Alvarus, Thomas, Liber de triplici motu, 1509

Table of contents

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[4.] Capitulū q̈rtū / in quo principalr̄ q̄rit̄̄ penes quid attendi intenſio qualitatis difformis debeat.
Page: 263
[5.] Capitulum quintum inquirens penes quid gradus ſummi inductio ſit attendenda.
Page: 277
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              <div xml:id="N1D065" level="4" n="2" type="chapter" type-free="capitulum">
                <pb chead="De motu locali quo ad effectū ſcḋm ſubiectū difformi." file="0140" n="140"/>
                <p xml:id="N1DA51">
                  <s xml:id="N1DA52" xml:space="preserve">Octaua ↄ̨̨cluſio ꝓportio quadratorū
                    <lb/>
                  ꝑfectoꝝ et eque ꝓfundoꝝ vniformiter eſt ꝓp̄ortio co­
                    <lb/>
                  ſtaꝝ duplicata. </s>
                  <s xml:id="N1DA59" xml:space="preserve">Et voco quadratū pērfectū cuiꝰ oēs
                    <lb/>
                  coſte ſunt eq̈les et oēs anguli recti eq̈les. </s>
                  <s xml:id="N1DA5E" xml:space="preserve">Nõ intelli-
                    <lb/>
                  gas tñ /  velim dicere /  oēs coſte debent eſſe eq̈les
                    <lb/>
                  ſcḋm oēm dimēſionē: ſed ſatis eſt ſcḋm latitudinem
                    <lb/>
                  et lõgitudinē. </s>
                  <s xml:id="N1DA67" xml:space="preserve">Exēplū / vt ſi ſit vnū q̈dratū pedaliter
                    <lb/>
                  longū, pedaliter latū et pedaliter ꝓfundū: et aliud
                    <lb/>
                  bipedaliter longū bipedaliter latū et ſolū pedali-
                    <lb/>
                  ter ꝓfundū / tūc dico /  vnū eſt q̈druplū ad alteꝝ: qm̄
                    <lb/>
                  coſte ſe habēt in proportio dupla et magnitudi-
                    <lb/>
                  nes ſe habebãt in ꝓportiõe dupla ad duplã cuiuſ-
                    <lb/>
                  modi eſt q̈drupla ꝓportio. </s>
                  <s xml:id="N1DA76" xml:space="preserve">Probat̄̄ hec cõcluſio et
                    <lb/>
                  capio duo q̈drata ꝑfecta eq̈liter ꝓfunda vniformiṫ /
                    <lb/>
                  q̇ꝝ minꝰ ſit a. et maiꝰ c. et habeat ſe coſta ipſius c. ad
                    <lb/>
                  coſtã ipſiꝰ a. in ꝓportione f. / tūc dico /  ipſiꝰ c. ad ip­
                    <lb/>
                  ſum a. eſt ꝓportio duplicata ad ꝓportionē ipſiꝰ f.
                    <lb/>
                  </s>
                  <s xml:id="N1DA82" xml:space="preserve">Quod ꝓbo ſic et capio vnū aliud corpꝰ puta b. /  ſit
                    <lb/>
                  eque ꝓfundū et eque latū ſicut a. vniformiter et in f.
                    <lb/>
                  ꝓportiõe lõgiꝰ et manifeſtū eſt /  ipſiꝰ b. ad ipſuꝫ a.
                    <lb/>
                  eſt ꝓportio f. / vt ptꝫ ex prima ſuppoſitiõe: et ipſius c.
                    <lb/>
                  ad ipſum b. eſt etiã f. ꝓportio: vt ptꝫ ex ſcḋa ſuppo-
                    <lb/>
                  ſitione: qm̄ cū ipſū c. (vt ponit̄̄ in caſu) ſit in f. ꝓpor-
                    <lb/>
                  tione latiꝰ quã ipſum b. et eſt eque longū et eque ꝓ-
                    <lb/>
                  fundū ſicut ipſum b. / igr̄ eſt in f. ꝓportiõe maiꝰ ipſo
                    <lb/>
                  b. / vt oſtēdit p̄dicta ſcḋa ſuppoſitio: igr̄ ipſius c. ad
                    <lb/>
                  ipſuꝫ a. eſt ꝓportio duplicata ad ꝓportionē f. </s>
                  <s xml:id="N1DA97" xml:space="preserve">Ptꝫ
                    <lb/>
                  hec ↄ̨ña ex cõcluſione octaua ſexti capitis ſcḋe par­
                    <lb/>
                  tis / qm̄ ibi ſunt .3. termini cõtinuo ꝓportiõales f. ꝓ-
                    <lb/>
                  portionē / qm̄ b. ad a. eſt ꝓportio f. et c. ad b. eſt ꝓpor­
                    <lb/>
                  tio f. / igr̄ c. ad a. eſt proportio duplicata ſiue dupla
                    <lb/>
                  ad ꝓportionē f. / vt clare oſtendit p̄dicta octaua con­
                    <lb/>
                  cluſio allegata.
                    <note position="left" xlink:href="note-0140-01a" xlink:label="note-0140-01" xml:id="N1DB3B" xml:space="preserve">1. correĺ.</note>
                  </s>
                  <s xml:id="N1DAAB" xml:space="preserve">¶ Ex hac cõcluſione ſequit̄̄ tale cor-
                    <lb/>
                  relariū /  ꝓportio duoꝝ corpoꝝ cuboꝝ ſiue ꝑfecte
                    <lb/>
                  quadratoruꝫ ſimpliciter cuiuſmodi ſunt data ſiue
                    <lb/>
                  taxilli quoꝝ lõgitudo eſt eq̈lis latitudini et ꝓfūdi-
                    <lb/>
                  tati: ē ꝓportio coſtaꝝ triplicata. </s>
                  <s xml:id="N1DAB6" xml:space="preserve">Exēplū / vt ſi fuerit
                    <lb/>
                  vnū corpꝰ cubū pedaliṫ ꝓfundū et aliud corpꝰ cubū
                    <lb/>
                  bipedaliter ꝓfundū dico /  illud bipedalittr ꝓfun­
                    <lb/>
                  dū eſt octuplū ad illud pedaliter ꝓfundū / qm̄ coſte
                    <lb/>
                  ad coſtã ē ꝓportio dupla / igr̄ ex correlario oꝫ ꝓpor­
                    <lb/>
                  tionē magnitudīs eē triplã ad ꝓportionē duplã: et
                    <lb/>
                  illa ē octupla / vt pꝫ ex ſcḋa ꝑte: igr̄. </s>
                  <s xml:id="N1DAC5" xml:space="preserve">Probat̄̄ hoc cor­
                    <lb/>
                  relariū et capio duo corꝑa cuba quoꝝ latera ſiue co­
                    <lb/>
                  ſte ſe habeãt in f. ꝓportiõe et ſit minꝰ illoꝝ a. et maiꝰ
                    <lb/>
                  illoꝝ d. / deīde capio b. corpꝰ /  ſit eq̄ ꝓfnndū et eque
                    <lb/>
                  latū ſicut a. et in f. ꝓportiõe lõgiꝰ: deīde capio q̈rtū
                    <lb/>
                  corpꝰ puta c. / qḋ ſit eq̄ longū et eq̄ ꝓfundū ſicut b. et ī
                    <lb/>
                  f. ꝓportiõe latiꝰ: et arguo ſic / d. ad c. eſt f. ꝓportio / vt
                    <lb/>
                  pꝫ ex ſcḋa ſuppõe et b. ad a. ē f. ꝓportio / vt pꝫ ex ṗma /
                    <lb/>
                  igr̄ d. ad a. eſt triplicata ꝓportio ſiue tripla ad ꝓ-
                    <lb/>
                  portionē f. / vt ptꝫ ex .8. ↄ̨cluſiõe ſexti capitꝪ ſcḋe ꝑtꝪ /
                    <lb/>
                  qḋ fuit ꝓbãdū.
                    <note position="left" xlink:href="note-0140-02a" xlink:label="note-0140-02" xml:id="N1DB41" xml:space="preserve">2. correĺ.</note>
                  </s>
                  <s xml:id="N1DAE1" xml:space="preserve">¶ Ex q̊ ſequit̄̄ /  datꝪ duobꝰ q̈drãgu­
                    <lb/>
                  lis cubis quoꝝ coſte ſe hñt in ꝓportiõe ſexq̇altera:
                    <lb/>
                  maiorꝪ q̈drãguli ad mīorē ē ꝓportio tripla ſuꝑtri­
                    <lb/>
                  partiēs octauas q̈lis .27. ad .8. </s>
                  <s xml:id="N1DAEA" xml:space="preserve">Probat̄̄ qm̄ / vt ptꝫ
                    <lb/>
                  ex p̄cedēti correlario ꝓportio duoꝝ cuboꝝ ſiue q̈dra­
                    <lb/>
                  toꝝ perfectorū ē ꝓportio coſtaꝝ triplata: ſꝫ ꝓpor-
                    <lb/>
                  tio tripla ſuꝑtriꝑtiēs .8. eſt tripla ad ꝓportionē ſex­
                    <lb/>
                  q̇alterã / q̄ ē īter coſtas datoꝝ q̈dratoꝝ: igr̄ talia q̄-
                    <lb/>
                  drata cuba ſe hñt ī ꝓportiõe tripla ſuꝑtriꝑtiēte .8.
                    <lb/>
                  </s>
                  <s xml:id="N1DAF8" xml:space="preserve">Maior ptꝫ cū ↄ̨ña: et ꝓbat̄̄ minor / qm̄ ꝓportior .27.
                    <lb/>
                  ad .8. ↄ̨ponit̄̄ ex tribꝰ ſexq̇alterꝪ. </s>
                  <s xml:id="N1DAFD" xml:space="preserve">Sint em̄ īter illos
                    <lb/>
                  nūeros .4. termini cõtinuo ꝓportiõales ꝓportione
                    <lb/>
                  ſexq̇altera. </s>
                  <s xml:id="N1DB04" xml:space="preserve">Nã .27. ad .18. eſt ꝓportio ſexq̇altera et
                    <lb/>
                  18. ad .12. eſt ꝓportio ſexq̇altera et .12. ad .8. ſexq̇alte­
                    <lb/>
                  ra.
                    <note position="left" xlink:href="note-0140-03a" xlink:label="note-0140-03" xml:id="N1DB47" xml:space="preserve">3. correĺ.</note>
                  </s>
                  <s xml:id="N1DB10" xml:space="preserve">¶ Seq̇tur vlteriꝰ /  datis duobꝰ q̈dratis cubicis
                    <lb/>
                  quoꝝ latera ſe hñt in ꝓportiõe tripla: īter maius et
                    <cb chead="De motu locali quo ad effectū ſcḋm ſubiectū difformi."/>
                  minꝰ reperit̄̄ ꝓportio vicecupla ſeptupla: qualis eſt
                    <lb/>
                  ꝓportio .27. ad vnū. </s>
                  <s xml:id="N1DB1A" xml:space="preserve">Ptꝫ hoc correlariū ex primo
                    <lb/>
                  correlario hoc addito /  proportio vicecupla ſe-
                    <lb/>
                  ptupla ex tribꝰ triplis cõponit̄̄ / qḋ facile eſt ꝓſpice­
                    <lb/>
                  re. </s>
                  <s xml:id="N1DB23" xml:space="preserve">Nã .27. ad .9. eſt ꝓportio tripla: et .9. ad .3. eſt pro­
                    <lb/>
                  portio tripla: et .3. ad vnū ſimiliter tripla ꝓportio
                    <lb/>
                  </s>
                  <s xml:id="N1DB29" xml:space="preserve">Iſto modo ꝓcedēdo aliquãtula primeditatiõe et cõ­
                    <lb/>
                  ſideratione cõpoſitionis ꝓportionū: īfinita corre-
                    <lb/>
                  laria ex p̄dicto primo correlario īferri valent et ſi-
                    <lb/>
                  militer ex cõcluſione. </s>
                  <s xml:id="N1DB32" xml:space="preserve">ſed differantur vſ ad mate-
                    <lb/>
                  riam de augmentatione.</s>
                </p>
                <note position="right" xml:id="N1DB4D" xml:space="preserve">Cõcluſio
                  <lb/>
                brauarḋ</note>
                <p xml:id="N1DB53">
                  <s xml:id="N1DB54" xml:space="preserve">Nona cõcluſio. </s>
                  <s xml:id="N1DB57" xml:space="preserve">Scḋm opinionē q̄ po­
                    <lb/>
                  nit velocitatē motus difformiter difformis quo ad
                    <lb/>
                  ſubiectū attendi debere penes gradū ſummū: ꝓpor­
                    <lb/>
                  tio motus duaꝝ ſperaꝝ ſiue duoꝝ orbiū: pariter
                    <lb/>
                  duoꝝ circuloꝝ in equali tēpore ceteris paribꝰ circū­
                    <lb/>
                  giratoꝝ eſt ſicut ꝓportio ſuoꝝ diametroꝝ. </s>
                  <s xml:id="N1DB64" xml:space="preserve">Proba­
                    <lb/>
                  tur hec cõcluſio / qm̄ ꝓportio perimetroꝝ circulorū
                    <lb/>
                  eſt ſicut ꝓportio diametroꝝ: et quãto vna diameter
                    <lb/>
                  eſt maior altera tanto maiorē lineã deſcribit eius
                    <lb/>
                  punctꝰ maxime a centro diſtans: igr̄ cõcluſio vera.
                    <lb/>
                  </s>
                  <s xml:id="N1DB70" xml:space="preserve">¶ Hic tñ aduerte /  ad inducendã hanc concluſionē
                    <lb/>
                  proceſſu mathematico oportet maiori apparatu
                    <lb/>
                  vti quã p̄ſens exigat opus: ſatis eſt em̄ in iſtis. </s>
                  <s xml:id="N1DB77" xml:space="preserve">Eu-
                    <lb/>
                  clidi et mathematicoꝝ ṗmoribꝰ fidē exhibere. </s>
                  <s xml:id="N1DB7C" xml:space="preserve">In
                    <lb/>
                  hac em̄ cõſideratione phiſica mathematice ſcien­
                    <lb/>
                  tie ſubalternari nõ dedignatur: quēadmodū in ſci­
                    <lb/>
                  entia de iride ſubalternata perſpectiue dinoſcitur
                    <lb/>
                  teſte philoſopho primo poſteriorum.</s>
                </p>
                <note position="right" xml:id="N1DB87" xml:space="preserve">Pḣs pri-
                  <lb/>
                mo po-
                  <lb/>
                ſteriorū.</note>
                <p xml:id="N1DB8F">
                  <s xml:id="N1DB90" xml:space="preserve">Decima cõcluſio. </s>
                  <s xml:id="N1DB93" xml:space="preserve">Proportio motuū
                    <lb/>
                  duaꝝ ſpheraꝝ ſolidaꝝ eſt ſicut ꝓportio diametro-
                    <lb/>
                  rum. </s>
                  <s xml:id="N1DB9A" xml:space="preserve">Et hoc ſcḋm oēm opinionē. </s>
                  <s xml:id="N1DB9D" xml:space="preserve">Probat̄̄ ex priori
                    <lb/>
                  ̄tum ad opinionē / q̄ dicit velocitatē attendi debe-
                    <lb/>
                  re penes punctū velociſſime motū. </s>
                  <s xml:id="N1DBA4" xml:space="preserve">Sed ̄tū ad aliã
                    <lb/>
                  opinionē ptꝫ / qm̄ ſcḋm aliã velocitas ſpere ſolide
                    <lb/>
                  debet attendi ſcḋm lineã deſcriptã a pūcto medio
                    <lb/>
                  ſemidiametri īter centrū et circūferentiã: et ꝑ ↄ̨ñs a
                    <lb/>
                  puncto deſcripto ab vna quarta ſemidiametri: ſed
                    <lb/>
                  in quacū ꝓportione vna diameter eſt maior alte­
                    <lb/>
                  ra in eadē vna quarta eſt maior vna quarta alteriꝰ /
                    <lb/>
                  ergo ſcḋm hãc opinionē in quacū ꝓportionē dia­
                    <lb/>
                  meter vniꝰ ſpere ſolide erit maior diametro alteriꝰ
                    <lb/>
                  in eadē ꝓportiõe maiorē lineã deſcribet punctꝰ me­
                    <lb/>
                  dius ſemidiametri: et per ↄ̨ñs ꝓportio motus erit
                    <lb/>
                  ſicut ꝓportio diametrorum / quod fuit probandū.</s>
                </p>
                <p xml:id="N1DBBD">
                  <s xml:id="N1DBBE" xml:space="preserve">Undecima ↄ̨̨cluſio </s>
                  <s xml:id="N1DBC1" xml:space="preserve">Proportio motuū
                    <lb/>
                  duaꝝ ſperaꝝ īequaliū in eode tēpore circūgirataꝝ
                    <lb/>
                  dūmodo ſint ſolide eſt ſubtripla ad ꝓportionē ſpe­
                    <lb/>
                  raꝝ īter ſe. </s>
                  <s xml:id="N1DBCA" xml:space="preserve">Probat̄̄ hec cõcluſio / qm̄ ꝓportio mo-
                    <lb/>
                  tuū duaꝝ ſperaꝝ eſt ꝓportio diametroꝝ taliū ſpe­
                    <lb/>
                  raꝝ / vt ptꝫ ex priori: ſꝫ ꝓportio ſperaꝝ īequaliū eſt
                    <lb/>
                  ꝓportio diametroꝝ triplata ſiue eſt tripla ad pro­
                    <lb/>
                  portionē diametroꝝ / qḋ idē eſt / vt patꝫ ex vltīa decī
                    <lb/>
                  elemētoꝝ. </s>
                  <s xml:id="N1DBD7" xml:space="preserve">Euclidis
                    <gap/>
                  g̊ ꝓportio diametroꝝ eſt ſubtri­
                    <lb/>
                  pla ad ꝓportionē ſperaꝝ et talis ē ꝓportio motuū /
                    <lb/>
                  igr̄ ꝓportio motuuū duaꝝ ſperaꝝ ineq̈liū etc. ē ſub­
                    <lb/>
                  tripla ꝓportio ad proportionem ſperarū inter ſe.
                    <lb/>
                    <note position="right" xlink:href="note-0140-04a" xlink:label="note-0140-04" xml:id="N1DC1F" xml:space="preserve">1. correĺ.</note>
                  </s>
                  <s xml:id="N1DBE9" xml:space="preserve">¶ Ex quo ſequit̄̄ /  ſi vna ſpera eſt in octuplo maior
                    <lb/>
                  altera /  mouet̄̄ preciſe in duplo velociꝰ altera: et ſi
                    <lb/>
                  vna ſpera fuerit in triplo ſupertripartiēti octauas
                    <lb/>
                  maior altera ipſa mouet̄̄ in ſexq̇altero velociꝰ alte­
                    <lb/>
                  ra. </s>
                  <s xml:id="N1DBF4" xml:space="preserve">Ptꝫ hoc correlariū / q̊ ad primã ꝑtē qm̄ ꝓportio
                    <lb/>
                  octupla eſt tripla ad duplã: g̊ ſi ſpere ſe habēt ī octu­
                    <lb/>
                  pla ꝓportione motꝰ earū ſe habebūt in dupla q̄ eſt
                    <lb/>
                  ſubtripla ad octuplã: ptꝫ ↄ̨ña ex immediate prece-
                    <lb/>
                  dēte ↄ̨cluſione. </s>
                  <s xml:id="N1DBFF" xml:space="preserve">Eodē mõ ptꝫ / q̄ ad ſcḋaꝫ partē qm̄ ſi
                    <lb/>
                  ſpere ſe habent in ꝓportione tripla ſuꝑtripartiēti </s>
                </p>
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