173 citra centrū mūdi efficit̄̄ minor et pars vltra centrū
mūdi maior) / igr̄ perpetuo aliqua pars aggregati
ex oībus partibꝰ ꝓpotiõalibꝰ ipſiꝰ d. partis deſcē-
det / qḋ fuit ꝓbandū. Conſequētia ꝓbat̄̄ / q2 ſi q̄libet
pars ꝓportionalis cõtinuo ipſius d. partis diuiſe
ꝓportione dupla deſcēderet ſiue moueret̄̄ a ꝓpor-
tione a qua ipſa īcipit deſcendere: ꝑpertuo aliqua
pars aggregati ex oībus partibꝰ ꝓportionalibus
ipſiꝰ d. partis deſcēderet / g̊ ſi q̄libet pars ꝓportio-
nalis ipſiꝰ d. partis cõtinuo deſcēderet et mouetur
a ꝓportione minori ꝙ̄ ſit illa a qua īcipit deſcēde-
re: ꝑpetuo aliqua pars aggregati ex oībus parti-
bus ꝓportionalibꝰ ipſiꝰ d. partis deſcendit / qḋ fuit
ꝓbandū. Cõſequētia ptꝫ cū ãtecedēte ex deductione
ſecūdi argumēti ſexti capitis primi tractatꝰ huius
partis: hoc addito / ille partes cõtinuo ſe habent
in ꝓportione dupla: et in tꝑe in quo adequate deſcē
dit aliqua pars ſcḋm ſe vel aliquid eiꝰ p̄tereundo
centrū mūdi ipſa pars deſcribit tm̄ ſpaciū quanta
ipſamet pars eſt / vt ptꝫ intuenti caſum. Sed iã ꝓbo
ſcḋam partē maioris vcꝫ ſecūda pars ꝓportona
lis ipſiꝰ d. partis īcipit deſcēdere a ꝓportione ſub
dupla ad ꝓportionē f. vĺ minori: q2 cū primū ṗma
pars proportionalis ipſius d. partis eſt totaliter
vltra centrū mūdi, pars citra centrū mundi perdit
quartã partē exceſſus quo excedit partē vltra cen-
trū mūdi: et illã acquirit pars vltra centrū mūdi / vt
cõſtat: g̊ tūc ꝓportio f. partis citra centrū ad partē
vltra centrū ꝑdit pluſ̄ medietatē ſui: et pluſ̄ ad
ſubduplū ſui diminuit̄̄: ptꝫ ↄ̨ña ex ſecūda ſuppoſi-
tione huiꝰ notabilis hoc addito pars citra cen-
trū eſt terminꝰ maior proportionis f. et pars vltra
centrū eſt terminꝰ minor. Et ab illa proportione q̄
eſt minor quã ſubdupla ad f. īcipit ſecūda pars ꝓ-
portionalis ipſiꝰ d. partis deſcēdere / vt cõſtat: g̊ ꝓ-
poſitū. Et iſto modo ꝓbabis tertra īcipit deſcen
dere a ꝓportione ſubdupla vel mīori ſubdupla ad
proportionē a qua incipit deſcendere ſecūda: et ſic
ↄ̨ñter de aliis partibꝰ. Sed iã ꝓbo maiorē vcꝫ cū
primū prima pars proportionalis ipſiꝰ d. partis
eſt totaliter vltra centrū pars citra centrū mundi
perdit quartã partē exceſſus quo ipſa excedit par-
tem vltra centrū mūdi: q2 ipſa d. pars eſt medietas
exceſſus quo pars citra centrū excedit partē vltra
centrū / vt ptꝫ ex prima ſuppoſitione huiꝰ notabilis /
g̊ prima pars proportionalis proportione dupla
ipſiꝰ d. partis eſt quarta pars totiꝰ exceſſus: et illã
ꝑdit pars citra centrū mūdi primū ipſa eſt tota
liter vltra centrū: g̈ propoſitū. Ptꝫ g̊ maior: et totū
añs / et ꝑ ↄ̨ñs cõcluſio q̄ fuerat probanda. 111. correĺ. ¶ Ex his
infero aliqua correlaria. Primū in caſu huiꝰ de-
monſtrationis īmediate poſt inſtãs / qḋ eſt preſens
aſcendet aliq̇d īmediate poſt illud deſcēdet : et tñ
nichil īmediate poſt hoc aſcendet qḋ īmediate poſt
hoc deſcendet. Probat̄̄ prima pars / q2 quocun
inſtanti dato illiꝰ tꝑis in quo deſcēdet tale quadra
tū q̄libet pars illiꝰ quadrati q̄ eſt citra centrū īme-
diate poſt tale inſtans deſcendet / vt ſatis conſtat et
īmediate poſt idē inſtans aliqua talis pars aſcen-
det: igr̄ in caſu demonſtrationis, īmediate poſt in-
ſtans / qḋ eſt preſens aliq̇d aſcēdet qḋ īmediate poſt
idē inſtans deſcendet ſcḋa pars ptꝫ ex falſitate ſue
cõtradictorie. Ad hoc em̄ aliquid aſcendat non
ſufficit aliquã partē eiꝰ aſcendere: ſed requiritur /
maior pars ꝙ̄ eiꝰ medietas aſcendat. Conſimiliter
dicat̄̄ de deſcensu. 222. correĺ. ¶ Scḋm correlariū. Immediate
poſt inſtans qḋ eſt preſens aſcendet aliq̇d qḋ p̄ſens
aſcendet aliq̇d qḋ īmediate poſt idē inſtans deſcen
det: et tñ nõ īmediate poſt inſtãs qḋ eſt p̄ſens deſcen
det aliq̇d qḋ īmediate poſt idem inſtans aſcendet.
Ptꝫ prima pars huiꝰ ex priori correlario. Et ſcḋa
ꝓbat̄̄ / q2 ↄ̨tradictoria illiꝰ eſt falſa / vt ptꝫ ꝑ falſitatē
prime exponētis q̄ eſt iſta poſt inſtans quod eſt pre
ſens deſcendet aliquid quod īmediate poſt idem in
ſtans aſcendet q2 nulla pars illiꝰ corporis quadra
ti que poſt inſtans quod eſt preſens deſcendit īme-
diate poſt idem inſtans aſcendet. 333. correĺ. ¶ Tertiū correla
riū. Immediate poſt inſtans quod eſt p̄ſens aſcen-
det aliq̇d qḋ īmediate poſt idē inſtans quod eſt pre
ſens deſcendet: et tñ nichil ſimul aſcendet, et deſcen-
det adequate diuiſiue capiendo ly. et ſicut ſtat ſor
tes īmediate poſt hoc erit albus, et immediate poſt
hoc erit niger: et tamen nõ ſimul erit albus et niger
Patet correlarium. ¶ Ex his tribus notabilibus
patet facile reſponſio ad queſtionem.
mūdi maior) / igr̄ perpetuo aliqua pars aggregati
ex oībus partibꝰ ꝓpotiõalibꝰ ipſiꝰ d. partis deſcē-
det / qḋ fuit ꝓbandū. Conſequētia ꝓbat̄̄ / q2 ſi q̄libet
pars ꝓportionalis cõtinuo ipſius d. partis diuiſe
ꝓportione dupla deſcēderet ſiue moueret̄̄ a ꝓpor-
tione a qua ipſa īcipit deſcendere: ꝑpertuo aliqua
pars aggregati ex oībus partibꝰ ꝓportionalibus
ipſiꝰ d. partis deſcēderet / g̊ ſi q̄libet pars ꝓportio-
nalis ipſiꝰ d. partis cõtinuo deſcēderet et mouetur
a ꝓportione minori ꝙ̄ ſit illa a qua īcipit deſcēde-
re: ꝑpetuo aliqua pars aggregati ex oībus parti-
bus ꝓportionalibꝰ ipſiꝰ d. partis deſcendit / qḋ fuit
ꝓbandū. Cõſequētia ptꝫ cū ãtecedēte ex deductione
ſecūdi argumēti ſexti capitis primi tractatꝰ huius
partis: hoc addito / ille partes cõtinuo ſe habent
in ꝓportione dupla: et in tꝑe in quo adequate deſcē
dit aliqua pars ſcḋm ſe vel aliquid eiꝰ p̄tereundo
centrū mūdi ipſa pars deſcribit tm̄ ſpaciū quanta
ipſamet pars eſt / vt ptꝫ intuenti caſum. Sed iã ꝓbo
ſcḋam partē maioris vcꝫ ſecūda pars ꝓportona
lis ipſiꝰ d. partis īcipit deſcēdere a ꝓportione ſub
dupla ad ꝓportionē f. vĺ minori: q2 cū primū ṗma
pars proportionalis ipſius d. partis eſt totaliter
vltra centrū mūdi, pars citra centrū mundi perdit
quartã partē exceſſus quo excedit partē vltra cen-
trū mūdi: et illã acquirit pars vltra centrū mūdi / vt
cõſtat: g̊ tūc ꝓportio f. partis citra centrū ad partē
vltra centrū ꝑdit pluſ̄ medietatē ſui: et pluſ̄ ad
ſubduplū ſui diminuit̄̄: ptꝫ ↄ̨ña ex ſecūda ſuppoſi-
tione huiꝰ notabilis hoc addito pars citra cen-
trū eſt terminꝰ maior proportionis f. et pars vltra
centrū eſt terminꝰ minor. Et ab illa proportione q̄
eſt minor quã ſubdupla ad f. īcipit ſecūda pars ꝓ-
portionalis ipſiꝰ d. partis deſcēdere / vt cõſtat: g̊ ꝓ-
poſitū. Et iſto modo ꝓbabis tertra īcipit deſcen
dere a ꝓportione ſubdupla vel mīori ſubdupla ad
proportionē a qua incipit deſcendere ſecūda: et ſic
ↄ̨ñter de aliis partibꝰ. Sed iã ꝓbo maiorē vcꝫ cū
primū prima pars proportionalis ipſiꝰ d. partis
eſt totaliter vltra centrū pars citra centrū mundi
perdit quartã partē exceſſus quo ipſa excedit par-
tem vltra centrū mūdi: q2 ipſa d. pars eſt medietas
exceſſus quo pars citra centrū excedit partē vltra
centrū / vt ptꝫ ex prima ſuppoſitione huiꝰ notabilis /
g̊ prima pars proportionalis proportione dupla
ipſiꝰ d. partis eſt quarta pars totiꝰ exceſſus: et illã
ꝑdit pars citra centrū mūdi primū ipſa eſt tota
liter vltra centrū: g̈ propoſitū. Ptꝫ g̊ maior: et totū
añs / et ꝑ ↄ̨ñs cõcluſio q̄ fuerat probanda. 111. correĺ. ¶ Ex his
infero aliqua correlaria. Primū in caſu huiꝰ de-
monſtrationis īmediate poſt inſtãs / qḋ eſt preſens
aſcendet aliq̇d īmediate poſt illud deſcēdet : et tñ
nichil īmediate poſt hoc aſcendet qḋ īmediate poſt
hoc deſcendet. Probat̄̄ prima pars / q2 quocun
inſtanti dato illiꝰ tꝑis in quo deſcēdet tale quadra
tū q̄libet pars illiꝰ quadrati q̄ eſt citra centrū īme-
diate poſt tale inſtans deſcendet / vt ſatis conſtat et
īmediate poſt idē inſtans aliqua talis pars aſcen-
det: igr̄ in caſu demonſtrationis, īmediate poſt in-
ſtans / qḋ eſt preſens aliq̇d aſcēdet qḋ īmediate poſt
idē inſtans deſcendet ſcḋa pars ptꝫ ex falſitate ſue
cõtradictorie. Ad hoc em̄ aliquid aſcendat non
ſufficit aliquã partē eiꝰ aſcendere: ſed requiritur /
maior pars ꝙ̄ eiꝰ medietas aſcendat. Conſimiliter
dicat̄̄ de deſcensu. 222. correĺ. ¶ Scḋm correlariū. Immediate
poſt inſtans qḋ eſt preſens aſcendet aliq̇d qḋ p̄ſens
aſcendet aliq̇d qḋ īmediate poſt idē inſtans deſcen
det: et tñ nõ īmediate poſt inſtãs qḋ eſt p̄ſens deſcen
det aliq̇d qḋ īmediate poſt idem inſtans aſcendet.
Ptꝫ prima pars huiꝰ ex priori correlario. Et ſcḋa
ꝓbat̄̄ / q2 ↄ̨tradictoria illiꝰ eſt falſa / vt ptꝫ ꝑ falſitatē
prime exponētis q̄ eſt iſta poſt inſtans quod eſt pre
ſens deſcendet aliquid quod īmediate poſt idem in
ſtans aſcendet q2 nulla pars illiꝰ corporis quadra
ti que poſt inſtans quod eſt preſens deſcendit īme-
diate poſt idem inſtans aſcendet. 333. correĺ. ¶ Tertiū correla
riū. Immediate poſt inſtans quod eſt p̄ſens aſcen-
det aliq̇d qḋ īmediate poſt idē inſtans quod eſt pre
ſens deſcendet: et tñ nichil ſimul aſcendet, et deſcen-
det adequate diuiſiue capiendo ly. et ſicut ſtat ſor
tes īmediate poſt hoc erit albus, et immediate poſt
hoc erit niger: et tamen nõ ſimul erit albus et niger
Patet correlarium. ¶ Ex his tribus notabilibus
patet facile reſponſio ad queſtionem.
Ad rationes ante oppoſitū.
Ad primã
reſponſum eſt ibi vſ ad replicam ad quam reſpo-
deo negando ſequelam, et ad probationem dico /
illud correlarium ibi adductum ad probationem
illiꝰ ſequele nõ eſt ad propoſitū, q2 ſupponit ꝓpor
tionē tēpoꝝ excedere proportionē velocitatū. Cuiꝰ
oppoſitū ī caſu argumēti eſt verū. Cõmēſurãda em̄
eſt vtra velocitas, et qua illud corpus mouet̄̄ cir-
culariter, et qua mouetur motu rarefactiõis pūcto
eiꝰ a quo debet ſumi velocitas totiꝰ motus cõtinuo
acquirente maiorē et maiorē diſtantiam a centro / vt
ptꝫ ex deductione eiuſdē replice. 441. correĺ ¶ Ex quo ſequitur /
poſſibile eſt aliquod corpꝰ circulare cõtinuo vni-
formiter et eque velociter moueri: et tñ ipſū ↄ̨tinuo
rarefieri et effici maiꝰ. Probat̄̄ ponēdo / vna rota
incipiat moueri circulariter pūcto medio ſemidia-
metri incipiēte moueri velocitate vt .4. et volo / ſiĺ
incipiat rarefieri illud corpus acquirendo in hora
pedalē diſtantiã adequate a centro ſupra diſtan-
tiã p̄habitã, eo tñ modo moueat̄̄ ille punctꝰ medius
ſemidiametri nun̄ ꝑtranſeat ſiue deſcribat ma
iorē lineã in aliquo tꝑe ꝙ̄ nata ſit deſcribi a veloci-
tate vt .4. in eodē tꝑe quo poſito ſequit̄̄ correlariū.
552. correĺ. ¶ Sequit̄̄ ſecūdo / ſi aliqua rota in hora moueat̄̄
circulariter puncto medio ſemidiametri continuo
motu circulari mouēte vniformiter, motu vero ra-
refactionis cõtinuo intendente motū ſuū in q̈libet
parte proportionali hore ꝓportione dupla ſequē-
te in duplo velociꝰ rarefiente ꝙ̄ in īmediate p̄cedēti
tūc ſpaciū deſcriptū a tali puncto eſt infinitū. Ptꝫ
hoc correlariū ex ſexta concluſione ꝑcedētꝪ capitis
reſponſum eſt ibi vſ ad replicam ad quam reſpo-
deo negando ſequelam, et ad probationem dico /
illud correlarium ibi adductum ad probationem
illiꝰ ſequele nõ eſt ad propoſitū, q2 ſupponit ꝓpor
tionē tēpoꝝ excedere proportionē velocitatū. Cuiꝰ
oppoſitū ī caſu argumēti eſt verū. Cõmēſurãda em̄
eſt vtra velocitas, et qua illud corpus mouet̄̄ cir-
culariter, et qua mouetur motu rarefactiõis pūcto
eiꝰ a quo debet ſumi velocitas totiꝰ motus cõtinuo
acquirente maiorē et maiorē diſtantiam a centro / vt
ptꝫ ex deductione eiuſdē replice. 441. correĺ ¶ Ex quo ſequitur /
poſſibile eſt aliquod corpꝰ circulare cõtinuo vni-
formiter et eque velociter moueri: et tñ ipſū ↄ̨tinuo
rarefieri et effici maiꝰ. Probat̄̄ ponēdo / vna rota
incipiat moueri circulariter pūcto medio ſemidia-
metri incipiēte moueri velocitate vt .4. et volo / ſiĺ
incipiat rarefieri illud corpus acquirendo in hora
pedalē diſtantiã adequate a centro ſupra diſtan-
tiã p̄habitã, eo tñ modo moueat̄̄ ille punctꝰ medius
ſemidiametri nun̄ ꝑtranſeat ſiue deſcribat ma
iorē lineã in aliquo tꝑe ꝙ̄ nata ſit deſcribi a veloci-
tate vt .4. in eodē tꝑe quo poſito ſequit̄̄ correlariū.
552. correĺ. ¶ Sequit̄̄ ſecūdo / ſi aliqua rota in hora moueat̄̄
circulariter puncto medio ſemidiametri continuo
motu circulari mouēte vniformiter, motu vero ra-
refactionis cõtinuo intendente motū ſuū in q̈libet
parte proportionali hore ꝓportione dupla ſequē-
te in duplo velociꝰ rarefiente ꝙ̄ in īmediate p̄cedēti
tūc ſpaciū deſcriptū a tali puncto eſt infinitū. Ptꝫ
hoc correlariū ex ſexta concluſione ꝑcedētꝪ capitis
Ad ſecundã rationē reſponſum eſt ibi
vſ ad replicã: ad quã reſpõdeo negando ſequelã
et ad probationē nego nullū ſit īpedimentū. īmo
cõtra motio nauis eſt ſorti īpedimento. Fatigat̄̄ tñ
ſortes nõ ꝑ motū quo deſcribat aliquod ſpacium
fixū: ſed q2 deſcribit aliquod ſpaciū nõ fixū ad cuiꝰ
deſcriptionē nõ ſequit̄̄ ſortē proprie moueri. Ma-
net enim ſortes in eodem loco fixo.
vſ ad replicã: ad quã reſpõdeo negando ſequelã
et ad probationē nego nullū ſit īpedimentū. īmo
cõtra motio nauis eſt ſorti īpedimento. Fatigat̄̄ tñ
ſortes nõ ꝑ motū quo deſcribat aliquod ſpacium
fixū: ſed q2 deſcribit aliquod ſpaciū nõ fixū ad cuiꝰ
deſcriptionē nõ ſequit̄̄ ſortē proprie moueri. Ma-
net enim ſortes in eodem loco fixo.
Ad ṫciã rationē rñdeo negãdo añs: et
ad ꝓbationē ↄ̨cedo maiorē, et nego mīorē et ad ꝓba
nē diſtinguo ſeq̄lã aut ſi tale corpꝰ ſit taliter diſpo
ſitū partes eiꝰ proportionales ꝓportiõe dupla
ita ſe habeant ſcḋm eã dimenſionē ſcḋm quã de-
ſcendūt cõtinuo ſe habet in ꝓportione dupla oībꝰ
aliis iuuamentis et īpedimētis deductis: et ſic cõce-
do ſequelã. Si vero partes eiꝰ proportionales pro
portione dupla ſe habuerīt in maiori proportione
quã ſit ꝓportio dupla et hoc quantū ad dimēſionē
ſcḋm quã deſcendūt, et ſic nõ oportet. Nego igitur
illo modo ſeq̄lã. ¶ Ex q̊ ſequit̄̄ / ita põt aliqḋ cor-
ad ꝓbationē ↄ̨cedo maiorē, et nego mīorē et ad ꝓba
nē diſtinguo ſeq̄lã aut ſi tale corpꝰ ſit taliter diſpo
ſitū partes eiꝰ proportionales ꝓportiõe dupla
ita ſe habeant ſcḋm eã dimenſionē ſcḋm quã de-
ſcendūt cõtinuo ſe habet in ꝓportione dupla oībꝰ
aliis iuuamentis et īpedimētis deductis: et ſic cõce-
do ſequelã. Si vero partes eiꝰ proportionales pro
portione dupla ſe habuerīt in maiori proportione
quã ſit ꝓportio dupla et hoc quantū ad dimēſionē
ſcḋm quã deſcendūt, et ſic nõ oportet. Nego igitur
illo modo ſeq̄lã. ¶ Ex q̊ ſequit̄̄ / ita põt aliqḋ cor-
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