Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N22CA5">
                  <s xml:id="N22CD1" xml:space="preserve">
                    <pb chead="De motu rarefactionis  condenſationis." file="0192" n="192"/>
                  Qm̄ ſi materia corporis minoris ꝑderet ꝓportio-
                    <lb/>
                  nē ſexquitertiã ſue materie ſtante quantitate: tunc
                    <lb/>
                  maius et minꝰ eſſent eque denſa / vt ptꝫ ex quarta cõ­
                    <lb/>
                  cluſione. </s>
                  <s xml:id="N22CDE" xml:space="preserve">In ea em̄ ꝓportione qua minꝰ eſt minꝰ in
                    <lb/>
                  ea minꝰ ↄ̨tineret de materia. </s>
                  <s xml:id="N22CE3" xml:space="preserve">Sed modo illud corpꝰ
                    <lb/>
                  minꝰ in ſexq̇tertio plus de materia cõtinet denſius
                    <lb/>
                  quã tūc: et tunc erat ita denſum ſicut modo eſt illud
                    <lb/>
                  bipedale: g̊ modo in ſexq̇tertio eſt denſiꝰ illo bipe-
                    <lb/>
                  dali: et ꝓportio ſexquitertia eſt illa ꝑ quã ꝓportio
                    <lb/>
                  quãtitatis maioris ad quantitatē minoris excedit
                    <lb/>
                  ꝓportionē materie maioris ad materiã minoris: g̊
                    <lb/>
                  ꝑ ↄ̨ñs minꝰ eſt denſius maiore in ꝓportione ꝑ quantuꝫ
                    <lb/>
                  ꝓportio quantitatis maioris ad quantitatē mino­
                    <lb/>
                  ris excedit ꝓportionē materie maioris ad materiã
                    <lb/>
                  minoris. </s>
                  <s xml:id="N22CFA" xml:space="preserve">Et ſic ꝓbabis q̇buſcū duabꝰ ꝓportiõibꝰ
                    <lb/>
                  ̄titatū et materieꝝ īeq̈libꝰ ꝓpoſitꝪ ī caſu ↄ̨cluſiõis</s>
                </p>
                <p xml:id="N22CFF">
                  <s xml:id="N22D00" xml:space="preserve">Ultima cõcluſio. </s>
                  <s xml:id="N22D03" xml:space="preserve">Si duoꝝ corporum
                    <lb/>
                  inequaliū ꝓportio quantitatis ad quantitatē ſiue
                    <lb/>
                  materie ad materiã fuerit irrationalis: tūc ꝓpor-
                    <lb/>
                  tio raritatis vniꝰ et denſitatis ſimiliter ad denſita­
                    <lb/>
                  tem et raritatē alteriꝰ eſt irratiõalis. </s>
                  <s xml:id="N22D0E" xml:space="preserve">Probat̄̄ / ſicut
                    <lb/>
                  concluſio qm̄ ꝓportio quantitatis vniꝰ ad quan-
                    <lb/>
                  titatē alteriꝰ nõ denoīatur ab aliquo certo numero
                    <lb/>
                  ita etiã diſtantia punctoꝝ nõ denoīatur ab aliquo
                    <lb/>
                  certo numero: et ꝑ ↄ̨ñs iam ꝓportio raritatis vnius
                    <lb/>
                  ad raritatē alteriꝰ eſt irratiõalis / ptꝫ ↄ̨ña ꝑ diffini-
                    <lb/>
                  tioneꝫ ꝓportiõis irratiõalis in ṗma ꝑte huiꝰ oꝑis.</s>
                </p>
                <p xml:id="N22D1D">
                  <s xml:id="N22D1E" xml:space="preserve">Notãnda eſt quarto / q̄dã diuiſio dēſita­
                    <lb/>
                  tū partibꝰ alicuiꝰ ſubiecti inherentiū q̄ diuiſio huic
                    <lb/>
                  materie multū claritatis et vtilitatis affert: ex qua
                    <lb/>
                  ꝓpoſitiones nõ nulle deducūtur: ex quibꝰ ꝓpoſiti-
                    <lb/>
                  onibus quedã cõcluſiones huiꝰ materie ſubtilitatē
                    <lb/>
                  cõprehendētes naſcūtur. </s>
                  <s xml:id="N22D2B" xml:space="preserve">Diuiſio vero ſub his ver-
                    <lb/>
                  bis deſcribetur. </s>
                  <s xml:id="N22D30" xml:space="preserve">¶ Denſitates per diuerſas partes
                    <lb/>
                  ſubiecti diſtribute qñ ſūt equales in gradu: ſepiꝰ
                    <lb/>
                  o īequales. </s>
                  <s xml:id="N22D37" xml:space="preserve">Exemplū primi: vt ſi vtra medietas
                    <lb/>
                  vniꝰ pedalis ſit denſa vt .4. </s>
                  <s xml:id="N22D3C" xml:space="preserve">Exemplū ſecūdi: vt ſi al­
                    <lb/>
                  tera medietas ſit vt .8. et altera vt .4. </s>
                  <s xml:id="N22D41" xml:space="preserve">Itē ſi ſūt equa­
                    <lb/>
                  les in gradu ipſe denſitates, aut extendūtur parti­
                    <lb/>
                  bus ſubiecti equalibꝰ, aut īequalibus. </s>
                  <s xml:id="N22D48" xml:space="preserve">Exempla in
                    <lb/>
                  prõptu ſunt. </s>
                  <s xml:id="N22D4D" xml:space="preserve">Itē ſi ſunt inequales in gradu: aut per
                    <lb/>
                  partes equales ſubiecti extendūtur, aut ꝑ īequales
                    <lb/>
                  </s>
                  <s xml:id="N22D53" xml:space="preserve">Preterea ſi denſitates inequales inequalibꝰ par-
                    <lb/>
                  tibus ſubiecti inhereãt: hoc cõtinget dupliciter: q2
                    <lb/>
                  aut maior denſitas maiori parti inheret, aut mino­
                    <lb/>
                  ri. </s>
                  <s xml:id="N22D5C" xml:space="preserve">Exemplū primi / vt ſi denſitas vt .4. inhereat ſiue
                    <lb/>
                  coextendatur medietati pedalis: et dēſitas vt .3. vni
                    <lb/>
                  q̈rte eiuſdē pedalis. </s>
                  <s xml:id="N22D63" xml:space="preserve">Prepoſtero ordine denſitates
                    <lb/>
                  illis partibus diſtribuendo. </s>
                  <s xml:id="N22D68" xml:space="preserve">exemplum ſecūdi mē-
                    <lb/>
                  bri patebit. </s>
                  <s xml:id="N22D6D" xml:space="preserve">Itē ſi ītenſior dēſitas parti ſubiecti mi­
                    <lb/>
                  nori aſſcribitur et remiſſior denſitas maiori parti:
                    <lb/>
                  hoc tripliciter euenire ſolet: q2 aut ꝓportio illarū
                    <lb/>
                  partiū ſubiecti ꝓportionē illaꝝ denſitatū excedit,
                    <lb/>
                  aut ꝓportio denſitatū proportionē partiū ſubiecti
                    <lb/>
                  excedit. </s>
                  <s xml:id="N22D7A" xml:space="preserve">aut ꝓportio illaꝝ partiū eſt equalis ꝓpor-
                    <lb/>
                  tioni denſitatū. </s>
                  <s xml:id="N22D7F" xml:space="preserve">Exemplū primi / vt ſi in vna medie-
                    <lb/>
                  tate pedalis ponat̄̄ denſitas vt .8. et in vna quarta
                    <lb/>
                  denſitas vt .12. tūc ꝓportio partiū eſt maior ꝓpor-
                    <lb/>
                  tione denſitatū. </s>
                  <s xml:id="N22D88" xml:space="preserve">Nã hec ſexquialtera eſt, illa auteꝫ
                    <lb/>
                  dupla. </s>
                  <s xml:id="N22D8D" xml:space="preserve">Exemplum ſecūdi / vt ſi in medietate ſubiecti
                    <lb/>
                  ponatur denſitas vt .4. et in quarta ponat̄̄ dēſitas
                    <lb/>
                  vt .12. tunc ꝓportio denſitatū excedit ꝓportionem
                    <lb/>
                  partiū ſubiecti: </s>
                  <s xml:id="N22D96" xml:space="preserve">Nã hec dupla eſt: illa vero tripla vt
                    <lb/>
                  conſtat. </s>
                  <s xml:id="N22D9B" xml:space="preserve">Exemplū tertii / vt ſi in vna tertia ponatur
                    <lb/>
                  denſitas vt .6. et in vna ſexta denſitas vt .12. tūc ea-
                    <lb/>
                  dem eſt ꝓportio illaꝝ partiū, et etiã illaꝝ denſita-
                    <lb/>
                  tum. </s>
                  <s xml:id="N22DA4" xml:space="preserve">Utra em̄ dupla eſt. </s>
                  <s xml:id="N22DA7" xml:space="preserve">Hac partitione ſiue diui-
                    <cb chead="De motu rarefactionis  condenſationis."/>
                  ſione exacta at conſūmata: reſtat quaſdē ꝓpoſi-
                    <lb/>
                  tiones preambulas ſequentiū cõcluſionū probare</s>
                </p>
                <p xml:id="N22DAF">
                  <s xml:id="N22DB0" xml:space="preserve">Prima ꝓpoſitio. </s>
                  <s xml:id="N22DB3" xml:space="preserve">Si denſitates eque
                    <lb/>
                  intenſe ſiue gradu equales (quod idē eſt) partibus
                    <lb/>
                  eiuſdē ſubiecti extendatur equalibus: ipſe equali-
                    <lb/>
                  ter totū denominãt. </s>
                  <s xml:id="N22DBC" xml:space="preserve">Si o partibus ſubiecti ineq̈-
                    <lb/>
                  libus aſſcribant̄̄: tūc illa deuſitas q̄ maiori parti
                    <lb/>
                  ſubiecti aſſcribit̄̄ plus totū ipſuꝫ ſubiectū denoīat
                    <lb/>
                  in ꝓportione in qua ſe hñt ille partes ſubiecti ad ī-
                    <lb/>
                  uicē: vt ſi denſitas vt .4. ſit in vna medietate alicuiꝰ
                    <lb/>
                  ſubiecti: et tanta denſitas intenſiue ſit in vna quar-
                    <lb/>
                  ta eiuſdē ſubiecti: tūc in duplo plus denomīat totū
                    <lb/>
                  ilud ſubiectū denſitas ī medietate quã denſitas in
                    <lb/>
                  quarta: q2 medietatis ad quartã eſt ꝓportio dupla
                    <lb/>
                  </s>
                  <s xml:id="N22DD0" xml:space="preserve">Probatur tñ ſecūda pars huiꝰ ꝓpoſitionis (quia
                    <lb/>
                  prima ex ſe ptꝫ) qm̄ ex poſitione quã iam ſuſtinemꝰ
                    <lb/>
                  et p̄cedenti notabili recitauimꝰ / ptꝫ /  denſitas exi-
                    <lb/>
                  ſtens in parte ſubiecti in ea ꝓportione minꝰ deno-
                    <lb/>
                  minat ſuū ſubiectū in qua eſt in minori parte ſubie­
                    <lb/>
                  cti: igr̄ in quacū ꝓportione aliq̈ denſitas per ma-
                    <lb/>
                  iorem partem alicuius ſubiecti extenditur quã alia
                    <lb/>
                  em̄ equalis in gradu: in eadē ꝓportione plus ſuum
                    <lb/>
                  ſubiectū denominat / quod fuit probandum.</s>
                </p>
                <p xml:id="N22DE3">
                  <s xml:id="N22DE4" xml:space="preserve">Scḋa ꝓpoſitio. </s>
                  <s xml:id="N22DE7" xml:space="preserve">Qñ inequales denſi­
                    <lb/>
                  tates equalibus partibus ſubtecti inherent: tūc in­
                    <lb/>
                  tenſior denſitas in ea ꝓportione plus denominat
                    <lb/>
                  totū ſubiectū in qua eſt intenſior. </s>
                  <s xml:id="N22DF0" xml:space="preserve">Probat̄̄ / qm̄ ſi il-
                    <lb/>
                  le denſitas eſſent equales in gradu cum inhereant
                    <lb/>
                  partibus equalibus ipſum equaliter totū denſum
                    <lb/>
                  denominarēt: vt docet prior pars p̄cedentis cõclu-
                    <lb/>
                  ſionis: ſed modo vna illaꝝ denſitatū eſt intēſior in
                    <lb/>
                  f. ꝓportione exempli gratia et ſicut eſt intenſior ita
                    <lb/>
                  plus denoīat ceteris paribus: igr̄ in f. ꝓportione
                    <lb/>
                  plus denoīat ꝙ̄ reliqua, et in f. ꝓportione eſt inten-
                    <lb/>
                  ſior / vt ponitur: igr̄ in ea ꝓportiõe in qua intenſior
                    <lb/>
                  plus totū ſubiectū denoīat / quod fuit probandum.</s>
                </p>
                <p xml:id="N22E05">
                  <s xml:id="N22E06" xml:space="preserve">Tertia ꝓpoſitio. </s>
                  <s xml:id="N22E09" xml:space="preserve">Si inequales den-
                    <lb/>
                  ſitates in gradu partibus eiuſdē ſubiecti inequali­
                    <lb/>
                  bus accõmodant̄̄, et intenſior maiori parti depute­
                    <lb/>
                  tur remiſſior vero minori: tunc intenſior denſitas
                    <lb/>
                  plus denominant totū ꝙ̄ remiſſior in ꝓportione cõ-
                    <lb/>
                  poſita ex ꝓportione partis maioris ad partē mi-
                    <lb/>
                  norē, et denſitatis intenſioris ad denſitatē remiſſi-
                    <lb/>
                  orē. </s>
                  <s xml:id="N22E1A" xml:space="preserve">Exemplū / vt ſi in vna medietate pedalis ponat̄̄
                    <lb/>
                  denſitas vt .4. et in quarta eiuſdē ponat̄̄ denſitas
                    <lb/>
                  vt .2. / tūc dico intenſionē exiſtentē in medietate ſub-
                    <lb/>
                  iecti in quadruplo plus denominare illud ſubiectū
                    <lb/>
                  denſitate exiſtente in quarta eiuſdē ſubiecti: qm̄ ꝓ-
                    <lb/>
                  portio illaꝝ partiū et etiã denſitatū eſt dupla et ſic
                    <lb/>
                  cõpoſita ex illis duplis eſt quadrupla: vt ptꝫ. </s>
                  <s xml:id="N22E29" xml:space="preserve">Pro­
                    <lb/>
                  batur tñ hec ꝓpoſitio vniuerſaliter: et ſit a. dēſitas
                    <lb/>
                  intenſior ꝑ maiorē partē extenſa b.o remiſſior ꝑ
                    <lb/>
                  minorē partē extenſa: tūc a. denſitas denoīat ſub-
                    <lb/>
                  iectū totale pluſ̄ b. denſitas in ꝓportione cõpoſi-
                    <lb/>
                  ta ex ꝓportione partis in qua eſt a. ad partē in qua
                    <lb/>
                  eſt b. q̄ ꝓportio ſit c. et ex ꝓportiõe denſitatis a. ad
                    <lb/>
                  dēſitatē b. q̄ ꝓportio ſit d. </s>
                  <s xml:id="N22E3A" xml:space="preserve">Qḋ ſic oſtenditur / q2 ſi a.
                    <lb/>
                  denſitas eſſet equalis b. denſitati tūc a. plus deno-
                    <lb/>
                  minaret ſubiectū ꝙ̄ b. in ꝓportione c. q̄ eſt ꝓportio
                    <lb/>
                  partiū. </s>
                  <s xml:id="N22E43" xml:space="preserve">vt pꝫ ex ſecūda parte prime cõcluſionis: ſꝫ
                    <lb/>
                  modo a. eſt intenſior denſitas quam tunc eſſet in d.
                    <lb/>
                  ꝓportione q̄ eſt ꝓportio illaꝝ denſitatū: igr̄ modo
                    <lb/>
                  in d. ꝓportione plus denoīat totū quã tūc. </s>
                  <s xml:id="N22E4C" xml:space="preserve">Ptꝫ tñ
                    <lb/>
                  hec ↄ̨ña / q2 quãto aliqua denſitas eſt intenſior cete­
                    <lb/>
                  ris paribus exiſtēs in aliqua parte ſubiecti, tanto
                    <lb/>
                  plꝰ facit ad denoīationē ſui ſubiecti vt tenet hec po­
                    <lb/>
                  ſitio: igr̄ nūc a. denſitas plus facit ad denoīationē </s>
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