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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                  <s xml:id="N22B33" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0191" n="191"/>
                  us minoris ipſum eſt denſius illo minori. </s>
                  <s xml:id="N22B3D" xml:space="preserve">Pro quo
                    <lb/>
                  intelligendo in ſuo fundamento: et radice ponã ali-
                    <lb/>
                  quas concluſiones: quadam diuiſione prepoſita q̄
                    <lb/>
                  talis eſt. </s>
                  <s xml:id="N22B46" xml:space="preserve">¶ Corporum ꝓportionabi­
                    <lb/>
                  liū ad inuicem in raritate et denſitate: quedam ſunt
                    <lb/>
                  equalia: quedam inequalia. </s>
                  <s xml:id="N22B4D" xml:space="preserve">Item equalium que-
                    <lb/>
                  dã cõtinēt equaliter de materia: quedam inequali-
                    <lb/>
                  ter. </s>
                  <s xml:id="N22B54" xml:space="preserve">Corporum inequalium quedam cõtinent equa-
                    <lb/>
                  liter de materia quedã vero nõ. </s>
                  <s xml:id="N22B59" xml:space="preserve">Exēplū / vt ſi ſint duo
                    <lb/>
                  corpora quorum vnū eſt pedale et aliud ſemipeda-
                    <lb/>
                  le poſſibile eſt  vnū tm̄ contineat de materia ſicut
                    <lb/>
                  aliud vel vnum cõtineat plus de materia ꝙ̄ aliud.
                    <lb/>
                  </s>
                  <s xml:id="N22B63" xml:space="preserve">Item corporum inequalium inequaliter contenen-
                    <lb/>
                  tiū de materia: quedam ita ſe habent  minus con­
                    <lb/>
                  tinet minus de materia: quedã ita ſe habent  mi-
                    <lb/>
                  nus continet magis de materia. </s>
                  <s xml:id="N22B6C" xml:space="preserve">Item minorum cõ­
                    <lb/>
                  tinentium minus quã maius: quoddam cõtinet mi-
                    <lb/>
                  nus in ea ꝓportione qua eſt minus: quoddã in ma-
                    <lb/>
                  iori ꝓportione: quoddã vero in minori. </s>
                  <s xml:id="N22B75" xml:space="preserve">Exemplum /
                    <lb/>
                  vt ſi ſint duo corpora quorum vnū eſt pedale aliud
                    <lb/>
                  ſemipedale poſſibile eſt  ſemipedale cõtineat ma­
                    <lb/>
                  teriam in duplo minorem: in triplo maiorem: et in
                    <lb/>
                  ſexquialtero minorē quã ↄ̨tineat pedale. </s>
                  <s xml:id="N22B80" xml:space="preserve">Itē corpo­
                    <lb/>
                  rum inequaliū quorū minus continet plus de mate­
                    <lb/>
                  ria ꝙ̄ maius. </s>
                  <s xml:id="N22B87" xml:space="preserve">quoddã cõtinet plus de materia quaꝫ
                    <lb/>
                  maius in equali ꝓportione qua eſt minus. </s>
                  <s xml:id="N22B8C" xml:space="preserve">quoddã
                    <lb/>
                  in maiori quoddã vero in minori ꝓportione quã ē
                    <lb/>
                  minus: </s>
                  <s xml:id="N22B93" xml:space="preserve">Exēelū / vt captis pedali et ſemipedali poſſi-
                    <lb/>
                  bile eſt  ſemipedale continet in duplo plus de ma­
                    <lb/>
                  teria quam pedale: </s>
                  <s xml:id="N22B9A" xml:space="preserve">Poſſibile ē  in triplo: poſſi­
                    <lb/>
                  bile eſt etiam  in ſexquialtero. </s>
                  <s xml:id="N22B9F" xml:space="preserve">His diuiſionibꝰ po­
                    <lb/>
                  ſitis pono aliquas concluſiones quarum</s>
                </p>
                <p xml:id="N22BA4">
                  <s xml:id="N22BA5" xml:space="preserve">Prima cõcluſio eſt hec. </s>
                  <s xml:id="N22BA8" xml:space="preserve">Corpora equa­
                    <lb/>
                  lia equaliter continentia de materia ſunt equaliter
                    <lb/>
                  rara et equaliter dēſa dūmõ ſint rara et denſa. </s>
                  <s xml:id="N22BAF" xml:space="preserve">Hec
                    <lb/>
                  concluſio patet ex diffinitionibus rari et denſi.</s>
                </p>
                <p xml:id="N22BB4">
                  <s xml:id="N22BB5" xml:space="preserve">Secunda concluſio </s>
                  <s xml:id="N22BB8" xml:space="preserve">Si aliqua duo in
                    <lb/>
                  equalia equaliter contineant de materia: minus il­
                    <lb/>
                  lorum in eadem ꝓportione eſt denſius in qua ē mi-
                    <lb/>
                  nus. </s>
                  <s xml:id="N22BC1" xml:space="preserve">Probat̄̄ hec concluſio et capio duo corpora in
                    <lb/>
                  equalia gratia exempli pedale et ſemipedale habē-
                    <lb/>
                  tia equaliter de materia / et volo /  ſemipedale rare­
                    <lb/>
                  fiat quovſ ſit pedale ſine acquiſitione aut deper-
                    <lb/>
                  ditione materie. </s>
                  <s xml:id="N22BCC" xml:space="preserve">quo poſito in fine illa duo corpora
                    <lb/>
                  ſunt eque rara et denſa / vt patet ex prima concluſio-
                    <lb/>
                  ne: et illud quod antea erat minus perdidit propor­
                    <lb/>
                  tionem duplam denſitatis cum acquiſiuerit duplã
                    <lb/>
                  raritatem / vt patet per duplam punctorum diſtan-
                    <lb/>
                  tiam ſine acquiſitione aut deperditione materie:
                    <lb/>
                  igitur antea erat in duplo denſius quã ſit modo: et
                    <lb/>
                  per conſequens in duplo denſius quolibet equali
                    <lb/>
                  modo in denſitate. </s>
                  <s xml:id="N22BDF" xml:space="preserve">quoniam in quacun ꝓportio-
                    <lb/>
                  ne aliquid excedit aliud in eadeꝫ ꝓportione excedit
                    <lb/>
                  quolibet equale illi: igitur concluſio vera:</s>
                </p>
                <p xml:id="N22BE6">
                  <s xml:id="N22BE7" xml:space="preserve">Tertia concluſio </s>
                  <s xml:id="N22BEA" xml:space="preserve">Si fuerint duo cor-
                    <lb/>
                  pora inequalia: et minus illorum cõtinet plus ḋ ma­
                    <lb/>
                  teria quã maius: tunc minus eſt denſius in propor-
                    <lb/>
                  tione compoſita ex proportione qua maius excedit
                    <lb/>
                  minus: et ex proportione qua materia minoris ex-
                    <lb/>
                  dit materiam maioris: </s>
                  <s xml:id="N22BF7" xml:space="preserve">Probatur et capio pedale
                    <lb/>
                  et ſemipedale quod cõtinet in duplo maigs de ma-
                    <lb/>
                  teria quã pedale: et volo /  illud ſemipedale rarefi-
                    <lb/>
                  at quouſ ſit bipedale: quo poſito arguitur ſic in fi­
                    <lb/>
                  ne talis rarefactionis illud corpꝰ quod antea erat
                    <lb/>
                  ſemipedale eſt eque denſum adequate cum alio cor­
                    <lb/>
                  pore pedali cū ſubdupla quãtitate duplã maṫiã cõ­
                    <lb/>
                  tiuet: et ipſum eſt in quadruplo minus denſum quã
                    <lb/>
                  erat antea cum modo puncta in quadruplo plꝰ di-
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                  ſtent etc. / igitur ipſum erat antea in quadruplo deu­
                    <lb/>
                  ſius quã ſit modo: et per conſequens in quadruplo
                    <lb/>
                  denſius quolibet quod eſt modo equale ei in den-
                    <lb/>
                  ſitate: igitur ipſum antea cum eſſet ſemipedale erat
                    <lb/>
                  in quadruplo denſius illo pedali: et proportio qua­
                    <lb/>
                  drupla eſt ꝓportio compoſita ex ꝓportione quãti-
                    <lb/>
                  tatis qua maius excedit minus puta dupla: et ex ꝓ-
                    <lb/>
                  portione qua materia minoris excedit materiam
                    <lb/>
                  maioris ſimiliter dupla / vt patet ex ſecunda parte
                    <lb/>
                  huius operis: igitur intentum. </s>
                  <s xml:id="N22C1F" xml:space="preserve">ſic enim vniuerſali-
                    <lb/>
                  ter probabis.</s>
                </p>
                <p xml:id="N22C24">
                  <s xml:id="N22C25" xml:space="preserve">Quarta concluſio </s>
                  <s xml:id="N22C28" xml:space="preserve">Si ſint duo corpo-
                    <lb/>
                  ra inequalia inequaliter continentia de materia.
                    <lb/>
                  </s>
                  <s xml:id="N22C2E" xml:space="preserve">ita  ī q̈cū ꝓportiõe minꝰ minus eſt ī eadē ꝓpor-
                    <lb/>
                  tione continet minus de materia. </s>
                  <s xml:id="N22C33" xml:space="preserve">talia corpora ſūt
                    <lb/>
                  equaliter denſa. </s>
                  <s xml:id="N22C38" xml:space="preserve">Patet hec concluſio de ſe quoniã
                    <lb/>
                  capto corpore pedali vniformiter denſo / manifeſtū
                    <lb/>
                  eſt /  medietas eius eſt eque denſa ſicut totum: et ſi-
                    <lb/>
                  cut medietas eſt in duplo minor ita in duplo minus
                    <lb/>
                  continet de materia. </s>
                  <s xml:id="N22C43" xml:space="preserve">Et iſto modo vniuerſaliter ꝓ-
                    <lb/>
                  babis de quibuſcun aliis proportionibus ſiue ra­
                    <lb/>
                  tionalibus ſiue non rationalibus</s>
                </p>
                <p xml:id="N22C4A">
                  <s xml:id="N22C4B" xml:space="preserve">Quinta concluſio </s>
                  <s xml:id="N22C4E" xml:space="preserve">Si ſint duo corpo-
                    <lb/>
                  ra inequalia: et minus contineat minus de materia
                    <lb/>
                  quam maius in maiore proportione quam ma-
                    <lb/>
                  ius excedat minus: tunc maiꝰ eſt deſius minore ī ea
                    <lb/>
                  ꝓportione qua ꝓportio materie ad materiam exce­
                    <lb/>
                  dit ꝓportionē quantitatū: </s>
                  <s xml:id="N22C5B" xml:space="preserve">Uel ſub aliis verbis ea-
                    <lb/>
                  dē rententa ſententia. </s>
                  <s xml:id="N22C60" xml:space="preserve">Si duorū corporum inequa-
                    <lb/>
                  liū ꝓportio materie maioris ad materiam mino-
                    <lb/>
                  ris excedit ꝓportionē quãtitatis ad quantitatem:
                    <lb/>
                  maius illorum eſt denſius in ꝓportione ꝑ quã pro-
                    <lb/>
                  portio materie maioris ad materiã minoris exce-
                    <lb/>
                  dit ꝓportionē quantitatū. </s>
                  <s xml:id="N22C6D" xml:space="preserve">Probat̄̄ hec concluſio
                    <lb/>
                  et capio duo corpora ſe habentia in ꝓportione du­
                    <lb/>
                  pla / et volo /  materia maioris ſit tripla ad materi­
                    <lb/>
                  am minoris quo poſito maius eſt denſius in ꝓpor­
                    <lb/>
                  tione ſexquialtera ꝑ quã ꝓportio tripla excedit du­
                    <lb/>
                  plam: igr̄ cõcluſio vera. </s>
                  <s xml:id="N22C7A" xml:space="preserve">Añs ꝓbatur: et pono /  cor­
                    <lb/>
                  pus maius condenſetur quovſ ſit equale minori
                    <lb/>
                  puta ad ſubduplū / quo poſito argr̄ ſic. </s>
                  <s xml:id="N22C81" xml:space="preserve">Illud corpꝰ
                    <lb/>
                  quod antea erat maius eſt in triplo denſius altero
                    <lb/>
                  corpore quod antea erat minus eo: et ꝑ talē cõdēſa-
                    <lb/>
                  tionē p̄ciſe acquiſiuit duplam denſitatem: ergo ſe-
                    <lb/>
                  quitur /  antea habebat ſexquialteram: igitur ip-
                    <lb/>
                  ſum erat ãtea in ꝓportione ſexq̇altera dēſiꝰ / qḋ fuit
                    <lb/>
                  ꝓbandū. </s>
                  <s xml:id="N22C90" xml:space="preserve">Sequela tamē ꝓbatur / q2 qñ aliq̇d efficit̄̄
                    <lb/>
                  in aliqua ꝓportiõe maiꝰ reſpectu alterius: et tūc ac­
                    <lb/>
                  quirit preciſe vnã partē talis ꝓportionis ſequitur /
                    <lb/>
                   iã antea habebat alterã ꝑtem: ſed tale corpꝰ acq̇-
                    <lb/>
                  ſiuit ꝓportionē triplã id eſt effectū eſt denſius ī pro­
                    <lb/>
                  portione tripla: et nõ acq̇ſiuit niſi duplã: ergo ſequi­
                    <lb/>
                  tur /  iã antea habebat adequate ſexquialterã: qm̄
                    <lb/>
                  tripla ex dupla et ſexquialtera cõponit̄̄ adequate.
                    <lb/>
                  </s>
                  <s xml:id="N22CA2" xml:space="preserve">Et iſto mõ ꝓbabis de q̇buſcū aliis ꝓportiõibus.</s>
                </p>
                <p xml:id="N22CA5">
                  <s xml:id="N22CA6" xml:space="preserve">Sexta concluſio </s>
                  <s xml:id="N22CA9" xml:space="preserve">Si fuerint duo cor­
                    <lb/>
                  pora inequalia: et ꝓportio quantitatū fuerit ma-
                    <lb/>
                  ior proportione materie maioris ad materiã mi-
                    <lb/>
                  noris. </s>
                  <s xml:id="N22CB2" xml:space="preserve">tunc minus eſt denſius maiori in ꝓportione
                    <lb/>
                  qua proportio quantitatis excedit ꝓportionē ma-
                    <lb/>
                  terie. </s>
                  <s xml:id="N22CB9" xml:space="preserve">Probat̄̄ hec concluſio: et volo /  ſint duo cor-
                    <lb/>
                  pora puta pedale et bipedale: et bipedale in ſexqui-
                    <lb/>
                  altero plus cõtineat de materia ꝙ̄ pedale: tūc dico /
                    <lb/>
                   pedale eſt denſius bipedali in ꝓportione ſexqui­
                    <lb/>
                  tertia. </s>
                  <s xml:id="N22CC4" xml:space="preserve">quoniam ꝑ talem ꝓportionē ſexquitertiam
                    <lb/>
                  ꝓportio quãtitatis maioris ad quãtitatē minoris
                    <lb/>
                  q̄ ē dupla excedit ꝓportionē maṫie maiorꝪ ad maṫi­
                    <lb/>
                  am minoris q̄ ē ſexq̇altera / vt ↄ̨ſtat </s>
                  <s xml:id="N22CCD" xml:space="preserve">Probat̄̄ hoc ſic
                    <lb/>
                  </s>
                  <s xml:id="N22CD1" xml:space="preserve"/>
                </p>
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