Alvarus, Thomas, Liber de triplici motu, 1509

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                <p xml:id="N1F334">
                  <s xml:id="N1F335" xml:space="preserve">
                    <pb chead="De motu locali quo ad effectū tempore difformi." file="0156" n="156"/>
                  caſus concluſionis per illas partes proportiona-
                    <lb/>
                  les et ſit proportio totius hore diuiſe per partes
                    <lb/>
                  proportionales proportione g. ad primam parteꝫ
                    <lb/>
                  proportionalem f. / tunc dico /  tota illa velocitas
                    <lb/>
                  totius hore ſe habet in proportione f. ad propor-
                    <lb/>
                  tionem prime partis proportionalis. </s>
                  <s xml:id="N1F34B" xml:space="preserve">Quod pro-
                    <lb/>
                  bo ſic: quia velocitas equalis velocitate prime par­
                    <lb/>
                  tis proportionalis extenſa per illam horam ali-
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                  quid facit ad intenſionem totius velocitatis: quia
                    <lb/>
                  eſt pars eius / vt oſtendit ſuppoſitio p̄cedens: et tan­
                    <lb/>
                  ta velocitas ſicut illa ſuperaddita preexiſtenti ex-
                    <lb/>
                  tenditur per totum reſiduum a prima parte pro-
                    <lb/>
                  portionali proportione g. / vt etiam dicit ſuppoſi-
                    <lb/>
                  tio: igitur illa in g. proportione minus facit / quia
                    <lb/>
                  eſt equalis alteri extenſe per totum, et eſt in tempo­
                    <lb/>
                  re in g. proportione minori / vt dicit prima conclu-
                    <lb/>
                  ſio, quia tempus diuiditur proportione g. / ergo to­
                    <lb/>
                  tum ſe habet ad reſiduum a prima parte propor-
                    <lb/>
                  tionali in g. proportione. </s>
                  <s xml:id="N1F368" xml:space="preserve">Item per totum reſiduū
                    <lb/>
                  a prima parte proportionali et ſecunda extenditur
                    <lb/>
                  iterum tanta velocitas non communicans cum a-
                    <lb/>
                  liqua precedentium: et illud tempus reſiduum a pri­
                    <lb/>
                  ma et ſecunda ſe habet in g. proportione ad totum
                    <lb/>
                  reſiduum a prima: igitur illa velocitas ei coextēſa
                    <lb/>
                  in g. proportione minus denominat quam prece-
                    <lb/>
                  dens velocitas equalis ei coextenſa ſubiecto in g.
                    <lb/>
                  proportione maiori / et ſic conſequenter: igitur de-
                    <lb/>
                  nominatio totius illius velocitatis componitur ex
                    <lb/>
                  infinitis continuo ſe habentibus in proportione g:
                    <lb/>
                  ergo illa denominatio totius velocitatis ſiue illa
                    <lb/>
                  tota velocitas (quod pro eodem capio) ſe habet ad
                    <lb/>
                  primam illarum denominationum ſiue velocitatū
                    <lb/>
                  que eſt prime partis proportiõalis et etiam totius
                    <lb/>
                  reſidui a prima, in proportione f. / quod fuit infercn­
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                  dum. </s>
                  <s xml:id="N1F38B" xml:space="preserve">Patet hec conſequentia: quia ſemper quan-
                    <lb/>
                  do aliquid diuiditur proportione g. ipſum ſe ha-
                    <lb/>
                  bet ad primã partē proportionalem in ꝓportione
                    <lb/>
                  f. / vt poſitum eſt. </s>
                  <s xml:id="N1F394" xml:space="preserve">Et ex hoc patet /  in caſu concluſio­
                    <lb/>
                  nis tota velocitas ſe habet ad velocitatē prime par­
                    <lb/>
                  tis proportiõalis in ea proportione in qua habet
                    <lb/>
                  totum tempus in ordine od primam partem pro-
                    <lb/>
                  portionalē proportione qua diuiditur ipſum tem-
                    <lb/>
                  pus / quod fuit probandum.</s>
                </p>
                <p xml:id="N1F3A1">
                  <s xml:id="N1F3A2" xml:space="preserve">Tertia cõcluſio. </s>
                  <s xml:id="N1F3A5" xml:space="preserve">Diuiſa hora vel tem­
                    <lb/>
                  pore aliquo quauis proportiõe f. volueris: et in pri-
                    <lb/>
                  ma parte proportionali talis proportionis mobi­
                    <lb/>
                  le aliquod moueatur adequate certa velocitate, et
                    <lb/>
                  aliud mobile vĺ idē in tota illa hora vel tēpore mo-
                    <lb/>
                  ueatur eadem velocitate: tunc in quacun propor-
                    <lb/>
                  tione ſe habuerit tempus ad primam partem pro-
                    <lb/>
                  portionalem: in ea proportione ſe habebit ſpaciū
                    <lb/>
                  abſolutum ſiue pertranſitum in toto tempore ad
                    <lb/>
                  ſpacium pertranſitum in prima parte proportio-
                    <lb/>
                  nali: vt ſi aliquod mobile moueatur velocitate vt .2.
                    <lb/>
                  in prima parte proportiõali hore proportione tri-
                    <lb/>
                  pla, et aliud vel idem mobile moueatur in tota ho-
                    <lb/>
                  ra adequate eadem velocitate vt .2. / tūc dico /  illud
                    <lb/>
                  mobile quod mouetur iu tota hora velocitate vt: 2.
                    <lb/>
                  vel correſpondente ei: ſexquialterum ſpacium per-
                    <lb/>
                  tranſit ad ſpacium pertranſitum velocitate vt .2. in
                    <lb/>
                  prima parte proportionali quoniam omne totum
                    <lb/>
                  diuiſum per partes proportionales proportione
                    <lb/>
                  tripla ſe habet ad primam partem proportiona-
                    <lb/>
                  lem in proportione ſexquialtera / vt patet ex primo
                    <lb/>
                  correlario ſecunde concluſionis quinti capitis pri­
                    <lb/>
                  me partis. </s>
                  <s xml:id="N1F3D4" xml:space="preserve">Probatur tamen facile hec concluſio:
                    <lb/>
                  quoniam quãdo velocitas eſt vniformis in aliquo
                    <lb/>
                  tempore, ipſa diuiditur in eaſdem partes propor­
                    <lb/>
                  tionales in quas diuiditur tempus / vt patet in phi­
                    <cb chead="De motu locali quo ad effectū tempore difformi."/>
                    <note position="right" xlink:href="note-0156-01a" xlink:label="note-0156-01" xml:id="N1F40F" xml:space="preserve">pḣus .6.
                      <lb/>
                    phiſicoꝝ.</note>
                  loſopho ſexto phiſicorū vbi inquit ꝓ motus et ma-
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                  gnitudo pertranſita perinde at tempus diuidi-
                    <lb/>
                  tur: ergo quancun proportionem habebit totum
                    <lb/>
                  tempus ad primam partem proportionalem: ean-
                    <lb/>
                  dem habet velocitas: et per conſequens totum ſpa-
                    <lb/>
                  cium pertranſitum in toto tempore ad ſpaciū per-
                    <lb/>
                  tranſitum in prima parte. </s>
                  <s xml:id="N1F3F1" xml:space="preserve">Patet hec conſequen-
                    <lb/>
                  tia ex prima concluſione ſecundi notabilis. </s>
                  <s xml:id="N1F3F6" xml:space="preserve">In ca-
                    <lb/>
                  ſu enim velocitas equales inequalibus coexten-
                    <lb/>
                  duntur temporibus / ergo ſpacia ſe habent in pro-
                    <lb/>
                  portione temporum: ſed minus tempus eſt prima
                    <lb/>
                  pars proportionalis, et tempus maius eſt totum
                    <lb/>
                  diuiſum in partes proportionales: ergo ſpacium
                    <lb/>
                  pertranſirum in toto tempore ſe habet ad ſpacium
                    <lb/>
                  pertranſituꝫ in prima parte proportionali ſicut ſe
                    <lb/>
                  habet totum tempus ad primam partem propor-
                    <lb/>
                  tionalem eius / quod fuit probandum.</s>
                </p>
                <p xml:id="N1F417">
                  <s xml:id="N1F418" xml:space="preserve">Quarta concluſio. </s>
                  <s xml:id="N1F41B" xml:space="preserve">Diuiſa hora qua-
                    <lb/>
                  uis proportione volueris in partes proportiona-
                    <lb/>
                  les: et in prima illarum partium proportionalium
                    <lb/>
                  mobile aliquod aliquanta velocitate moueatur, et
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                  in ſecunda in duplo maiori velocitate ꝙ̄ in prima:
                    <lb/>
                  et in tertia in triplo maiori ꝙ̄ in prima, et ſic con-
                    <lb/>
                  ſequenter: tunc illo caſu totalis velocitas ſe habe-
                    <lb/>
                  bit ad velocitatem prime partis proportionalis
                    <lb/>
                  in ea proportione in qua ſe habebit totum tempus
                    <lb/>
                  ad primam partem proportionalem eius: et ſpa-
                    <lb/>
                  cium in toto tempore adequate pertranſitum ſe
                    <lb/>
                  habebit ad ſpaciū abſolutum in prima parte pro-
                    <lb/>
                  portionali in proportione duplicata. </s>
                  <s xml:id="N1F436" xml:space="preserve">Uolo dicere /
                    <lb/>
                   ſi hora diuidatur modo poſito in concluſione et
                    <lb/>
                  exempli gratia diuidatur proportione ſexquialte-
                    <lb/>
                  ra: et moueatur mobile per illas partes propor-
                    <lb/>
                  tionales proportione ſexquialtera / vt dicit caſus
                    <lb/>
                  concluſionis: tunc totalis velocitas talis motus
                    <lb/>
                  ſe habebit ad velocitatem prime partis proporti-
                    <lb/>
                  onalis in proportione tripla: quia ſic ſe habet to-
                    <lb/>
                  tum diuiſum proportione ſexquialtera ad primaꝫ
                    <lb/>
                  partem proportionalem / vt patet ex quarta conclu­
                    <lb/>
                  ſione quinti capitis prime partis: et ſpacium per-
                    <lb/>
                  tranſitum in tota hora ad ſpacium pertranſitum
                    <lb/>
                  in prima parte proportiõali ſe habet in ꝓportio-
                    <lb/>
                  ne dupla ad triplam: quia tripla eſt proportio ve-
                    <lb/>
                  locitatum. </s>
                  <s xml:id="N1F455" xml:space="preserve">Modo illa proportio tripla ad duplaꝫ
                    <lb/>
                  eſt noncupla / vt patet ex octaua concluſione ſexti
                    <lb/>
                  capitis ſecūde partis. </s>
                  <s xml:id="N1F45C" xml:space="preserve">Et ſic ſi ꝑtranſit vnū pedale
                    <lb/>
                  in ṗma parte ꝓportiõali: nouē ꝑtrãſit in tota hora
                    <lb/>
                  </s>
                  <s xml:id="N1F462" xml:space="preserve">Demõſtratur concluſio ſic: ſit vnum mobile quod
                    <lb/>
                  adequate moueatur velocitate prime partis pro-
                    <lb/>
                  pprtionalis per primam partem proportionalem
                    <lb/>
                  dumtaxat, et tranſeat ſpacium c. et aliud mobile
                    <lb/>
                  moueatur per totam horam velocitate prime par-
                    <lb/>
                  tis proportionalis. </s>
                  <s xml:id="N1F46F" xml:space="preserve">et pertranſeat ſpacium b. et
                    <lb/>
                  tertiū mobile moueatur per totam horam totali
                    <lb/>
                  illa velocitate ſicut ponitur in caſu concluſiõis que
                    <lb/>
                  ſe habet in f. proportione ad velocitatē prime par-
                    <lb/>
                  tis proportionalis: in qua f. proportione ſe habet
                    <lb/>
                  totum tempus ad primam partē eius proportio-
                    <lb/>
                  nalē / vt dicit ſecunda concluſio et prima pars hu-
                    <lb/>
                  ius concluſionis: et pertranſeat ſpacium a. / et argui­
                    <lb/>
                  tur ſic / ſpacii a. ad ſpacium b. eſt f. proportio: quo-
                    <lb/>
                  niã tempora in quibus pertranſeuntur ſunt equa-
                    <lb/>
                  lia: et velocitas qua pertranſitur a. in f. proporti-
                    <lb/>
                  one eſt maior velocitate qua pertraſitur b. / vt patet
                    <lb/>
                  ex caſu. </s>
                  <s xml:id="N1F48A" xml:space="preserve">Et etiam ſpaci b. ad ſpacium c. eſt propor-
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                  tio f. et a. eſt ſpacium pertranſitum in tota hora
                    <lb/>
                  in caſu concluſionis: et c. pertranſitum in prima
                    <lb/>
                  parte proportionali: igitux propoſitum. </s>
                  <s xml:id="N1F493" xml:space="preserve">Maior
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                  patet ex ſecunda propoſitione ſecundi notabilis </s>
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