Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secundi tractatus
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huius capitis. </
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<
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N1F49D
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xml:space
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">Et minor ex ſecunda parte prime
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propoſitionis eiuſdem notabilis.</
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</
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<
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<
s
xml:id
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N1F4A3
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xml:space
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preserve
">¶ Alio modo et breuiꝰ demonſtratur concluſio ſic:
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velocitatis totius hore ad velocitatem prime par-
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tis proportionalis eſt proportio f. et temporis to-
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tius hore quod eſt maius ad tempus prime partis
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proportionalis eſt etiam f. proportio: ergo ſpacii
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pertranſiti in tota hora ad ſpacium pertranſitum
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in prima parte proportionali eſt proportio com-
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poſita ex duplici proportione f. / et per conſequens
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ſpacium pertranſitum in tota hora ad ſpaciū per
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tranſitū in prima parte proportionali eſt propor-
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tio dupla ad proportionem velocitatum que eſt f.
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</
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<
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xml:space
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">Patet tamen conſequentia ex tertia propoſitione
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ſecundi notabilis huius capitis.</
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>
</
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<
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left
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xml:space
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">1. correĺ.</
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<
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<
s
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xml:space
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preserve
">¶ Ex his concluſionibus ſequitur primo: diuiſa
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hora per partes proportionales proportione mul
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tiplici, ſiue dupla, ſiue tripla, ſiue quadrupla, ſiue
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quauis alia multiplici: et in prima parte proporti-
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onali aliquod mobile moueatur aliquantulum, et
<
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ī ſcḋa in duplo maiori velocitate ꝙ̄ in ṗma: et ī ṫcia
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in triplo ꝙ̄ in prima / vt precedentis theorematis
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caſus oſtendit: totius illius velocitatis ad velo-
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citatem prime partis proportionalis erit propor-
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tio dupla, ſi diuiſio facta fuerit proportiõe dupla:
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et ſexquialtera ſi tripla: et ſexquitertia ſi quadru-
<
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pla: et ſic in infinitum aſcendendo ſeriatim per ſpe-
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cies proportiõis ſuperparticularis et multiplicis.
<
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</
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<
s
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N1F4E1
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xml:space
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preserve
">et ſpacli pertranſiti in tota hora ad ſpacium per-
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tranſituꝫ in prima parte eſt proportio quadrupla
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que eſt dupla ad duplam et hoc ſi fiat diuiſio par-
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tium proportionalium proportione dupla: ſi vero
<
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fiat proportione tripla: ſpacii pertranſiti in tota
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hora ad ſpacium pertranſitum in prima parte erit
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proportio dupla ad ſexquialteram que eſt dupla
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ſexquiquarta: ſi vero fiat diuiſio proportione qua
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drupla: tunc ſpacii pertranſiti in tota hora ad ſpa
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cium pertranſitum in prima parte proportionali
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erit proportio dupla ad ſexquitertiam que eſt ſu-
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pra ſeptipartiens nonas: et ſi fiat diuiſio proporti-
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one quintupla: tunc totius ſpacii ad ſpacium per-
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tranſitū in prima parte proportionali eſt propor-
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tio dupla ad proportioneꝫ ſexquiquartam que eſt
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proportio ſupra nonipartiens ſexdecimas: et ſic in
<
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infinitum duplicando proportionem velocitatum.
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</
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<
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xml:space
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">Prima pars huius correlarii patet ex ſecūda con
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cluſione manifeſte et ſecunda pars eiuſdem ex quar
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ta: et applica ſi potes
<
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xml:id
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xml:space
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">2. correĺ.</
note
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</
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<
s
xml:id
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N1F511
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xml:space
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">¶ Sequitur ſecundo particu-
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lariter / diuiſa hora per partes proportionales
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proportione ſextupla: et in prima illarū moueatur
<
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aliquod mobile aliquanta velocitate, et in ſecunda
<
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in duplo maiori, et in tertia in triplo, modo ſepi-
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us recitato: tunc totius velocitatis ad velocitatem
<
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prime partis proportionalis eſt proportio ſexqui
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quinta: et ſpacii pertranſiti in tota hora ad ſpaciū
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pertranſitū in prima parte proportionali eſt pro-
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portio ſupra vndecimpartiens viceſimas quintas
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</
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<
s
xml:id
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xml:space
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preserve
">Probatur prima pars huius correlarii: quia velo
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citate ita ſe habente vt ponitur: totalis velocitas
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ex omniū partium velocitatibus conſurgens ſe ha
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bet ad velocitateꝫ prime partis proportionalis in
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proportione in qua ſe habet totum tempus ad pri
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mam partem proportionalem / vt patet ex ſecunda
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concluſione: ſed hora diuiſa per partes proporti-
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onales proportione ſextupla ſe habet ad primam
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partem proportionalē in proportione ſexquiquin
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ta / vt docet quītum capitulum prime partis huius
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operis: igitur tota illa velocitas ſe habet ad velo-
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Capitulū tertiū.
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citatē prime patis proportionalis in proportione
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ſexquiquinta / quod fuit probandum. </
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<
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">Sed iam pro
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batur ſecunda pars: quia proportio ſupra vndecī-
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patiēs viceſimas quintas eſt dupla ad proportio-
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nem ſexquiquītam / vt patet in his terminis .36.30.
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25. iuuamine ſexti capitis ſecunde partis huiꝰ ope-
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ris: igitur ſpacium pertranſitum in tota hora ad
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ſpacium pertranſitum in parte ꝓportionali ſe ha-
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bet in proportione ſupra vndecimpartiente viceſi-
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maſquintas. </
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<
s
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">Patet hec conſequentia ex quarta cõ
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cluſione.
<
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right
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xml:id
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xml:space
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">3. correĺ.</
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</
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<
s
xml:id
="
N1F560
"
xml:space
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">¶ Sequitur tertio / diuiſa hora per par
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tes proportionales proportione octupla: et in eiſdē
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moueatur aliquod mobile modo pluries reſūpto
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totius velocitatis ad velocitatē prime partis pro-
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portionalis eſt ꝓportio ſexquiſeptima: et ſpacii ꝑ-
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tranſiti in tota hora ad ſpacium pertranſitum in
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prima parte proportionnali erit proportio dupla
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ad ſexquiſeptima que eſt ſuper quindecimpartiens
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quadrageſimas: cuiuſmodi eſt .9. cū ſeptima ad .7.
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et .64. ad .49. </
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<
s
xml:id
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N1F575
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xml:space
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preserve
">Probatur prima pars correlarii:
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quia hora ſic diuiſa per partes proportiõales pro
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portione octupla ſe habet ad primaꝫ partem pro-
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portionalem in proportione ſexquiſeptima / vt ptꝫ
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ex quīto capite prime partis huiꝰ operis: et in eadē
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proportione ſe debet habere velocitas totiꝰ ad ve-
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locitatem prime partis / vt dicit ſecunda concluſio:
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igitur propoſitum. </
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<
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">Secunda pars probatur: quia
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proportio ſupra quindecimpartiens quadrageſi-
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maſnonas eſt dupla ad proportionem ſexquiſepti
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mam / vt patet in his terminis .64.56. et .49. patro
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cinio ſexti capitis ſecunde partis: igitur in ſupra-
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quīdecimpartiens quadrageſimaſnonas ſe habet
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ſpacium pertranſitū in tota hora ad ſpacium per-
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tranſitum in prima parte proportiõali / quod fuit
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probandum. </
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>
<
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xml:id
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N1F599
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xml:space
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">Patet tamen conſequentia: ex quar-
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ta concluſione. </
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<
s
xml:id
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N1F59E
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xml:space
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">¶ Ex hoc modo poteris inferre in-
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nita correlaria ſimilia retento caſu velocitatis et
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variando continuo diuiſionē hore, que omnia cor-
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relaria ſuffragantibus ſeēunda et quarta conclu-
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ſionibus facilem ſortiuntur demonſtrationem.</
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</
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<
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xml:space
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">Quinta concluſio generi proportiõis
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ſuperparticularis ſpeciebuſ eius deſeruiens. </
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<
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xml:space
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">Di
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uiſa hora per partes proportionales proportiõe
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ſuperparticulari ſexquialtera, ſexquiquarta, ſeu
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quauis alia ſuperparticulari: diſtributa veloci-
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tate partibus illis proportionalibus ita vt mobi-
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le in prima illarum moueatur aliqnantulum, et in
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ſecunda in duplo velocius, et in tertia in triplo ve-
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locius ꝙ̄ in prima, et ſic conſequenter in caſu ſepiꝰ
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repetito: tunc tota velocitas ſe habet ad velocita-
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tem prime partis proportionalis in proportione
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tripla ſi fuerit hora diuiſa in proportione ſexqui-
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altera. </
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<
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xml:space
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">ſi vero fuerit diuiſa in proportione ſexqui-
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tertia: in proportione quadrupla: ſi in proportio-
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ne ſexquiquarta: in proportione quintupla. </
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<
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xml:id
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xml:space
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">et ſic cõ
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ſequenter aſcendendo ſeriatim per ſpecies propor
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tionis ſuperparticularis et multiplicis. </
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<
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xml:space
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">Et ſpacia
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pertranſita in totali tempore ad ſpacia prime par
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tis proportionalis ſe habent in proportione du-
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plicata (duplicata inquam ad triplam ſiue dupla
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ad triplam: ſi fuerit diuiſio facta in proportione
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ſexquialtera: et quadrupla ſi fuerit facta diuiſio
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in proportione ſexquitertia: et ſic conſequenter).</
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<
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<
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xml:id
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xml:space
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">Probatur hec concluſio / que infinitas habet par-
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tes in termino illo / et ſic cõſequenter incluſas et pri
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mo probatur eius prima pars que eſt de ꝓportiõe
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velocitatum ex ſecunda concluſione: hoc addito /
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totum diuiſum proportione ſexquialtera ſe habet </
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