: et ſimiliter ipſa quã­
                    <lb/>
                  titas certas portiones quas iſta opinio gradꝰ ap-
                    <lb/>
                  pellat: vt ſi dicamus quartã partem vnius pedalis
                    <lb/>
                  vnū gradum quantitatis eſſe, et medietatē quarte
                    <lb/>
                  mediū gradum quantitatis, et ſic cõſequenter: tunc
                    <lb/>
                  recte dicemus pedale quatuor gradus quãtitatis
                    <lb/>
                  cõtinere, et bipedale octo, et ſic cõſequēter, et pari in­
                    <lb/>
                  duſtria nõ abs re aſſignauerit hec opinio ipſa ma-
                    <lb/>
                  terie gradus: vt ſi dicamꝰ mariam exiſtentē in vna
                    <lb/>
                  octaua parte pedalis terre exñtis in ſua naturali
                    <lb/>
                  diſpoſitiõe eſſe vnū gradū materie, et medietatem
                    <lb/>
                  illiꝰ materie vnū mediū gradū, et ſic ↄ̨ñter diuiden-
                    <lb/>
                  do ex ↄ̨ñti manifeſtū nobis eſſet vnū pcdale terre in
                    <lb/>
                  ſua naturali, et optima diſpoſitione exiſtēs .8. gra­
                    <lb/>
                  dus materie ↄ̨tinere, et bipedale terre decē et ſex, et
                    <lb/>
                  ſic ↄ̨ñter aſcēdendo: et iſto mõ aſſignãdo g̈dus et ip̄i
                    <lb/>
                  materie et quãtitati facile erit inſpicere qñ gradus
                    <lb/>
                  quãtitatis excedunt gradꝰ materie: aut econtra, et
                    <lb/>
                  ſic iuidicare: vtrū tale corp° debeat dici dēſum, aut
                    <lb/>
                  nõ. </s>
                  <s xml:id="N2268F" xml:space="preserve">Nã ſcḋm hanc opinionē nullū dēſum eſt rarum
                    <lb/>
                  nec rarū eſt dēſum. </s>
                  <s xml:id="N22694" xml:space="preserve">Qḋ ſic patꝫ manifeſte. </s>
                  <s xml:id="N22697" xml:space="preserve">Si em̄ a.
                    <lb/>
                  eſt dēſum gradꝰ materie ipſiꝰ a. exuperant gradus
                    <lb/>
                  quãtitatis eiꝰ. </s>
                  <s xml:id="N2269E" xml:space="preserve">Si vero ip̄m a. ſit rarū iam gradus
                    <lb/>
                  quãtitatis gradꝰ materie exuperãt: ſed īpoſſibile ē
                    <lb/>
                   idē ſit maiꝰ altero: et ecõtra. </s>
                  <s xml:id="N226A5" xml:space="preserve">Ideo nõ eſt poſſibile
                    <lb/>
                  huic opinioni adherēdo idē ſimul fater rarū et dē-
                    <lb/>
                  ſum vel ſaltē in eodē loco etc̈. </s>
                  <s xml:id="N226AC" xml:space="preserve">Sequit̄̄ ſecūdo iuxta
                    <lb/>
                  hanc opinionē  nullū infinitarū vbi eſt infinitum de
                    <lb/>
                  materia eſt rarū aut denſum. </s>
                  <s xml:id="N226B3" xml:space="preserve">Patet / q2 ibi, nec ma­
                    <lb/>
                  teria exuperat quantitatē, nec ab ea ſuperatur: vt
                    <lb/>
                  conſtat. </s>
                  <s xml:id="N226BA" xml:space="preserve">Sequitur tertio /  aliquod finitū eſt quod
                    <cb chead="Capitulū primum."/>
                  nec eſt rarū, nec denſum: et tamen habet materiam
                    <lb/>
                  </s>
                  <s xml:id="N226C1" xml:space="preserve">Patet de pedali habēte quatuor gradus materie
                    <lb/>
                  eſto /  quarta pedalis ſit vnus gradus quantitatꝪ
                    <lb/>
                  </s>
                  <s xml:id="N226C7" xml:space="preserve">In tali enim pedali, nec quantitas excedit mate-
                    <lb/>
                  riam, nec ab ea exceditur.</s>
                </p>
                <p xml:id="N226DC">
                  <s xml:id="N226DD" xml:space="preserve">Aduertendum eſt ſecundo /  diuerſi-
                    <lb/>
                  mode hec opinio, et communis qui ī ſequenti no-
                    <lb/>
                  tabili declarabitur cenſent raritatem duplari, tri­
                    <lb/>
                  plari: aut in aliqua alia proportione augeri. </s>
                  <s xml:id="N226E6" xml:space="preserve">Nam
                    <lb/>
                  opinio cõmunis aſſeuerat ad duplationem quan-
                    <lb/>
                  titatis ſequi duplationem raritatis: et econtra ad
                    <lb/>
                  duplationem raritatis ſequi duplationem quan-
                    <lb/>
                  titatis. </s>
                  <s xml:id="N226F1" xml:space="preserve">Hec vero opinio oppoſitum dicit. </s>
                  <s xml:id="N226F4" xml:space="preserve">Ali-
                    <lb/>
                  quando enim ad duplationem raritatis dupla-
                    <lb/>
                  tur quantitas, aliquando vero efficitur in ſexqui-
                    <lb/>
                  altero maior dumtaxat. </s>
                  <s xml:id="N226FD" xml:space="preserve">vt ſecundum huius prin-
                    <lb/>
                  cipalis queſtionis argumentum oſtendit. </s>
                  <s xml:id="N22702" xml:space="preserve">Unum ta­
                    <lb/>
                  men certum habet hec opinio: dicit enī ſemper ad
                    <lb/>
                  dupla tionem raritatis ſequi duplationē propor-
                    <lb/>
                  tionis quantitatis ad materiam: vt ſi ipſa propor­
                    <lb/>
                  tio quantitatis ad materiã fuerit dupla: duplata
                    <lb/>
                  raritate erit quadrupla: et ſi fuerit quadrupla: du-
                    <lb/>
                  plata raritate erit ſexdecupla. </s>
                  <s xml:id="N22711" xml:space="preserve">Si autem tripla du­
                    <lb/>
                  plata raritate erit nonocupla. </s>
                  <s xml:id="N22716" xml:space="preserve">ſi vero fuerit ſexqui­
                    <lb/>
                  altera: duplata raritate erit dupla ſexquiquarta:
                    <lb/>
                  et ſic in aliis exemplificandum eſt.</s>
                </p>
                <note position="right" xml:id="N2271D" xml:space="preserve">1. correĺ.</note>
                <p xml:id="N22721">
                  <s xml:id="N22722" xml:space="preserve">¶ Ex quo educitur clare /  ſi quantitatis ad ma-
                    <lb/>
                  teriam fuerit proportio minor dupla: duplata ra-
                    <lb/>
                  ritate nequaquam duplabitur quantitas: ſed mi-
                    <lb/>
                  nus quam ad duplam augebitur: quemadmo-
                    <lb/>
                  dum promptum eſt in proportione ſexquitertia
                    <lb/>
                  intueri. </s>
                  <s xml:id="N2272F" xml:space="preserve">Si veruo fuerit proportio maior dupla
                    <lb/>
                  neceſſum erit quantitatem pluſ̄ ad duplum au-
                    <lb/>
                  geri. </s>
                  <s xml:id="N22736" xml:space="preserve">Si autem fuerit dupla dumtaxat quanti-
                    <lb/>
                  tatis ad materiam proportio: raritate dupla-
                    <lb/>
                  ta quantitas ipſa dupla euadet dumtaxat. </s>
                  <s xml:id="N2273D" xml:space="preserve">Patet
                    <lb/>
                  hoc correlarium in ſingulis inducenti. </s>
                  <s xml:id="N22742" xml:space="preserve">Ipſum enim
                    <lb/>
                  correlariū mathematico ordine et apparatu oſten­
                    <lb/>
                  dere ſiue demõſtrare maiori ſollicitudini eſſet quã
                    <lb/>
                  huic opinioni adiumento. </s>
                  <s xml:id="N2274B" xml:space="preserve">Radix tamen et baſis hu­
                    <lb/>
                  ius opinionis eſt: ex qua baſi facile ea que ab hac
                    <lb/>
                  opinione aſſeuerantur claram ſortiuntur demon-
                    <lb/>
                  ſtrationem. </s>
                  <s xml:id="N22754" xml:space="preserve">Eſt em̄ hoc fundamentum: cuilibet pro­
                    <lb/>
                  portioni quantitatis ad materiam determinati
                    <lb/>
                  gradus raritatis correſpondent: itidem et cuilibet
                    <lb/>
                  proportioni materie ad quantitatem determinati
                    <lb/>
                  gradus denſitatis correſpondent: perinde at in
                    <lb/>
                  motus velocitate certe proportioni potentie ad re­
                    <lb/>
                  ſiſtentiam certa motuum velocitas correſpondet:
                    <lb/>
                  et duple proportioni dupla motus velocitas: et ſex­
                    <lb/>
                  quialtere proportioni ſexquialtera velocitas aſcri­
                    <lb/>
                  bitur: volo dicere /  ſecundum hanc opinionē pro-
                    <lb/>
                  portioni duple quantitatis ad materiam corre-
                    <lb/>
                  ſpondent certi gradus raritatis qui gratia exem­
                    <lb/>
                  pli ſint duo, ita videlicet  vbicun ſiue in magno
                    <lb/>
                  corpore ſiue in paruo dupla proportio quantita-
                    <lb/>
                  tis ad materiam reperiatur iudicabitur tale cor-
                    <lb/>
                  pus rarum adequate vt duo: et vbicun reperietur
                    <lb/>
                  proportio quadrupla quantitatis ad materiam
                    <lb/>
                  raritas erit vt .4. quoniam proportio quadrupla
                    <lb/>
                  dupla eſt ad ipſam duplam: et ſic conſequenter tu
                    <lb/>
                  poteris exemplicare in aliis proportionum ſpe-
                    <lb/>
                  ciebus et generibus.</s>
                </p>
                <p xml:id="N2277F">
                  <s xml:id="N22780" xml:space="preserve">¶ Ex quo ſequitur /  raritas proueniens a pro-
                    <lb/>
                  portione tripla non ſe habet in aliqua proportio-
                    <lb/>
                  ne rationali ad raritatem prouenientem a propor­
                    <lb/>
                  tione dupla. </s>
                  <s xml:id="N22789" xml:space="preserve">Quod ptꝫ / q2 ꝓportio dupla et tripla
                    <lb/>
                  nõ ſe hñt ī in ꝓportiõe rõnali / igitur nec raritas pro-
                    <lb/>
                  ueniens a ꝓportione dupla ad raritatē ꝓuenieutē </s>
                </p>
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