Alvarus, Thomas, Liber de triplici motu, 1509

Table of contents

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            <div xml:id="N20F21" level="3" n="3" type="other" type-free="tractatus">
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                  <s xml:id="N22FE2" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0194" n="194"/>
                  denſitas in partes ꝓportionales / vt ponit̄̄ in pre-
                    <lb/>
                  cedēti correlario: ita  prima ſit aliqualiter dēſa
                    <lb/>
                  ſcḋa in duplo, tertia in triplo, et ſic ↄ̨ſequēter: tunc
                    <lb/>
                  totum eſt in duplo denſius ſua prima parte ꝓpor-
                    <lb/>
                  tionali. </s>
                  <s xml:id="N22FF4" xml:space="preserve">Probat̄̄ / q2 totū diuiſum per partes ꝓpor­
                    <lb/>
                  tionales ꝓportiõe dupla eſt duplum ad primã par­
                    <lb/>
                  tē ꝓportionalē eius / vt ptꝫ ex quinto capite prealle­
                    <lb/>
                  gato prime partis huius libri: igitur ꝑ cõcluſionē
                    <lb/>
                  primã immediate p̄cedentē illud eſt denſius prima
                    <lb/>
                  parte ꝓportionali in ꝓportione dupla.
                    <note position="left" xlink:href="note-0194-01a" xlink:label="note-0194-01" xml:id="N230F5" xml:space="preserve">3. correĺ.</note>
                  </s>
                  <s xml:id="N23006" xml:space="preserve">¶ Sequit̄̄
                    <lb/>
                  tertio /  diuiſo corpore ſi ꝑ partes ꝓportionales
                    <lb/>
                  ꝓportione dupla / vt ponit̄̄ in antecedēti correlario
                    <lb/>
                  totum eſt ita denſum ſicut ſcḋa pars proportiona­
                    <lb/>
                  lis eius. </s>
                  <s xml:id="N23011" xml:space="preserve">Probat̄̄ / q2 in duplo denſius prima vt ſe-
                    <lb/>
                  cundū correlarium aſſerit: et ſcḋa pars ꝓportiona­
                    <lb/>
                  lis eſt etiã in duplo denſior prima: g̊ totū eſt ita dē­
                    <lb/>
                  ſum ſicut ſecūda pars ꝓportionalis / quod fuit pro­
                    <lb/>
                  bandū. </s>
                  <s xml:id="N2301C" xml:space="preserve">Patet conſequētia ꝑ hanc maximã </s>
                  <s xml:id="N2301F" xml:space="preserve">Oīa ha-
                    <lb/>
                  bentia equalē ꝓportionē ad vnū tertiū ſunt equa-
                    <lb/>
                  lia: ſꝫ totius denſitas et denſitas ſecūde partis ꝓ-
                    <lb/>
                  portionalis habent equalem proportionē ad den-
                    <lb/>
                  ſitatē prime partis ꝓportiõis puta duplã: igit̄̄ dē-
                    <lb/>
                  ſitas totius et ſecūde partis ꝓportionalis ſūt equa­
                    <lb/>
                  les / quod erat inducendū.
                    <note position="left" xlink:href="note-0194-02a" xlink:label="note-0194-02" xml:id="N230FB" xml:space="preserve">4. correĺ.</note>
                  </s>
                  <s xml:id="N23033" xml:space="preserve">¶ Sequit̄̄ quarto /  ſi ali­
                    <lb/>
                  quod corpus diuidat̄̄ ꝑ partes ꝓportiõales ꝓpor-
                    <lb/>
                  portione ſexquialtera: et prīa pars ꝓportionalis
                    <lb/>
                  ſit aliqualiter denſa: et ſecūda ī duplo: et tertia ī tri­
                    <lb/>
                  plo ꝙ̄ prima: et ſic cõſequēter vt ponitur in caſu pri­
                    <lb/>
                  me cõcluſionis et correlarii: totū eſt in triplo dēſius
                    <lb/>
                  prima parte ꝓportionali. </s>
                  <s xml:id="N23042" xml:space="preserve">Et ſi diuidatur ꝓportio­
                    <lb/>
                  ne ſexquitertia: totū erit denſius prima parte pro-
                    <lb/>
                  portionali in quadruplo. </s>
                  <s xml:id="N23049" xml:space="preserve">Et ſi in ſexquiquarta: to­
                    <lb/>
                  tum erit denſius prima parte ꝓportionali in ꝓpor­
                    <lb/>
                  tione quītupla. </s>
                  <s xml:id="N23050" xml:space="preserve">et ſic ↄ̨ſequēter ꝓcedēdo ꝑ ſpecies
                    <lb/>
                  ꝓportionis ſuperparticularis in diuiſione corpo­
                    <lb/>
                  is: et per ſpecies proportionis multiplicis ex parte
                    <lb/>
                  denſitatis. </s>
                  <s xml:id="N23059" xml:space="preserve">Probatur hoc corolariuꝫ / quia totum
                    <lb/>
                  diuiſum ꝑ partes porportionales proportione ſex­
                    <lb/>
                  quialtera eſt tripluꝫ ad primã ꝑtē eiꝰ ꝓportionalē
                    <lb/>
                  et ſexquitertia quadruplū: et ſexquiquarta quītu-
                    <lb/>
                  plum. </s>
                  <s xml:id="N23064" xml:space="preserve">vt pꝫ ex prima parte huiꝰ operis: g̊ in eiſdem
                    <lb/>
                  ꝓportionibus ſe habēt denſitates totius ad denſi­
                    <lb/>
                  tatem prime partis ꝓportionalis. </s>
                  <s xml:id="N2306B" xml:space="preserve">igit̄̄ correlariuꝫ
                    <lb/>
                  verum.
                    <note position="left" xlink:href="note-0194-03a" xlink:label="note-0194-03" xml:id="N23101" xml:space="preserve">5. correĺ.</note>
                  </s>
                  <s xml:id="N23075" xml:space="preserve">¶ Sequitur quito /  ſi diuidatur corpus vt
                    <lb/>
                  dicitur in p̄cedenti correlario vt puta ꝓportiõe ſex­
                    <lb/>
                  quialtera: et prima pars ſit aliqualiter denſa: et ſe­
                    <lb/>
                  cunda in duplo: et tertia in triplo .etc̈. totum eſt ita
                    <lb/>
                  denſum ſicut tertia pars ꝓportionalis eius. </s>
                  <s xml:id="N23080" xml:space="preserve">Et ſi
                    <lb/>
                  ſexq̇tertia ſicut quarta pars ꝓportiõalis eiꝰ. </s>
                  <s xml:id="N23085" xml:space="preserve">Et ſi
                    <lb/>
                  ſexquiquarta ſicut quīta pars ꝓportionalis eius.
                    <lb/>
                  </s>
                  <s xml:id="N2308B" xml:space="preserve">Et ſexquiquinta: ſicut ſexta pars ꝓportionalis eiꝰ /
                    <lb/>
                  et ſic cõſequēter aſcendēdo ꝑ partes ꝓportionales
                    <lb/>
                  et per ſpecies ꝓportiõis ſuꝑparticularis in infini­
                    <lb/>
                  tum. </s>
                  <s xml:id="N23094" xml:space="preserve">Probat̄̄ / qm̄ ſi corpus ſit diuiſum ꝓportione
                    <lb/>
                  ſexquialtera ipſum eſt in triplo denſius prīa par-
                    <lb/>
                  te ꝓportionali / vt ptꝫ ex precedenti correlario et ter­
                    <lb/>
                  tia pars ꝓportionalis eſt etiã in triplo denſior pri­
                    <lb/>
                  ma / vt ptꝫ ex caſu. </s>
                  <s xml:id="N2309F" xml:space="preserve">g̊ eſt ita denſum tale corpus ſicut
                    <lb/>
                  tertia pars ꝓportionalis. </s>
                  <s xml:id="N230A4" xml:space="preserve">Itē ſi diuidatur ꝓportio­
                    <lb/>
                  ne ſexquitertia ip̄m eſt in quadruplo denſius ṗma
                    <lb/>
                  eius parte ꝓportionali / vt pꝫ ex p̄cedenti correlario
                    <lb/>
                  et etiã quarta pars ꝓportionalis eiꝰ eſt in quadru­
                    <lb/>
                  plo denſior ṗma / vt pꝫ ex caſu. </s>
                  <s xml:id="N230AF" xml:space="preserve">igit̄̄ illud corpus ita
                    <lb/>
                  diuiſum ꝑ partes ꝓportionales ꝓportione ſexqui­
                    <lb/>
                  tertia eſt ita denſum ſicut quarta pars proportio-
                    <lb/>
                  nalis eius. </s>
                  <s xml:id="N230B8" xml:space="preserve">Et iſto mõ probabis ceteras ꝑticulas
                    <lb/>
                  correlarii.
                    <note position="left" xlink:href="note-0194-04a" xlink:label="note-0194-04" xml:id="N23107" xml:space="preserve">6. correĺ.</note>
                  </s>
                  <s xml:id="N230C2" xml:space="preserve">¶ Sequit̄̄ ſexto /  ſi aliquod corpꝰ diui­
                    <lb/>
                  datur ꝑ partes ꝓportionales proportiõe ſuperbi­
                    <lb/>
                  patiente tertias
                    <gap/>
                  et partes eius ſint ita denſe / vt ſe-
                    <cb chead="Capitulū primum."/>
                  pius dictum eſt in p̄cedētibus correlariis: totū erit
                    <lb/>
                  denſius ṗma parte ꝓportionali in ꝓportione du-
                    <lb/>
                  pla ſexquialtera: ita  ſi ṗma eſt denſa vt .2. totuꝫ
                    <lb/>
                  erit denſum vt .5. </s>
                  <s xml:id="N230D4" xml:space="preserve">Probat̄̄ correlariū / qm̄ totū erit
                    <lb/>
                  denſius ṗma parte proportionali in tali caſu in ꝓ­
                    <lb/>
                  portione qua ſe habet totū diuiſum ꝑ partes ꝓpor­
                    <lb/>
                  tionales ꝓportione ſuperbipartiēte tertias ad ſu­
                    <lb/>
                  am primã partē proportionalē / vt ptꝫ ex cõcluſione
                    <lb/>
                  ſed talis eſt ꝓportio dupla ſexquialtera / vt patꝫ ex
                    <lb/>
                  capĺo q̇nto prime partis huius operis: igit̄̄ corre-
                    <lb/>
                  larium verum.</s>
                </p>
                <p xml:id="N2310D">
                  <s xml:id="N2310E" xml:space="preserve">Secūda cõcluſio </s>
                  <s xml:id="N23111" xml:space="preserve">Diuiſo corpore per
                    <lb/>
                  ꝑtes ꝓportiõales quauis ꝓportiõe et ī quacū pro­
                    <lb/>
                  portiõe ſe habuerīt ꝑtes ꝓportionales ī eadē vĺ ma­
                    <lb/>
                  iori ſe habuerit dēſitas mīoris ad dēſitatē maiorꝪ
                    <lb/>
                  totū illud corpꝰ eſt īfinite dēſum. </s>
                  <s xml:id="N2311C" xml:space="preserve">patet hec cõcluſio
                    <lb/>
                  ex ꝓbatione ſexte cõcluſionis octaui capitis ſecūdi
                    <lb/>
                  tractatus huius partis.
                    <note position="right" xlink:href="note-0194-05a" xlink:label="note-0194-05" xml:id="N2314D" xml:space="preserve">1. correĺ.</note>
                  </s>
                  <s xml:id="N23128" xml:space="preserve">¶ Ex hac cõcluſione ſeq̇tur
                    <lb/>
                  primo /  partito aliquo corꝑe proportiõe ſexquial­
                    <lb/>
                  tera et prima pars ſit aliqualiter denſa: et ſecunda
                    <lb/>
                  in duplo et tertia ī duplo ꝙ̄ ſecūda: et quarta ꝙ̄ ter­
                    <lb/>
                  tia: totum eſt infinite denſum. </s>
                  <s xml:id="N23133" xml:space="preserve">¶ Sequit̄̄ ſecundo / 
                    <lb/>
                  diuiſo corpore per partes proportionales ꝓpor-
                    <lb/>
                  tione ſexquitertia et ṗma ſit aliqualiter denſa et ſe­
                    <lb/>
                  cunda in ſexquialtero plus et tertia in ſexquialtero
                    <lb/>
                  quã ſecunda / et ſic conſequēter: totum corpus eſt in­
                    <lb/>
                  finite denſum </s>
                  <s xml:id="N23140" xml:space="preserve">Hec correlaria ex ſecunda cõcluſione
                    <lb/>
                  patent: qm̄ in vtro illoꝝ proportio denſitatū cõ­
                    <lb/>
                  tinuo eſt maior ꝓportione partiū / ergo ſubiecta il-
                    <lb/>
                  la ſunt infinite denſa.</s>
                </p>
                <p xml:id="N23153">
                  <s xml:id="N23154" xml:space="preserve">Tertia
                    <reg norm="concluſio" type="context">cõcluſio</reg>
                  </s>
                  <s xml:id="N23157" xml:space="preserve">Diuiſo aliquo corpo­
                    <lb/>
                  re per partes
                    <reg norm="proportionales" type="simple">ꝓportionales</reg>
                  quauis proportiõe et
                    <lb/>
                  in certa proportiõe quelibet pars
                    <reg norm="praecedens" type="simple context">p̄cedēs</reg>
                  ſit
                    <reg norm="denſior" type="context">dēſior</reg>
                    <lb/>
                  immediate
                    <reg norm="ſequenti" type="context">ſequēti</reg>
                  : totius denſitatís ad
                    <reg norm="denſitatet" type="simple">denſitatꝫ</reg>
                    <lb/>
                  ſiue denoīationē qua
                    <reg norm="totum" type="context">totū</reg>
                    <reg norm="denominabitur" type="simple">denominabit̄̄</reg>
                  a denſita­
                    <lb/>
                  te prime partis
                    <reg norm="proportionalis" type="simple">ꝓportionalis</reg>
                  eſt illa
                    <reg norm="proportio" type="simple">ꝓportio</reg>
                  qua
                    <lb/>
                  ſe habet totum diuiſum in proportione ↄ̨poſita ex
                    <lb/>
                  proportione partis
                    <reg norm="proportionalis" type="simple">ꝓportionalis</reg>
                    <reg norm="precedententis" type="context">precedentētis</reg>
                  ad
                    <reg norm="im­ mediate" type="context">ī­
                      <lb/>
                    mediate</reg>
                    <reg norm="ſequentem" type="context">ſequētem</reg>
                  : et denſitatis
                    <reg norm="praecedentis" type="simple context">p̄cedētis</reg>
                  ad denſi-
                    <lb/>
                  tatem immediate
                    <reg norm="ſequentis" type="context">ſequētis</reg>
                  ad
                    <reg norm="primam" type="simple context">ṗmã</reg>
                  eius
                    <reg norm="partem" type="context">partē</reg>
                    <reg norm="pro- portionalem" type="context">pro-
                      <lb/>
                    portionalē</reg>
                  . </s>
                  <s xml:id="N2316E" xml:space="preserve">Patet hec et cluſio
                    <reg norm="cum" type="context">cū</reg>
                  multis
                    <reg norm="ſimilibus" type="simple">ſimilibꝰ</reg>
                  ex
                    <lb/>
                  probatione octaue
                    <reg norm="concluſionis" type="context">cõcluſionis</reg>
                  tertii capitis
                    <reg norm="ſecundi" type="context">ſecūdi</reg>
                    <lb/>
                  tractatus huius tertie partis videas ibi.</s>
                </p>
                <p xml:id="N23175">
                  <s xml:id="N23176" xml:space="preserve">Quarta
                    <reg norm="concluſio" type="context">cõcluſio</reg>
                  </s>
                  <s xml:id="N23179" xml:space="preserve">Diuiſo corpore per
                    <lb/>
                  partes
                    <reg norm="proportionales" type="simple">ꝓportionales</reg>
                  aliqua proportiõe multipli-
                    <lb/>
                  ci: et in prima parte proportionali ſit aliquantula
                    <lb/>
                  denſitas. </s>
                  <s xml:id="N23182" xml:space="preserve">et in ſecunda in ſexquialtero maior et in
                    <lb/>
                  tertia in ſexquitertia maior denſitas
                    <reg norm="quam" type="context">quã</reg>
                  in
                    <reg norm="prima" type="simple">ṗma</reg>
                  /
                    <lb/>
                  et ſic ↄ̨ſequenter
                    <reg norm="procedendo" type="context">procedēdo</reg>
                  per ſpecies
                    <reg norm="proportionis" type="simple context">ꝓportiõis</reg>
                    <lb/>
                  ſuperparticularis: totius corporis denſitas
                    <reg norm="cenſen­ da" type="context">cēſen­
                      <lb/>
                    da</reg>
                  eſt
                    <reg norm="incommenſurabilis" type="context">incõmenſurabilis</reg>
                  proportione rationali
                    <reg norm="den­ ſitati" type="context">dē­
                      <lb/>
                    ſitati</reg>
                  prime partis proportionalis et denoīationi
                    <lb/>
                  qua
                    <reg norm="ipſa" type="wordlist">ip̄a</reg>
                  denſitas exiſtens in
                    <reg norm="prima" type="simple">ṗma</reg>
                  parte proportio­
                    <lb/>
                  nali totum denominat. </s>
                  <s xml:id="N23193" xml:space="preserve">vel
                    <reg norm="ſaltem" type="context">ſaltē</reg>
                  ſi
                    <reg norm="commenſurabilis" type="context">cõmenſurabilis</reg>
                    <lb/>
                  eſt pro ſtatu iſto a nobis
                    <reg norm="capacitatem" type="context">capacitatē</reg>
                    <reg norm="finitam" type="context">finitã</reg>
                  habenti-
                    <lb/>
                  bus nequa̄
                    <reg norm="commenſurari" type="context context">cõmēſurari</reg>
                  poteſt. </s>
                  <s xml:id="N2319A" xml:space="preserve">Probatur /
                    <reg norm="quam" type="wordlist">qm̄</reg>
                  ille
                    <lb/>
                  denſitates
                    <reg norm="continuo" type="context">cõtinuo</reg>
                  ſe habent in alia et alia
                    <reg norm="propor- tione" type="simple">ꝓpor-
                      <lb/>
                    tione</reg>
                  : et
                    <reg norm="non" type="wordlist">nõ</reg>
                  eſt poſſibile omnes tales proportiones
                    <lb/>
                    <reg norm="commenſurari" type="context">cõmenſurari</reg>
                  ab intellectu finito cum ſint infinite: et
                    <lb/>
                    <reg norm="continuo" type="context">cõtinuo</reg>
                  alie et alie: igitur concluſio propoſita vera
                    <lb/>
                  </s>
                  <s xml:id="N231A6" xml:space="preserve">Non
                    <reg norm="tamem" type="context">tamē</reg>
                  puto hanc
                    <reg norm="concluſionem" type="context context">cõcluſionē</reg>
                  demonſtraſſe aut
                    <lb/>
                  ſufficienter
                    <reg norm="oſtendiſſe" type="context">oſtēdiſſe</reg>
                  :
                    <reg norm="sed" type="wordlist">ſꝫ</reg>
                  eam
                    <reg norm="probabiliter" type="simple">ꝓbabiliter</reg>
                  pono.
                    <note position="right" xlink:href="note-0194-06a" xlink:label="note-0194-06" xml:id="N231E4" xml:space="preserve">1.
                      <reg norm="corollarium" type="wordlist">correĺ.</reg>
                    </note>
                  </s>
                  <s xml:id="N231B0" xml:space="preserve">¶ Ex
                    <lb/>
                  hac concluſione
                    <reg norm="ſequitur" type="simple">ſequit̄̄</reg>
                  primo /  ſi aliquod corpus
                    <lb/>
                  diuidatur
                    <reg norm="per" type="wordlist">ꝑ</reg>
                  partes proportionales proportione
                    <lb/>
                  dupla: et prima ſit aliqualiter denſa: et ſecunda in
                    <lb/>
                  ſexquitertio pluſ̄ prima et tertia in ſexquiquinta
                    <lb/>
                  pluſ̄ prima et
                    <reg norm="quarta" type="wordlist">q̈rta</reg>
                  in ſexquiſeptimo pluſ̄
                    <reg norm="prima" type="simple">ṗma</reg>
                  / </s>
                </p>
              </div>
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