Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N21003">
                  <s xml:id="N21010" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0175" n="175"/>
                  tionabiliter ſicut raritas augetur ita et dēſitas di­
                    <lb/>
                  minuit̄̄ / g̊ raritas et denſitas nõ ſunt poſitiua. </s>
                  <s xml:id="N2101A" xml:space="preserve">Con­
                    <lb/>
                  ſequētia eſt nota cū minori, et argr̄ maior / q2 ſi illḋ
                    <lb/>
                  eſſet poſſibile de aliquibꝰ poſitiuis: hoc maxīe eſſet
                    <lb/>
                  qñ maiꝰ diminuit̄̄ et minꝰ auget̄̄ ſicut dictū eſt in ſo-
                    <lb/>
                  lutione: ſed hoc nõ: igr̄. </s>
                  <s xml:id="N21025" xml:space="preserve">Probat̄̄ minor / q2 vel illud
                    <lb/>
                  minꝰ qḋ augetur ſemꝑ in augmentatione manebit
                    <lb/>
                  minꝰ altero, vel aliqñ deueniet ad equalitatē: ſi cõ-
                    <lb/>
                  tinuo illud qḋ augetur erit minꝰ illo qḋ diminuitur
                    <lb/>
                  et ita velociter diminuit̄̄ maiꝰ ſicut augetur minus
                    <lb/>
                  ſequit̄̄ /  cõtinuo in toto illo tꝑe in quo erit minus
                    <lb/>
                  ipſum velociꝰ ꝓportiõabiliter augebit̄̄ ꝙ̄ aliud di-
                    <lb/>
                  minuitur volo dicere in quolibet inſtãti intrinſeco
                    <lb/>
                  illiꝰ tēporis: ptꝫ hec ↄ̨ña regulã geometricã. </s>
                  <s xml:id="N21038" xml:space="preserve">Qñcū­
                    <lb/>
                   aliqua latitudo demit̄̄ a maiori et addit̄̄ minori
                    <lb/>
                  ipſo manēte minori ꝙ̄ illud ad quo demit̄̄ illa latitu­
                    <lb/>
                  do cõtinuo maiorē ꝓportionē acquirit illud minꝰ
                    <lb/>
                  ̄ deꝑdat illud maiꝰ. </s>
                  <s xml:id="N21043" xml:space="preserve">Quod ptꝫ / q2 ſi poſt̄ illa la­
                    <lb/>
                  titudo eſt addita minori addat̄̄ tanta latitudo illi
                    <lb/>
                  maiori a quo fuit dēpta, minorē ꝓportionē acq̇ret
                    <lb/>
                  illud maiꝰ ꝙ̄ deꝑdet illud minꝰ: g̊ qñ maius deꝑdat
                    <lb/>
                  illã latitudinē et minꝰ acq̇rit eandē maiorē ꝓporti-
                    <lb/>
                  onē acq̇rit minꝰ ꝙ̄ deꝑdat maiꝰ, cū nõ deꝑdat niſi
                    <lb/>
                  illã quã acq̇ſiuit: igr̄ illa regula eſt vera. </s>
                  <s xml:id="N21052" xml:space="preserve">Si aūt illa
                    <lb/>
                  ꝑueniant ad equalitatē, iã nõ eque velociter et eque
                    <lb/>
                  ꝓbationabiliter vnū illoꝝ augebitur ſicut aliud di­
                    <lb/>
                  minuitur / vt ꝓbatū eſt in argumento
                    <note position="left" xlink:href="note-0175-01a" xlink:label="note-0175-01" xml:id="N210E0" xml:space="preserve">1. confir-
                      <lb/>
                    matio</note>
                  </s>
                  <s xml:id="N21060" xml:space="preserve">¶ Cõfirmatur
                    <lb/>
                  </s>
                  <s xml:id="N21064" xml:space="preserve">Quia raritas et denſitas inter ſe nõ differūt cū idē
                    <lb/>
                  ſit ꝓpinquitas punctoꝝ et diſtantia eorūdē: igr̄ ille
                    <lb/>
                  nõ ſunt qualitates poſitiue.
                    <note position="left" xlink:href="note-0175-02a" xlink:label="note-0175-02" xml:id="N210E8" xml:space="preserve">2. confir-
                      <lb/>
                    matio</note>
                  </s>
                  <s xml:id="N21070" xml:space="preserve">¶ Cõfirmat̄̄ ſcḋo. </s>
                  <s xml:id="N21073" xml:space="preserve">Q2
                    <lb/>
                  ſi eſſent qualitates eſſent cõtrarie: ſed hoc eſt falſū /
                    <lb/>
                  q2 tūc nullū rarū eſſet denſum et eocõtra et aliquid
                    <lb/>
                  eſſet qḋ nõ eſſet rarū ne dēſum : q2 rarū et denſum
                    <lb/>
                  eſſent termini cõtrarii.
                    <note position="left" xlink:href="note-0175-03a" xlink:label="note-0175-03" xml:id="N210F0" xml:space="preserve">3. confir-
                      <lb/>
                    matio</note>
                  </s>
                  <s xml:id="N21083" xml:space="preserve">¶ Cõfirmatur tertio. </s>
                  <s xml:id="N21086" xml:space="preserve">Quia
                    <lb/>
                  tūc ſeq̇tur /  poſſibile eſt dare rarū vniformiter dif­
                    <lb/>
                  forme a certo gradu vſ ad nõ gradū, vt ab octa-
                    <lb/>
                  uo vſ ad nõ gradū: ſed ↄ̨ñs eſt falſū: g̊ et illud ex q̊
                    <lb/>
                  ſeq̇tur. </s>
                  <s xml:id="N21091" xml:space="preserve">Cõſequētia ꝓbatur: q2 oīs qualitas corpo-
                    <lb/>
                  rea poteſt eſſe vniformiter difformis a certo gradu
                    <lb/>
                  vſ ad nõ gradū: ſed raritas eſt huiuſmodi per te
                    <lb/>
                  igr̄. </s>
                  <s xml:id="N2109A" xml:space="preserve">Maior ptꝫ / q2 vbicū eſt qualitas vniformis:
                    <lb/>
                  ibi eſt vna medietas intenſiua vniformiter diffor-
                    <lb/>
                  mis a maximo gradu quē hꝫ illa qualitas vſ ad
                    <lb/>
                  nõ gradū: vt ptꝫ iutuenti. </s>
                  <s xml:id="N210A3" xml:space="preserve">Sed iam argr̄ falſitas cõ­
                    <lb/>
                  ſequentis / q2 ſit illud a. / et arguo ſic / illud eſt vniformi­
                    <lb/>
                  ter difformiter rarū ab octauo vſ ad nõ gradū: g̊
                    <lb/>
                  prima pars ꝓportionalis eiꝰ eſt aliqualiter rara et
                    <lb/>
                  ſcḋa in duplo minꝰ rara, et tertia in duplo minꝰ ra-
                    <lb/>
                  ra ꝙ̄ ſcḋa, et ſic ↄ̨ñter / vt ptꝫ de albedine vniformiter
                    <lb/>
                  difformi ab octauo vſ ad nõ g̈dū, et ꝑ ↄ̨ñs ṗma ꝑs
                    <lb/>
                  ꝓportiõalis eſt aliq̈liṫ denſa, et ſcḋa in duplo den­
                    <lb/>
                  ſior, et tertia in duplo denſior ꝙ̄ ſcḋa. etc̈. / igr̄ a. eſt
                    <lb/>
                  infinite denſū q2 infinitã materiã cõtinet ſub finita
                    <lb/>
                  quãtitate, nã q̄libet pars ꝓportionalis cõtinet tan­
                    <lb/>
                  tam materiã ſicut prima: q2 in quacū ꝓportione
                    <lb/>
                  aliqua pars ꝓportiõalis eſt minor prima in eadē
                    <lb/>
                  eſt denſior prima, et vltra a. eſt īfinite denſum: g̊ nõ
                    <lb/>
                  eſt rarū, et ſic nõ eſt vniformiter difformiter rarum /
                    <lb/>
                  quod eſt oppoſitū cõceſſi.
                    <note position="left" xlink:href="note-0175-04a" xlink:label="note-0175-04" xml:id="N210F8" xml:space="preserve">4. confir­
                      <lb/>
                    matio.</note>
                  </s>
                  <s xml:id="N210C9" xml:space="preserve">¶ Cõfirmatur quarto / q2
                    <lb/>
                  rarū eſt qḋ ſub magna quãtitate cõtinet parum de
                    <lb/>
                  materia, denſum vero eſt /  ſub parua quãtitate cõ­
                    <lb/>
                  tinet multū de materia: et hoc deſcribendo rarū et
                    <lb/>
                  denſum: g̊ dato  a. nullã qualititatē haberet et ſub
                    <lb/>
                  finita quantitate finitam materiam contineret ad
                    <lb/>
                  huc illud eſſet rarum et denſum, vt facile deducitur
                    <lb/>
                  ex deſcriptione rari et denſi: igitur raritas et den-
                    <lb/>
                  ſitas nõ ſunt qualitates nec poſitiue ſe habent.</s>
                </p>
                <p xml:id="N21100">
                  <s xml:id="N21101" xml:space="preserve">Scḋo prīcipaliter. </s>
                  <s xml:id="N21104" xml:space="preserve">Tangēdo penes
                    <lb/>
                  quid maioritas raritatis et dēſitatis attēdat̄̄ argr̄
                    <cb chead="Capitulū primū."/>
                  ſic. </s>
                  <s xml:id="N2110C" xml:space="preserve">Si raritas et denſitas eſſent poſſibiles vel ī qua­
                    <lb/>
                  cun ꝓportione raritas efficitur maior: ꝓportio
                    <lb/>
                  quãtitatis ad materiã efficiret̄̄ maior, et nõ quãti-
                    <lb/>
                  tas in illa ꝓportione, vel in quacū ꝓportione ra­
                    <lb/>
                  ritas efficit̄̄ maior: quãtitas efficit̄̄ maior. </s>
                  <s xml:id="N21117" xml:space="preserve">Sed neu­
                    <lb/>
                  trū iſtoꝝ eſt dicendū: igr̄ raritas et denſitas nõ ſūt
                    <lb/>
                  poſſibiles. </s>
                  <s xml:id="N2111E" xml:space="preserve">Minor ptꝫ / q2 iſte due ſunt famate opi-
                    <lb/>
                  niones quas maior tangit de maioritate et minori­
                    <lb/>
                  tate raritatis et nõ plures ꝓ nūc practicantur. </s>
                  <s xml:id="N21125" xml:space="preserve">Sed
                    <lb/>
                  iam ꝓbatur minor: et primo  nõ in quacū ꝓpor­
                    <lb/>
                  tione raritas efficitur maior proportio quãtitatis
                    <lb/>
                  ad materiã efficitur maior: q2 tūc ſeq̄retur /  ad du­
                    <lb/>
                  plationē raritatis nõ ſeq̄retur duplatio quãtita-
                    <lb/>
                  tis q2 aliqñ ſequitur magis ꝙ̄ duplatio quantita-
                    <lb/>
                  tes, et aliqñ minꝰ, et aliqñ adequata duplatio: igr̄.
                    <lb/>
                  </s>
                  <s xml:id="N21135" xml:space="preserve">ſed ↄ̨ñs eſt falſū: igr̄. </s>
                  <s xml:id="N21138" xml:space="preserve">Falſitas ↄ̨ñtis argr̄ / q2 rarum
                    <lb/>
                  eſt qḋ ſub magna quãtitate cõtinet modicū de ma-
                    <lb/>
                  teria, ergo illud erit in duplo magis rarum quod
                    <lb/>
                  ſubdupla maiori quantitate ↄ̨tinet equale de ma-
                    <lb/>
                  teria, et ſic ſemꝑ ad duplationē raritatis ſequitur
                    <lb/>
                  duplatio quãtitatꝪ. </s>
                  <s xml:id="N21145" xml:space="preserve">Sed iam ꝓbo ſequelã: et capio
                    <lb/>
                  vnū pedale cuiꝰ quãtitatis ad materiã ſit ꝓportio
                    <lb/>
                  ſexq̇altera et volo /  dupletur eiꝰ raritas quo po-
                    <lb/>
                  ſito argr̄ ſic / quãtitas illiꝰ pedalis nõ efficitur in du­
                    <lb/>
                  plo maior: ſed p̄ciſe in ſexq̇altero maior: igr̄ ꝓpoſi­
                    <lb/>
                  tum. </s>
                  <s xml:id="N21152" xml:space="preserve">Probat̄̄ añs, q2 in fine ꝓportio quantitatis
                    <lb/>
                  ad materiã erit dupla ad ſexquialterã puta dupla
                    <lb/>
                  ſexquiq̈rta: g̊ ſeq̇tur,  p̄ciſe quãtitas acq̇ſiuit pro­
                    <lb/>
                  portionē ſexq̇alterã et nõ duplã. </s>
                  <s xml:id="N2115B" xml:space="preserve">Ptꝫ ↄ̨ña / q2 ꝓpor-
                    <lb/>
                  tio quãtitatis ad materiã in fine cõponitur ex dua­
                    <lb/>
                  bus ſexq̇alteris: et iã quãtitas ad materiã habebat
                    <lb/>
                  ꝓportionē ſexquialterã: g̊ modo p̄ciſe acq̇ſiuit ſexq̇al­
                    <lb/>
                  terã ſupra ſe. </s>
                  <s xml:id="N21166" xml:space="preserve">Probat̄̄ ſcḋa / q2 ſi acq̇ſiuiſſet duplaꝫ
                    <lb/>
                  ꝓportionē ſupra ſe in fine ꝓportio quantitatis ad
                    <lb/>
                  materiã fuiſſet tripla q̄ ex dupla et ſexq̇altera cõpo­
                    <lb/>
                  nitur et ſic nõ ad duplationē raritatis fuiſſet ſequu­
                    <lb/>
                  ta duplatio ꝓportionis cū tripla ſit maior ꝙ̄ du-
                    <lb/>
                  pla ad ſexq̇alterã / vt pꝫ ex ſecūda parte huiꝰ operꝪ
                    <lb/>
                  et ſic ſeq̇tur /  ad duplationē raritatis aliqñ ſeq̇tur
                    <lb/>
                  minꝰ ꝙ̄ duplatio quãtitatis. </s>
                  <s xml:id="N21177" xml:space="preserve">Q, vero aliqñ
                    <lb/>
                  ſeq̈tur p̄ciſe duplatio quãtitatis ꝓbatur ponendo
                    <lb/>
                   ꝓportio quãtitatis ad materiã ſit dupla, et  du­
                    <lb/>
                  pletur raritas, et ſic habebitur intentū. </s>
                  <s xml:id="N21180" xml:space="preserve">Nã tūc ꝓ-
                    <lb/>
                  portio quãtitatis ad materiã efficeretur quadru-
                    <lb/>
                  pla q̄ eſt dupla ad duplã, et iã antea ꝓportio ̄tita­
                    <lb/>
                  tis ad materiã fuit dupla adequate: g̊ modo acq̇ſi-
                    <lb/>
                  uit aliquã ꝓportionē duplã, et ſic ſeq̇tur /  quãtitas
                    <lb/>
                  acq̇ſiuit duplã ꝓportionē ſupra ſe: qm̄ tantã acq̇ſi-
                    <lb/>
                  uit ſupra ſe quantam ſupra ſuam materiam. </s>
                  <s xml:id="N2118F" xml:space="preserve">Sed
                    <lb/>
                  iam ꝓbo /  nõ in quacū ꝓportione raritas effici­
                    <lb/>
                  tur maior quãtitas efficitur maior: q2 alias ſeq̄ret̄̄ /
                    <lb/>
                   poſſet dari infinite rarū: ſed ↄ̨ñs eſt falſum: igr̄ et
                    <lb/>
                  illud ex quo ſeq̇tur. </s>
                  <s xml:id="N2119A" xml:space="preserve">Seq̄la ꝓbatur et capio vnū pe-
                    <lb/>
                  dale vniforme ꝑ totū et volo /  rarefiat in īfinitum
                    <lb/>
                  quo poſito illud erit īfinite raꝝ qm̄ ad duplationē
                    <lb/>
                  eiꝰ ſeq̇tur duplatio raritatis et ad triplationē quã­
                    <lb/>
                  tatis ſeq̇tur triplatio raritatis / et ſic cõſequenter:
                    <lb/>
                  et acquiret̄̄ quãtitas īfinita: g̊ raritas īfinita. </s>
                  <s xml:id="N211A7" xml:space="preserve">Sed
                    <lb/>
                  falſitas ↄ̨ñtis argr̄ et ſi illud eſt infinite rarū: ſequit̄̄ /
                    <lb/>
                   nullã materiã cõtinet, et vltra nullã materiã cõti­
                    <lb/>
                  net. </s>
                  <s xml:id="N211B0" xml:space="preserve">g̊ nec eſt rarū nec eſt dēſum. </s>
                  <s xml:id="N211B3" xml:space="preserve">Conſequētia ptꝫ et
                    <lb/>
                  argr̄ ſeq̄la qm̄ vt ſuppono ipſum eſt vniforme, et
                    <lb/>
                  vniformiter rarefactū: ſi igr̄ hꝫ aliquã materiaꝫ in
                    <lb/>
                  aliqua parte ſui cū ipſum ſit vniforme: ſeq̇tur /  in
                    <lb/>
                  qualibet tanta ſui parte hꝫ tantaꝫ ſicut ipſa eſt: et
                    <lb/>
                  ſunt infinite partes illi parti equales: g̊ ſeq̇tur / 
                    <lb/>
                  hꝫ īfinitã materiã, et ſic eſt īfinite raꝝ / qḋ fuit ꝓbandū</s>
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