Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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                <p xml:id="N1E963">
                  <s xml:id="N1E97D" xml:space="preserve">
                    <pb chead="De motu locali quo ad effectū ſcḋm ſubiectū difformi." file="0150" n="150"/>
                  que dicta ſunt in ſecunda parte huius operis capi­
                    <lb/>
                  te quarto. </s>
                  <s xml:id="N1E991" xml:space="preserve">Infinita alia correlaria poſſūt ex hac ꝓ­
                    <lb/>
                  poſitione inferri. </s>
                  <s xml:id="N1E996" xml:space="preserve">Sed iſta ſufficiant pro praxi pro­
                    <lb/>
                  poſitionis habenda.</s>
                </p>
                <p xml:id="N1E99B">
                  <s xml:id="N1E99C" xml:space="preserve">Quīta ꝓpoſitio. </s>
                  <s xml:id="N1E99F" xml:space="preserve">Si maior velocitas
                    <lb/>
                  tēpori minori coextendat̄̄ et minor maiori, et ꝓpor-
                    <lb/>
                  tio velocitatis maioris ad velocitatem minoris ſit
                    <lb/>
                  equalis ꝓportioni tēporis maioris ad tēpus minꝰ
                    <lb/>
                  tūc illa mobilia equalia ſpacia ꝑtranſeūt. </s>
                  <s xml:id="N1E9AA" xml:space="preserve">Exēplū /
                    <lb/>
                  vt ſi a. mobile per mediã horã moueatur velocitate
                    <lb/>
                  vt .4. et b. mobile per horã velocitate vt .2. tunc quia
                    <lb/>
                  proportio tēporis ad tēpus eſt dupla / et velocitatis
                    <lb/>
                  etiã ad velocitatē dupla ſequitur /  a. et b. equalia
                    <lb/>
                  ſpacia ꝑtranſeūt. </s>
                  <s xml:id="N1E9B7" xml:space="preserve">Probat̄̄ hec ꝓpoſitio: ſit a. mo-
                    <lb/>
                  bile qḋ moueatur ꝑ aliqḋ tēpus: et b. mouetur ꝑ tē-
                    <lb/>
                  pus in f. ꝓportione maius: et in f. ꝓportione minori
                    <lb/>
                  velocitate: tūc ibi ꝓportio velocitatū et tēpoꝝ ſunt
                    <lb/>
                  equales q2 vtra f. / igit̄̄ ſi a. moueat̄̄ equali veloci-
                    <lb/>
                  tate cū b. tunc in f. ꝓportione b. maius ſpaciū per-
                    <lb/>
                  tranſit quã a. q2 in ꝓportiõe tēporis / vt ptꝫ ex ſcḋa
                    <lb/>
                  parte prime propoſitionis: ſed modo a. mouet̄̄ in f.
                    <lb/>
                  ꝓportione velocius quã tunc: ergo in f. ꝓportione
                    <lb/>
                  maius ſpacium ꝑtranſit quã tunc in eodē tēpore: vt
                    <lb/>
                  ptꝫ ex ſecūda ꝓpoſitione: ergo tantū ſicut b. </s>
                  <s xml:id="N1E9CE" xml:space="preserve">Patet
                    <lb/>
                  ↄ̨ña per hanc maximã quãdo aliqua duo ſe habent
                    <lb/>
                  in aliqua ꝓportione vt puta f. </s>
                  <s xml:id="N1E9D5" xml:space="preserve">Si minꝰ illoꝝ acqui­
                    <lb/>
                  rit illã ꝓportionē f. ſupra ſe, efficitur equale alteri /
                    <lb/>
                  quod erat maius: vt ſi quaternariꝰ ad quē octona-
                    <lb/>
                  rius habet ꝓportionē duplã acquirat ſupra ſe ꝓ-
                    <lb/>
                  portionē duplã efficiet̄̄ equalis octauario / vt ptꝫ de
                    <lb/>
                  ſe: et ſic ptꝫ ꝓpoſitio.
                    <note position="left" xlink:href="note-0150-01a" xlink:label="note-0150-01" xml:id="N1EA02" xml:space="preserve">Correĺ.</note>
                  </s>
                  <s xml:id="N1E9E7" xml:space="preserve">¶ Ex hac ꝓpoſitione ſequitur /
                    <lb/>
                   ſi a. mobile moueatur per horã velocitate vt .4.
                    <lb/>
                  et b. mobile per duas tertias hore velocitate vt ſex
                    <lb/>
                  b. et a. equalia ſpacia ꝑtranſeūt. </s>
                  <s xml:id="N1E9F0" xml:space="preserve">Probatio / q2 qua­
                    <lb/>
                  lis eſt ꝓportio tꝑis maioris ad tempus minꝰ: talis
                    <lb/>
                  eſt ꝓportio velocitatis fluentis per tēpus minꝰ ad
                    <lb/>
                  velocitatem per maius tēpus labentem. </s>
                  <s xml:id="N1E9F9" xml:space="preserve">(Utrobi
                    <lb/>
                  enim ſexquialtera proportio reperitur.</s>
                </p>
                <p xml:id="N1EA08">
                  <s xml:id="N1EA09" xml:space="preserve">Quīta ꝓpoſitio. </s>
                  <s xml:id="N1EA0C" xml:space="preserve">Si maior velocitas
                    <lb/>
                  tēpori et extendatur minori, et minor velocitas ma­
                    <lb/>
                  iori tēpori: ꝓportio velocitatis tēporis ꝓportio­
                    <lb/>
                  nē exuperet: tūc mobile minori tēpore motū maius
                    <lb/>
                  ſpaciū deſcribet ꝙ̄ mobile motū in maiori tēpore
                    <lb/>
                  in ea ꝓportione per quã velocitatū ꝓportio tēpoꝝ
                    <lb/>
                  ꝓportionē excedit. </s>
                  <s xml:id="N1EA1B" xml:space="preserve">Exemplū / vt ſi a. mobile moueat̄̄
                    <lb/>
                  per horã velocitate vt .2. et b. mobile per mediã ho-
                    <lb/>
                  ram velocitate vt .8. tunc b. mobile maius ſpacium
                    <lb/>
                  ꝑtranſit quã a. mobile in ea ꝓportione per quã pro­
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                  portio quadrupla velocitatū excedit ꝓportitioneꝫ
                    <lb/>
                  duplã tēpoꝝ. </s>
                  <s xml:id="N1EA28" xml:space="preserve">Et q2 quadrupla velocitatū duplam
                    <lb/>
                  tēpoꝝ per duplã antecedit notū euadet ſpaciū a b.
                    <lb/>
                  mobili pertrãſitū ad ſpaciū ab a. mobili ꝑtranſi-
                    <lb/>
                  tū duplū eſſe. </s>
                  <s xml:id="N1EA31" xml:space="preserve">Uniuerſalitcr tamen mathematico
                    <lb/>
                  ordine hanc quintã ꝓpoſitiouē inducamus. </s>
                  <s xml:id="N1EA36" xml:space="preserve">Sit em̄
                    <lb/>
                  a. mobile quod per aliquod tēpus aliqua velocita­
                    <lb/>
                  te moueatur: et b. mobile moueatur per tēpus in f.
                    <lb/>
                  ꝓportione minus: et velocitate in g. ꝓportione ma-
                    <lb/>
                  iori quã velocitas qua mouetur a. ſit g. ꝓportio
                    <lb/>
                  maior f. excedat g. ꝓportio ꝓportionē f. per h. ꝓ-
                    <lb/>
                  portionē. </s>
                  <s xml:id="N1EA45" xml:space="preserve">quibꝰ ſtructis ſic argr̄: ſi ꝓportio veloci-
                    <lb/>
                  tatis b. ad velocitatē a. eſſet equalis ꝓportiõi tēpo-
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                  ris tn quo mouet̄̄ a. ad tēpus in quo mouetur b. que
                    <lb/>
                  eſt f.a. et b. equalia ſpacia ꝑtranſirent in illis tēpo-
                    <lb/>
                  ribus in equallbꝰ / vt p̄cedens ꝓpoſitio demonſtrat
                    <lb/>
                  puta quarta. </s>
                  <s xml:id="N1EA52" xml:space="preserve">Sed modo velocitas qua mouet̄̄ b. eſt
                    <lb/>
                  in h. ꝓportione mãior velocitate / qua tunc moueret̄̄ /
                    <lb/>
                  ergo in h. ꝓportione maius ſpaciū pertranſit mo-
                    <cb chead="De motu locali quo ad effectū ſcḋm ſubiectū difformi."/>
                  do b. quã tunc: qm̄ ſicut ſe habent velocitates in a-
                    <lb/>
                  liquo tēpore: ita ſpacia pertranſita in eodē / vt ptꝫ
                    <lb/>
                  ex ſecūda ꝓpoſitione: et ex conſequenti ſequitur / ­
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                  modo b. in h. proportione maius ſpaciū pertrãſit ̄
                    <lb/>
                  a. qm̄ a. et b. tunc equalia ſpacia pertranſirent: et h.
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                  ꝓportio eſt ꝓportio per quã g. ꝓportio velocitatū
                    <lb/>
                  excedit f. proportionē tēpoꝝ: igr̄ b. mouetur velocius
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                  ipſo a. in ꝓportione per quã ꝓportio velocitatum
                    <lb/>
                  temporum proportionem excedit: quod fuit ꝓban-
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                  dum: et ſic patet propoſitio.</s>
                </p>
                <note position="right" xml:id="N1EA6E" xml:space="preserve">Correĺ.</note>
                <p xml:id="N1EA72">
                  <s xml:id="N1EA73" xml:space="preserve">¶ Ex hac propoſitione ſequitur /  ſi a. mobile mo-
                    <lb/>
                  ueatur per horã velocitate vt duo: et b. mobile per
                    <lb/>
                  mediã horam, velocitate vt .6.  b. mobile in ſexqui­
                    <lb/>
                  altero maius ſpaciū pertranſit quã a. vt ſi a. per-
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                  trãſit bipedale b. tripedale ꝑtrãſit. </s>
                  <s xml:id="N1EA7E" xml:space="preserve">Probat̄̄ / q2 ibi
                    <lb/>
                  velocitates inequales in equalibus temporibus co­
                    <lb/>
                  extenduntur: et mbior velocitas maiori tempori co­
                    <lb/>
                  extenditur / vt notū eſt: et proportio velocitatū que
                    <lb/>
                  tripla eſt, proportionē tēporum duplã per ꝓporti-
                    <lb/>
                  onem ſexquialterã antecedit. </s>
                  <s xml:id="N1EA8B" xml:space="preserve">Hec igitur ſignum eſt
                    <lb/>
                  et fidem facit auxilio precedentis propoſitionis b.
                    <lb/>
                  mobile in ſuo tēpore quo mouetur ſexquialterum
                    <lb/>
                  ſpaciuꝫ ad ſpaciū ab a. exactū abſoluiſſe: quod ab
                    <lb/>
                  iniitio ꝓpoſitū fuit </s>
                  <s xml:id="N1EA96" xml:space="preserve">¶ Inferas tuo marte mĺta huic
                    <lb/>
                  ſimilia correlaria que ex hac quīta propoſitiõe
                    <lb/>
                  ſuã demonſtrationem facile ſortiūtur. </s>
                  <s xml:id="N1EA9D" xml:space="preserve">Hoc em̄ cor-
                    <lb/>
                  relariū: ideo poſitū eſt: quia neceſſe intelligentem
                    <lb/>
                  particularia fantaſmata ſpeculari.
                    <note position="right" xlink:href="note-0150-02a" xlink:label="note-0150-02" xml:id="N1EAB6" xml:space="preserve">phdot;us .2.
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                    de aīa</note>
                  </s>
                  <s xml:id="N1EAA9" xml:space="preserve">teſte philoſo-
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                  pho ſecūdo de aīa: nichil eſt in īteliectu quin priꝰ
                    <lb/>
                  quodammodo ſingulariter preceſſerit in ſenſu de
                    <lb/>
                  ſenſu et ſeaſato aſſerente philoſopho.</s>
                </p>
                <p xml:id="N1EABE">
                  <s xml:id="N1EABF" xml:space="preserve">Sexta ꝓpoſitio. </s>
                  <s xml:id="N1EAC2" xml:space="preserve">Ubicun maior ve­
                    <lb/>
                  locitas tꝑri coaſſiſtit mīori, mīor o maiori eſt ꝓ­
                    <lb/>
                  portio velocitatū tp̄m ꝓportiõe īferior et mīor, tūc
                    <lb/>
                  mobile qḋ maiori velocitate mouent̄̄ mīori tp̄e mīo­
                    <lb/>
                  rem magnitudinē deſcribet quã mobile motū ma-
                    <lb/>
                  iori tēpore in ea ꝓportione per quã temporū ꝓpor­
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                  tio velocitatū ꝓportioni effertur. </s>
                  <s xml:id="N1EAD1" xml:space="preserve">Exemplū / vt ſi a.
                    <lb/>
                  mobile per horan moueatur velocitate vt duo ade-
                    <lb/>
                  quate, et b. per mediã horam moueatur velocitate
                    <lb/>
                  vt .3. tunc b. minꝰ ſpaciū pertranſit quã a (minꝰ in-
                    <lb/>
                  quam) in ꝓportione ſexquitertia per quã ſexquiter­
                    <lb/>
                  tiam ꝓportio dupla tēporuꝫ ꝓportionē ſexq̇alterã
                    <lb/>
                  velocitatū excedit: ſi igitur a. pedale pertrãſeat: b.
                    <lb/>
                  tres quartas deſcribet. </s>
                  <s xml:id="N1EAE2" xml:space="preserve">Generaliṫ tñ iudicat̄̄ ↄ̨clu-
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                  ſio iſto modo. </s>
                  <s xml:id="N1EAE7" xml:space="preserve">Sit a. mobile per aliquod tēpus mo­
                    <lb/>
                  tum aliqua velocitate, b. vero per tēpus in g. ꝓpor-
                    <lb/>
                  tione minus, et moueatur b. in f. ꝓportione minori
                    <lb/>
                  tamen g. velocius ipſo a. excedat g. ꝓportio pro-
                    <lb/>
                  portionē f. per h. ꝓportionē: tunc a. maius ſpaciuꝫ
                    <lb/>
                  pertranſit in h. ꝓportione ꝙ̄ b. </s>
                  <s xml:id="N1EAF4" xml:space="preserve">Quod ꝓbatur ſic.
                    <lb/>
                  </s>
                  <s xml:id="N1EAF8" xml:space="preserve">quia ſi proportio velocitatis qua moueatur b. mobi­
                    <lb/>
                  le per tempus minus eſſet equalis proportioni tē-
                    <lb/>
                  porum: tunc b: equale ſpaciū pertranſiret adequa-
                    <lb/>
                  te in tempore in quo mouetur ſpacio pertrãſito ab
                    <lb/>
                  a. in tempore in quo a. mouetur, vt patet ex quarta
                    <lb/>
                  prppoſitione: ſed modo mouetur b. velocitate in h.
                    <lb/>
                  proportione minori quam tunc: igitur b. pertran-
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                  ſit modo ſpacium in eodem tempore in h. propor-
                    <lb/>
                  tione minus quam tunc / vt patet ex ſecunda propo-
                    <lb/>
                  ſitione, et ex conſequenti ſequitur /  medo pertran­
                    <lb/>
                  ſit b. ſpacium in h. proportione minus quam a. qm̄
                    <lb/>
                  a. pertranſit tantum ſicut tūc pertranſibat b. / quod
                    <lb/>
                  fuit probandum. </s>
                  <s xml:id="N1EB13" xml:space="preserve">Sed iam probo illam minorem:
                    <lb/>
                  videlicet  b. modo mouetur velocitate in h. ꝓpor-
                    <lb/>
                  tione minori quam tunc, per hanc maximã. </s>
                  <s xml:id="N1EB1A" xml:space="preserve">Quan­
                    <lb/>
                  docun duo numeri inequales habent duas pro-
                    <lb/>
                  portiones ad vnum tertium: tunc in </s>
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