Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N236A4">
                  <s xml:id="N236A8" xml:space="preserve">
                    <pb chead="De motu rarefactionis  cõdenſationis." file="0198" n="198"/>
                  acquirit vel deperdit de quantitate.</s>
                </p>
                <p xml:id="N236B6">
                  <s xml:id="N236B7" xml:space="preserve">Tertium. </s>
                  <s xml:id="N236BA" xml:space="preserve">Si ſint duo inequalia in
                    <lb/>
                  quantitate et denſitate et ſicut vnum eſt alio maius
                    <lb/>
                  ita ſit eo denſius que eque velociter acquirant vel
                    <lb/>
                  deperdant de denſitate: eque velocitrr acquirunt
                    <lb/>
                  vel deperdunt de quantitate.</s>
                </p>
                <p xml:id="N236C5">
                  <s xml:id="N236C6" xml:space="preserve">Quartum notabile. </s>
                  <s xml:id="N236C9" xml:space="preserve">Si ſint duo ine-
                    <lb/>
                  qualia et inequaliter denſa ita tamen  maior ſit
                    <lb/>
                  proportio quantitatis vnius ad quantitatem al-
                    <lb/>
                  terius ꝙ̄ denſitatis vnius ad denſitatem alterius
                    <lb/>
                  que eque velociter acquirant vel deperdãt de den-
                    <lb/>
                  ſitate: velocius acquirit vel deperdit de quantita-
                    <lb/>
                  te maius quam minus.</s>
                </p>
                <p xml:id="N236D8">
                  <s xml:id="N236D9" xml:space="preserve">Quintum. </s>
                  <s xml:id="N236DC" xml:space="preserve">Si ſint duo inequalia in
                    <lb/>
                  quantitate et in denſitate, et minor ſi proportio
                    <lb/>
                  quãtitatis denſioris ad quantitatem alterius quã
                    <lb/>
                  denſitatis vnius ad denſitatem alterius que eque
                    <lb/>
                  velociter acquirant vel deperdãt de denſitate: den­
                    <lb/>
                  ſius tardius acquiret vel deperdet de quantitate
                    <lb/>
                  quam rarius. </s>
                  <s xml:id="N236EB" xml:space="preserve">His notabilibus poſitis pono ali-
                    <lb/>
                  quas propoſitiones.</s>
                </p>
                <p xml:id="N236F0">
                  <s xml:id="N236F1" xml:space="preserve">Prima propoſitio: ſecūdum notabile
                    <lb/>
                  eſt falſum
                    <note position="left" xlink:href="note-0198-01a" xlink:label="note-0198-01" xml:id="N23751" xml:space="preserve">ↄ̈ calcuĺ.</note>
                  </s>
                  <s xml:id="N236FB" xml:space="preserve">Probatur / quia eſt vna cõditionalis cu­
                    <lb/>
                  ius eſt antecedens eſt verum et conſequens falſum: er-
                    <lb/>
                  go illud notabile eſt falſum. </s>
                  <s xml:id="N23702" xml:space="preserve">Probatur antecedēs /
                    <lb/>
                  et volo /  ſint duo pedalia quorum vnum ſit denſuꝫ
                    <lb/>
                  vt .8. et aliud vt .4. et vtrum illorum eque velociter
                    <lb/>
                  acquirat duos gradus denſitatis: tunc illud quod
                    <lb/>
                  eſt minus denſum deperdit vnam tertiam, et aliud
                    <lb/>
                  vnam quintam / vt patet. </s>
                  <s xml:id="N2370F" xml:space="preserve">Sed vnius tertie ad vnaꝫ
                    <lb/>
                  quintam non eſt proportio dupla qualis eſt pro-
                    <lb/>
                  portio inter illorum pedalium denſitates: ergo nõ
                    <lb/>
                  in ea proportione qua vnuꝫ eſt minus denſum alio
                    <lb/>
                  in ea proportione velocius deperdit de quantita-
                    <lb/>
                  te: et ſic in hoc caſu anteccdens illius conditiona-
                    <lb/>
                  lis eſt verum, et conſequens falſum: quod fuit pro-
                    <lb/>
                  bandum. </s>
                  <s xml:id="N23720" xml:space="preserve">Sed tu diceres /  iſta ratio nõ impugnat
                    <lb/>
                  notabile quoniam in notabile habetur que eque ve­
                    <lb/>
                  lociter acquirant vel deperdant de denſitate pro-
                    <lb/>
                  portionali: modo in caſu argumenti non eque pro­
                    <lb/>
                  portionalem denſitatem deperdunt illa duo peda­
                    <lb/>
                  lia. </s>
                  <s xml:id="N2372D" xml:space="preserve">Sed hoc nichil eſt dicere. </s>
                  <s xml:id="N23730" xml:space="preserve">Nam ſi eque propor-
                    <lb/>
                  tionalem denſitatem acquirerent vel deperderent
                    <lb/>
                  cum ſint equalia: ipſa equalem quantitatem oīno
                    <lb/>
                  acquirerunt aut deperderent quod eſt contra no-
                    <lb/>
                  tabile. </s>
                  <s xml:id="N2373B" xml:space="preserve">Nec probatio qua calculator intēdit illud
                    <lb/>
                  notabile probare aliquid valet: quia antecedens
                    <lb/>
                  eius eſt falſum: videlicet hoc in qua proportione
                    <lb/>
                  vnum eſt minus denſum alio in ea proportione ve-
                    <lb/>
                  locius proportionabiliter acquirit vĺ deperdit de
                    <lb/>
                  denſitate. </s>
                  <s xml:id="N23748" xml:space="preserve">Falſitas enim eius patet ex caſu argu-
                    <lb/>
                  menti contra illud notabile.</s>
                </p>
                <p xml:id="N23757">
                  <s xml:id="N23758" xml:space="preserve">Secunda propoſitio. </s>
                  <s xml:id="N2375B" xml:space="preserve">Tertium nota­
                    <lb/>
                  bile eſt ſimiliter falſum.
                    <note position="left" xlink:href="note-0198-02a" xlink:label="note-0198-02" xml:id="N23799" xml:space="preserve">Impuḡ-
                      <lb/>
                    tur terti-
                      <lb/>
                    um nöbi-
                      <lb/>
                    le calcuĺ.</note>
                  </s>
                  <s xml:id="N23765" xml:space="preserve">Probatur / quia eſt vna cõ-
                    <lb/>
                  ditionalis cuius antecedens eſt verū , et conſquens
                    <lb/>
                  falſum: ergo illud notabile eſt falſū. </s>
                  <s xml:id="N2376C" xml:space="preserve">Arguitur an-
                    <lb/>
                  tecedens / quia capto quadrupedali denſo vt .4. et
                    <lb/>
                  pedali denſo vt vnum, et acquirat quadrupedale
                    <lb/>
                  4. gradus denſitatis, et pedale etiam eque veloci-
                    <lb/>
                  ter: tunc antecedens illius conditionalis eſt verum /
                    <lb/>
                  vt conſtat: et conſequens falſum : ergo propoſitum.
                    <lb/>
                  </s>
                  <s xml:id="N2377A" xml:space="preserve">Iam probo falſitatem conſequentis in illo caſu
                    <lb/>
                  quoniã illud quadrupedale efficitur in duplo den-
                    <lb/>
                  ſius, et per conſequens in duplo minus: et ſic perdit
                    <lb/>
                  bipedale: pedale vero non perdit bipedale / vt con-
                    <lb/>
                  ſtat cum non ſit niſi pedale: ergo tunc illa duo non
                    <lb/>
                  eque velociter acquirunt vel deperdunt de denſi-
                    <cb chead="De motu rarefactionis  cõdenſationis."/>
                  tate et ſic antecedens eſt verū: et conſequens falſum /
                    <lb/>
                  quod fuit probandū. </s>
                  <s xml:id="N2378C" xml:space="preserve">Nec valet fugere ad id qḋ cal-
                    <lb/>
                  culator dicit in illo notabili tertio pro hoc inſtan­
                    <lb/>
                  ti quoniam pro inſtanti nulla fit acquiſitio quan-
                    <lb/>
                  titatis: et ideo illud nullo modo iuuat.</s>
                </p>
                <note position="right" xml:id="N237A5" xml:space="preserve">īpugnat̄̄
                  <lb/>
                4. nõbile</note>
                <p xml:id="N237AB">
                  <s xml:id="N237AC" xml:space="preserve">Tertia propoſitio. </s>
                  <s xml:id="N237AF" xml:space="preserve">Quartum notabi­
                    <lb/>
                  le non eſt verum.
                    <note position="right" xlink:href="note-0198-03a" xlink:label="note-0198-03" xml:id="N2381A" xml:space="preserve">Calcuĺ.</note>
                  </s>
                  <s xml:id="N237B9" xml:space="preserve">Probatur / quia eſt vna conditio-
                    <lb/>
                  nalis: cuius antecedens in caſu eſt verum: et conſe-
                    <lb/>
                  quens falſum: ergo. </s>
                  <s xml:id="N237C0" xml:space="preserve">Probatur antecedens, et ca-
                    <lb/>
                  pio pedale et ſemipedale, et pedale ſit denſum vt .6
                    <lb/>
                  ſemipedale vero vt .4. et deperdat vtrum illorum
                    <lb/>
                  duos gradus denſitatis in hora eque velociter.
                    <lb/>
                  </s>
                  <s xml:id="N237CA" xml:space="preserve">Quo poſito antecedens eſt verum. </s>
                  <s xml:id="N237CD" xml:space="preserve">Nam illa ſunt
                    <lb/>
                  inequalia in quantitate et dēſitate maior et eſt pro-
                    <lb/>
                  portio quantitatis proportione denſitatis. </s>
                  <s xml:id="N237D4" xml:space="preserve">Nam
                    <lb/>
                  illa eſt dupla hec vero ſexquialtera: et illa duo eque
                    <lb/>
                  velociter deperdunt vel acquirunt de denſitate. </s>
                  <s xml:id="N237DB" xml:space="preserve">Et
                    <lb/>
                  tamen conſequens eſt falſum / quoniam maius illo­
                    <lb/>
                  rum non velocius acquirit de quantitate quã mi-
                    <lb/>
                  nus: immo equaliter. </s>
                  <s xml:id="N237E4" xml:space="preserve">Nam vtrum illorum acqui­
                    <lb/>
                  rit ſemipedale / vt conſtat: ergo illud notabile falſū /
                    <lb/>
                  quod fuit probandum. </s>
                  <s xml:id="N237EB" xml:space="preserve">Et aduerte  aliquãdo da-
                    <lb/>
                  ta veritate antecedentis: maius illorum equaliter
                    <lb/>
                  acquirit vt in caſu poſito. </s>
                  <s xml:id="N237F2" xml:space="preserve">Aliquãdo maius acqui-
                    <lb/>
                  rit maiorem quantitatem quam minus: vt poſito
                    <lb/>
                  quadrupedali denſo vt .6. et pedali denſo vt .4. et
                    <lb/>
                  equaliter deperdat vtrum duos gradus denſita­
                    <lb/>
                  tis: tunc quadrupedale acquirit bipedale: pedale
                    <lb/>
                  vero vnū pedale preciſe. </s>
                  <s xml:id="N237FF" xml:space="preserve">Aliquando maius deper-
                    <lb/>
                  dit minus de quantitate: vt videlicet poſito  a. ſit
                    <lb/>
                  9. pedum b.4.a. denſum vt .8.b. vero vt .4. et deper­
                    <lb/>
                  dat vtrum illorum eque velociter vnum gradum
                    <lb/>
                  denſitatis: tūc quadrupedale acquirit pedale cum
                    <lb/>
                  tertia. </s>
                  <s xml:id="N2380C" xml:space="preserve">Aliud vero corpus maius acquirit pedale
                    <lb/>
                  cum duabus ſeptimis: modo plus eſt pedale cum
                    <lb/>
                  tertia quã cū duabus ſeptimis. </s>
                  <s xml:id="N23813" xml:space="preserve">Ptꝫ hoc calculãti.</s>
                </p>
                <note position="right" xml:id="N23820" xml:space="preserve">īpugnat̄̄
                  <lb/>
                .5. nöbile</note>
                <p xml:id="N23826">
                  <s xml:id="N23827" xml:space="preserve">Quarta propoſitio. </s>
                  <s xml:id="N2382A" xml:space="preserve">Quiutum nota-
                    <lb/>
                  bile eſt falſum.
                    <note position="right" xlink:href="note-0198-04a" xlink:label="note-0198-04" xml:id="N2384C" xml:space="preserve">calcuĺ.</note>
                  </s>
                  <s xml:id="N23834" xml:space="preserve">Probatur: quoniã dato  ſit vnuꝫ
                    <lb/>
                  ſextipedale denſum vt octo, et vnū bipedale denſū
                    <lb/>
                  vt .2. et vtrum illorum acquirat .4. gradus denſi-
                    <lb/>
                  tatis eque velociter: tunc antecedens illius conditi­
                    <lb/>
                  onalis eſt verum, et conſequens falſum. </s>
                  <s xml:id="N2383F" xml:space="preserve">Nam tunc
                    <lb/>
                  denſius deperdit duo pedalia, et minus denſum nõ
                    <lb/>
                  perdit tantum quia tunc efficeretur non quantum /
                    <lb/>
                  g̊ illud notabile quintū eſt falſū / qḋ fuit ꝓbandum.</s>
                </p>
                <p xml:id="N23852">
                  <s xml:id="N23853" xml:space="preserve">Sit ergo concluſio reſponſiua ad du­
                    <lb/>
                  bium quodlibet illorum notabilium dēpto primo
                    <lb/>
                  eſt falſum. </s>
                  <s xml:id="N2385A" xml:space="preserve">Patet hec concluſio per quatuor predi-
                    <lb/>
                  ctas concluſiones. </s>
                  <s xml:id="N2385F" xml:space="preserve">Sed quia poſſunt poni et demõ-
                    <lb/>
                  ſtari .4. notabilia conformia .4. his notabilibus
                    <lb/>
                  falſis impugnatis que plurimum ſubtilitatis ha­
                    <lb/>
                  bent. </s>
                  <s xml:id="N23868" xml:space="preserve">Ideo huic loco ea interſerenduꝫ non ī merito
                    <lb/>
                  optaui illorum demonſtrationibus breuiitatis cau­
                    <lb/>
                  ſa et quadaꝫ alia occulta cauſa omiſſis. </s>
                  <s xml:id="N2386F" xml:space="preserve">Sit igitur
                    <lb/>
                    <note position="right" xlink:href="note-0198-05a" xlink:label="note-0198-05" xml:id="N238B4" xml:space="preserve">1. nöbile</note>
                  </s>
                  <s xml:id="N23879" xml:space="preserve">Primū illorum .4. notabilium. </s>
                  <s xml:id="N2387C" xml:space="preserve">¶ Si ſint duo ine-
                    <lb/>
                  qualiter denſa equalia tamen in quantitate que
                    <lb/>
                  eque velociter acquirant vel deperdant de denſita­
                    <lb/>
                  te: tunc in ea proportione minus denſum plus ac-
                    <lb/>
                  quirit vel deperdit de quantitate in qua ſe habet
                    <lb/>
                  denſitas denſioris ad denſitatem minus denſi in
                    <lb/>
                  fine depertionis vel acquiſitionis talis denſitatis.
                    <lb/>
                  </s>
                  <s xml:id="N2388C" xml:space="preserve">et nolo dicere /  per totum tempus in ea proporti-
                    <lb/>
                  one velocius acquirit: ſed in toto tempore cathego­
                    <lb/>
                  rematice. </s>
                  <s xml:id="N23893" xml:space="preserve">Exēplum / vt ſi duo pedalia quoꝝ vnū eſt
                    <lb/>
                  denſum vt .8. et aliud vt .4. perdant duos gradus
                    <lb/>
                  denſitatis eque velociter dico /  pedale minꝰ den-
                    <lb/>
                  ſum in triplo maiorem quantitatem acquiſiuit
                    <lb/>
                  quam magis denſum quia proportio denſitatum </s>
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