Alvarus, Thomas, Liber de triplici motu, 1509

Table of contents

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                  <s xml:id="N16254" xml:space="preserve">
                    <pb chead="Primi tractatus" file="0063" n="63"/>
                  tiam vt .2. aliquanta velocitate neceſſe eſt eandem
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                  potentiam vt octo natam eſſe mouere duplam re-
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                  ſiſtentiaꝫ in ſubdupla velocitate. </s>
                  <s xml:id="N16262" xml:space="preserve">et potentia vt .8
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                  eſt aliqua potentia: et reſiſtentia vt duo aliqua re­
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                  ſiſtentia: igitur. </s>
                  <s xml:id="N16269" xml:space="preserve">Si aliqua potētia moueat aliquã
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                  reſiſtentiã in aliquo tempore ali̄ta velocitate: ea­
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                  dem mouebit duplam reſiſtentiã in ſubdupla ve-
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                  locitate / quod eſt oppoſitum regule. </s>
                  <s xml:id="N16272" xml:space="preserve">Patet hec cõ­
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                  ſequentia ab inferiori ad ſuuꝫ ſuperius.</s>
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                <p xml:id="N16277">
                  <s xml:id="N16278" xml:space="preserve">Quarto contra ſeptimam arguitur
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                  ſic / quoniã ſi potētia vt ſex moueat reſiſtentiaꝫ vt
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                  quatuor et potentia vt .8. moueat reſiſtentiã etiaꝫ
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                  vt .4. diuiſim ille potentie coniuncte non mouebūt
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                  eaſdem potentias coniunctas in duplo velocius.
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                  </s>
                  <s xml:id="N16284" xml:space="preserve">igitur regula falſa. </s>
                  <s xml:id="N16287" xml:space="preserve">Probatur antecendens / quoni­
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                  am proportio reſultans ex illis duabus potētiis
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                  ſimul ſumptis et duabus reſiſtentiis etiam ſimul
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                  ſumptis eſt proportio .14. ad .8. que eſt minor du-
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                  pla. eſt enim proportio ſupertripartiēs quartas.
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                  </s>
                  <s xml:id="N16293" xml:space="preserve">Modo illa eſt minor dupla / vt ptꝫ ex tertia ſuppo­
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                  ſitiõe ſuperiꝰ allegati q̈rti capitis / g̊ ſequit̄̄ /  nõ
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                  eque velociter manebit talis proportio ſicut ãtea
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                  mouebat dupla que eſt .8. ad .4.</s>
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                <p xml:id="N1629C">
                  <s xml:id="N1629D" xml:space="preserve">Ad iſta reſpondetur ꝑ ordinē ad pri-
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                  ma duo argumenta reſpondet paulus venetus et
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                  brauardinus  ille regule philoſophi intelligun­
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                  tur preciſe de proportione dupla: modo inſtantie
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                  fuerunt adducte in alia ſpecie proportionis </s>
                  <s xml:id="N162A8" xml:space="preserve">¶ Ad
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                  tertium reſpondeo /  non eſt ad propoſitum ma-
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                  terie non valet eni3 conſequentia ab inferiori ad
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                  ſuum ſuperius cum dictione illatiua. </s>
                  <s xml:id="N162B1" xml:space="preserve">Adduxi ta-
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                  men illud argumentum / qm̄ ſemper tenet in pro-
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                  portione quadrupla. </s>
                  <s xml:id="N162B8" xml:space="preserve">¶ Ad quartuꝫ reſpondeo / 
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                  regula philoſophi ſeptima intelligitur dūmodo
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                  ille proportiões ſint equales. </s>
                  <s xml:id="N162BF" xml:space="preserve">Que aūt ſunt equa­
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                  les patet ex tertia ſuppoſitione quarti capitis ſe­
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                  cunde partis. </s>
                  <s xml:id="N162C6" xml:space="preserve">Sed quia ex ſolutione quã dat bra-
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                  uardinus ad primū argumentū / ſequitur philoſo­
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                  phum poſuiſſe regulas ſatis inſufficientes: que p̄­
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                  ciſe in vna ſpecie proportionis tenerent.
                    <note position="left" xlink:href="note-0063-01a" xlink:label="note-0063-01" xml:id="N1630F" xml:space="preserve">Qūo in-
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                    telligunt̄̄
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                    regule
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                    phī.</note>
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                  <s xml:id="N162D4" xml:space="preserve">Ideo di­
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                  co aliter /  philoſophus capit potentiaꝫ pro pro­
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                  portione maioris inequalitatis. </s>
                  <s xml:id="N162DB" xml:space="preserve">Et iſto modo ca-
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                  piendo regule habēt veritatem in omni genere ꝓ­
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                  protionum. </s>
                  <s xml:id="N162E2" xml:space="preserve">Et argumentum nichil concludit / qm̄
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                  oportet quando duplatur potentia duplare pro-
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                  portionem: et non curare de potentia: ita  ſit ſen­
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                  ſus prime regule ſi aliqua potētia moueat aliquã
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                  reſiſtentiã per aliquod ſpacium in aliquo tempo-
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                  re etc. eadem mouebit ſubduplam reſiſtentiam etc.
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                  id eſt ſi aliqua virtus moueat aliquã reſiſtentiam
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                  ab aliqua proportione eadem virtus mouebit re-
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                  ſiſtentiam ad quam habet proportionem duplaꝫ
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                  ad aliam proportionem .i. ad quam habet ꝓpor-
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                  tionē duplicatã in duplo velocius. </s>
                  <s xml:id="N162F9" xml:space="preserve">Et ſenſus huiꝰ
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                  regule eſt ſi aliqua potentia moueat aliquam reſi­
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                  ſtentiam in aliquo tempore etc. dupla virtus mo-
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                  uebit eandem reſiſtentiam in duplo velocius hoc ē
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                  ſi aliqua virtus moueat aliquam reſiſtentiam ab
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                  aliqua proportione: dupla proportio mouebit in
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                  duplo velocius. </s>
                  <s xml:id="N16308" xml:space="preserve">Et ſic intelliguntur alie regule.</s>
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                <note position="left" xml:id="N1631B" xml:space="preserve">1. correl.</note>
                <p xml:id="N1631F">
                  <s xml:id="N16320" xml:space="preserve">¶ Ex quo ſequitur /  ſi virtus ſe habens ad aliquã
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                  reſiſtentiam in proportione irrationali diametri
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                  ad coſtam moueat ali̄tum velociter: proportio
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                  dupla ad eandē reſiſtentiã mouebit in duplo velo­
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                  cius.
                    <note position="left" xlink:href="note-0063-02a" xlink:label="note-0063-02" xml:id="N16343" xml:space="preserve">2. correl.</note>
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                  <s xml:id="N16330" xml:space="preserve">¶ Secundo igitur /  non oportet q̄rere in q̈-
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                  libet proportione proportionem rationalem ī du­
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                  plo tardius mouentem eam reſiſtentiam: ſed ſa-
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                  tis eſt /  detur ꝓportio rationalis vel irrationa-
                    <cb chead="Capitulum quintum"/>
                  lis. </s>
                  <s xml:id="N1633C" xml:space="preserve">et hec de regulis philoſophi.</s>
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              <div xml:id="N16349" level="4" n="5" type="chapter" type-free="capitulum">
                <head xml:id="N1634E" xml:space="preserve">Capitulum quintum / in quo ponuntur
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                regule ſiue concluſiones velocitatis et tar­
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                ditatis motus penes proportionem pro­
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                portionum conformiter ad intentionem
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                calculatoris.</head>
                <p xml:id="N16359">
                  <s xml:id="N1635A" xml:space="preserve">AD inducendas ſeriatim ma­
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                  thematico more concluſiones docētes
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                  velocitatem et tarditatē motus penes
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                  cauſam iuxta opinionem quartam ſit.</s>
                </p>
                <p xml:id="N16363">
                  <s xml:id="N16364" xml:space="preserve">Prima ſuppoſitio / ab equalibus pro­
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                  portionibus equales velocitates proueniunt: et ab
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                  inequalibus inequales. </s>
                  <s xml:id="N1636B" xml:space="preserve">et a rationalibus rationa­
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                  les: et ab incõmēſurabilibus īcõmēſurabiles </s>
                  <s xml:id="N16370" xml:space="preserve">Pa­
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                  tet hec ſuppoſitio ex opinione que ponit velocita­
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                  tem ſequi proportionem ꝓproportionum.</s>
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                <p xml:id="N16377">
                  <s xml:id="N16378" xml:space="preserve">Secundua ſuppoſitio ab equalibꝰ pro­
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                  portionibus que ſunt partes aliarum proportio­
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                  num ſiue equalium ſiue inequalium equales velo­
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                  citates proueniunt. </s>
                  <s xml:id="N16381" xml:space="preserve">Declaro hanc ſuppoſitionem
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                  et capio proportionem triplam et duplam: et ma­
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                  nifeſtum eſt:  vtriuſ proportio ſexquialtera eſt
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                  pars. </s>
                  <s xml:id="N1638A" xml:space="preserve">dico tunc /  quãtam velocitatē producit ſex­
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                  quialtera que eſt pars duple tantam velocitatem
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                  ꝓducit ſexquialtera que eſt pars triple. </s>
                  <s xml:id="N16391" xml:space="preserve">Proba-
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                  tur ex priori ſuppoſitione / quia ſexquialtera que
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                  eſt pars duple et ſexquialtera que eſt pars triple
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                  ſunt equales proportiones.</s>
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                <p xml:id="N1639A">
                  <s xml:id="N1639B" xml:space="preserve">Tertia ſuppoſitio / ꝑ additionē equa­
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                  lium proportionum ſuper proportiones equales
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                  vel inequales: velocitates equaliter intenduntur
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                  </s>
                  <s xml:id="N163A3" xml:space="preserve">Declaro hoc in terminis et capio proportionem
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                  duplam et quadruplam / et volo /  vtri addatur
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                  proportio ſexquialtera: qua addita dico /  equa­
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                  liter intendunt proportiones ille ſiue ille potentie
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                  motū ſuum intendunt / et tantam velocitatem acq̇-
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                  rit proportio maior ſicut et minor ſupra velocita­
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                  tem habitam ante additionem proportionis ſexq̇­
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                  altere. </s>
                  <s xml:id="N163B4" xml:space="preserve">Probatur hec ſuppoſitio ex ſecūda / quia il­
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                  la proportio ſexquialtera efficitur pars duaꝝ ꝓ-
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                  portionum inequalium / igitur cum vtra equalē
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                  velocitatem producet.</s>
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                <p xml:id="N163BD">
                  <s xml:id="N163BE" xml:space="preserve">Quarta ſuppoſitio / ꝑ decremētū dua­
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                  rum proportionū equalium que ſunt partes dua­
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                  rum proportionū ſiue equalium ſiue inequalium:
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                  equales velocitates perdētur. </s>
                  <s xml:id="N163C7" xml:space="preserve">¶ Declarat̄̄ hec ſup­
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                  poſitio et capio proportionem duplam et triplaꝫ /
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                  et volo /  vtra deperdat proportionem ſexqui-
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                  alterã / tunc dico /  ſi proportio dupla ꝑdat duos
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                  gradus velocitatis etiam duos adequate perdit
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                  proportio tripla. </s>
                  <s xml:id="N163D4" xml:space="preserve">Patet hec ſuppoſitio ex priori /
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                  quoniam ille due proportiones deperdite cū eēnt
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                  equales: equalē velocitatem producebant: igitur
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                  per decrementum illarum equales velocitates ꝑ-
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                  duntur / quia perduntur ipſemet quas ipſe produ­
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                  cebant.</s>
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                <p xml:id="N163E1">
                  <s xml:id="N163E2" xml:space="preserve">Quinta ſpupoſitio / ꝑ additionē equa­
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                  lis ̄titatis maiori et minori ̄titati maior ꝓpor­
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                  tio acquiritur minori ̄titati ꝙ̄ maiori. </s>
                  <s xml:id="N163E9" xml:space="preserve">¶ Hec eſt
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                  octaua ſuppoſitio quarti capitis ſecunde partis.</s>
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                <p xml:id="N163EE">
                  <s xml:id="N163EF" xml:space="preserve">Sexta ſuppoſitio eq̄ velociṫ intēde­
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                  re motum: eſt in equali tempore equales ꝑtes ade­
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                  quate acquirere: et eque proportionabiliter intē-
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                  dere eſt in equali tempore equales proportiones
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                  acquirere: </s>
                  <s xml:id="N163FA" xml:space="preserve">Et ſimiliter dicendum eſt de eque velo-
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                  citer remittere et eque proportionabiliter / vt ſi nu­ </s>
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