Alvarus, Thomas
,
Liber de triplici motu
,
1509
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div
xml:id
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level
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3
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n
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3
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type
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chapter
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type-free
="
capitulum
">
<
pb
chead
="
Prime partis
"
file
="
0009
"
n
="
9
"/>
<
head
xml:id
="
N10833
"
xml:space
="
preserve
">Capitulū tertiū / in quo oſtenditur: et de
<
lb
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mõſtratur: proportionem irrationalem
<
lb
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eſſe ponendam.</
head
>
<
p
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N1083A
">
<
s
xml:id
="
N1083B
"
xml:space
="
preserve
">AD demonſtrandum inter a-
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liquas magnitudines ꝓportionē irra
<
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tionalem inueniri: que nullo pacto ſit
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ſicut numeri ad numerum.</
s
>
</
p
>
<
p
xml:id
="
N10844
">
<
s
xml:id
="
N10845
"
xml:space
="
preserve
">Suppono primo / proportio qua-
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dratorum ſuperficialium: eſt proportio coſtarum
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dublicata. </
s
>
<
s
xml:id
="
N1084C
"
xml:space
="
preserve
">Hoc eſt ſi inter coſtas duorum quadra
<
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torum ſuperficialium: ſit aliqua proportio maio-
<
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ris inequalitatis: inter quadrata erit proportio
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/>
dupla: ad illã: que eſt inter coſtas ſignatorū qua-
<
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/>
dratorū: vt ſi inter coſtas duorū quadratorū ine-
<
lb
/>
qualiū ſuperficialiū: fuerit proportio dupla: inter
<
lb
/>
quadrata erit proportio q̈drupla </
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>
<
s
xml:id
="
N1085B
"
xml:space
="
preserve
">Hec ſuppoſitio
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/>
clare ꝓbatur: et demõſtratur: inferiꝰ. in tertia ꝑte
<
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/>
tractatu ſecūdo: capitulo .2. </
s
>
<
s
xml:id
="
N10862
"
xml:space
="
preserve
">Uideas eã ibi.</
s
>
</
p
>
<
p
xml:id
="
N10865
">
<
s
xml:id
="
N10866
"
xml:space
="
preserve
">Secunda ſuppoſitio. </
s
>
<
s
xml:id
="
N10869
"
xml:space
="
preserve
">Quadratum
<
lb
/>
diametri: ſe hꝫ ad q̈dratū coſte in ꝓportiõe dupla
<
lb
/>
</
s
>
<
s
xml:id
="
N1086F
"
xml:space
="
preserve
">Hoc eſt q̈dratū cuiꝰ q̈libet coſta. </
s
>
<
s
xml:id
="
N10872
"
xml:space
="
preserve
">eſt eq̈lis diametro
<
lb
/>
alicuiꝰ q̈drati ſe hꝫ in ꝓportiõe dupla: ad illud q̈-
<
lb
/>
dratū. </
s
>
<
s
xml:id
="
N10879
"
xml:space
="
preserve
">Probat̄̄ hec ſuppoſitio: et ſit vnū q̈dratum
<
lb
/>
magnū: cuiꝰ latꝰ ſit .d.c. et diameṫ ſit a.c. ſit aliḋ
<
lb
/>
paruū cū iſto cõicans cuiꝰ coſta ſit .c.f. et diameter
<
lb
/>
ſit .d.c et diuidat̄̄ q̈dratū maiꝰ: ꝑ duos diametros
<
lb
/>
in quatuor triãgulos equales: vt ptꝫ in hac figura /
<
lb
/>
<
figure
xlink:href
="
fig-0009-01a
"
xlink:label
="
fig-0009-01
"
xml:id
="
N108CE
"
number
="
3
">
<
image
file
="
0009-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YHKVZ7B4/figures/0009-01
"/>
</
figure
>
quo poſito argr̄ ſic / ma-
<
lb
/>
gnū q̈dratū ē duplū
<
lb
/>
ad paruū q̈dratū et
<
lb
/>
ipſū magnū q̈dratū
<
lb
/>
eſt quadratū diame
<
lb
/>
tri ipſius parui qua
<
lb
/>
drati. </
s
>
<
s
xml:id
="
N10897
"
xml:space
="
preserve
">vt ptꝫ manife
<
lb
/>
ſte / igit̄̄ quadratū di
<
lb
/>
ametti: ſe hꝫ ad q̈dra
<
lb
/>
tū coſte: in ꝓportiõe
<
lb
/>
dupla. </
s
>
<
s
xml:id
="
N108A2
"
xml:space
="
preserve
">Cõſeq̄ntia ptꝫ
<
lb
/>
cū mīore. </
s
>
<
s
xml:id
="
N108A7
"
xml:space
="
preserve
">et argr̄ maior. </
s
>
<
s
xml:id
="
N108AA
"
xml:space
="
preserve
">q2 q̈dratū magnū: cõtinet
<
lb
/>
q̈termedietatē parui q̈drati. </
s
>
<
s
xml:id
="
N108AF
"
xml:space
="
preserve
">adeq̈te igr̄ ipſū ma-
<
lb
/>
gnū q̈dratū: cõtinet bis adeq̈te: paruū q̈dratū. </
s
>
<
s
xml:id
="
N108B4
"
xml:space
="
preserve
">Cõ
<
lb
/>
ſequentia ptꝫ ex ſe: et ꝓbat̄̄ añs. </
s
>
<
s
xml:id
="
N108B9
"
xml:space
="
preserve
">q2 q̈dratū magnū
<
lb
/>
q̈ter ↄ̨tinet tm̄: ſicut ē triãgulꝰ .d.e.c. / vt ptꝫ. </
s
>
<
s
xml:id
="
N108BE
"
xml:space
="
preserve
">et ille tri
<
lb
/>
angulꝰ eſt medietas parui quadrati: vt manifeſte
<
lb
/>
ptꝫ in figura. </
s
>
<
s
xml:id
="
N108C5
"
xml:space
="
preserve
">igit̄̄ magnū quadratū: quater conti-
<
lb
/>
net adequate: mediante parui / qḋ fuit ꝓbandum.</
s
>
</
p
>
<
p
xml:id
="
N108D4
">
<
s
xml:id
="
N108D5
"
xml:space
="
preserve
">Terita ſuppoſitio. </
s
>
<
s
xml:id
="
N108D8
"
xml:space
="
preserve
">diametri ad coſtã
<
lb
/>
eſt ꝓportio: que eſt medietas duple. </
s
>
<
s
xml:id
="
N108DD
"
xml:space
="
preserve
">Probatur / q2
<
lb
/>
quadrati diametri ad quadratū coſte eſt ꝓportio
<
lb
/>
dupla: vt ptꝫ ex ſcḋa ſuppoſitione. </
s
>
<
s
xml:id
="
N108E4
"
xml:space
="
preserve
">ergo diametri
<
lb
/>
ad coſtã: eſt ꝓportio ſubdupla ad duplã. </
s
>
<
s
xml:id
="
N108E9
"
xml:space
="
preserve
">et ꝑ conſe
<
lb
/>
quēs medietas duple. </
s
>
<
s
xml:id
="
N108EE
"
xml:space
="
preserve
">Patet cõſequētia ex prima
<
lb
/>
ſuppoſitione. </
s
>
<
s
xml:id
="
N108F3
"
xml:space
="
preserve
">Qm̄ ſemꝑ ꝓportio quadratorū: eſt
<
lb
/>
dupla ad ꝓportionē coſtaꝝ. </
s
>
<
s
xml:id
="
N108F8
"
xml:space
="
preserve
">Et ſic ꝓportio coſtaꝝ
<
lb
/>
eſt medietas ꝓportionis quadratoꝝ. </
s
>
<
s
xml:id
="
N108FD
"
xml:space
="
preserve
">Cum igitur
<
lb
/>
proportio quadratoruꝫ fuerit dupla: coſtaꝝ pro-
<
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/>
portio erit medietas duple.</
s
>
</
p
>
<
note
position
="
left
"
xml:id
="
N10904
"
xml:space
="
preserve
">Numeri
<
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primi.</
note
>
<
p
xml:id
="
N1090A
">
<
s
xml:id
="
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"
xml:space
="
preserve
">Quarta ſuppoſitio cuinſlibet ꝓpor
<
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tionis ſuprapartientis alter primorū numeroruꝫ
<
lb
/>
eſt impar. </
s
>
<
s
xml:id
="
N10912
"
xml:space
="
preserve
">Sunt autē primi numeri alicuius ꝓpor
<
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tionis: qui in ea ꝓportiõe ſunt numeri: vt tria et .2.
<
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/>
ſunt primi numeri ꝓportionis ſexquialtere: quia
<
lb
/>
in naturali ſerie numeroruꝫ: inter nullos minores
<
cb
chead
="
Capitulum tertiū.
"/>
ꝓportio ſexquialtera inuenit̄̄: </
s
>
<
s
xml:id
="
N1091E
"
xml:space
="
preserve
">Probatur ſuppoſi
<
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tio. </
s
>
<
s
xml:id
="
N10923
"
xml:space
="
preserve
">q2 ſi non: detur oppoſitū. </
s
>
<
s
xml:id
="
N10926
"
xml:space
="
preserve
">videlicet / vter ſit
<
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/>
numerus par. </
s
>
<
s
xml:id
="
N1092B
"
xml:space
="
preserve
">et arguitur ſic. </
s
>
<
s
xml:id
="
N1092E
"
xml:space
="
preserve
">vter iſtorꝝ eſt nume
<
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/>
rus par. </
s
>
<
s
xml:id
="
N10933
"
xml:space
="
preserve
">ergo ſequitur / vter illoꝝ eſt medietas /
<
lb
/>
vt patet ex diffinitione numeri paris: et proportio
<
lb
/>
medietatū: eſt eadē cū ꝓportione totoꝝ: vt conſtat
<
lb
/>
et inferius ꝓbabis: igitur illi non erant primi nu-
<
lb
/>
meri talis ꝓportiõis. </
s
>
<
s
xml:id
="
N1093E
"
xml:space
="
preserve
">q2 nõ erant minimi illiꝰ pro
<
lb
/>
portionis: cū ſue medietates ſint numeri minores
<
lb
/>
et ꝑ ↄ̨ñs: nõ dediſti ṗmos nūeros: talis ꝓpoſitiõis</
s
>
</
p
>
<
p
xml:id
="
N10945
">
<
s
xml:id
="
N10946
"
xml:space
="
preserve
">Quīta ſuppoſitio. </
s
>
<
s
xml:id
="
N10949
"
xml:space
="
preserve
">Omne quadratū
<
lb
/>
numeri īparis: eſt īpar. </
s
>
<
s
xml:id
="
N1094E
"
xml:space
="
preserve
">Probatur: q2 oē quadra-
<
lb
/>
tum numeri īparis: eſt ille numerꝰ: qui reſultat ex
<
lb
/>
ductu numeri īparis: in ſeipſum ſemel. </
s
>
<
s
xml:id
="
N10955
"
xml:space
="
preserve
">vt patet ex
<
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/>
ſcḋo arithmetice nichomachi. </
s
>
<
s
xml:id
="
N1095A
"
xml:space
="
preserve
">ſed oīs numerꝰ: re-
<
lb
/>
ſultãs ex ductu numeri īparis in ſeipſum: eſt īpar /
<
lb
/>
igitur oē quadratū numeri īparis: eſt īpar. </
s
>
<
s
xml:id
="
N10961
"
xml:space
="
preserve
">Pro-
<
lb
/>
batur minor: q2 ſi numerꝰ īpar: multiplicetur per
<
lb
/>
numeꝝ parē immediate precedentē ipſum vt .5. per
<
lb
/>
4. / tunc reſultaret numerꝰ par: ſed quãdo multipli
<
lb
/>
catur per ſeipſum: ſiue dicetur ī ſeipſum ſemel (qḋ
<
lb
/>
ideꝫ ē) adhuc illi nūero pari: qui reſultabat ex mul
<
lb
/>
tiplicatione numeri paris: immediate preceden-
<
lb
/>
tis: additur numerꝰ īpar: vt patet intelligenti. </
s
>
<
s
xml:id
="
N10972
"
xml:space
="
preserve
">igr̄
<
lb
/>
totū reſultans: erit nūerꝰ īpar. </
s
>
<
s
xml:id
="
N10977
"
xml:space
="
preserve
">Patet cõſequētia:
<
lb
/>
qm̄ ſi numerꝰ īpar: addatur numero pari: reſulta
<
lb
/>
bit numerꝰ īpar. </
s
>
<
s
xml:id
="
N1097E
"
xml:space
="
preserve
">Exemplū / vt ſi ternariꝰ: multipli
<
lb
/>
cetur per numeꝝ parem: īmediate precedentē: puta
<
lb
/>
binariū: reſultabit numerꝰ par: puta ſenariꝰ. </
s
>
<
s
xml:id
="
N10985
"
xml:space
="
preserve
">et ſi
<
lb
/>
vlteriꝰ addatur numerꝰ teruariꝰ: ſupra ſenariū re
<
lb
/>
ſultabit nouenarius: qui eſt numerꝰ īpar reſultãs
<
lb
/>
ex ductu ternarii in ſeipſum ſemel.</
s
>
</
p
>
<
p
xml:id
="
N1098E
">
<
s
xml:id
="
N1098F
"
xml:space
="
preserve
">Sexta ſuppoſitio. </
s
>
<
s
xml:id
="
N10992
"
xml:space
="
preserve
">nullus numerus
<
lb
/>
impar: eſt duplas ad aliquē numerū. </
s
>
<
s
xml:id
="
N10997
"
xml:space
="
preserve
">Probatur:
<
lb
/>
q2 ſi eſſet duplus ad aliquē numerū: iã ille numerꝰ
<
lb
/>
eſſet ſua medietas adequate: et ſic diuideret̄̄ in du-
<
lb
/>
as medietates: et ꝑ cõſequēs nõ eſſet impar.</
s
>
</
p
>
<
p
xml:id
="
N109A0
">
<
s
xml:id
="
N109A1
"
xml:space
="
preserve
">Hīs iactis ſuppoſitiõibus: ſit prima
<
lb
/>
cõcluſio. </
s
>
<
s
xml:id
="
N109A6
"
xml:space
="
preserve
">Nulla ꝓportio diametri ad coſtã: ē mĺti
<
lb
/>
plex, aut mĺtiplex ſuꝑparticularis: aut multiplex
<
lb
/>
ſuprapartiēs. </
s
>
<
s
xml:id
="
N109AD
"
xml:space
="
preserve
">Probat̄̄ hec cõcluſio: oīs ꝓportio
<
lb
/>
mĺtiplex, aut mĺtiplex ſuꝑparticĺaris, aut mĺti-
<
lb
/>
plex ſuprapartiēs eſt dupla aut maior dupla: ſed
<
lb
/>
nulla ꝓportio diametri ad coſtã: ē dupla aut ma-
<
lb
/>
ior dupla: igit̄̄ nulla ꝓportio diametri ad coſtam
<
lb
/>
eſt mĺtiplex: aut mĺtiplex ſuꝑparticularꝪ, aut mĺ-
<
lb
/>
tiplex ſuprapartiēs. </
s
>
<
s
xml:id
="
N109BC
"
xml:space
="
preserve
">Ptꝫ ↄ̨ña in ſcḋo ſcḋe et maior
<
lb
/>
ſimiliter: q2 oīs proportio multiplex: eſt dupla: vĺ
<
lb
/>
mior: et oīs ꝓportio multiplex ſuperparticularis
<
lb
/>
aut multiplex ſuprapartiens: eſt maior dupla: vt
<
lb
/>
patebit ex cſḋa parte: igitur oīs proportio multi
<
lb
/>
plex: aut multiplex ſuꝑparticularis: aut mĺtiplex
<
lb
/>
ſuprapartiens: eſt dupla: vel maior dupla. </
s
>
<
s
xml:id
="
N109CB
"
xml:space
="
preserve
">Iã ꝓ-
<
lb
/>
batur minor. </
s
>
<
s
xml:id
="
N109D0
"
xml:space
="
preserve
">q2 oīs proportio diametri ad coſtã:
<
lb
/>
eſt medietas duple: ſiue ſubdupla ad duplã (quod
<
lb
/>
idē eſt) adequate: ergo nulla proportio diametri
<
lb
/>
ad coſtã: eſt ipſa tota dupla: vel maior dupla </
s
>
<
s
xml:id
="
N109D9
"
xml:space
="
preserve
">Pa
<
lb
/>
tet antecedēs. </
s
>
<
s
xml:id
="
N109DE
"
xml:space
="
preserve
">ex tertia ſuppoſitione: et probat̄̄ cõ
<
lb
/>
ſequētia. </
s
>
<
s
xml:id
="
N109E3
"
xml:space
="
preserve
">q2 alias medietas eſſet equalis ſuo toti:
<
lb
/>
vel maior. </
s
>
<
s
xml:id
="
N109E8
"
xml:space
="
preserve
">quod nõ eſt poſibile: deductis ſophiſta
<
lb
/>
rum quiſquiliis.</
s
>
</
p
>
<
p
xml:id
="
N109ED
">
<
s
xml:id
="
N109EE
"
xml:space
="
preserve
">Secunda concluſio. </
s
>
<
s
xml:id
="
N109F1
"
xml:space
="
preserve
">nulla proportio
<
lb
/>
diametri ad coſtã: eſt aliqua proportio ſuꝑparti-
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