Alvarus, Thomas
,
Liber de triplici motu
,
1509
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<
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Tertii tractatus
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file
="
0213
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n
="
213
"/>
et tūc manifeſtū eſt / viceſima prima efficitur eq̈-
<
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lis illi viceſime vel maior qm̄ tot proportiones ac-
<
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quiſiuit viceſima prima ſicut viceſima et cū hoc ac-
<
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quiſiuit bis ꝓportionē diuiſionis vel maiorē ea: g̊
<
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effecta eſt maior viceſima parte. </
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>
<
s
xml:id
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N2507C
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xml:space
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preserve
">et ſic ꝓbabis de vi-
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ceſima ſcḋa reſpectu viceſime prime.
<
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">Correĺ.</
note
>
</
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<
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xml:id
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xml:space
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">¶ Ex quo ſe-
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quit̄̄ / diuiſo corpore quauis ꝓportiõe volueris
<
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vt ponit̄̄ in caſu ↄ̨cluſiõis nõ eſt poſſibile tale cor-
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pꝰ ſucceſſiue ī tali caſu augmentari. </
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<
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xml:space
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</
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<
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<
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xml:space
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">Quarta ↄ̨̨cluſio. </
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>
<
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xml:id
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xml:space
="
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">Diuiſo corpore qua-
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uis optata ꝓportione, et prima pars ꝓportiõalis
<
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/>
talis corporis in hora aliqualiter augeat̄̄: et ſcḋa
<
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/>
velociꝰ prima in ꝓportione in qua eſt minor ea vel
<
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maiori: et tertia etiã velociꝰ prima in qua eſt minor
<
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/>
ea vel maiori, et ſic ↄ̨ñter ↄ̨tinuo totū illud corpus
<
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infinite velociter auget̄̄ in illa hora et ſubito efficit̄̄
<
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infinite magnū. </
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<
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xml:space
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">Probat̄̄ hec ↄ̨cluſio et volo / di-
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uidat̄̄ aliqḋ corpꝰ ꝓportione a. et incipiant partes
<
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augmentari: vt ponit̄̄ in ↄ̨cluſione. </
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>
<
s
xml:id
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xml:space
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">Tunc argr̄ ſic.
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</
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<
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xml:space
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">Quocun inſtanti dato poſt hoc dabit̄̄ vna pars
<
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ꝓportionalis equalis īmediate p̄cedenti vĺ maior
<
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et q̄libet ſequēs equalis illi vĺ maior: ergo quocū
<
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/>
inſtanti dato poſt hoc illud corpꝰ erit infinite ma-
<
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gnū. </
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>
<
s
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N250C7
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xml:space
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">Cõſequētia ptꝫ et argr̄ añs. </
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>
<
s
xml:id
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N250CA
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xml:space
="
preserve
">Qm̄ ſignato aliq̊
<
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/>
inſtanti poſt hoc aliqua eſt ꝓportio acq̇ſita prime
<
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/>
parte ꝓportionali q̄ ſit h. et in īfinitū maior acq̇ſi-
<
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ta eſt alicui parti / vt ptꝫ ex caſu: qm̄ in īfinitū mīor
<
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/>
prima eſt aliqua pars, capio igr̄ vnã / que acq̇ſiuit
<
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/>
vnã ꝓportionē a. vel maiorē vltra ꝓportionē acq̇-
<
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ſitã parti īmediate p̄cedenti: et ſequit̄̄ / illa eſt eq̈-
<
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lis vel maior īmediate p̄cedenti: qm̄ acq̇ſiuit tantã
<
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/>
ꝓportionē ſicut īmediate p̄cedens: et inſuꝑ illã per
<
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/>
quã excedebat̄̄ ab īmediate p̄cedenti vel maiorē: et
<
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/>
ꝑ ↄ̨ñs eſt equalis vel maior / vt ptꝫ ex maxīa ſcḋe cõ-
<
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/>
cluſionis. </
s
>
<
s
xml:id
="
N250E3
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xml:space
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preserve
">et ſiĺr pars ſequēs illã eſt eq̈lis īmediate
<
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p̄cedēti vel maior: qm̄ acquiſiuit tantã ꝓportionē
<
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/>
quantã īmediate p̄cedens et cū hoc vnã ꝓportionē
<
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/>
maiorē ꝙ̄ ſic proportio a. per quã excedebatur ab
<
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/>
īmediate p̄cedēti: et ſic ↄ̨ñter ꝓbabis de q̈libet alia</
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</
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<
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">Quīta ↄ̨̨cluſio. </
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<
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">Diuiſo corꝑe quacū-
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proportione volueris: et tn aliquo tꝑe prima pars
<
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/>
ꝓportionalis acquirat aliquã ꝓportionē: et q̄libet
<
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/>
ſequēs tantã in eodē tꝑe: tūc oēs ille partes manēt
<
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/>
in eadē ꝓportione in qua antea ſe habebãt: et totū
<
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acquirit illã ꝓportionē quã acquirit prima eiꝰ ꝑs
<
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</
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<
s
xml:id
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xml:space
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">Probat̄̄ prima pars ↄ̨cluſiõis: qm̄ q̄libet due par
<
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tes īmediate ita ſe hñt quanttã ꝓportionē acq̇ſi-
<
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uit maior tantã acq̇ſiuit mīor: et ſic manent in eadē
<
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ꝓportiõe in qua ſe habebãt ãtea. </
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<
s
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">Ptꝫ ↄ̨ña ex ſcḋa
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parte: et ꝑ ↄ̨ñs oēs ille partes ꝓportiõales ſe hñt in
<
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ea ꝓportiõe in qua ſe habebant antea. </
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>
<
s
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xml:space
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">Scḋa pars
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ꝓbat̄̄ / et ſit h. ꝓportio acq̇ſita primi parti ꝓportio
<
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nali. </
s
>
<
s
xml:id
="
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xml:space
="
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">et argr̄ ſic. </
s
>
<
s
xml:id
="
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xml:space
="
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">Quelꝫ ꝑs ꝓportionalis iſtiꝰ corꝑis
<
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demõſtrato corꝑe ſic diuiſo et augmētato eſt in h.
<
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ꝓportiõe maior ꝙ̄ ãtea: g̊ totū corpꝰ eſt in h. ꝓpor-
<
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/>
tiõe maiꝰ: et illa eſt ꝓportio quã acq̇ſiuit ṗma ꝑs ꝓ-
<
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portiõalis: igr̄ in caſu ↄ̨cluſiõis totū corpꝰ effectuꝫ
<
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ē maiꝰ in ꝓportiõe quã acq̇ſiuit ṗma ꝑs ꝓportiõa-
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lis. </
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<
s
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xml:space
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">Probat̄̄ ↄ̨ña: et ſit illud corpꝰ a. in fine augmē-
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tatiõis: et b. in prīcipio. </
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>
<
s
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xml:space
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">Et argr̄ ſic / ṗme ꝑtꝪ ꝓpor-
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tiõalis ipſiꝰ a .h. ꝓportiõe ad primã ipſiꝰ b. eſt ꝓpor
<
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tio h. </
s
>
<
s
xml:id
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xml:space
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">Et ſcḋe ipſiꝰ a. ad ſcḋaꝫ ipſiꝰ b. eſt etiã ꝓportio
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h. / Et ṫtie ipſiꝰ a. ad ṫtiã ipſiꝰ b. eſt ꝓportio h. / et ſic
<
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ↄ̨ñter: igr̄ oīm partiū ꝓportionaliū ipſiꝰ a. ad oēs
<
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ꝑtes ꝓportiõales ipſiꝰ b. eſt ꝓportio h. / ptꝫ hec ↄ̨ña
<
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qm̄ eadē eſt ꝓpor° ↄ̨iūtoꝝ et diuiſorū / vt ptꝫ ex ſcḋo
<
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capite ſecunde partis. </
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<
s
xml:id
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xml:space
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">Et ex conſequenti totum a.
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eſt in h. proportione maius ipſo b.</
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</
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<
cb
chead
="
Capitulū ſecundum.
"/>
<
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<
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<
s
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">Partito corpore per
<
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partes ꝓportionales quacun ꝓportione volue-
<
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ris et in aliquo tēpore prima pars proportionalis
<
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/>
acquirat aliquã ꝓportionē et ſecunda acquirat in
<
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/>
aliqua certa ꝓportione in eodē tēpore ꝓportioneꝫ
<
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/>
minorē: et tertia in eadē ꝓportione minorē ſecūda:
<
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et quarta in eadem ꝓportione minorē tertia: et ſic
<
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ↄ̨ñter. </
s
>
<
s
xml:id
="
N2515E
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xml:space
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">tūc ꝓportio inter primã partē et ſcḋam effi-
<
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citur maior ꝑ primã partē ꝓportionalē ꝓportiõis
<
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/>
acquiſite prime diuiſe in ea ꝓportione qua ſecūda
<
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/>
tardius augmētaã prima: et tertia ꝙ̄ ſcḋa: et ſic cõ-
<
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/>
ſequēter: et ꝓportio īter ſcḋaꝫ et tertiã efficit̄̄ maior
<
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/>
ꝑ ſcḋam partē ꝓportionalē ꝓportiõis acq̇ſite ṗme
<
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/>
et ꝓportio īter tertiã et quartã efficit̄̄ maior ꝑ tertiã
<
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/>
partē ꝓportionalē ꝓportionis acq̇ſite prime: r ſic
<
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/>
ↄ̨ñter: et (vt opinor) nõ valet finita intellectꝰ capaci
<
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/>
tas cõmenſurare ꝓportionē toti corpori acquiſitã
<
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</
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<
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">Exemplū / vt ſi aliqḋ corpꝰ diuidat̄̄ ꝑ partes ꝓpor-
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tionales ꝓportiõe dupla: et prima pars ꝓportiõa
<
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lis acq̇rat ꝓportionē ſexquialterã: et ſcḋa ſubqua-
<
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druplã: et tertia ſubquadruplã ad acquiſitã ſecūde
<
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/>
et quarta ſubquadruplã ad acquiſitã tertie: et ſic cõ
<
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ſequenter: tūc dico / ꝓportio inter primã partem
<
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/>
ꝓportionalē et ſcḋam acquiſiuit primã partē ꝓpor
<
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/>
tionis ſexq̇altere diuiſe ꝑ partes ꝓportiõales pro
<
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portione quadrupla. </
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>
<
s
xml:id
="
N25187
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xml:space
="
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">Et ꝓportio inter ſcḋaꝫ et ter-
<
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tiã acq̇ſiuit ſcḋam partē ꝓportionalē ꝓportionis
<
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/>
ſexq̇altere. </
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>
<
s
xml:id
="
N2518E
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xml:space
="
preserve
">Et ꝓportio inter tertiã et quartã tertiã
<
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partē ꝓportionalē ꝓportionis ſexq̇altere. </
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>
<
s
xml:id
="
N25193
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xml:space
="
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">Et ſi cõ
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ſequēter. </
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>
<
s
xml:id
="
N25198
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xml:space
="
preserve
">Probat̄̄ / ſit a. proportio acq̇ſita prime
<
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parti ꝓportionali: et ſit f. ꝓportio in qua velocius
<
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auget̄̄ prima ꝙ̄ ſcḋa. </
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>
<
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xml:id
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="
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">Et argr̄ ſic. </
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>
<
s
xml:id
="
N251A2
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xml:space
="
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">Proportiões ac-
<
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quiſite partibꝰ huiꝰ corporis cõtinuo ſe habent in
<
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/>
ꝓportione f. / vt ptꝫ ex caſu: ergo exceſſus quibꝰ cõti
<
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/>
nuo ſe excedūt etiã ſe hñt ↄ̨tinuo in ꝓportione f. et
<
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/>
ꝑ primū illoꝝ exceſſuū ꝓportio inter primã et ſcḋaꝫ
<
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/>
partē efficit̄̄ maior: et ꝑ ſcḋm ꝓportio inter ſcḋaꝫ et
<
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/>
tertiaꝫ efficit̄̄ maior: et ꝑ tertiū ꝓportio inter tertiã
<
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/>
et quartã efficit̄̄ maior: et ſic ↄ̨ñter: et primꝰ illoꝝ ex-
<
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/>
ceſſuū eſt prima pars ꝓportionalis ipſiꝰ a. ꝓpor-
<
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/>
tionis diuiſe ꝓportiõe f. et ſcḋus ſcḋa: et tertiꝰ tertia /
<
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/>
et ſic ↄ̨ñter: igr̄ ꝓportio inter primã et ſcḋam partē
<
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/>
efficit̄̄ maior ꝑ primã partē ꝓportionalē ipſius a.
<
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/>
ꝓportione f. et ſcḋa ꝑ ſcḋaꝫ: et tertia per tertiã: et ſic
<
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/>
ↄ̨ñter / qḋ fuit ꝓbandū. </
s
>
<
s
xml:id
="
N251BF
"
xml:space
="
preserve
">Ptꝫ tamē prima ↄ̨ña ꝑ hãc
<
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regulã q̄ ſuperiꝰ demõſtrata eſt in quacun ꝓpor-
<
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/>
tione ſe hñt aliqua ↄ̨tinuo in eadē ↄ̨tinuo ſe habēt
<
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/>
exceſſus eoꝝ. </
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>
<
s
xml:id
="
N251C8
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xml:space
="
preserve
">Sed iã ꝓbo / ꝑ primū illorū exceſſuū
<
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/>
ꝓportio inter primã et ſcḋam efficitur maior: et per
<
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/>
ſcḋm proportio inter ſcḋam et tertiã etc̈. / et hoc ꝑ hãc
<
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/>
maximã. </
s
>
<
s
xml:id
="
N251D1
"
xml:space
="
preserve
">Quãdocū due quãtitates inequales ac
<
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/>
quirūt aliquas ꝓportiones: et maior illaꝝ acqui-
<
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/>
rit maiorē ꝓportionē ꝙ̄ minor: tunc ꝓportio inter
<
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/>
illas quantitates efficitur maior per exceſſum quo
<
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/>
ꝓportio acquiſita maiori excedit ꝓportionē acqui
<
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/>
ſitam mīori vt in capĺo .8. ſcḋe partis oſtēſum eſt
<
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/>
ſed ſic eſt in ꝓpoſito: igr̄. </
s
>
<
s
xml:id
="
N251E0
"
xml:space
="
preserve
">Sed iam ꝓbat̄̄ / primꝰ
<
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/>
illoꝝ exceſſuū eſt prima pars ꝓportionalis ipſius
<
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/>
a. ꝓportione f. q2 a. ſe habet ad ꝓportionē acquiſi
<
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/>
tam prime partiū ꝓportionali in ꝓportione f. / ergo
<
lb
/>
exceſſus quo a. excedit ꝓportionē acquiſitã ſecunde
<
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/>
parti ꝓportionali eſt prima pars proportionalis
<
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/>
ipſius a. ꝓportione f. </
s
>
<
s
xml:id
="
N251EF
"
xml:space
="
preserve
">Ptꝫ ↄ̨ña per hanc regulam.
<
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/>
</
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>
<
s
xml:id
="
N251F3
"
xml:space
="
preserve
">Quandocū aliquod totū excedit aliquid in cer-
<
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/>
ta ꝓportione tūc excedit illud per primã ſui partē
<
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/>
ꝓportionalē tali ꝓportione: vt ſi vnū pedale exce-
<
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/>
dat aliam quantitatē in proportione ſexquialtera
<
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/>
illud pedale excedit aliud per primã ſui partē pro-
<
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/>
portionalē ꝓportione ſexquialtera quia per vnaꝫ </
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>
</
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>
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