Alvarus, Thomas, Liber de triplici motu, 1509

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                <pb chead="Inductio gradus ſūmi cõſideratio." file="0281" n="281"/>
                <p xml:id="N2D08C">
                  <s xml:id="N2D08D" xml:space="preserve">Octaua concluſio. </s>
                  <s xml:id="N2D090" xml:space="preserve">Si duo equalia.
                    <lb/>
                  </s>
                  <s xml:id="N2D094" xml:space="preserve">in quali. tantū termi. ad ſū. alterentur equali lati-
                    <lb/>
                  tu. alterationis vniformi per totū: velocius conti-
                    <lb/>
                  nuo inducitur gra. ſū. in maiori in ea proportione
                    <lb/>
                  qua eſt maius. </s>
                  <s xml:id="N2D09D" xml:space="preserve">Sit a. maius b. in f. proportione cuꝫ
                    <lb/>
                  ceteris poſitis in concluſione. </s>
                  <s xml:id="N2D0A2" xml:space="preserve">Et arguitur ſic / eque
                    <lb/>
                  cito a. et b. erunt ſumma. </s>
                  <s xml:id="N2D0A7" xml:space="preserve">et a. eſt in f. proportione
                    <lb/>
                  maius ipſo b. ex hypotheſi: et vniformiter gradus
                    <lb/>
                  ſū. inducitur continuo in a. et in b. / ergo in f. propor­
                    <lb/>
                  tiõe velocius inducitur in a. ꝙ̄ in b. </s>
                  <s xml:id="N2D0B0" xml:space="preserve">Patet conſequē­
                    <lb/>
                  tia ex .4. ꝓpõe. </s>
                  <s xml:id="N2D0B5" xml:space="preserve">et ↄ̨tinuo notabilis huius c. et tn f.
                    <lb/>
                  proportionc a. eſt maius b. / igitur conclu. vera. </s>
                  <s xml:id="N2D0BA" xml:space="preserve">Sed
                    <lb/>
                   eque cito erūt ſūma a. et b. probatur / a quia eque
                    <lb/>
                  cito inducitur in extrema ipſorum a.b. gra. ſū. cum
                    <lb/>
                  equaliter diſtent a ſū. et equaliter continuo per idē
                    <lb/>
                  tempus alterentur. </s>
                  <s xml:id="N2D0C5" xml:space="preserve">igitur eque cito / a. et b. erunt ſū­
                    <lb/>
                  ma patet conſequentia. </s>
                  <s xml:id="N2D0CA" xml:space="preserve">quia eque cito / erunt ſū. cuꝫ
                    <lb/>
                  ſuis extremis remiſſioribus et non ante: vt conſtat
                    <lb/>
                  nec poſt cum continuo inducatur vniformiter par-
                    <lb/>
                  tibiliter ex prima conclu.
                    <note position="left" xlink:href="note-0281-01a" xlink:label="note-0281-01" xml:id="N2D154" xml:space="preserve">1. corre.</note>
                  </s>
                  <s xml:id="N2D0D8" xml:space="preserve">¶ Ex hac conclu. ſequitur
                    <lb/>
                  primo /  ſi a. in caſu conclu. alteretur maiori alte-
                    <lb/>
                  ratione ꝙ̄ b. in ipſum velocius inducitur gra. ſum.
                    <lb/>
                  ̄ in b. in proportione compoſita ex proportione
                    <lb/>
                  quantitatis a. ad quantitatē b. et alterationis ip̄iꝰ
                    <lb/>
                  a. ad alterationem ipſius b. </s>
                  <s xml:id="N2D0E5" xml:space="preserve">Probatur et ſit g. pro­
                    <lb/>
                  portio alterationum et h. compoſita ex f. et g. / et ar-
                    <lb/>
                  guo ſic / ſi a. alteraretur equeuelociter cum b. in f. ꝓ­
                    <lb/>
                  portione velocius inducetur gra. ſum. in a. ꝙ̄ in b. /
                    <lb/>
                  vt patet ex hac .8. conclu. </s>
                  <s xml:id="N2D0F0" xml:space="preserve">Sed modo in g. propor-
                    <lb/>
                  tione velocius adhuc inducitur gra. ſum. in a. ꝙ̄ tūc /
                    <lb/>
                  vt patet ex .3. conclu. / ergo modo in duabus propor­
                    <lb/>
                  tionibus v3 g. et f. velocius inducitur gradus ſum.
                    <lb/>
                  in a. ꝙ̄ in b. </s>
                  <s xml:id="N2D0FB" xml:space="preserve">Et g. et f. ſunt h. / igitur in h. proportione
                    <lb/>
                  velocius inducitur gra. ſum. in a. ꝙ̄ in b. </s>
                  <s xml:id="N2D100" xml:space="preserve">Et ſic patꝫ
                    <lb/>
                  corre.
                    <note position="left" xlink:href="note-0281-02a" xlink:label="note-0281-02" xml:id="N2D15A" xml:space="preserve">1. Corre.</note>
                  </s>
                  <s xml:id="N2D10A" xml:space="preserve">¶ Sequitur .2. / ſi  in caſu predicte conclu. b. al­
                    <lb/>
                  teratur alteratione maiori ꝙ̄ illa qua alteratur ī ea
                    <lb/>
                  proportione qua a. eſt maius b. </s>
                  <s xml:id="N2D111" xml:space="preserve">Tunc equeuelociter
                    <lb/>
                  continuo inducitur gra. ſum. in b. ſicut in a. </s>
                  <s xml:id="N2D116" xml:space="preserve">Probat̄̄ /
                    <lb/>
                  quia ſi a. et b. equali alteratione alterarentur: in b. ī
                    <lb/>
                  f. ꝓportione continuo tardius induceretur gra. ſum.
                    <lb/>
                  ̄ in a. ex hac octaua conclu. </s>
                  <s xml:id="N2D11F" xml:space="preserve">Sed modo in f. propor-
                    <lb/>
                  tione inducitur gra. ſum. velocius in b. ꝙ̄ tunc ex .3. cõ­
                    <lb/>
                  clu. / ergo equeuelociter modo inducitur in b. ſicut in
                    <lb/>
                  a. / quod fuit probandum.
                    <note position="left" xlink:href="note-0281-03a" xlink:label="note-0281-03" xml:id="N2D160" xml:space="preserve">3. Corre.</note>
                  </s>
                  <s xml:id="N2D12D" xml:space="preserve">¶ Sequitur .3. /  ſi in caſu cõ­
                    <lb/>
                  cluſionis b. alteretur velocius a. in maiori ꝓportiõe
                    <lb/>
                  ̄ f. </s>
                  <s xml:id="N2D134" xml:space="preserve">Tunc gra. ſum. velius inducitur in b. ꝙ̄ in a. in
                    <lb/>
                  ea proportione per quam ꝓportio alterationum ex-
                    <lb/>
                  cedit ꝓportionē f. quantitatum. </s>
                  <s xml:id="N2D13B" xml:space="preserve">Et ſi b. maiori altera­
                    <lb/>
                  tione alteretur ꝙ̄ a. que alteratio ipſius b. ſit maior
                    <lb/>
                  altera. ipſius a. in mīori ꝓportione ꝙ̄ ſit f. </s>
                  <s xml:id="N2D142" xml:space="preserve">Tūc gra.
                    <lb/>
                  ſum. tardius inducitur in b. ꝙ̄ in a. in ꝓportione per
                    <lb/>
                  quam proportio quantitatum f. excedit proportionē
                    <lb/>
                  alterationum. </s>
                  <s xml:id="N2D14B" xml:space="preserve">Hoc corre. facile ex priori auxiliante .3.
                    <lb/>
                  conclu. demonſtrationem admittit.</s>
                </p>
                <p xml:id="N2D166">
                  <s xml:id="N2D167" xml:space="preserve">Nona concluſio. </s>
                  <s xml:id="N2D16A" xml:space="preserve">Si duo vni. diff. ad
                    <lb/>
                  ſum. termi. in equalia in quãti. et qualitate et maius
                    <lb/>
                  vtro modo excedit minꝰ: et equali alteratiõe ꝑ totū
                    <lb/>
                  alterant̄̄ </s>
                  <s xml:id="N2D173" xml:space="preserve">Tūc in maius velocius īducitur ↄ̨tinuo gra.
                    <lb/>
                  ſum. ꝙ̄ in minus in ꝓportione cõpoſita ex ꝓportione
                    <lb/>
                  exceſſuū quibus gradus ſum. excedit extrema illorū
                    <lb/>
                  remiſſa et ex ꝓportione quanti. maioris ad quanti.
                    <lb/>
                  minoris. </s>
                  <s xml:id="N2D17E" xml:space="preserve">probatur. </s>
                  <s xml:id="N2D181" xml:space="preserve">Sit a. maius in f. proportione ip­
                    <lb/>
                  ſo b. </s>
                  <s xml:id="N2D186" xml:space="preserve">Et exceſſus quo gra. ſum. excedit extremuꝫ b. ad
                    <lb/>
                  exceſſum quo excedit extremū ipſius a. ſit g. propor-
                    <lb/>
                  tio. </s>
                  <s xml:id="N2D18D" xml:space="preserve">Et cõpoſita ex hiis ſit h. </s>
                  <s xml:id="N2D190" xml:space="preserve">Tūc dico /  gradus ſū. ī
                    <lb/>
                  h. proportione velocius inducitur continuo in a. ꝙ̄ in
                    <lb/>
                  b. </s>
                  <s xml:id="N2D197" xml:space="preserve">Probatur / quia ſi a. eſſet equale ī qualitate ip̄i .b.
                    <cb chead="Inductio gradus ſūmi cõſideratio."/>
                  in f. proportione in ipſum velocius induceretur gra-
                    <lb/>
                  dus ſum. ꝙ̄ in b. ex .8. conclu. </s>
                  <s xml:id="N2D19F" xml:space="preserve">Sed modo in g. ꝓporti-
                    <lb/>
                  one exceſſuū inducitur adhuc velocius in ipſum a. ̄
                    <lb/>
                  tunc ex .4. conclu. / ergo modo in ꝓportionibus f. et g.
                    <lb/>
                  ſimul velocius inducitur gra. ſum. in a. ꝙ̄ in b. </s>
                  <s xml:id="N2D1A8" xml:space="preserve">Et f. et
                    <lb/>
                  g. ſunt h. proportio ex hypoteſi: igitur in h. proporti­
                    <lb/>
                  one gradus ſum. velocius inducitur cõtinuo in a. ꝙ̄ ī
                    <lb/>
                  b. / quod fuit probandum.
                    <note position="right" xlink:href="note-0281-04a" xlink:label="note-0281-04" xml:id="N2D257" xml:space="preserve">1. Corre.</note>
                  </s>
                  <s xml:id="N2D1B6" xml:space="preserve">¶ Sequitur .1. /  ſi a. cum to­
                    <lb/>
                  to reſiduo caſus .9. conclu: alteretur ītenſiori altera-
                    <lb/>
                  tione vni. per totum ꝙ̄ b. </s>
                  <s xml:id="N2D1BD" xml:space="preserve">Tunc in ipſum a. velocius
                    <lb/>
                  continuo inducitur gradus ſūmus in proportione cõ­
                    <lb/>
                  poſita ex ꝓportione quantitatum, et proportione ex­
                    <lb/>
                  ceſſum quibus gradus ſum. excedit extrema illorum
                    <lb/>
                  remiſſa: et ex proportione alterationum. </s>
                  <s xml:id="N2D1C8" xml:space="preserve">Probatur
                    <lb/>
                  </s>
                  <s xml:id="N2D1CC" xml:space="preserve">Sit proportio alterationum e. cum reſiduo hypote-
                    <lb/>
                  ſis conclu. 9. et compoſita ex e. et f. et g. ſit h. </s>
                  <s xml:id="N2D1D1" xml:space="preserve">Tunc di­
                    <lb/>
                  co /  gradus ſummus cõtinuo inducitur velocius in
                    <lb/>
                  a. ꝙ̄ in b. in h. proportione. </s>
                  <s xml:id="N2D1D8" xml:space="preserve">Quod ſic oſtenditur / quia
                    <lb/>
                  ſi a alteretur equali alteratiõe cum ipſo b. in ipſum
                    <lb/>
                  a. velocius inducerētur continuo gradus ſummus ̄
                    <lb/>
                  in b. in ꝓportione cõpoſita ex f. et g. ex 9. conclu. </s>
                  <s xml:id="N2D1E1" xml:space="preserve">Sed
                    <lb/>
                  modo adhuc velocius iuducitur ꝙ̄ tunc in e. propor-
                    <lb/>
                  tione alterationum ex .3. conclu. / ergo modo velocius
                    <lb/>
                  inducitur gra. ſum. in a. ꝙ̄ in b. ꝓportionibus e.f.g.
                    <lb/>
                  </s>
                  <s xml:id="N2D1EB" xml:space="preserve">Et proportione e.f.g. ſunt h. ꝓportio: igitur modo
                    <lb/>
                  gra. ſum. velocius continuo inducitur non a. ꝙ̄ in b. in
                    <lb/>
                  h. proportione. </s>
                  <s xml:id="N2D1F2" xml:space="preserve">quod fuit probãdum.
                    <note position="right" xlink:href="note-0281-05a" xlink:label="note-0281-05" xml:id="N2D25D" xml:space="preserve">2. Corre.</note>
                  </s>
                  <s xml:id="N2D1FA" xml:space="preserve">¶ Sequitur .2. /
                    <lb/>
                   ſi cum toto reſiduo caſus conclu. 9 .b. alteretur al-
                    <lb/>
                  teratiõe vni. per totum maiori ꝙ̄ alteratio ipſius a.
                    <lb/>
                  in proportione compoſita ex ꝓportione quanti. et ex­
                    <lb/>
                  ceſſum quibus gra. ſum. excedit etc̈. </s>
                  <s xml:id="N2D205" xml:space="preserve">Tunc in b. equeue­
                    <lb/>
                  lociter continuo inducitur gra. ſum. ſicut in ipſum a.
                    <lb/>
                  </s>
                  <s xml:id="N2D20B" xml:space="preserve">Probatur / quia ſi a. et b. equali alteratione alterarē­
                    <lb/>
                  tur: gra. ſum. induceretur tardius in b. ꝙ̄ in a. in pro-
                    <lb/>
                  portione h. compoſita ex proportione quanti. et exceſ­
                    <lb/>
                  ſuū. </s>
                  <s xml:id="N2D214" xml:space="preserve">vt patet ex .9. conclu. </s>
                  <s xml:id="N2D217" xml:space="preserve">Sed modo in h. proportio­
                    <lb/>
                  ne intenſiori alteratione alteratur per totum ipſum
                    <lb/>
                  b. ꝙ̄ tunc: ergo modo in h. proportione velocius indu­
                    <lb/>
                  citur gra. ſum. in b. ꝙ̄ tunc / vt pꝫ ex .3. conclu. </s>
                  <s xml:id="N2D220" xml:space="preserve">Et tam
                    <lb/>
                  velociter inducitur in ipſum a. / ergo in b. eque velociṫ
                    <lb/>
                  continuo inducitur gradus ſum. ſicuit in ipſum a. / qḋ
                    <lb/>
                  fuit probandum
                    <note position="right" xlink:href="note-0281-06a" xlink:label="note-0281-06" xml:id="N2D263" xml:space="preserve">3. Corre.</note>
                  </s>
                  <s xml:id="N2D22E" xml:space="preserve">¶ Sequitur .3. /  ſi cum toto reſiduo
                    <lb/>
                  caſus b. alteretur alteratione vni. maiori alteratione
                    <lb/>
                  ̄ a. in maiore proportione ꝙ̄ ſit ꝓportio compoſita
                    <lb/>
                  ex ꝓportione exceſſuū et quantitatum que eſt g. </s>
                  <s xml:id="N2D237" xml:space="preserve">Tūc
                    <lb/>
                  in b. velocius continuo inducitur gra. ſum. ꝙ̄ in a. ī ea
                    <lb/>
                  proportione per quam ꝓportio alterationum exce-
                    <lb/>
                  dit ꝓportionem h. </s>
                  <s xml:id="N2D240" xml:space="preserve">Et ſi talis ꝓportio qua alteratio
                    <lb/>
                  b. excedit alterationem ipſius a. ſit minor ꝙ̄ ꝓpor-
                    <lb/>
                  tio h. </s>
                  <s xml:id="N2D247" xml:space="preserve">Tunc tardius inducetur gra ſum. in b. ꝙ̄ in a.
                    <lb/>
                  in ꝓportione per quam ꝓportio h excedit proportio­
                    <lb/>
                  nem alterationum. </s>
                  <s xml:id="N2D24E" xml:space="preserve">Hoc facile patet ex priori auxi-
                    <lb/>
                  lio .3. conclu.</s>
                </p>
                <p xml:id="N2D269">
                  <s xml:id="N2D26A" xml:space="preserve">Decima cõcluſio. </s>
                  <s xml:id="N2D26D" xml:space="preserve">Si ſint duo īequa-
                    <lb/>
                  lia vtro modo vni diff. termi. ad ſū. </s>
                  <s xml:id="N2D272" xml:space="preserve">Et minus exce­
                    <lb/>
                  dit ī qualitate ipſuꝫ maius: </s>
                  <s xml:id="N2D277" xml:space="preserve">Et equali alteratione in
                    <lb/>
                  qua vnum eſt maius in ea extremum remiſſius illius
                    <lb/>
                  per maiorem latitudinem diſtat a ſū. ꝙ̄ extremum re­
                    <lb/>
                  miſſius ipſius minoris. </s>
                  <s xml:id="N2D280" xml:space="preserve">Tunc per illa continuo eque­
                    <lb/>
                  uelociter inducitur gra. ſū. </s>
                  <s xml:id="N2D285" xml:space="preserve">Probatur. </s>
                  <s xml:id="N2D288" xml:space="preserve">Sit propor-
                    <lb/>
                  tio exceſſuum .f. que etiam eſt proportio quantitatuꝫ
                    <lb/>
                  a. maioris ad .b. minus. </s>
                  <s xml:id="N2D28F" xml:space="preserve">Et arguo ſic / in .f. proportio­
                    <lb/>
                  ne citius gra. ſū. veniet ad extremum remiſſius ipſius
                    <lb/>
                  b. ꝙ̄ ipſius .a. cum illa extrema equeuelociter cõtinuo
                    <lb/>
                  alterentur: et extremum remiſſius ipſius .b. per mino­
                    <lb/>
                  rem latitudinem in .f. proportione diſtat a ſū. ex caſu
                    <lb/>
                  ̄ extremum remiſſius ipſius .a. </s>
                  <s xml:id="N2D29C" xml:space="preserve">Et vniformiṫ in vtrū­ </s>
                </p>
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