Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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                <p xml:id="N1E680">
                  <s xml:id="N1E685" xml:space="preserve">
                    <pb chead="De motu locali quo ad effectū ſcḋm tempus difformi." file="0148" n="148"/>
                  Oīs potentia mouēs vniformiter difformiter lati­
                    <lb/>
                  tudine terminata ad nõ gradū: in triplo plus ꝑtrã­
                    <lb/>
                  ſit ī medietate in qua mouet̄̄ intēſius ꝙ̄ ī medietate
                    <lb/>
                  tēporis in qua mouetur remiſſius: vt ſi in medieta-
                    <lb/>
                  te in qua mouetur remiſſius ꝑtranſit vnū pedale: in
                    <lb/>
                  alia ꝑtranſit tripedale. </s>
                  <s xml:id="N1E696" xml:space="preserve">Probatur hec propoſitio
                    <lb/>
                  facile ex priori: qm̄ motꝰ fluens in medietate in qua
                    <lb/>
                  mouetur velocius eſt triplus ad motū factū in me-
                    <lb/>
                  dietate tēporis in qua mouetur remiſſiꝰ / vt dicit pre­
                    <lb/>
                  cedens: igit̄̄ ꝑtrãſitū in medietate in qua mouetur
                    <lb/>
                  velocius erit triplū ad ꝑtranſitū in reliqua medie-
                    <lb/>
                  tate. </s>
                  <s xml:id="N1E6A5" xml:space="preserve">Cõſequentia ptꝫ / q2 tēporibꝰ exiſtentibus equa­
                    <lb/>
                  libus et velocitatibus in equalibus ſpacia ꝑtranſi-
                    <lb/>
                  ta ſe habent in ea ꝓportione in qua ſe habent velo­
                    <lb/>
                  citates: vt facile induci poteſt ex definitione velocio­
                    <lb/>
                  ris et tardioris data ſexto phiſicoꝝ </s>
                  <s xml:id="N1E6B0" xml:space="preserve">¶ Ex quo ſequi­
                    <lb/>
                  tur /  ſi a. mobile moueatur ꝑ horam vniformiter
                    <lb/>
                  difformiter incipiendo a non gradu vſ ad certum
                    <lb/>
                  gradū et in prima medietate vnã leucã ꝑtranſit: in
                    <lb/>
                  ſecūda medietate triū leucarū ſpaciū abſoluet. </s>
                  <s xml:id="N1E6BB" xml:space="preserve">Et
                    <lb/>
                  ſi ordine prepoſtero moueri incepiſſet puta ab illo
                    <lb/>
                  dato gradu vſ ad nõ gradū in prima medietate
                    <lb/>
                  hore tribus abſolutis leucis: vna dumtaxat reſta-
                    <lb/>
                  ret tranſeunda in ſecunda tēporis medietate.</s>
                </p>
                <p xml:id="N1E6C6">
                  <s xml:id="N1E6C7" xml:space="preserve">Quinta ꝓpoſitio. </s>
                  <s xml:id="N1E6CA" xml:space="preserve">Si aliquod mobile
                    <lb/>
                  moueatur vniformiter difformiter a nõ gradu vſ
                    <lb/>
                  ad certū gradū in aliquo tēpore: ipſum adequate
                    <lb/>
                  ſubduplū ſpaciū ꝑtranſit ad ſpaciū natū ꝑtranſiri
                    <lb/>
                  illo gradu intenſiori ꝑ idem tēpus cõtinuato. </s>
                  <s xml:id="N1E6D5" xml:space="preserve">Pro­
                    <lb/>
                  batur / q2 totalis velocitas illius motus eſt ſubdu-
                    <lb/>
                  pla ad velocitatē illius gradus iutenſioris eiuſdē
                    <lb/>
                  latitudinis: igitur ſubduplū ſpaciū ꝑtranſibitur
                    <lb/>
                  mediante vna illaꝝ ad ſpaciū ꝑtranſitū ab illa que
                    <lb/>
                  eſt in duplo intenſior dūmodo tēpora ſint equalia
                    <lb/>
                  ſi ſpaciorum proportio proportionem velocitatū
                    <lb/>
                  eodem tempore ſequitur / vt oportet. </s>
                  <s xml:id="N1E6E6" xml:space="preserve">Ex hac ſequit̄̄.</s>
                </p>
                <p xml:id="N1E6E9">
                  <s xml:id="N1E6EA" xml:space="preserve">Sexta ꝓpoſitio que talis eſt. </s>
                  <s xml:id="N1E6ED" xml:space="preserve">Omne
                    <lb/>
                  mobile motū vniformiter difformiter a certo gra-
                    <lb/>
                  du vſ ad certū gradū in aliquo tēpore maiꝰ ſpa-
                    <lb/>
                  ciū quã ſubduplū ꝑtranſit in eodem tēpore ad ſpa­
                    <lb/>
                  ciū natū ꝑtranſiri mediante extremo intenſiori il-
                    <lb/>
                  lius latitudinis ꝑ idem tēpus cõtinuato. </s>
                  <s xml:id="N1E6FA" xml:space="preserve">Probat̄̄ /
                    <lb/>
                  quia ſi talis latitudo inctperet a gradu ſuo inten-
                    <lb/>
                  ſiori et terminaretur ad nõ gradū: p̄ciſe illud mobi­
                    <lb/>
                  le ꝑtranſiret in illo tēpore ſubduplū ſpaciū ad ſpa­
                    <lb/>
                  ciū natū ꝑtranſiri mediante extremo intenſiori il­
                    <lb/>
                  lius latitudinis ꝑ idem tēpus cõtinuato / vt patꝫ ex
                    <lb/>
                  priori: ſed modo illa latitudo ab illo gradu incipi­
                    <lb/>
                  ens et ad gradū terminata eſt intenſior / vt ptꝫ ex ſe­
                    <lb/>
                  cunda / ergo in equali tēpore maiꝰ ſpaciū quã illud
                    <lb/>
                  ſubduplum pertranſibit / quod fuit probandum.</s>
                </p>
                <p xml:id="N1E70F">
                  <s xml:id="N1E710" xml:space="preserve">Septima ꝓpoſitio. </s>
                  <s xml:id="N1E713" xml:space="preserve">Si aliqḋ mobile
                    <lb/>
                  vniformiter difformiter moueat̄̄ a certo gradu in-
                    <lb/>
                  tēſiori ad cetū gradū remiſſiorē ī hora: ipſū in pri­
                    <lb/>
                  ma medietate hore minus quã triplū ſpaciū ꝑtran­
                    <lb/>
                  ſit ad ſpaciū ꝑtranſitū in ſecunda medietate hore
                    <lb/>
                  in qua tardiꝰ mouetur. </s>
                  <s xml:id="N1E720" xml:space="preserve">Probatur / quia ſi talis la-
                    <lb/>
                  titudo motus diuidatur ꝑ partes proportionales
                    <lb/>
                  ꝓportione dupla ſecundū partes tēporis: ille par-
                    <lb/>
                  tes nõ cõtinue ſe habebūt in ꝓportione dupla ſicut
                    <lb/>
                  ſe habent tales partes in latitudine terminata ad
                    <lb/>
                  nõ gradū: igr̄ reſiduū oīm partiū a prima non eſt
                    <lb/>
                  ſubtriplū ad velocitatē prime ſed maius quã ſub-
                    <lb/>
                  triplū: et ꝑ conſequens ſpaciū ꝑtranſitum in oībus
                    <lb/>
                  partibus a prima puta in ſecūda medietate eſt ma­
                    <lb/>
                  ius quã ſubtriplum ad ſpacium pertranſitū in pri­
                    <cb chead="De motu locali quo ad effectū ſcḋm tempus difformi."/>
                  ma. </s>
                  <s xml:id="N1E738" xml:space="preserve">Antecedens patet intuenti et conſequentia pro­
                    <lb/>
                  batur / quia quanto proportio aliqua in qua ſe ha­
                    <lb/>
                  bent cõtinuo aliqua infinita eſt minor tanto aggre­
                    <lb/>
                  gatum ex omnibus ſequentibus primū eſt maius.
                    <lb/>
                  </s>
                  <s xml:id="N1E742" xml:space="preserve">Item patet predicta propoſitio exemplariter / qm̄
                    <lb/>
                  capta latitudine incipiente a duodecim et termina­
                    <lb/>
                  ta ad quatuor gradus medius medietatis intenſi­
                    <lb/>
                  oris eſt vt decem: et gradus medius medietatis re-
                    <lb/>
                  miſſioris eſt vt .6. modo gradus ſextus nõ eſt ſub-
                    <lb/>
                  triplus ad duodenarium: et ſic in omni alia lati-
                    <lb/>
                  tudine inuenies predicte propoſitionis certitudinē
                    <lb/>
                    <note position="right" xlink:href="note-0148-01a" xlink:label="note-0148-01" xml:id="N1E76B" xml:space="preserve">Queſtio</note>
                  </s>
                  <s xml:id="N1E758" xml:space="preserve">¶ Et ſi queras quomodo cognoſcēdum ſit in omni
                    <lb/>
                  latitudine motus vtrim ad graduꝫ terminata in
                    <lb/>
                  qua proportione ſe habeat extremuꝫ intenſius ad
                    <lb/>
                  gradum mediuꝫ eiuſdem latitudinis: et in qua pro-
                    <lb/>
                  portione plus pertrãſitur mediante medietate in-
                    <lb/>
                  tenſiori talis latitudinis quam mediante medieta­
                    <lb/>
                  te remiſſiori.</s>
                </p>
                <p xml:id="N1E771">
                  <s xml:id="N1E772" xml:space="preserve">Rſpõdeo /  in hac materia nulla põt
                    <lb/>
                  dari certa et vniuerſalis regula. </s>
                  <s xml:id="N1E777" xml:space="preserve">Quoniã ſecundū /
                    <lb/>
                  quod extremum intenſius et remiſſius ſe habent in
                    <lb/>
                  alia et alia ꝓportiõe ad īuicē: ita ſe habet gxadꝰ me­
                    <lb/>
                  dius ad extremū intenſius talis latitudinis in alia
                    <lb/>
                  et alia ꝓportiõe: tamen poſſent ſiguari peculiares
                    <lb/>
                  regule certis ſpeciebus proportionum accõmode
                    <lb/>
                  </s>
                  <s xml:id="N1E785" xml:space="preserve">Si enim extrema ſe habeant in proportiõe dupla
                    <lb/>
                  gradus medius eſt ſubſexquitertius ad extremum
                    <lb/>
                  intenſius. </s>
                  <s xml:id="N1E78C" xml:space="preserve">Si vero extrema ſe habent in proporti-
                    <lb/>
                  one tripla: tunc gradus medius erit ſubſexquial-
                    <lb/>
                  terus ad extremum intenſius. </s>
                  <s xml:id="N1E793" xml:space="preserve">Si vero ſe habent in
                    <lb/>
                  proportione quadrupla: tunc gradus medius eſt
                    <lb/>
                  ſubſupertripartiens quintas ad extremum inten-
                    <lb/>
                  ſius. </s>
                  <s xml:id="N1E79C" xml:space="preserve">Si vero ſe habeant in proportione ſextupla:
                    <lb/>
                  gradus medius eſt ſuperquintipartiens ſeptimas
                    <lb/>
                  ad gradum intenſiorem. </s>
                  <s xml:id="N1E7A3" xml:space="preserve">et ſic diuerſis proportioni­
                    <lb/>
                  bus diuerſe regule aſſignatur.
                    <note position="right" xlink:href="note-0148-02a" xlink:label="note-0148-02" xml:id="N1E7B8" xml:space="preserve">Queſtio</note>
                  </s>
                  <s xml:id="N1E7AD" xml:space="preserve">¶ Quereret tamē
                    <lb/>
                  aliquis vlterius quo tramite et menſura poſſet fa-
                    <lb/>
                  cile inueſtigari gradus medius in omni latitudīe.</s>
                </p>
                <p xml:id="N1E7BE">
                  <s xml:id="N1E7BF" xml:space="preserve">Reſpondeo /  per hanc regulam quia
                    <lb/>
                  aut latitudo illa terminatur ad nõ gradū / tūc diui­
                    <lb/>
                  datur extremum intenſius per medium: et vna me-
                    <lb/>
                  dietas eſt gradus medius. </s>
                  <s xml:id="N1E7C8" xml:space="preserve">Si vero incipit a gradu
                    <lb/>
                  et terminatur ad gradum: tunc ſubduplum ad ag-
                    <lb/>
                  gregatum ex extremo intenſiori et remiſſiori eſt gra­
                    <lb/>
                  dus medius inter illa extrema. </s>
                  <s xml:id="N1E7D1" xml:space="preserve">Exemplum primi /
                    <lb/>
                  vt ſi aliqua latitudo incipiati ab octauo et termina­
                    <lb/>
                  tur ad non gradum: quoniam medietas ipſorum
                    <lb/>
                  8. eſt .4. ideo gradus quartus eſt gradus medius.
                    <lb/>
                  </s>
                  <s xml:id="N1E7DB" xml:space="preserve">Exemplum ſecundi / vt ſi aliqua latitudo incipiat
                    <lb/>
                  ab octauo et terminatur ad quartum. </s>
                  <s xml:id="N1E7E0" xml:space="preserve">dico /  gra-
                    <lb/>
                  dus ſextus eſt gradus mediꝰ qui eſt ſubduplus ad
                    <lb/>
                  aggregatum ex 8. et .4. </s>
                  <s xml:id="N1E7E7" xml:space="preserve">Illud enim aggregatum eſt
                    <lb/>
                  vt duodecim: et ſic vniuerſaliter reperies omni ſe-
                    <lb/>
                  cluſa exceptione.</s>
                </p>
                <p xml:id="N1E7EE">
                  <s xml:id="N1E7EF" xml:space="preserve">Notandum eſt ſecundo /  motum ve-
                    <lb/>
                  locitates quando ſunt equales quãdo inequa-
                    <lb/>
                  les intenſiue: et ſi equales, aut coextenſe partibus
                    <lb/>
                  temporis equalibus, aut inequalibus. </s>
                  <s xml:id="N1E7F8" xml:space="preserve">Si vero in­
                    <lb/>
                  equales idem etiam contingit, quia aut extendun-
                    <lb/>
                  tur per tempora equalia, aut per inequalia. </s>
                  <s xml:id="N1E7FF" xml:space="preserve">Si
                    <lb/>
                  ſint inequales inequalibus coextenſe temporibus /
                    <lb/>
                  hoc contingit dupliciter quia aut maior velocitas
                    <lb/>
                  coextenditur tempori maiori aut minori. </s>
                  <s xml:id="N1E808" xml:space="preserve">Exemplū
                    <lb/>
                  primi / vt ſi velocitas vt .4. coextendatur vni hore:
                    <lb/>
                  hoc eſt mobile moueatur vt .4. per vnam horam et
                    <lb/>
                  vt duo per dimidiam. </s>
                  <s xml:id="N1E811" xml:space="preserve">Exemplum ſecundi / vt ſi
                    <lb/>
                  aliquod mobile moueatur velocitate vt quatuor </s>
                </p>
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