Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

< >
[Note]
Page: 50
[Note]
Page: 50
[Note]
Page: 50
[Note]
Page: 50
[Note]
Page: 51
[Note]
Page: 51
[Note]
Page: 51
[Note]
Page: 52
[Note]
Page: 52
[Note]
Page: 52
[Note]
Page: 52
[Note]
Page: 53
[Note]
Page: 53
[Note]
Page: 53
[Note]
Page: 53
[Note]
Page: 53
[Note]
Page: 54
[Note]
Page: 54
[Note]
Page: 54
[Note]
Page: 54
[Note]
Page: 55
[Note]
Page: 55
[Note]
Page: 55
[Note]
Page: 55
[Note]
Page: 56
[Note]
Page: 57
[Note]
Page: 57
[Note]
Page: 57
[Note]
Page: 57
[Note]
Page: 57
< >
page |< < of 290 > >|
    <echo version="1.0">
      <text xml:lang="la">
        <div xml:id="N10132" level="1" n="1" type="body">
          <div xml:id="N15C17" level="2" n="3" type="other" type-free="pars">
            <div xml:id="N20F21" level="3" n="3" type="other" type-free="tractatus">
              <div xml:id="N20F2E" level="4" n="1" type="chapter" type-free="capitulum">
                <p xml:id="N22E05">
                  <s xml:id="N22E4C" xml:space="preserve">
                    <pb chead="Tertii tractatus" file="0193" n="193"/>
                  ſui ſubiecti quã b. in c. proportione partium, et in d.
                    <lb/>
                  ꝓportione intenſionū illaꝝ denſitatū ſimul: igitur
                    <lb/>
                  plus denoīat a. quã b. ſuū ſubiectū in proportione
                    <lb/>
                  q̄ adequate cõponitur ex proportione c. partiū et d.
                    <lb/>
                  intenſionū illaꝝ dēſitatum: quod fuit probandum.</s>
                </p>
                <p xml:id="N22E64">
                  <s xml:id="N22E65" xml:space="preserve">Quarta ꝓpoſitio. </s>
                  <s xml:id="N22E68" xml:space="preserve">Si intenſior denſi­
                    <lb/>
                  tas parti extendatur minori: et remiſſior maiori: ſit­
                    <lb/>
                   equalis ꝓportio partiū ad inuicē: et etiã denſita-
                    <lb/>
                  tum: tunc ille denſitates equaliter ad totius deno-
                    <lb/>
                  minationē faciūt. </s>
                  <s xml:id="N22E73" xml:space="preserve">Exemplū / vt ſi in vna medietate
                    <lb/>
                  ponatur denſitas vt .4. et in vna quarta vt .8. quia
                    <lb/>
                  tunc inter partes et inter denſitates eſt proportio
                    <lb/>
                  dupla. </s>
                  <s xml:id="N22E7C" xml:space="preserve">Ideo tm̄ adequate facit ad denoīationē to-
                    <lb/>
                  tius ſubiecti denſitas vt .8. in vna quarta quantuꝫ
                    <lb/>
                  denſitas vt .4. in vna medietate: q2 vtra facit vt
                    <lb/>
                  duo vt ptꝫ calculanti et aſpicienti attentius. </s>
                  <s xml:id="N22E85" xml:space="preserve">Pro-
                    <lb/>
                  batur tñ generaliter / et ſit a. denſitas intenſior per
                    <lb/>
                  minorē partē extenſa et b. remiſſior extenſa ꝑ ma-
                    <lb/>
                  iorē partē, et ſit f. ꝓportio inter illas partes et etiã
                    <lb/>
                  ſi .f. proportio inter illas denſitates a .b. / tunc dico
                    <lb/>
                   b. deuſitas equaliter denoīat totū ſuū ſubiectuꝫ
                    <lb/>
                  cū ipſa a. denſitate. </s>
                  <s xml:id="N22E94" xml:space="preserve">Quod ſic argr̄ / ſi a. dēſitas exi­
                    <lb/>
                  ſtens in minori parte quã b. eſſet equalis in gradu
                    <lb/>
                  ipſi b. tunc in f. ꝓportione minꝰ denoīaret totum ̄
                    <lb/>
                  b. modo denoīat / vt ptꝫ clare ex ſecūda parte prime
                    <lb/>
                  ꝓpoſitionis: ſed modo in f. ꝓportiõe plus denoīat
                    <lb/>
                  quã tunc: q2 in f. ꝓportione eſt intenſior ceteris pa-
                    <lb/>
                  ribus: igitur modo tantū denominat ſicut b. / quod
                    <lb/>
                  fuit probandum.</s>
                </p>
                <p xml:id="N22EA5">
                  <s xml:id="N22EA6" xml:space="preserve">Quinta ꝓpoſitio. </s>
                  <s xml:id="N22EA9" xml:space="preserve">Si intenſior denſi­
                    <lb/>
                  tas parti ſubiecti extendatur minori, et remiſſior
                    <lb/>
                  maiori parti eiuſdē ſubiecti īhereat, et ꝓportio in-
                    <lb/>
                  tenſionū illaꝝ deuſitatū excedat ꝓportionē partiū
                    <lb/>
                  tunc denſitas exiſtēs in miuore parte ſubiecti ipſū
                    <lb/>
                  totū ſubiectū denſius denoīabit ꝙ̄ denſitas exiſtēs
                    <lb/>
                  in maiori parte in ea ꝓportione ꝑ quã ꝓportio in-
                    <lb/>
                  tenſionū illaꝝ denſitatū excedit ꝓportionē partiū
                    <lb/>
                  in quibus ſunt ille denſitates. </s>
                  <s xml:id="N22EBC" xml:space="preserve">Exemplū / vt ſi in vna
                    <lb/>
                  medietate pedalis ponatur denſitas vt duo, et in
                    <lb/>
                  quarta eiuſdē denſitas vt .8. q2 ꝓportio partiū ex-
                    <lb/>
                  ceditur a ꝓportione quadrupla illaꝝ denſitatum
                    <lb/>
                  et quadrupla ex duplã per duplã. </s>
                  <s xml:id="N22EC7" xml:space="preserve">Ideo in du-
                    <lb/>
                  plo plus denoīat denſitas vt .8. quã denſitas vt .2.
                    <lb/>
                  illud totale ſubiectū denoīet q2 illa vt .2. denoīat
                    <lb/>
                  vt vnū, alia vero vt .8. denoīat vt .2. / vt ptꝫ calculã-
                    <lb/>
                  ti. </s>
                  <s xml:id="N22ED2" xml:space="preserve">Probat̄̄ tñ vniuerſaliter ſit a. denſitas intēſior
                    <lb/>
                  b. vero remiſſior exiſtens in maiore parte ſubiecti
                    <lb/>
                  quã a. ſit ꝓportio partiū c. ꝓportio vero intenſi-
                    <lb/>
                  onū illaꝝ denſitatū d. q̄ ſit maior et excedãt d. ꝓpor­
                    <lb/>
                  tio ipſam c. ꝓportionē ꝑ f. ꝓportionē: tunc a. denſi-
                    <lb/>
                  tas denoīat ſubiectū in f. ꝓportiõe denſius quã b.
                    <lb/>
                  </s>
                  <s xml:id="N22EE0" xml:space="preserve">Quod ſic argr̄ / q2 ſi ꝓportio intenſionū illaꝝ den-
                    <lb/>
                  ſitatū eſſet equalis ꝓportioni c. illaꝝ partiū ſubie­
                    <lb/>
                  cti: tūc eq̈liṫ a. faceret ad totius ſubiecti denoīatio­
                    <lb/>
                  nē / vt pꝫ ex preredenti ꝓportione ſed modo a. eſt iu
                    <lb/>
                  f. ꝓportione intenſior denſitas quam tunc / g̊ modo
                    <lb/>
                  in f. ꝓportione plus facit ad totius denoīationem
                    <lb/>
                  ̄ tunc: et ꝑ ↄ̨ñs in f. ꝓportiõe modo plus facit ̄
                    <lb/>
                  b. / quod fuit ꝓbandū </s>
                  <s xml:id="N22EF1" xml:space="preserve">Ptꝫ ↄ̨ña / q2 tm̄ facit b. modo
                    <lb/>
                  ſicut tunc a. / vt ptꝫ. </s>
                  <s xml:id="N22EF6" xml:space="preserve">Q, vero a. denſitas ſit nunc in f.
                    <lb/>
                  ꝓportione intenſior / ꝙ̄ tunc ptꝫ per hanc maximã.
                    <lb/>
                  </s>
                  <s xml:id="N22EFC" xml:space="preserve">Quãdo due ꝓportiones ſunt equales ad hoc 
                    <lb/>
                  vna illaꝝ excedat alterã per f. proportionē requiri­
                    <lb/>
                  tur  numerus maior acquirat illã f. ꝓportionē ſu­
                    <lb/>
                  pra ſe, ſi numerus minor debet manere inuariatus /
                    <lb/>
                  vt ptꝫ facile in numeris: et ſic ptꝫ propoſitio.</s>
                </p>
                <p xml:id="N22F07">
                  <s xml:id="N22F08" xml:space="preserve">Sexta ꝓpoſitio. </s>
                  <s xml:id="N22F0B" xml:space="preserve">Ubicū maior den­
                    <cb chead="Capitulū primū."/>
                  ſitas parti ſubiecti minori inheret, et remiſſior den­
                    <lb/>
                  ſitas maiori parti, eſt inter partes maior ꝓpor-
                    <lb/>
                  tio quã inter illaꝝ denſitatū intenſiones: tunc den­
                    <lb/>
                  ſitas remiſſior plus facit ad totius denoīationem
                    <lb/>
                  quã intenſior in ea proportione per quã proportio
                    <lb/>
                  partiū ꝓportionē denſitatū exuperat. </s>
                  <s xml:id="N22F1B" xml:space="preserve">Exemplum
                    <lb/>
                  eſt facile. </s>
                  <s xml:id="N22F20" xml:space="preserve">Probat̄̄ hec ꝓpoſitio generaliter / ſit a.
                    <lb/>
                  denſitas intenſior ī minore parte exiſtēs, b. vero re­
                    <lb/>
                  miſſior in maiore parte exiſtēs et ſi proportio par­
                    <lb/>
                  tiū c, et denſitatū d. et c. proportio partium excedat
                    <lb/>
                  d. ꝓportionē denſitatū per f. / tunc argr̄ ſic / ſi ꝓpor-
                    <lb/>
                  tio partiū puta partis maioris ad partē minoreꝫ
                    <lb/>
                  diminueretur per f. ꝓportionē tūc b. denſitas equa­
                    <lb/>
                  liter denoīaret totū ſicut a. denſitas: ſed modo eſt
                    <lb/>
                  in parte in f. ꝓportioue maiore quã tunc eſſet cete-
                    <lb/>
                  ris paribus: g̊ modo in f. ꝓportione b. plus deno-
                    <lb/>
                  minat quã tuuc: et per cõſequēs modo in f. ꝓportiõe
                    <lb/>
                  b. plus denoīat totū ſubiectū quã a. denſitas. </s>
                  <s xml:id="N22F39" xml:space="preserve">Ptꝫ
                    <lb/>
                  cõſequētia / q2 denoīatio qua modo denoīat a. den­
                    <lb/>
                  ſitas, et qua tunc denoīaret b. dēſitas ſunt equales
                    <lb/>
                  </s>
                  <s xml:id="N22F41" xml:space="preserve">Q, o tunc b. equaliter denoīaret cū ipſa a. denſi-
                    <lb/>
                  tate ptꝫ ex quarta ꝓpoſitione. </s>
                  <s xml:id="N22F46" xml:space="preserve">Et ſic ptꝫ /  in ea ꝓ-
                    <lb/>
                  portione denſitas remiſſior plus facit ad denoīa-
                    <lb/>
                  tionē totius per quam proportio partium excedit
                    <lb/>
                  ꝓportionē denſitatū / quod fuit ꝓbandū.
                    <note position="right" xlink:href="note-0193-01a" xlink:label="note-0193-01" xml:id="N22F6C" xml:space="preserve">2. pars q̄­
                      <lb/>
                    ſtionis.</note>
                  </s>
                  <s xml:id="N22F54" xml:space="preserve">¶ Abſolu­
                    <lb/>
                  tis notabilibꝰ, prima parte huiꝰ q̄ſtionis expedi­
                    <lb/>
                  ta: reſtat ad ſecundã partē ſiue articulū huiꝰ q̄ſtio-
                    <lb/>
                  nis accedere qui articulus cõcluſionibus quibuſdã
                    <lb/>
                  ex p̄dictis ꝓpoſitonibus ſequentibꝰ accõmodatur
                    <lb/>
                  </s>
                  <s xml:id="N22F60" xml:space="preserve">His em̄ ſequētibꝰ cõcluſionibꝰ p̄ſentis q̄ſtionis dif­
                    <lb/>
                  ficultas notatur at abſoluitur. </s>
                  <s xml:id="N22F65" xml:space="preserve">Sit igitur.</s>
                </p>
                <p xml:id="N22F74">
                  <s xml:id="N22F75" xml:space="preserve">Prima concluſio. </s>
                  <s xml:id="N22F78" xml:space="preserve">Diuiſo aliquo cor-
                    <lb/>
                  pore benſo per partes ꝓportionales quauis pro-
                    <lb/>
                  portione, et prima pars ꝓportionalis ſit aliquali­
                    <lb/>
                  ter denſa, et ſecūda in duplo plus, et tertia in triplo
                    <lb/>
                  plus ꝙ̄ prima, et ſic in infinitū: tunc totū corpus eſt
                    <lb/>
                  denſus prima parte ꝓportionali in ea ꝓportione
                    <lb/>
                  qua ſe hꝫ totū ſic diuiſum ad primã partē eiꝰ ꝓpor­
                    <lb/>
                  tionalē. </s>
                  <s xml:id="N22F89" xml:space="preserve">Ptꝫ hec cõcluſio ex ꝓbatione ſecūde cõclu­
                    <lb/>
                  ſionis tertii capitis ſecūdi tractatus huius tertie
                    <lb/>
                  partis vbi et ꝓbationē et exemplū eiꝰ īueniens.
                    <note position="right" xlink:href="note-0193-02a" xlink:label="note-0193-02" xml:id="N230E9" xml:space="preserve">1. correĺ.</note>
                  </s>
                  <s xml:id="N22F95" xml:space="preserve">¶ Ex
                    <lb/>
                  hac cõcluſione ſequitur primo /  ſi aliquod corpus
                    <lb/>
                  diuidatur ꝓportione tripla, et prima pars ꝓpor-
                    <lb/>
                  tionalis eiꝰ ſit aliquantulū denſa, et ſecūda in du-
                    <lb/>
                  plo, et tertia in triplo quã prima, et ſic cõſequenter:
                    <lb/>
                  tunc totum eſt in ſexquialtero denſius prima parte
                    <lb/>
                  </s>
                  <s xml:id="N22FA3" xml:space="preserve">Et ſi diuidatur corpus ꝓportione quadrupla: totū
                    <lb/>
                  eſt denſius prima parte proportionali in ſexq̇ter-
                    <lb/>
                  tio. </s>
                  <s xml:id="N22FAA" xml:space="preserve">et ſi proportiõe quītupla: totū erit denſius pri-
                    <lb/>
                  ma parte proportionali in ꝓportione ſexquiquar­
                    <lb/>
                  ta. </s>
                  <s xml:id="N22FB1" xml:space="preserve">Et ſi in proportiõe ſextupla: in proportiõe ſexq̇-
                    <lb/>
                  quīta. </s>
                  <s xml:id="N22FB6" xml:space="preserve">Et fi proportiõe ſeptupla: in proportiõe ſex-
                    <lb/>
                  quiſexta: et ſic cõſequēter ꝓcedendo per ſpecies pro­
                    <lb/>
                  portionis multiplicis ſuperparticularis. </s>
                  <s xml:id="N22FBD" xml:space="preserve">Proba­
                    <lb/>
                  tur hoc longū correlariū / q2 corpus diuiſum ꝓpor-
                    <lb/>
                  tione tripla ſe hꝫ ad primã partē proportionalem
                    <lb/>
                  eiꝰ in proportiõe ſexq̇altera: et diuiſum proportiõe
                    <lb/>
                  quadrupla in proportiõe ſexq̇tertia: et diuiſum quī­
                    <lb/>
                  tupla ſe hꝫ ad primã partē proportiõalē in propor­
                    <lb/>
                  tione ſexquiquarta / et ſic cõſequēter / vt ptꝫ ex prima
                    <lb/>
                  parte huiꝰ operis capitulo quīto et ſexto: igr̄ in ca-
                    <lb/>
                  ſu correlarii ſequit̄̄ /  ſi diuidat̄̄ proportiõe tripla
                    <lb/>
                  ipſum erit denſius prima parte proportionali in
                    <lb/>
                  ſexquialtero, et ſi quadrupla in proportione ſexq̇-
                    <lb/>
                  tertia, et ſi quītupla in ſexquiquarta, et ſic cõſequē-
                    <lb/>
                  ter. </s>
                  <s xml:id="N22FD8" xml:space="preserve">Ptꝫ hec cõſequentia per cõcluſionē precedentē
                    <lb/>
                    <note position="right" xlink:href="note-0193-03a" xlink:label="note-0193-03" xml:id="N230EF" xml:space="preserve">2. correĺ.</note>
                  </s>
                  <s xml:id="N22FE2" xml:space="preserve">¶ Sequit̄̄ ſecundo /  ſi diuidat̄̄ corpus per partes
                    <lb/>
                  proportionales proportiõe dupla, diſtribuatur </s>
                </p>
              </div>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum