69
Sed illã ꝑtranſibit arguitur: q2 quãlibet par-
tem eius proportionalē ꝓportione dupla mino-
ribus terminatis verſus extremū intenſius per-
tranſibit: igitur totã reſiſtentiã pertranſibit. Cõ-
ſequentia patet: q2 oēs partes ꝓportionales pro
portione dupla illius reſiſtentie totã illam reſi-
ſtentiã conſtituūt. Sed iam reſtat ꝓbare pro ꝓba
tione alterius partis / nun̄ ad finē deueniet: q2
nõ ſufficit in tēpore finito tranſire illã reſiſtentiã:
igitur nun̄ deueniet ad finē illius reſiſtentie. Ar-
guitur antecedens et capio vnã aliam reſiſtentiam
difformiter difformē diuiſam per partes ꝓpor-
tionales ꝓportione dupla: cuius prima pars pro
portionales ſit vniformis vt duo et ſecūda vt tria
et tertia vt .3. cū dimidio et quarta vt tria cū dimi-
dio et dimidio dimidii / et ſic ↄ̨ſequenter aſcenden-
do: ita quelibet pars ꝓportionalis tali ꝓpor-
tionis duple diuiſione / ſit vniformiter intenſa in
iſta reſiſtentia difformiter difformi ſicut punctus
initiatiuus conſimilis partis in reſiſtētia vnifor-
miter difformi: et ſint tales reſiſtentie equales ex-
tenſiue / quo poſito ſic argumentor iſta potētia vt
4. non ſufficit ꝑtranſire iſtã reſiſtentiã difformem
in tēpore finito / et iſta reſiſtentia minꝰ reſiſtit quã
alia vniformiter difformis / vt conſtat reſpiciēdo
ad reſiſtentiã partiū ꝓportionaliū vniꝰ et alteriꝰ:
igitur talis potentia vt .4. nõ ſufficit pertranſire
talē reſiſtentiã vniformiter difformē a ſecūdo gra
du vſ ad quartū / quod fuit ꝓbandū. Cõſequētia
eſt nota cū minore et maior arguitur / q2 aliquantū
tēpus requirit illa potentia ad pertranſeundum
primã partē ꝓportionalē: et tantū vel maiꝰ requi
rit ad ꝑtrãſeundū ſcḋam: et iterū tantū vel maius
ad ꝑtranſeundū tertiã: et ſic cõſequenter: et ſunt in
finite partes ꝓportionales: igitur in nullo tēpo-
re finito ſufficit talis potentia illã reſiſtentiã dif-
formiter difformē ꝑtranſire. Conſequentia patet /
et ꝓbatur antecedēs / qm̄ tranſeundo primã partē
ꝓportionalē / que eſt vt duo mouetur a ꝓportione
dupla: et tranſeundo ſcḋam / que eſt vt .3. mouetur a
ꝓportione ſexquitertia: et tranſeundo tertiã / que
eſt vt .3. cū dimidio mouetur a ꝓportione ſexqui-
ſeptima / et ſic conſequenter ſemꝑ a minori ꝓpor-
tione quã ſubdupla ad precedentē: igitur cõtinuo
tranſeundo partē ꝓportionalē ſequentē, requirit
maiꝰ tēpus quã trãſeūdo partē precedentē. Patet
cõſequentia / qm̄ ſi cotinuo moueretur a ſubdupla
ꝓportione in parte ꝓportionali ſequenti ad pro-
portionē qua mouebatur in parte īmediate pre-
cedenti: ſemꝑ adequate tantū tēpus requireret ad
tranſeundū partē ſequentē ſicut īmediate prece-
dentē: q2 partes continuo ſe habent in ꝓportione
dupla et ſimiliter ꝓportiones ſe tunc haberent in
ꝓportione dupla: ſed modo cõtinuo in parte ſe-
quenti mouetur a minori ꝓportione quã ſubdu-
pla ad ꝓportionē / qua mouetur in parte īmediate
precedenti: igitur continuo maius tēpus requirit
ad pertranſeundū partē ſequentē quã precedentē
Sed cõtinuo moueatur a minori ꝓportiõe quã
ſubdupla in parte ſequenti quã in parte īmediate
precedenti patet / q2 in prima mouetur a ꝓportiõe
dupla / et in ſecūda a ꝓportiõe ſexquitertia modo
ſexquitertia minor eſt quã ſubdupla duple / vt ptꝫ
ex ꝓbatione tertie cõcluſiõis quarti capitis ſcḋe
partis et ſexta ſuppoſitione capitis eiuſdē. Itē in
tertia mouet̄̄ a ꝓportiõe ſexquiſeptīa: modo ſexq̇
ſeptīa minor eſt quã ſubdupla ſexq̇tertie / et ſic cõ-
ſequenter / vt patet ex ſexta ſuppoſitione quarti ca
pitis preallegati: igitur.
tem eius proportionalē ꝓportione dupla mino-
ribus terminatis verſus extremū intenſius per-
tranſibit: igitur totã reſiſtentiã pertranſibit. Cõ-
ſequentia patet: q2 oēs partes ꝓportionales pro
portione dupla illius reſiſtentie totã illam reſi-
ſtentiã conſtituūt. Sed iam reſtat ꝓbare pro ꝓba
tione alterius partis / nun̄ ad finē deueniet: q2
nõ ſufficit in tēpore finito tranſire illã reſiſtentiã:
igitur nun̄ deueniet ad finē illius reſiſtentie. Ar-
guitur antecedens et capio vnã aliam reſiſtentiam
difformiter difformē diuiſam per partes ꝓpor-
tionales ꝓportione dupla: cuius prima pars pro
portionales ſit vniformis vt duo et ſecūda vt tria
et tertia vt .3. cū dimidio et quarta vt tria cū dimi-
dio et dimidio dimidii / et ſic ↄ̨ſequenter aſcenden-
do: ita quelibet pars ꝓportionalis tali ꝓpor-
tionis duple diuiſione / ſit vniformiter intenſa in
iſta reſiſtentia difformiter difformi ſicut punctus
initiatiuus conſimilis partis in reſiſtētia vnifor-
miter difformi: et ſint tales reſiſtentie equales ex-
tenſiue / quo poſito ſic argumentor iſta potētia vt
4. non ſufficit ꝑtranſire iſtã reſiſtentiã difformem
in tēpore finito / et iſta reſiſtentia minꝰ reſiſtit quã
alia vniformiter difformis / vt conſtat reſpiciēdo
ad reſiſtentiã partiū ꝓportionaliū vniꝰ et alteriꝰ:
igitur talis potentia vt .4. nõ ſufficit pertranſire
talē reſiſtentiã vniformiter difformē a ſecūdo gra
du vſ ad quartū / quod fuit ꝓbandū. Cõſequētia
eſt nota cū minore et maior arguitur / q2 aliquantū
tēpus requirit illa potentia ad pertranſeundum
primã partē ꝓportionalē: et tantū vel maiꝰ requi
rit ad ꝑtrãſeundū ſcḋam: et iterū tantū vel maius
ad ꝑtranſeundū tertiã: et ſic cõſequenter: et ſunt in
finite partes ꝓportionales: igitur in nullo tēpo-
re finito ſufficit talis potentia illã reſiſtentiã dif-
formiter difformē ꝑtranſire. Conſequentia patet /
et ꝓbatur antecedēs / qm̄ tranſeundo primã partē
ꝓportionalē / que eſt vt duo mouetur a ꝓportione
dupla: et tranſeundo ſcḋam / que eſt vt .3. mouetur a
ꝓportione ſexquitertia: et tranſeundo tertiã / que
eſt vt .3. cū dimidio mouetur a ꝓportione ſexqui-
ſeptima / et ſic conſequenter ſemꝑ a minori ꝓpor-
tione quã ſubdupla ad precedentē: igitur cõtinuo
tranſeundo partē ꝓportionalē ſequentē, requirit
maiꝰ tēpus quã trãſeūdo partē precedentē. Patet
cõſequentia / qm̄ ſi cotinuo moueretur a ſubdupla
ꝓportione in parte ꝓportionali ſequenti ad pro-
portionē qua mouebatur in parte īmediate pre-
cedenti: ſemꝑ adequate tantū tēpus requireret ad
tranſeundū partē ſequentē ſicut īmediate prece-
dentē: q2 partes continuo ſe habent in ꝓportione
dupla et ſimiliter ꝓportiones ſe tunc haberent in
ꝓportione dupla: ſed modo cõtinuo in parte ſe-
quenti mouetur a minori ꝓportione quã ſubdu-
pla ad ꝓportionē / qua mouetur in parte īmediate
precedenti: igitur continuo maius tēpus requirit
ad pertranſeundū partē ſequentē quã precedentē
Sed cõtinuo moueatur a minori ꝓportiõe quã
ſubdupla in parte ſequenti quã in parte īmediate
precedenti patet / q2 in prima mouetur a ꝓportiõe
dupla / et in ſecūda a ꝓportiõe ſexquitertia modo
ſexquitertia minor eſt quã ſubdupla duple / vt ptꝫ
ex ꝓbatione tertie cõcluſiõis quarti capitis ſcḋe
partis et ſexta ſuppoſitione capitis eiuſdē. Itē in
tertia mouet̄̄ a ꝓportiõe ſexquiſeptīa: modo ſexq̇
ſeptīa minor eſt quã ſubdupla ſexq̇tertie / et ſic cõ-
ſequenter / vt patet ex ſexta ſuppoſitione quarti ca
pitis preallegati: igitur.
Reſpõdeo ad argumentum breuiter /
negando ſequelã: et ad ꝓbationē dico / illa ↄ̨ña
nichil valet: quãlibet partē ꝓportionalē ſecundū
hanc diuiſionē hoc mobile ꝑtranſibit: ergo totuꝫ
ſpaciū ſiue reſiſtentiã ꝑtranſibit: īmo ſicut ꝓbat
argumentū ſi mobile et illa reſiſtentia ſimul ma-
nerent ꝑ infinitū tēpus: ꝑ īfinitū tēpus mobile mo
ueret̄̄ ſupra reſiſtentiã / et nū̄ veniret ad terminū.
negando ſequelã: et ad ꝓbationē dico / illa ↄ̨ña
nichil valet: quãlibet partē ꝓportionalē ſecundū
hanc diuiſionē hoc mobile ꝑtranſibit: ergo totuꝫ
ſpaciū ſiue reſiſtentiã ꝑtranſibit: īmo ſicut ꝓbat
argumentū ſi mobile et illa reſiſtentia ſimul ma-
nerent ꝑ infinitū tēpus: ꝑ īfinitū tēpus mobile mo
ueret̄̄ ſupra reſiſtentiã / et nū̄ veniret ad terminū.
Sed ↄ̨̨tra q2 poſſibile eſt / potētia vt
4. ꝑtranſeat reſiſtentiã difformē in tꝑe finito, cuiꝰ
ṗma pars ꝓportiõalis eſt vniformiter difformis
a duobꝰ vſ ad tertiū, et ſecūda etiã vniformiter
difformis a tertio vſ ad tertiū cū dimidio, et ſic
cõſequenter vſ ad quartū excluſiue: igit̄̄ poſſibi-
le eſt potentiã vt .4. ꝑtranſire reſiſtentiã vniformi-
ter difformē a duobꝰ vſ ad quartū: et per conſe-
quens male negatū eſt hoc. Arguit̄̄ antecedens: et
pono / ſit vna reſiſtentia pedalis diuiſa per par
tes ꝓportionales ꝓportione quadrupla: cuiꝰ pri-
ma pars ꝓportionalis ſit vniformiter difformis
a ſecūdo vſ ad tertiū, et ſecunda a tertio vſ ad
tertiū cū dimidio, et ſic cõſequēter vſ ad quarū
excluſiue: deinde capio vnã aliã reſiſtentiã ſimili-
ter pedalē: diuiſam per partes ꝓportionales ꝓ-
portione quadrupla: cuiꝰ prima pars ꝓportiona
lis ſit vniformis vt .3. et ſecūda vt .3. cū dimidio, et
tertia vt .3. cū dimidio et dimidio dimidii, et ſic cõ-
ſequēter: ita quelibet pars ꝓportiõalis in tali
reſiſtentia ſit vniformiter intēſa ſicut gradus in-
ſiſſimꝰ in parte cõſimili ſiue correſpondēte in alia
reſiſtentia pedali cuiꝰ partes ꝓportionales ſunt
vniformiter difformes: quo poſito ſic argumētor
iſta ſecūda reſiſtentia cuiꝰ partes ꝓportiõales ſūt
vniformes eſt maioris reſiſtentie quã altera: vt ſa
tis facile ptꝫ intelligenti reſiſtentiã partiū ꝓpor-
tionabiliū in vna et in altera: et tamen potentia vt
4. ſufficit in tēpore finito ꝑtranſire iſtã ſecundam
reſiſtentiã: igit̄̄ et alterã cuiꝰ partes ꝓportionales
ſunt vniformiter difformes. Cõſequētia ptꝫ ꝑ locū
a maiori et maior ſimiliter: et minor ꝓbat̄̄: ſuppo-
nendo / oīs ꝓportio ſuꝑparticularis diuidit̄̄ in
duas ꝓportiones / quaꝝ vna eſt medii numeri ad
minimū et alia maximi ad mediū: et illa que eſt ma
ximi ad mediuꝫ, eſt maior quã tertia pars totius
ꝓportionis ſuꝑparticularis: vt ptꝫ ex decimo cor
relario tertie cõcluſionis quarti capitis ſecunde
partis. Hoc ſuppoſito ſic arguo potentia / vt .4. in
aliquo tēpore ꝑtranſit prima partē ꝓportionalē
talis reſiſtentie: et in ſubſexquitertio tēpore ꝑtran
ſit ſcḋam: et ſic cõſequēter ita quãlibet ſequentē
ꝑtranſit in ſubſexq̇tertio tēpore ad tēpus in quo
ꝑtrãſit īmediate p̄cedentē: igit̄̄ totū tēpus in quo
pertranſit oēs partes alias a prima eſt triplū ad
tempus in quo pertranſit primã: vt patet intelli-
genti quintum caput prime partis: et tempus in
quo pertrãſit primam eſt finitū: igitur totū tēpus
aggregatū eſt finituꝫ. Sed iam probo antecedens /
quoniam in aliquo tempore pertranſit primam:
ſignetur igitur illud tempus et ſit vna hora gra-
tia exempli: et in illa hora per illam partem con-
tinuo mouetur a proportione ſexquitertia: quia
reſiſtentia eſt vt .3. et potentia vt .4. et tranſeundo
ſecundam partem proportionalem / que eſt vt .3. cū
dimidio mouetur a proportione ſexquiſeptima:
que vt patet ex ſuppoſitione non eſt ſubtripla ad
ſexquitertiam ſed maior quam ſubtripla: ſed ſi
illa eſſet ſubtripla tranſiret ſecundam partem
ꝓportiõalē in ſubſexquitertio tēpore / ergo modo
4. ꝑtranſeat reſiſtentiã difformē in tꝑe finito, cuiꝰ
ṗma pars ꝓportiõalis eſt vniformiter difformis
a duobꝰ vſ ad tertiū, et ſecūda etiã vniformiter
difformis a tertio vſ ad tertiū cū dimidio, et ſic
cõſequenter vſ ad quartū excluſiue: igit̄̄ poſſibi-
le eſt potentiã vt .4. ꝑtranſire reſiſtentiã vniformi-
ter difformē a duobꝰ vſ ad quartū: et per conſe-
quens male negatū eſt hoc. Arguit̄̄ antecedens: et
pono / ſit vna reſiſtentia pedalis diuiſa per par
tes ꝓportionales ꝓportione quadrupla: cuiꝰ pri-
ma pars ꝓportionalis ſit vniformiter difformis
a ſecūdo vſ ad tertiū, et ſecunda a tertio vſ ad
tertiū cū dimidio, et ſic cõſequēter vſ ad quarū
excluſiue: deinde capio vnã aliã reſiſtentiã ſimili-
ter pedalē: diuiſam per partes ꝓportionales ꝓ-
portione quadrupla: cuiꝰ prima pars ꝓportiona
lis ſit vniformis vt .3. et ſecūda vt .3. cū dimidio, et
tertia vt .3. cū dimidio et dimidio dimidii, et ſic cõ-
ſequēter: ita quelibet pars ꝓportiõalis in tali
reſiſtentia ſit vniformiter intēſa ſicut gradus in-
ſiſſimꝰ in parte cõſimili ſiue correſpondēte in alia
reſiſtentia pedali cuiꝰ partes ꝓportionales ſunt
vniformiter difformes: quo poſito ſic argumētor
iſta ſecūda reſiſtentia cuiꝰ partes ꝓportiõales ſūt
vniformes eſt maioris reſiſtentie quã altera: vt ſa
tis facile ptꝫ intelligenti reſiſtentiã partiū ꝓpor-
tionabiliū in vna et in altera: et tamen potentia vt
4. ſufficit in tēpore finito ꝑtranſire iſtã ſecundam
reſiſtentiã: igit̄̄ et alterã cuiꝰ partes ꝓportionales
ſunt vniformiter difformes. Cõſequētia ptꝫ ꝑ locū
a maiori et maior ſimiliter: et minor ꝓbat̄̄: ſuppo-
nendo / oīs ꝓportio ſuꝑparticularis diuidit̄̄ in
duas ꝓportiones / quaꝝ vna eſt medii numeri ad
minimū et alia maximi ad mediū: et illa que eſt ma
ximi ad mediuꝫ, eſt maior quã tertia pars totius
ꝓportionis ſuꝑparticularis: vt ptꝫ ex decimo cor
relario tertie cõcluſionis quarti capitis ſecunde
partis. Hoc ſuppoſito ſic arguo potentia / vt .4. in
aliquo tēpore ꝑtranſit prima partē ꝓportionalē
talis reſiſtentie: et in ſubſexquitertio tēpore ꝑtran
ſit ſcḋam: et ſic cõſequēter ita quãlibet ſequentē
ꝑtranſit in ſubſexq̇tertio tēpore ad tēpus in quo
ꝑtrãſit īmediate p̄cedentē: igit̄̄ totū tēpus in quo
pertranſit oēs partes alias a prima eſt triplū ad
tempus in quo pertranſit primã: vt patet intelli-
genti quintum caput prime partis: et tempus in
quo pertrãſit primam eſt finitū: igitur totū tēpus
aggregatū eſt finituꝫ. Sed iam probo antecedens /
quoniam in aliquo tempore pertranſit primam:
ſignetur igitur illud tempus et ſit vna hora gra-
tia exempli: et in illa hora per illam partem con-
tinuo mouetur a proportione ſexquitertia: quia
reſiſtentia eſt vt .3. et potentia vt .4. et tranſeundo
ſecundam partem proportionalem / que eſt vt .3. cū
dimidio mouetur a proportione ſexquiſeptima:
que vt patet ex ſuppoſitione non eſt ſubtripla ad
ſexquitertiam ſed maior quam ſubtripla: ſed ſi
illa eſſet ſubtripla tranſiret ſecundam partem
ꝓportiõalē in ſubſexquitertio tēpore / ergo modo