Alvarus, Thomas
,
Liber de triplici motu
,
1509
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De motu augmentationis.
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ſi aliquod corpus diuidatur ꝑ partes proportio-
<
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nales proportiõe dupla: et in aliquo tēpore puta ī
<
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/>
hora prīa pars proportionalis augeatur aliquã-
<
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tulū velociter: et ſecūda in duplo velocius: et tertia
<
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/>
in triplo ꝙ̄ prima: et ſic ↄ̨ſequēter ſequeret̄̄ / totuꝫ
<
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/>
illud corpus in fine tēporis eſſet infinite magnū: et
<
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per cõſequēs illud corpus infinite velociter augmen-
<
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taretur: ſꝫ cõſequēs eſt falſum: igit̄̄ et añs. </
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<
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xml:space
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<
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cõſequētis arguit̄̄: et pono caſum / ſit vnū corpus
<
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/>
diuiſum ꝑ partes proportionales proportiõe du-
<
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/>
pla: et in hora prima pars proportionaiis acqui-
<
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/>
rat proportionē ſexquialterã: et ſecūda in eodē tē-
<
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pore acquirat duas ſexquialteras: et quarta .4. / et
<
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ſic cõſequēter. </
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<
s
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xml:space
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">Quo poſito arguit̄̄ ſic: prima pars
<
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proportionalis illiꝰ corporis aliqualiter auget̄̄: et
<
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/>
ſecūda in duplo magis: et tertia in triplo: et ſic cõſe
<
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/>
quēter: et tamē illud corpus in fine nõ erit infinituꝫ
<
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/>
ſed ſolū finitū / igit̄̄ in tali caſu nõ acq̇rit īfinitã pro
<
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/>
portionē: et ꝑ ↄ̨ſequēs illud illatū eſt falſum / q2 eſt
<
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vna cõditionalis cuius añs eſt verū / et cõſequēs fal
<
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ſum. </
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xml:space
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">Sed iã probo / illud in illo caſu erit finitum
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in fine hore q2 ī fine hore ille partes q̄ ante augmē-
<
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/>
tationē ſe habebãt in proportione dupla ſe habe-
<
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/>
bunt ↄ̨tinuo in proportiõe ſexquitertia: igit̄̄ aggre
<
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/>
gatū ex oībꝰ ſequētibiꝰ primã eſt triplū ad primaꝫ /
<
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/>
vt ptꝫ ītelligēti quītū caput prime partis: ſꝫ primū
<
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eſt finitū: ergo totū eſt finitū. </
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<
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xml:space
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preserve
">Sꝫ iã probo / ille ꝑ-
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tes cõtinuo ſe habēt in ꝓportiõe ſexquitertia: qm̄
<
lb
/>
prima et ſcḋa ſe habēt in ꝓportiõe ſexquitertia: et ſe
<
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/>
cunda et tertia: et ſic de q̇buſcū duabus immediatꝪ
<
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</
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xml:space
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">Quod ſic ꝓbat̄̄ / quoniã ſi prima et ſcḋa equalem ꝓ-
<
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portionē acq̇ſiuiſſent puta ſexquialteram: tunc ad
<
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/>
huc mãſiſſent in ꝓportiõe dupla ſicut antea vt con
<
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/>
ſtat: ſed modo ſecūda que eſt minor acquirit adhuc
<
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/>
ſexquialterã adequate: g̊ ꝓportio dupla que eſt in-
<
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/>
ter primã et ſcḋam ꝑdit ſexquialterã: et ſic manet ſex
<
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quitertia tantū inter primã et ſecundã. </
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xml:space
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">Itē ſi tertia
<
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pars ꝓportionalis acq̇ſiuiſſet duas ſexquialteras
<
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adequate ſicut ſecūda: ſecunda et tertia manſiſſent
<
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/>
in ꝓportõe dupla: ſꝫ modo tertia acquiſiuit adhuc
<
lb
/>
vnã ſexquialterã: igit̄̄ illam ſexq̇alteram deperdit
<
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/>
dupla q̄ eſt inter ſecundã et tertiam: et ꝑ cõſequens
<
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/>
manet ſexquitertia / vt patet intelligenti quartū ca
<
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/>
put ſecunde partis cū octauo: et ſit ꝓbabis de ter-
<
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/>
tia et quarta: et de oībus:
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xml:space
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">Maxīa.</
note
>
igit̄̄ ille partes cõtinuo ꝓ
<
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portionãtur ꝓportiõe ſexquitertia / qḋ fuit ꝓbandū
<
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tenet hec deductio ꝑ hanc maximã biꝑtitã. </
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<
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cū aliqui duo numeri vel ̄titates ſe habēt ī ali-
<
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qua ꝓportione et equales ꝓportiões acq̇runt ſem-
<
lb
/>
per manēt in eadē proportiõe et ſi numerus mīor
<
lb
/>
ſiue ̄titas minor acq̇rat aliquã ꝓportionē vltra
<
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/>
numerū ſiue ̄titatē maiorē ita tamē ſemꝑ ma-
<
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neat minor illã ꝓportionē deperdit ꝓportio que a
<
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principio erat inter numerū maiorē et minorē. </
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<
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maxīa claret ex quarta cõcluſiõe et ſecundo correla-
<
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rio ſexte ↄ̨̨cluſiõis octaui capitis ſecūde partꝪ. </
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>
<
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">Sꝫ
<
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iam probo ſeq̄lam prīcipalē argumēti: q2 ſi prima
<
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/>
pars ꝓportionalis talis corporis diuiſi ꝑ partes
<
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/>
ꝓportionales ꝓportiõe dupla acquireret duplã
<
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/>
et ſecūda duas duplas: et tertia tres duplas: et q̈rta
<
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/>
quatuor: et ſic ↄ̨ſequēter: tūc ī fine hore illud corpꝰ
<
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/>
manebit īfinite magnū: igit̄̄ īfinitã ꝓportionē ac-
<
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/>
quiſiuit in illo tꝑe et ſic infinite velociter augmēta-
<
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/>
bit̄̄: igit̄̄ ſi talis corporis diuiſi ꝑ partes ꝓportio-
<
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/>
nales ꝓportiõe dupla prīa pars ꝓportionalis ac-
<
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/>
quirat aliquã proportionē: et ſecūda duas tales: et
<
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/>
tertia tres: et quarta .4. / et ſic ↄ̨ſequēter: tūc tale cor
<
lb
/>
pus in illa hora infinitã proportionē acq̇rit: et ſic ī
<
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/>
finite velociter augmētat̄̄: quod fuit ꝓbãdū </
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<
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">Ptꝫ
<
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De motu augmentationis.
"/>
hec cõſequētia ab inferiori ad ſuꝑius. </
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<
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xml:space
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añs / q2 in fine hore quelibet illarū partiū propor-
<
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tionaliū erit equalis prime: et ſunt infinite / igit̄̄ illḋ
<
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corpus erit infinitū: </
s
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<
s
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xml:space
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">Probat̄̄ maior / q2 prīa et ſcḋa
<
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erūt equales in fiue: et ſecunda et tertia: et tertia et
<
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/>
quarta: et ſic de q̇buſcū aliis immediatis: quoni-
<
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/>
am ſi ſcḋa acq̇reret adequate vnam duplã ſicut pri
<
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/>
ma: tunc ṗma et ſcḋa adhuc manerēt in ꝓportione
<
lb
/>
dupla / vt pꝫ ex maxīa nuperrime poſita: ſꝫ modo ſe
<
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/>
cunda acq̇rit adhuc vnã duplã: et illã deꝑdit ꝓpor-
<
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/>
portio inter ṗmã et ſcḋam: igitur totalis ꝓportio
<
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/>
inter primã et ſcḋaꝫ deꝑditur q2 nõ erat niſi dupla
<
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et ſic prima et ſcḋa manēt equales. </
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<
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">Itē ſi tertia p̄ciſe
<
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acquireret duas duplas ſicut ſecūda adhuc inter
<
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/>
ſecūdam et tertiam maneret proportio dupla: ſed
<
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/>
modo illam duplã acquirit tertia: igit̄̄ ſecunda et
<
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tertia manent equales. </
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>
<
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xml:space
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plū auget̄̄ ad duplū efficit̄̄ duplo equale. </
s
>
<
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xml:id
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xml:space
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preserve
">et iſto mõ
<
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ꝓbabis de q̇buſcū aliis duabꝰ immediatis: igit̄̄
<
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/>
oēs ille partes in fine manebūt equales: et ꝑ conſe-
<
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quens illud corpus erit in fine infiuitū / qḋ fuit pro
<
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/>
bandū.
<
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xml:space
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<
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ſiue poſi
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tio.</
note
>
</
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<
s
xml:id
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xml:space
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preserve
">hec inductio gñaliter pꝫ ꝑ hanc maximam.
<
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</
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<
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xml:space
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">Quãdocū alique due ̄titates ſe habēt in aliq̈ ꝓ
<
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portiõe maioris inequalitatis et minor acq̇rit totã
<
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/>
illam proportionē que eſt inter ip̄am et maiorem q̄
<
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/>
maior etiã auget̄̄: et cū hoc illa minor acq̇rit etiam
<
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/>
illam ꝓportionē quã acq̇rit maior: tūc in fine mane
<
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/>
bunt equales. </
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>
<
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xml:space
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">Patet / q2 minor acq̇ſiuit totū quod
<
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deficiebat ei vt eſſet equalis alteri et cū hoc illḋ qḋ
<
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/>
illa maior acq̇ſiuit: ſꝫ ſic eſt in propoſito de his ꝑti
<
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/>
bus īmeditatis vt cõſtat: igitur in fine ille ꝑtes ma-
<
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nent equales.</
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note
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p
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<
s
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xml:space
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">Et confirmatur / quia ſi illa poſitio eēt
<
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/>
vera ſequeret̄̄ / ſi aliquod corpus diuideret̄̄ ī par-
<
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/>
tes ꝓportionales ꝓportione dupla: et prima pars
<
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/>
proportionalis in hora acq̇rat aliquã proportio-
<
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/>
ne ita augeat̄̄ aliquantulū velociter: et ſecūda ī
<
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/>
duplo velocius in eodē tꝑe: et tertia in duplo veloci
<
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/>
us ꝙ̄ ſecūda: et quarta in duplo velocius ꝙ̄ tertia
<
lb
/>
in eodē tēpore / et ſic ↄ̨ſeq̄nter tunc in fine illud cor-
<
lb
/>
pus manebit infinite magnū: et ſic in illo tēpore in
<
lb
/>
finite velociter augmētabitur: ſꝫ ↄ̨ſequēs eſt falſuꝫ /
<
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/>
igitur et añs. </
s
>
<
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xml:space
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">Falſitas cõſequētis probat̄̄: et capio
<
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vnū pedale diuiſum in partes proportionales pro
<
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/>
portione dupla: et volo / in vna hora prima pars
<
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/>
proportionalis acq̇rat vnã ſexquioctauã: et in eodē
<
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/>
tēpore ſecunda acquirat duas ſexquioctauas: et ter
<
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/>
tia. quatuor: et quarta .8. et q̇nta .16. / et ſic ↄ̨ſequen-
<
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/>
ter duplando. Quo poſito ſic arguo prima pars
<
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/>
illius corporis proportione dupla in hora aliquã
<
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/>
tulū auget̄̄: et ſecūda in duplo velocius: et tertia in
<
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/>
duplo velocius ꝙ̄ ſecūda: et ſic ↄ̨ſequēter / et tñ in fine
<
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/>
illud corpus nõ erit infinite magnū nec tale corpꝰ
<
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/>
infinite velociter augetur: igit̄̄ illud ↄ̨ſequens fal-
<
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/>
ſum. </
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>
<
s
xml:id
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xml:space
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">Probat̄̄ añs / q2 ille partes proportionales q̄
<
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/>
ſunt minores cõtinuo manebūt minores: nec vnī
<
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/>
aliqua ſequēs erit equalis immediate precedēti in
<
lb
/>
tali caſu: igit̄̄ illud corpus in fine nõ erit infinitum
<
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/>
</
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<
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">Probatur añs / q2 ſecunda pars non erit equalis
<
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prime: nec tertia ſecunde: nec quarta tertia: et ſic cõ-
<
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/>
ſequenter vt apparet: igitur non dabunt̄̄ in tali ca
<
lb
/>
ſu due partes quarum vna ſit equalis immediate
<
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/>
precedēti. </
s
>
<
s
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xml:space
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">Sed iam probo ſeq̄lam principaleꝫ: qm̄
<
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/>
ſi quelibet pars proportionalis ſequens acquire
<
lb
/>
ret adequate tot proportiões ſicut īmediate prece
<
lb
/>
dens: tunc ille partes cõtinuo ſe haberēt in propor
<
lb
/>
tione dupla ſicut ſe habent in ṗncipio: ſed mõ aliq̈
<
lb
/>
pars ſequēs acquiret decē proportiones pluſ̄ im </
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