Alvarus, Thomas
,
Liber de triplici motu
,
1509
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211 - 240
241 - 270
271 - 290
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Tertii tractatus
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194
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denſitas in partes ꝓportionales / vt ponit̄̄ in pre-
<
lb
/>
cedēti correlario: ita prima ſit aliqualiter dēſa
<
lb
/>
ſcḋa in duplo, tertia in triplo, et ſic ↄ̨ſequēter: tunc
<
lb
/>
totum eſt in duplo denſius ſua prima parte ꝓpor-
<
lb
/>
tionali. </
s
>
<
s
xml:id
="
N22FF4
"
xml:space
="
preserve
">Probat̄̄ / q2 totū diuiſum per partes ꝓpor
<
lb
/>
tionales ꝓportiõe dupla eſt duplum ad primã par
<
lb
/>
tē ꝓportionalē eius / vt ptꝫ ex quinto capite prealle
<
lb
/>
gato prime partis huius libri: igitur ꝑ cõcluſionē
<
lb
/>
primã immediate p̄cedentē illud eſt denſius prima
<
lb
/>
parte ꝓportionali in ꝓportione dupla.
<
note
position
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left
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xlink:href
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note-0194-01a
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xlink:label
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note-0194-01
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xml:id
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N230F5
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xml:space
="
preserve
">3. correĺ.</
note
>
</
s
>
<
s
xml:id
="
N23006
"
xml:space
="
preserve
">¶ Sequit̄̄
<
lb
/>
tertio / diuiſo corpore ſi ꝑ partes ꝓportionales
<
lb
/>
ꝓportione dupla / vt ponit̄̄ in antecedēti correlario
<
lb
/>
totum eſt ita denſum ſicut ſcḋa pars proportiona
<
lb
/>
lis eius. </
s
>
<
s
xml:id
="
N23011
"
xml:space
="
preserve
">Probat̄̄ / q2 in duplo denſius prima vt ſe-
<
lb
/>
cundū correlarium aſſerit: et ſcḋa pars ꝓportiona
<
lb
/>
lis eſt etiã in duplo denſior prima: g̊ totū eſt ita dē
<
lb
/>
ſum ſicut ſecūda pars ꝓportionalis / quod fuit pro
<
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/>
bandū. </
s
>
<
s
xml:id
="
N2301C
"
xml:space
="
preserve
">Patet conſequētia ꝑ hanc maximã </
s
>
<
s
xml:id
="
N2301F
"
xml:space
="
preserve
">Oīa ha-
<
lb
/>
bentia equalē ꝓportionē ad vnū tertiū ſunt equa-
<
lb
/>
lia: ſꝫ totius denſitas et denſitas ſecūde partis ꝓ-
<
lb
/>
portionalis habent equalem proportionē ad den-
<
lb
/>
ſitatē prime partis ꝓportiõis puta duplã: igit̄̄ dē-
<
lb
/>
ſitas totius et ſecūde partis ꝓportionalis ſūt equa
<
lb
/>
les / quod erat inducendū.
<
note
position
="
left
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xlink:href
="
note-0194-02a
"
xlink:label
="
note-0194-02
"
xml:id
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N230FB
"
xml:space
="
preserve
">4. correĺ.</
note
>
</
s
>
<
s
xml:id
="
N23033
"
xml:space
="
preserve
">¶ Sequit̄̄ quarto / ſi ali
<
lb
/>
quod corpus diuidat̄̄ ꝑ partes ꝓportiõales ꝓpor-
<
lb
/>
portione ſexquialtera: et prīa pars ꝓportionalis
<
lb
/>
ſit aliqualiter denſa: et ſecūda ī duplo: et tertia ī tri
<
lb
/>
plo ꝙ̄ prima: et ſic cõſequēter vt ponitur in caſu pri
<
lb
/>
me cõcluſionis et correlarii: totū eſt in triplo dēſius
<
lb
/>
prima parte ꝓportionali. </
s
>
<
s
xml:id
="
N23042
"
xml:space
="
preserve
">Et ſi diuidatur ꝓportio
<
lb
/>
ne ſexquitertia: totū erit denſius prima parte pro-
<
lb
/>
portionali in quadruplo. </
s
>
<
s
xml:id
="
N23049
"
xml:space
="
preserve
">Et ſi in ſexquiquarta: to
<
lb
/>
tum erit denſius prima parte ꝓportionali in ꝓpor
<
lb
/>
tione quītupla. </
s
>
<
s
xml:id
="
N23050
"
xml:space
="
preserve
">et ſic ↄ̨ſequēter ꝓcedēdo ꝑ ſpecies
<
lb
/>
ꝓportionis ſuperparticularis in diuiſione corpo
<
lb
/>
is: et per ſpecies proportionis multiplicis ex parte
<
lb
/>
denſitatis. </
s
>
<
s
xml:id
="
N23059
"
xml:space
="
preserve
">Probatur hoc corolariuꝫ / quia totum
<
lb
/>
diuiſum ꝑ partes porportionales proportione ſex
<
lb
/>
quialtera eſt tripluꝫ ad primã ꝑtē eiꝰ ꝓportionalē
<
lb
/>
et ſexquitertia quadruplū: et ſexquiquarta quītu-
<
lb
/>
plum. </
s
>
<
s
xml:id
="
N23064
"
xml:space
="
preserve
">vt pꝫ ex prima parte huiꝰ operis: g̊ in eiſdem
<
lb
/>
ꝓportionibus ſe habēt denſitates totius ad denſi
<
lb
/>
tatem prime partis ꝓportionalis. </
s
>
<
s
xml:id
="
N2306B
"
xml:space
="
preserve
">igit̄̄ correlariuꝫ
<
lb
/>
verum.
<
note
position
="
left
"
xlink:href
="
note-0194-03a
"
xlink:label
="
note-0194-03
"
xml:id
="
N23101
"
xml:space
="
preserve
">5. correĺ.</
note
>
</
s
>
<
s
xml:id
="
N23075
"
xml:space
="
preserve
">¶ Sequitur quito / ſi diuidatur corpus vt
<
lb
/>
dicitur in p̄cedenti correlario vt puta ꝓportiõe ſex
<
lb
/>
quialtera: et prima pars ſit aliqualiter denſa: et ſe
<
lb
/>
cunda in duplo: et tertia in triplo .etc̈. totum eſt ita
<
lb
/>
denſum ſicut tertia pars ꝓportionalis eius. </
s
>
<
s
xml:id
="
N23080
"
xml:space
="
preserve
">Et ſi
<
lb
/>
ſexq̇tertia ſicut quarta pars ꝓportiõalis eiꝰ. </
s
>
<
s
xml:id
="
N23085
"
xml:space
="
preserve
">Et ſi
<
lb
/>
ſexquiquarta ſicut quīta pars ꝓportionalis eius.
<
lb
/>
</
s
>
<
s
xml:id
="
N2308B
"
xml:space
="
preserve
">Et ſexquiquinta: ſicut ſexta pars ꝓportionalis eiꝰ /
<
lb
/>
et ſic cõſequēter aſcendēdo ꝑ partes ꝓportionales
<
lb
/>
et per ſpecies ꝓportiõis ſuꝑparticularis in infini
<
lb
/>
tum. </
s
>
<
s
xml:id
="
N23094
"
xml:space
="
preserve
">Probat̄̄ / qm̄ ſi corpus ſit diuiſum ꝓportione
<
lb
/>
ſexquialtera ipſum eſt in triplo denſius prīa par-
<
lb
/>
te ꝓportionali / vt ptꝫ ex precedenti correlario et ter
<
lb
/>
tia pars ꝓportionalis eſt etiã in triplo denſior pri
<
lb
/>
ma / vt ptꝫ ex caſu. </
s
>
<
s
xml:id
="
N2309F
"
xml:space
="
preserve
">g̊ eſt ita denſum tale corpus ſicut
<
lb
/>
tertia pars ꝓportionalis. </
s
>
<
s
xml:id
="
N230A4
"
xml:space
="
preserve
">Itē ſi diuidatur ꝓportio
<
lb
/>
ne ſexquitertia ip̄m eſt in quadruplo denſius ṗma
<
lb
/>
eius parte ꝓportionali / vt pꝫ ex p̄cedenti correlario
<
lb
/>
et etiã quarta pars ꝓportionalis eiꝰ eſt in quadru
<
lb
/>
plo denſior ṗma / vt pꝫ ex caſu. </
s
>
<
s
xml:id
="
N230AF
"
xml:space
="
preserve
">igit̄̄ illud corpus ita
<
lb
/>
diuiſum ꝑ partes ꝓportionales ꝓportione ſexqui
<
lb
/>
tertia eſt ita denſum ſicut quarta pars proportio-
<
lb
/>
nalis eius. </
s
>
<
s
xml:id
="
N230B8
"
xml:space
="
preserve
">Et iſto mõ probabis ceteras ꝑticulas
<
lb
/>
correlarii.
<
note
position
="
left
"
xlink:href
="
note-0194-04a
"
xlink:label
="
note-0194-04
"
xml:id
="
N23107
"
xml:space
="
preserve
">6. correĺ.</
note
>
</
s
>
<
s
xml:id
="
N230C2
"
xml:space
="
preserve
">¶ Sequit̄̄ ſexto / ſi aliquod corpꝰ diui
<
lb
/>
datur ꝑ partes ꝓportionales proportiõe ſuperbi
<
lb
/>
patiente tertias
<
gap
/>
et partes eius ſint ita denſe / vt ſe-
<
cb
chead
="
Capitulū primum.
"/>
pius dictum eſt in p̄cedētibus correlariis: totū erit
<
lb
/>
denſius ṗma parte ꝓportionali in ꝓportione du-
<
lb
/>
pla ſexquialtera: ita ſi ṗma eſt denſa vt .2. totuꝫ
<
lb
/>
erit denſum vt .5. </
s
>
<
s
xml:id
="
N230D4
"
xml:space
="
preserve
">Probat̄̄ correlariū / qm̄ totū erit
<
lb
/>
denſius ṗma parte proportionali in tali caſu in ꝓ
<
lb
/>
portione qua ſe habet totū diuiſum ꝑ partes ꝓpor
<
lb
/>
tionales ꝓportione ſuperbipartiēte tertias ad ſu
<
lb
/>
am primã partē proportionalē / vt ptꝫ ex cõcluſione
<
lb
/>
ſed talis eſt ꝓportio dupla ſexquialtera / vt patꝫ ex
<
lb
/>
capĺo q̇nto prime partis huius operis: igit̄̄ corre-
<
lb
/>
larium verum.</
s
>
</
p
>
<
p
xml:id
="
N2310D
">
<
s
xml:id
="
N2310E
"
xml:space
="
preserve
">Secūda cõcluſio </
s
>
<
s
xml:id
="
N23111
"
xml:space
="
preserve
">Diuiſo corpore per
<
lb
/>
ꝑtes ꝓportiõales quauis ꝓportiõe et ī quacū pro
<
lb
/>
portiõe ſe habuerīt ꝑtes ꝓportionales ī eadē vĺ ma
<
lb
/>
iori ſe habuerit dēſitas mīoris ad dēſitatē maiorꝪ
<
lb
/>
totū illud corpꝰ eſt īfinite dēſum. </
s
>
<
s
xml:id
="
N2311C
"
xml:space
="
preserve
">patet hec cõcluſio
<
lb
/>
ex ꝓbatione ſexte cõcluſionis octaui capitis ſecūdi
<
lb
/>
tractatus huius partis.
<
note
position
="
right
"
xlink:href
="
note-0194-05a
"
xlink:label
="
note-0194-05
"
xml:id
="
N2314D
"
xml:space
="
preserve
">1. correĺ.</
note
>
</
s
>
<
s
xml:id
="
N23128
"
xml:space
="
preserve
">¶ Ex hac cõcluſione ſeq̇tur
<
lb
/>
primo / partito aliquo corꝑe proportiõe ſexquial
<
lb
/>
tera et prima pars ſit aliqualiter denſa: et ſecunda
<
lb
/>
in duplo et tertia ī duplo ꝙ̄ ſecūda: et quarta ꝙ̄ ter
<
lb
/>
tia: totum eſt infinite denſum. </
s
>
<
s
xml:id
="
N23133
"
xml:space
="
preserve
">¶ Sequit̄̄ ſecundo /
<
lb
/>
diuiſo corpore per partes proportionales ꝓpor-
<
lb
/>
tione ſexquitertia et ṗma ſit aliqualiter denſa et ſe
<
lb
/>
cunda in ſexquialtero plus et tertia in ſexquialtero
<
lb
/>
quã ſecunda / et ſic conſequēter: totum corpus eſt in
<
lb
/>
finite denſum </
s
>
<
s
xml:id
="
N23140
"
xml:space
="
preserve
">Hec correlaria ex ſecunda cõcluſione
<
lb
/>
patent: qm̄ in vtro illoꝝ proportio denſitatū cõ
<
lb
/>
tinuo eſt maior ꝓportione partiū / ergo ſubiecta il-
<
lb
/>
la ſunt infinite denſa.</
s
>
</
p
>
<
p
xml:id
="
N23153
">
<
s
xml:id
="
N23154
"
xml:space
="
preserve
">Tertia
<
reg
norm
="
concluſio
"
type
="
context
">cõcluſio</
reg
>
</
s
>
<
s
xml:id
="
N23157
"
xml:space
="
preserve
">Diuiſo aliquo corpo
<
lb
/>
re per partes
<
reg
norm
="
proportionales
"
type
="
simple
">ꝓportionales</
reg
>
quauis proportiõe et
<
lb
/>
in certa proportiõe quelibet pars
<
reg
norm
="
praecedens
"
type
="
simple context
">p̄cedēs</
reg
>
ſit
<
reg
norm
="
denſior
"
type
="
context
">dēſior</
reg
>
<
lb
/>
immediate
<
reg
norm
="
ſequenti
"
type
="
context
">ſequēti</
reg
>
: totius denſitatís ad
<
reg
norm
="
denſitatet
"
type
="
simple
">denſitatꝫ</
reg
>
<
lb
/>
ſiue denoīationē qua
<
reg
norm
="
totum
"
type
="
context
">totū</
reg
>
<
reg
norm
="
denominabitur
"
type
="
simple
">denominabit̄̄</
reg
>
a denſita
<
lb
/>
te prime partis
<
reg
norm
="
proportionalis
"
type
="
simple
">ꝓportionalis</
reg
>
eſt illa
<
reg
norm
="
proportio
"
type
="
simple
">ꝓportio</
reg
>
qua
<
lb
/>
ſe habet totum diuiſum in proportione ↄ̨poſita ex
<
lb
/>
proportione partis
<
reg
norm
="
proportionalis
"
type
="
simple
">ꝓportionalis</
reg
>
<
reg
norm
="
precedententis
"
type
="
context
">precedentētis</
reg
>
ad
<
reg
norm
="
im mediate
"
type
="
context
">ī
<
lb
/>
mediate</
reg
>
<
reg
norm
="
ſequentem
"
type
="
context
">ſequētem</
reg
>
: et denſitatis
<
reg
norm
="
praecedentis
"
type
="
simple context
">p̄cedētis</
reg
>
ad denſi-
<
lb
/>
tatem immediate
<
reg
norm
="
ſequentis
"
type
="
context
">ſequētis</
reg
>
ad
<
reg
norm
="
primam
"
type
="
simple context
">ṗmã</
reg
>
eius
<
reg
norm
="
partem
"
type
="
context
">partē</
reg
>
<
reg
norm
="
pro- portionalem
"
type
="
context
">pro-
<
lb
/>
portionalē</
reg
>
. </
s
>
<
s
xml:id
="
N2316E
"
xml:space
="
preserve
">Patet hec et cluſio
<
reg
norm
="
cum
"
type
="
context
">cū</
reg
>
multis
<
reg
norm
="
ſimilibus
"
type
="
simple
">ſimilibꝰ</
reg
>
ex
<
lb
/>
probatione octaue
<
reg
norm
="
concluſionis
"
type
="
context
">cõcluſionis</
reg
>
tertii capitis
<
reg
norm
="
ſecundi
"
type
="
context
">ſecūdi</
reg
>
<
lb
/>
tractatus huius tertie partis videas ibi.</
s
>
</
p
>
<
p
xml:id
="
N23175
">
<
s
xml:id
="
N23176
"
xml:space
="
preserve
">Quarta
<
reg
norm
="
concluſio
"
type
="
context
">cõcluſio</
reg
>
</
s
>
<
s
xml:id
="
N23179
"
xml:space
="
preserve
">Diuiſo corpore per
<
lb
/>
partes
<
reg
norm
="
proportionales
"
type
="
simple
">ꝓportionales</
reg
>
aliqua proportiõe multipli-
<
lb
/>
ci: et in prima parte proportionali ſit aliquantula
<
lb
/>
denſitas. </
s
>
<
s
xml:id
="
N23182
"
xml:space
="
preserve
">et in ſecunda in ſexquialtero maior et in
<
lb
/>
tertia in ſexquitertia maior denſitas
<
reg
norm
="
quam
"
type
="
context
">quã</
reg
>
in
<
reg
norm
="
prima
"
type
="
simple
">ṗma</
reg
>
/
<
lb
/>
et ſic ↄ̨ſequenter
<
reg
norm
="
procedendo
"
type
="
context
">procedēdo</
reg
>
per ſpecies
<
reg
norm
="
proportionis
"
type
="
simple context
">ꝓportiõis</
reg
>
<
lb
/>
ſuperparticularis: totius corporis denſitas
<
reg
norm
="
cenſen da
"
type
="
context
">cēſen
<
lb
/>
da</
reg
>
eſt
<
reg
norm
="
incommenſurabilis
"
type
="
context
">incõmenſurabilis</
reg
>
proportione rationali
<
reg
norm
="
den ſitati
"
type
="
context
">dē
<
lb
/>
ſitati</
reg
>
prime partis proportionalis et denoīationi
<
lb
/>
qua
<
reg
norm
="
ipſa
"
type
="
wordlist
">ip̄a</
reg
>
denſitas exiſtens in
<
reg
norm
="
prima
"
type
="
simple
">ṗma</
reg
>
parte proportio
<
lb
/>
nali totum denominat. </
s
>
<
s
xml:id
="
N23193
"
xml:space
="
preserve
">vel
<
reg
norm
="
ſaltem
"
type
="
context
">ſaltē</
reg
>
ſi
<
reg
norm
="
commenſurabilis
"
type
="
context
">cõmenſurabilis</
reg
>
<
lb
/>
eſt pro ſtatu iſto a nobis
<
reg
norm
="
capacitatem
"
type
="
context
">capacitatē</
reg
>
<
reg
norm
="
finitam
"
type
="
context
">finitã</
reg
>
habenti-
<
lb
/>
bus nequā
<
reg
norm
="
commenſurari
"
type
="
context context
">cõmēſurari</
reg
>
poteſt. </
s
>
<
s
xml:id
="
N2319A
"
xml:space
="
preserve
">Probatur /
<
reg
norm
="
quam
"
type
="
wordlist
">qm̄</
reg
>
ille
<
lb
/>
denſitates
<
reg
norm
="
continuo
"
type
="
context
">cõtinuo</
reg
>
ſe habent in alia et alia
<
reg
norm
="
propor- tione
"
type
="
simple
">ꝓpor-
<
lb
/>
tione</
reg
>
: et
<
reg
norm
="
non
"
type
="
wordlist
">nõ</
reg
>
eſt poſſibile omnes tales proportiones
<
lb
/>
<
reg
norm
="
commenſurari
"
type
="
context
">cõmenſurari</
reg
>
ab intellectu finito cum ſint infinite: et
<
lb
/>
<
reg
norm
="
continuo
"
type
="
context
">cõtinuo</
reg
>
alie et alie: igitur concluſio propoſita vera
<
lb
/>
</
s
>
<
s
xml:id
="
N231A6
"
xml:space
="
preserve
">Non
<
reg
norm
="
tamem
"
type
="
context
">tamē</
reg
>
puto hanc
<
reg
norm
="
concluſionem
"
type
="
context context
">cõcluſionē</
reg
>
demonſtraſſe aut
<
lb
/>
ſufficienter
<
reg
norm
="
oſtendiſſe
"
type
="
context
">oſtēdiſſe</
reg
>
:
<
reg
norm
="
sed
"
type
="
wordlist
">ſꝫ</
reg
>
eam
<
reg
norm
="
probabiliter
"
type
="
simple
">ꝓbabiliter</
reg
>
pono.
<
note
position
="
right
"
xlink:href
="
note-0194-06a
"
xlink:label
="
note-0194-06
"
xml:id
="
N231E4
"
xml:space
="
preserve
">1.
<
reg
norm
="
corollarium
"
type
="
wordlist
">correĺ.</
reg
>
</
note
>
</
s
>
<
s
xml:id
="
N231B0
"
xml:space
="
preserve
">¶ Ex
<
lb
/>
hac concluſione
<
reg
norm
="
ſequitur
"
type
="
simple
">ſequit̄̄</
reg
>
primo / ſi aliquod corpus
<
lb
/>
diuidatur
<
reg
norm
="
per
"
type
="
wordlist
">ꝑ</
reg
>
partes proportionales proportione
<
lb
/>
dupla: et prima ſit aliqualiter denſa: et ſecunda in
<
lb
/>
ſexquitertio pluſ̄ prima et tertia in ſexquiquinta
<
lb
/>
pluſ̄ prima et
<
reg
norm
="
quarta
"
type
="
wordlist
">q̈rta</
reg
>
in ſexquiſeptimo pluſ̄
<
reg
norm
="
prima
"
type
="
simple
">ṗma</
reg
>
/ </
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>