208 geat̄̄ ad ſexquialterã: ipſum efficiet̄̄ tripedale, et ſic
acq̇reret vnū pedale: et cū in ſcḋa parte ꝓportiõali
acq̇rit ꝓportionē ſexq̇tertiã, ipſum efficiet̄̄ quadru
pedale, et ſic adhuc acq̇rit vnū pedale, et in tertia
acq̇rit ꝓportionē ſexquiquartã, et ſic efficiet̄̄ quītu-
pedale, in quarta acquirit ſexquiquintã, et ſic effi
cietur ſextupedale, et ſic conſequēter. igr̄ in q̈libet
parte proportionali acquirit vnū pedale et ſic effi-
citur infinitū / qḋ fuit ꝓbandum. Idē aſſumptū ptꝫ
ex ſexto correla° q̈rte ↄ̨cĺonis q̈rti capitꝪ ſcḋe ꝑtꝪ
acq̇reret vnū pedale: et cū in ſcḋa parte ꝓportiõali
acq̇rit ꝓportionē ſexq̇tertiã, ipſum efficiet̄̄ quadru
pedale, et ſic adhuc acq̇rit vnū pedale, et in tertia
acq̇rit ꝓportionē ſexquiquartã, et ſic efficiet̄̄ quītu-
pedale, in quarta acquirit ſexquiquintã, et ſic effi
cietur ſextupedale, et ſic conſequēter. igr̄ in q̈libet
parte proportionali acquirit vnū pedale et ſic effi-
citur infinitū / qḋ fuit ꝓbandum. Idē aſſumptū ptꝫ
ex ſexto correla° q̈rte ↄ̨cĺonis q̈rti capitꝪ ſcḋe ꝑtꝪ
Quīto principaliṫ ad idē argr̄ ſic.
Si
illa poſitio eſſet vera ſeq̄ret̄̄: ſi aliqḋ corpꝰ in pri
ma parte ꝓportiõali ꝓportiõe dupla vniꝰ hore ali
quãtuluū velociter augeret̄̄, et in ſcḋa in duplo velo-
cius, et in tertia in triplo velocius ꝙ̄ in prima, et in
q̈rta in q̈druplo velocius ꝙ̄ in prima aſcendendo ꝑ
oēs ſpecies ꝓportiõis multiplicis: illud corpus in
fine eſſet īfinite magnū: ſed ↄ̨ñs eſt falſum: igr̄ illud
ex quo ſequit̄̄. Seq̄la ꝓbat̄̄: q2 nõ videt̄̄ cuiꝰ magni
tudinis illud corpꝰ in fine ſit niſi īfinite: igr̄. Itē ac
quirit īfinitas ꝓportiões tale corpꝰ eq̈les nõ ↄ̨mu-
nicãtes, qm̄ in prima parte ꝓportiõali acq̇rit ali-
quã, et in ſcḋa cū augmētet̄̄ in duplo velociꝰ acq̇rit
duplã et in alia q2 augmētat̄̄ in triplo velociꝰ acq̇-
rit triplã: igr̄ īfinitas acq̇rit eq̈les etc̈. Sed iã ꝓbat̄̄
falſitas ↄ̨ñtis: et volo / vnū pedale in prima parte
ꝓportiõali tꝑis acq̇rat ꝓportionē duplã, et in ſcḋa
parte augmētet̄̄ in duplo velociꝰ, et in tertia in tri-
plo, et ſic ↄ̨ñter. Tunc manifeſtū eſt / in ſcḋa parte
ꝓportiõali tantã acq̇rit ſicut in prima puta duplã
q2 auget̄̄ in duplo velociꝰ et tēpꝰ eſt ſubduplū, et in
tertia acq̇rit tres quartas vniꝰ duple, q2 auget̄̄ in
triplo velociꝰ et in q̈rta acq̇rit q̈tuor octauas vnius
duple q2 auget̄̄ in q̈druplo velociꝰ ꝙ̄ in prima: ſi eī
eq̄uelociter augmētaret̄̄ ſicut in prima cū q̈rta ꝑs
ſit in octuplo minor prima ſequit̄̄ / in illa acq̇re-
ret vnã octauã duple: ſed mõ auget̄̄ in q̈druplo ve-
lociꝰ in eadē q̈rta parte: ergo q̈tuor octauas acq̇rit
et ſic ꝓbabis / in quīta acq̇rit quī ſexdecīas vniꝰ
duple, et in ſexta ſex triceſimas ſecundas.
illa poſitio eſſet vera ſeq̄ret̄̄: ſi aliqḋ corpꝰ in pri
ma parte ꝓportiõali ꝓportiõe dupla vniꝰ hore ali
quãtuluū velociter augeret̄̄, et in ſcḋa in duplo velo-
cius, et in tertia in triplo velocius ꝙ̄ in prima, et in
q̈rta in q̈druplo velocius ꝙ̄ in prima aſcendendo ꝑ
oēs ſpecies ꝓportiõis multiplicis: illud corpus in
fine eſſet īfinite magnū: ſed ↄ̨ñs eſt falſum: igr̄ illud
ex quo ſequit̄̄. Seq̄la ꝓbat̄̄: q2 nõ videt̄̄ cuiꝰ magni
tudinis illud corpꝰ in fine ſit niſi īfinite: igr̄. Itē ac
quirit īfinitas ꝓportiões tale corpꝰ eq̈les nõ ↄ̨mu-
nicãtes, qm̄ in prima parte ꝓportiõali acq̇rit ali-
quã, et in ſcḋa cū augmētet̄̄ in duplo velociꝰ acq̇rit
duplã et in alia q2 augmētat̄̄ in triplo velociꝰ acq̇-
rit triplã: igr̄ īfinitas acq̇rit eq̈les etc̈. Sed iã ꝓbat̄̄
falſitas ↄ̨ñtis: et volo / vnū pedale in prima parte
ꝓportiõali tꝑis acq̇rat ꝓportionē duplã, et in ſcḋa
parte augmētet̄̄ in duplo velociꝰ, et in tertia in tri-
plo, et ſic ↄ̨ñter. Tunc manifeſtū eſt / in ſcḋa parte
ꝓportiõali tantã acq̇rit ſicut in prima puta duplã
q2 auget̄̄ in duplo velociꝰ et tēpꝰ eſt ſubduplū, et in
tertia acq̇rit tres quartas vniꝰ duple, q2 auget̄̄ in
triplo velociꝰ et in q̈rta acq̇rit q̈tuor octauas vnius
duple q2 auget̄̄ in q̈druplo velociꝰ ꝙ̄ in prima: ſi eī
eq̄uelociter augmētaret̄̄ ſicut in prima cū q̈rta ꝑs
ſit in octuplo minor prima ſequit̄̄ / in illa acq̇re-
ret vnã octauã duple: ſed mõ auget̄̄ in q̈druplo ve-
lociꝰ in eadē q̈rta parte: ergo q̈tuor octauas acq̇rit
et ſic ꝓbabis / in quīta acq̇rit quī ſexdecīas vniꝰ
duple, et in ſexta ſex triceſimas ſecundas.
Quibꝰ īſpectis argr̄ ſic.
Tale corpus
acq̇rit īfinitos ordines ꝓportionū qui q̇dē ordines
ↄ̨tinuo ſe habēt in ꝓportione dupla: et primꝰ illoꝝ
ordinū eſt vna ꝓportio q̈drupla: g̊ oēs illi ordines
ↄ̨ſtiuūt duas q̈druplas: et ꝑ ↄ̨ñs vnã ſexdecuplã: et
ſic illud corpꝰ nõ acq̇rit niſi ꝓportionē ſexdecuplã
in tali caſu: et nõ īfinitã. Probat̄̄ tñ ↄ̨ña facta. Qm̄
ſi illi ordines ꝓportionū ↄ̨tinuo ſe hñt in ꝓportiõe
dupla. manifeſtū eſt aggregatū ex oībꝰ ſequēti-
bus primū eſt eq̈le primo: vt ptꝫ ex quīto capite pri
me partis. Sed cū primꝰ ordo ſit ꝓportio q̈drupla
manifeſtū eſt oēs alii ſimĺ ſumpti ſūt etiã q̈dru-
pla: et ſic aggregatū ex oībꝰ ſimĺ eſt vna ſexdecupla /
vt ptꝫ ex ſexto capite ſcḋe partis. Sed iã reſtat ꝓba
re ibi ſunt īfiniti ordines cõtinuo ſe habētes in
ꝓportiõe dupla. Q2 ſic ꝓbat̄̄ / q2 capiēdo ꝓportio-
nē duplã quã acq̇rit in prima parte ꝓportionali, et
medietatem illiꝰ duple ꝙ̄ acq̇rit in ſcḋa ꝑte: et vnã
quartã duple ex illis q̈rtis q̈s acquirit in tertia, et
vnã octauã duple ex illis quas acq̇rit in q̈rta, et vnã
decimãſextã duple ex illis q̈s acq̇rit in quīta parte
ꝓportionali, et ſic ↄ̨ñter: tunc manifeſtū eſt ibi eſt
vnꝰ ordo ꝓportionū ↄ̨tinuo ſe habentiū in ꝓporti-
one ſubdupla et primū illiꝰ ordinis eſt vna ꝓportio
dupla: igr̄ totꝰ ille ordo ↄ̨ſtituit q̈druplã. Cõſequē
tia ptꝫ vt ſupra. Itē ab ↄ̨ſtituendū ſcḋm ordinē ca-
piat̄̄ alia medietas duple q̄ remanſit ex illa dupla
quã acq̇rebat corpꝰ in ſcḋa ꝑte ꝓportiõali et deīde
capiat̄̄ vna q̈rta duple ex illis duabꝰ remanētibꝰ et
acq̇ſitis in tertia parte ꝓportiõali: et deīde capias
vna octaua ex octauis remanētibꝰ et acq̇ſitis ī q̈rta /
et ſic ↄ̨ñter: et manifeſtū eſt ibi eſt alter ordo pro-
portionū ↄ̨tinuo ſe habentiū in ꝓportione dupla:
et primū illoꝝ eſt vna medietas duple: g̊ reſiduū a
prima eſt alia duple medietas: et ſic totus ſcḋus or
do eſt vna dupla. Itē ad īuiendū tertiū ordinem
incipias ab acq̇ſitis in tertia ꝑte ꝓportiõali et īue-
nies vnã quartã p̄ciſe duple: q2 alie dū ſunt poſite
in aliis duobꝰ ordinibꝰ: et capias illã ꝑ ṗma tertii
ordinis: deinde capias vnã ex duabꝰ octauis acq̇ſi
tis et remanētibꝰ in q̈rta parte ꝓportiõali: et deīde
ꝑ tertia parte illiꝰ ordinis capias vnã ex tribꝰ de-
cimiſſextis derilictis et acq̇ſitis in quīta parte pro
portiõali: et ſic ↄ̨ñter. Et ſic ad īuiendū quartum
ordinē incipias ab vna octaua derelicta et acq̇ſita
in quarta parte ꝓportionali. Et ad īuiendū quī
tum incipies ab vna ſexdecima derclicta et acq̇ſita
in quīta parte ꝓportionali: et ſic ↄ̨ñter īuenies infi
nitos ordines et iſti ordines ↄ̨tinuo ſe habēt in pro
portione dupla: ita q̄libet ſequēs ordo eſt ſubdu
plus ad īmediate p̄cedentē ordinē: igr̄ ibi ſunt īfini
ti ordines ↄ̨tinuo ſe habētes in ꝓportione dupla:
qḋ fuit ꝓbandū. Q, autē illi ordines ↄ̨tinuo ſe ha
bent in ꝓportione dupla. Ptꝫ / q2 quilibet illoꝝ or-
dinū cõponit̄̄ ex infinitis ↄ̨tinuo ſe habentibus in
ꝓportione dupla: et oīa prima oīm illoꝝ ordinum
ↄ̨tinuo ſe habent in ꝓportione dupla / vt cõſtat: igr̄
oēs illi ordines cõtinuo ſe habent in ꝓportiõe du-
pla. Ptꝫ ↄ̨ña: q2 cuiuſlibet ordinis primuꝫ eſt me-
dietas illius ordinis et reſiduum alia medietas
quia in quacū proportione ſe habēt medietates
aliquoꝝ in eadē proportione ſe habent et ipſa tota
quoꝝ ſunt medietates / vt ptꝫ ex vndecima ſuppoſi-
tione ſecūdi capitis ſecūde partis: ergo oēs illi or
dines cõtinuo ſe habent in ꝓportione dupla / quod
fuit ꝓbandū. 11Cõfir
matio. ¶ Et confirmat̄̄ / q2 ſi illa poſitio eſſet
vera ſequeret̄̄ / ſi aliquod corpus in prima parte
ꝓportionali alicuiꝰ hore augmētaret̄̄ aliquãtulū
velociter, et in ſecūda in duplo velocius et in tertia
in duplo velocius ꝙ̄ in ſecūda, et ſic ↄ̨ñter: tale cor-
pus in fine hore eſſet infinitū, et ſic illud corpꝰ in-
finite velociter augmētaretur. ſꝫ ↄ̨ſequens eſt falſū /
igr̄ illud ex quo ſequit̄̄. Sequela ꝓbatur / q2 ſi hora
ſit diuiſa per partes ꝓportionales ꝓportione du
pla, et aliquod corpꝰ in ṗma aliquãtulū velociter
augmentet̄̄ aliquã ꝓportionē acquirendo, et in ſe-
cunda in duplo velociꝰ: et in tertia in duplo velociꝰ
̄ in ſecūda: et ſic ↄ̨ñter. Tūc tale corpus in fine ac-
quiſiuit tnfinitã proportionē et nõ eſt maior ratio
quãdo diuidit̄̄ hora tali diuiſione quã aliqua alia
diuiſione: igr̄ ſi hora diuidat̄̄ aliqua diuiſiõe: et in
prima aliquod corpus aliqua velocitate augmen-
tetur: et in ſecūda in duplo velociꝰ: et in tertia in du
pio velociꝰ ꝙ̄ in ſecūda: et ſic ↄ̨ñter tale corpꝰ īfinitã
ꝓportionē acq̇ret: et augmētabit̄̄ īfinite velociṫ in
tali hora / qḋ fuit ꝓbãdū. Probr̄ tñ añs vcꝫ / qḋ ſi ho
ra diuidat̄̄ ꝓportiõe dupla etc̈. / qḋ illud corpꝰ acq̇rit
īfinitã ꝓportionē: q2 in q̈lꝫ ꝑte ꝓportiõali acquirit
tãtã ꝓportionē ſicut in ṗma. Nã in q̈cū ꝓportiõe
aliq̈ ꝑs eſt mīor in eadē ꝓportiõe velociꝰ augmētat̄̄
et ſūt īfinite: g̊ īfinitas eq̈les ꝓportiões acq̇rit / et per
ↄ̨ñs īfinitã ꝓportionē acq̇rit. Iã ꝓbo falſitatē ↄ̨ñtꝪ
et volo / hõ diuidit̄̄ ꝑ ꝑtes ꝓportiõales ꝓportiõe q̈
drupla, et in ṗma ꝑte augmētet̄̄ aliqḋ corpꝰ certe ve
lociṫ puta acq̇rēdo ꝓportionē duplã: et in ſcḋa ī du
plo velociꝰ et .3. ī duplo velociꝰ ꝙ̄ ī.2. / et ſic ↄ̨ñṫ. in po
ſitū eſt, q̊ poſito ar̄ ſic. illud corpꝰ auget̄̄ vt ponit̄̄ et
tñ nõ acquirit niſi ꝓportionē quadruplã tota illa ho
ra: igr̄ illud conſequens eſt vna conditiõalis falſa
acq̇rit īfinitos ordines ꝓportionū qui q̇dē ordines
ↄ̨tinuo ſe habēt in ꝓportione dupla: et primꝰ illoꝝ
ordinū eſt vna ꝓportio q̈drupla: g̊ oēs illi ordines
ↄ̨ſtiuūt duas q̈druplas: et ꝑ ↄ̨ñs vnã ſexdecuplã: et
ſic illud corpꝰ nõ acq̇rit niſi ꝓportionē ſexdecuplã
in tali caſu: et nõ īfinitã. Probat̄̄ tñ ↄ̨ña facta. Qm̄
ſi illi ordines ꝓportionū ↄ̨tinuo ſe hñt in ꝓportiõe
dupla. manifeſtū eſt aggregatū ex oībꝰ ſequēti-
bus primū eſt eq̈le primo: vt ptꝫ ex quīto capite pri
me partis. Sed cū primꝰ ordo ſit ꝓportio q̈drupla
manifeſtū eſt oēs alii ſimĺ ſumpti ſūt etiã q̈dru-
pla: et ſic aggregatū ex oībꝰ ſimĺ eſt vna ſexdecupla /
vt ptꝫ ex ſexto capite ſcḋe partis. Sed iã reſtat ꝓba
re ibi ſunt īfiniti ordines cõtinuo ſe habētes in
ꝓportiõe dupla. Q2 ſic ꝓbat̄̄ / q2 capiēdo ꝓportio-
nē duplã quã acq̇rit in prima parte ꝓportionali, et
medietatem illiꝰ duple ꝙ̄ acq̇rit in ſcḋa ꝑte: et vnã
quartã duple ex illis q̈rtis q̈s acquirit in tertia, et
vnã octauã duple ex illis quas acq̇rit in q̈rta, et vnã
decimãſextã duple ex illis q̈s acq̇rit in quīta parte
ꝓportionali, et ſic ↄ̨ñter: tunc manifeſtū eſt ibi eſt
vnꝰ ordo ꝓportionū ↄ̨tinuo ſe habentiū in ꝓporti-
one ſubdupla et primū illiꝰ ordinis eſt vna ꝓportio
dupla: igr̄ totꝰ ille ordo ↄ̨ſtituit q̈druplã. Cõſequē
tia ptꝫ vt ſupra. Itē ab ↄ̨ſtituendū ſcḋm ordinē ca-
piat̄̄ alia medietas duple q̄ remanſit ex illa dupla
quã acq̇rebat corpꝰ in ſcḋa ꝑte ꝓportiõali et deīde
capiat̄̄ vna q̈rta duple ex illis duabꝰ remanētibꝰ et
acq̇ſitis in tertia parte ꝓportiõali: et deīde capias
vna octaua ex octauis remanētibꝰ et acq̇ſitis ī q̈rta /
et ſic ↄ̨ñter: et manifeſtū eſt ibi eſt alter ordo pro-
portionū ↄ̨tinuo ſe habentiū in ꝓportione dupla:
et primū illoꝝ eſt vna medietas duple: g̊ reſiduū a
prima eſt alia duple medietas: et ſic totus ſcḋus or
do eſt vna dupla. Itē ad īuiendū tertiū ordinem
incipias ab acq̇ſitis in tertia ꝑte ꝓportiõali et īue-
nies vnã quartã p̄ciſe duple: q2 alie dū ſunt poſite
in aliis duobꝰ ordinibꝰ: et capias illã ꝑ ṗma tertii
ordinis: deinde capias vnã ex duabꝰ octauis acq̇ſi
tis et remanētibꝰ in q̈rta parte ꝓportiõali: et deīde
ꝑ tertia parte illiꝰ ordinis capias vnã ex tribꝰ de-
cimiſſextis derilictis et acq̇ſitis in quīta parte pro
portiõali: et ſic ↄ̨ñter. Et ſic ad īuiendū quartum
ordinē incipias ab vna octaua derelicta et acq̇ſita
in quarta parte ꝓportionali. Et ad īuiendū quī
tum incipies ab vna ſexdecima derclicta et acq̇ſita
in quīta parte ꝓportionali: et ſic ↄ̨ñter īuenies infi
nitos ordines et iſti ordines ↄ̨tinuo ſe habēt in pro
portione dupla: ita q̄libet ſequēs ordo eſt ſubdu
plus ad īmediate p̄cedentē ordinē: igr̄ ibi ſunt īfini
ti ordines ↄ̨tinuo ſe habētes in ꝓportione dupla:
qḋ fuit ꝓbandū. Q, autē illi ordines ↄ̨tinuo ſe ha
bent in ꝓportione dupla. Ptꝫ / q2 quilibet illoꝝ or-
dinū cõponit̄̄ ex infinitis ↄ̨tinuo ſe habentibus in
ꝓportione dupla: et oīa prima oīm illoꝝ ordinum
ↄ̨tinuo ſe habent in ꝓportione dupla / vt cõſtat: igr̄
oēs illi ordines cõtinuo ſe habent in ꝓportiõe du-
pla. Ptꝫ ↄ̨ña: q2 cuiuſlibet ordinis primuꝫ eſt me-
dietas illius ordinis et reſiduum alia medietas
quia in quacū proportione ſe habēt medietates
aliquoꝝ in eadē proportione ſe habent et ipſa tota
quoꝝ ſunt medietates / vt ptꝫ ex vndecima ſuppoſi-
tione ſecūdi capitis ſecūde partis: ergo oēs illi or
dines cõtinuo ſe habent in ꝓportione dupla / quod
fuit ꝓbandū. 11Cõfir
matio. ¶ Et confirmat̄̄ / q2 ſi illa poſitio eſſet
vera ſequeret̄̄ / ſi aliquod corpus in prima parte
ꝓportionali alicuiꝰ hore augmētaret̄̄ aliquãtulū
velociter, et in ſecūda in duplo velocius et in tertia
in duplo velocius ꝙ̄ in ſecūda, et ſic ↄ̨ñter: tale cor-
pus in fine hore eſſet infinitū, et ſic illud corpꝰ in-
finite velociter augmētaretur. ſꝫ ↄ̨ſequens eſt falſū /
igr̄ illud ex quo ſequit̄̄. Sequela ꝓbatur / q2 ſi hora
ſit diuiſa per partes ꝓportionales ꝓportione du
pla, et aliquod corpꝰ in ṗma aliquãtulū velociter
augmentet̄̄ aliquã ꝓportionē acquirendo, et in ſe-
cunda in duplo velociꝰ: et in tertia in duplo velociꝰ
̄ in ſecūda: et ſic ↄ̨ñter. Tūc tale corpus in fine ac-
quiſiuit tnfinitã proportionē et nõ eſt maior ratio
quãdo diuidit̄̄ hora tali diuiſione quã aliqua alia
diuiſione: igr̄ ſi hora diuidat̄̄ aliqua diuiſiõe: et in
prima aliquod corpus aliqua velocitate augmen-
tetur: et in ſecūda in duplo velociꝰ: et in tertia in du
pio velociꝰ ꝙ̄ in ſecūda: et ſic ↄ̨ñter tale corpꝰ īfinitã
ꝓportionē acq̇ret: et augmētabit̄̄ īfinite velociṫ in
tali hora / qḋ fuit ꝓbãdū. Probr̄ tñ añs vcꝫ / qḋ ſi ho
ra diuidat̄̄ ꝓportiõe dupla etc̈. / qḋ illud corpꝰ acq̇rit
īfinitã ꝓportionē: q2 in q̈lꝫ ꝑte ꝓportiõali acquirit
tãtã ꝓportionē ſicut in ṗma. Nã in q̈cū ꝓportiõe
aliq̈ ꝑs eſt mīor in eadē ꝓportiõe velociꝰ augmētat̄̄
et ſūt īfinite: g̊ īfinitas eq̈les ꝓportiões acq̇rit / et per
ↄ̨ñs īfinitã ꝓportionē acq̇rit. Iã ꝓbo falſitatē ↄ̨ñtꝪ
et volo / hõ diuidit̄̄ ꝑ ꝑtes ꝓportiõales ꝓportiõe q̈
drupla, et in ṗma ꝑte augmētet̄̄ aliqḋ corpꝰ certe ve
lociṫ puta acq̇rēdo ꝓportionē duplã: et in ſcḋa ī du
plo velociꝰ et .3. ī duplo velociꝰ ꝙ̄ ī.2. / et ſic ↄ̨ñṫ. in po
ſitū eſt, q̊ poſito ar̄ ſic. illud corpꝰ auget̄̄ vt ponit̄̄ et
tñ nõ acquirit niſi ꝓportionē quadruplã tota illa ho
ra: igr̄ illud conſequens eſt vna conditiõalis falſa