Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secunde partis
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file
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0040
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n
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40
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<
p
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N13D28
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<
s
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N13D29
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xml:space
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">Quarta ſuppoſitio ſiue petitio. </
s
>
<
s
xml:id
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N13D2C
"
xml:space
="
preserve
">Oē
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quod ſucceſſiue diminuitur vſ ad non gradū eſt
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latitudo diuiſibilis: et in duas medietates: et ī tres
<
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tertias, et in quatuor quartas, et ſic conſequenter
<
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</
s
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<
s
xml:id
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N13D36
"
xml:space
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preserve
">Diminuitur enim ad ſubduplum, ad ſubtriplum,
<
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ad ſubquadruplum: et ſic deinceps.</
s
>
</
p
>
<
p
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="
N13D3B
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<
s
xml:id
="
N13D3C
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xml:space
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preserve
">Quīta ſuppoſitio. </
s
>
<
s
xml:id
="
N13D3F
"
xml:space
="
preserve
">Latitudo propor-
<
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tionis maiores inequalitatis eſt ſucceſſiue dimi-
<
lb
/>
nuibilis vſ ad nõ gradum. </
s
>
<
s
xml:id
="
N13D46
"
xml:space
="
preserve
">Probatur tum pri-
<
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mo / quia maius extremum proportionis maioris
<
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īequalitatis ſucceſſiue valet diminui vſ ad equa
<
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litatē minoris extremi: et in tali diminutione pro-
<
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/>
portio maioris inequalitatis ſucceſſiue diminui-
<
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tur ad non gradum / vt conſtat: igitur in tali di-
<
lb
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minutione quelibet proportio minor illa ſignata
<
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dabitur. </
s
>
<
s
xml:id
="
N13D57
"
xml:space
="
preserve
">Tum ſecundo / q2 vt baſanus concedit ve
<
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locitas motus correſpondet magnitudini ꝓpor-
<
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tionis quo ad equalitatē: ſed ipſa velocitas mo-
<
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tus eſt diminuibilis continuo ſucceſſiue vſ ad nõ
<
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gradū: igitur et latitudo proportionis ſibi corre-
<
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ſpondens inequalitate. </
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>
<
s
xml:id
="
N13D64
"
xml:space
="
preserve
">¶ Ex hac ſequitur / que-
<
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libet latitudo proportionis maioris inequalita-
<
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tis diuidi poteſt in duas medietates, in tres ter-
<
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/>
tias, in quatuor quartas, et ſic deinceps. </
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>
<
s
xml:id
="
N13D6D
"
xml:space
="
preserve
">Patet
<
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hoc correlariū ex priore auxiliante quarta.</
s
>
</
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>
<
p
xml:id
="
N13D72
">
<
s
xml:id
="
N13D73
"
xml:space
="
preserve
">Sexta ſuppoſitio. </
s
>
<
s
xml:id
="
N13D76
"
xml:space
="
preserve
">Omne quod effi-
<
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citur ſubduplū ad id quod erat antea perdit me-
<
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/>
dietatem ſui: et id quod remanet eſt tantū quantuꝫ
<
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/>
eſt id quod perdidit / qm̄ perdidit aliã medietatem
<
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/>
et cuiuſlibet quanti medietates ſunt equales.</
s
>
</
p
>
<
p
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="
N13D81
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<
s
xml:id
="
N13D82
"
xml:space
="
preserve
">His ſuppoſitis aduertendū eſt / ba-
<
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ſanus volens defenſare quãlibet proportionalē ra-
<
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tionalē cuilibet alteri eſſe cõmenſurabilē aſtruxit
<
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/>
proportionū cõmenſurabilitatē ſiue ꝓportioneꝫ
<
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aſſumendã eſſe ex denominationū ꝓportionibus
<
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ponens talem concluſionē. </
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>
<
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">Proportionū propor
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tio eſt earū denominationū proportio: vt quadru
<
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pla eſt dupla ad duplã: q2 inter earum denomina
<
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/>
tiones ſiue numeros a quibus denominantur eſt
<
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/>
proportio dupla, a binario enim dupla: et a qua-
<
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/>
ternario quadrupla denomīatur. </
s
>
<
s
xml:id
="
N13D9C
"
xml:space
="
preserve
">Item dupla eſt
<
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/>
ſexquitertia ad ſexquialteram: q2 dupla a bina-
<
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rio ſexquialtera vero ab vnitate cū dimidio deno
<
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/>
minatur. </
s
>
<
s
xml:id
="
N13DA5
"
xml:space
="
preserve
">Conſtat autem binarii ad vnitatem cum
<
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dimidio proportionem ſexquitertiam eſſe.</
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>
</
p
>
<
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position
="
left
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xml:id
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N13DAA
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xml:space
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">Contra
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baſanū
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primo.</
note
>
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<
s
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xml:space
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">Sed contra hanc opinationem mea
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ſententia mathemathicis principiis derogantē et
<
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/>
contrariã: arguitur primo ſic. </
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>
<
s
xml:id
="
N13DBA
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xml:space
="
preserve
">Ex hac opinione ſe
<
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quitur octuplam eſſe duplaꝫ ad quadruplam: ſed
<
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conſequens eſt manifeſte falſū: igitur illud ex quo
<
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ſequitur. </
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>
<
s
xml:id
="
N13DC3
"
xml:space
="
preserve
">Sequela probatur / q2 illarū proportio-
<
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onū octuple videlicet et quadruple denominatio-
<
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/>
nes ſiue numeros a quibus denominãtur, duple
<
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/>
ꝓportiõis rationē habere conſtat .8. em̄ ad .4. du
<
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/>
pla ꝓportio eſt: igitur expoſitiõe octupla dupla
<
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eſt ad quadruplã. </
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>
<
s
xml:id
="
N13DD0
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xml:space
="
preserve
">Iã falſitatē cõſequentis oſten-
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damus ſuꝑeſt: qm̄ ſi octupla eſt dupla ad quadru-
<
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/>
plã: ſequitur / quadrupla eſt medietas ipſiꝰ octu
<
lb
/>
ple: vt ptꝫ ex prima ſuppoſitione: ſed cõſequēs eſt
<
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/>
falſum: igitur illud ex quo ſequitur: q2 tūc ſeque-
<
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/>
retur / octupla cõtineret quadruplã bis adequa
<
lb
/>
te: ſed hoc eſt falſuꝫ / q2 cõtinet quadruplã et duplã
<
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/>
adequate / vt ptꝫ in his terminis .8. ad .4. et .4. ad .1.
<
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/>
</
s
>
<
s
xml:id
="
N13DE2
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xml:space
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preserve
">Patet hec conſequentia ex ſecunda parte eiuſdē
<
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ſuppoſitionis.
<
note
position
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left
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xlink:href
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note-0040-01a
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note-0040-01
"
xml:id
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xml:space
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preserve
">Cõfirma
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tio ṗma.</
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>
</
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>
<
s
xml:id
="
N13DEC
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xml:space
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preserve
">¶ Et confirmatur / q2 omne dupluꝫ
<
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ad aliquod continet ipſum vel equale ei bis tantū
<
cb
chead
="
Capitulū quintū.
"/>
ſed octupla eſt dupla ad quadruplã per te / igitur
<
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/>
continet ipſum bis tantū: ſed cõſequens eſt falſuꝫ /
<
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/>
q2 ſexdecupla cõtinet quadruplã bis tantū. </
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>
<
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xml:id
="
N13DF8
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xml:space
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">Cõſe-
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quentia ptꝫ ex ſe: et minor eſt prima pars ſecunde
<
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ſuppoſitionis.
<
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position
="
right
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xlink:href
="
note-0040-02a
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="
note-0040-02
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xml:id
="
N13E75
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xml:space
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">Cõfirma
<
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/>
tio ſcḋa</
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>
</
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>
<
s
xml:id
="
N13E04
"
xml:space
="
preserve
">¶ Confirmatur ſecundo / q2 ſi poſi-
<
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tio eſſet vera / ſequeretur / dupla eſſet medietas
<
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/>
octuple: ſed hoc eſt falſum: igitur illud ex quo ſe-
<
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/>
quitur: q2 ſecundū iſtã opinionē octupla eſt qua-
<
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/>
drupla ad duplã / vt ptꝫ ex ꝓportione denoīationū
<
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/>
duple et octuple: et ſi octupla eſt quadrupla ad du
<
lb
/>
plã iam ſequitur / ipſa dupla eſt quarta octuple
<
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et nõ medietas. </
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>
<
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="
N13E15
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xml:space
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">Quodlibet em̄ eſt quadruplū ad
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ſui quartã: cum eã contineat quater adequate. </
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>
<
s
xml:id
="
N13E1A
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xml:space
="
preserve
">Iã
<
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probatur ſequela: et capio ꝓportionē octuplam: et
<
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volo / diminuatur quouſ fiat quadrupla ade-
<
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/>
quate: vt poſito / octo diminuãtur vſ ad quatu
<
lb
/>
or: et arguitur ſic: ipſa proportio octupla efficitur
<
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in duplo minor vĺ cõcedit poſitio. </
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>
<
s
xml:id
="
N13E27
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xml:space
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">Efficitur enim
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q̈drupla que eſt ſubdupla ad octuplã: igitur ipſa
<
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proportio octupla perdit adequate medietatem
<
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/>
ſui / vt ptꝫ ex tertia ſuppoſitione: et non perdit niſi
<
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/>
duplã adequate / vt conſtat / igitur dupla eſt medie
<
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/>
tas octuple / qḋ fuit inferendū.
<
note
position
="
right
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xlink:href
="
note-0040-03a
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xlink:label
="
note-0040-03
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xml:id
="
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xml:space
="
preserve
">3. confir-
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matio</
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>
</
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<
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xml:space
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">¶ Et ↄ̨firmat̄̄ tertio /
<
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q2 ſi iſta poſitio eſſet vera / ſeq̄ret̄̄ / dupla eſſet eq̈-
<
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lis quadruple. </
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>
<
s
xml:id
="
N13E40
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xml:space
="
preserve
">Cõſequēs eſt falſum et cõtra opi-
<
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nantem / igitur illud ex quo ſequitur. </
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>
<
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xml:id
="
N13E45
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xml:space
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">Sequela ar-
<
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guitur / et volo / potentia vt octo moueat reſiſten
<
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/>
tiam vt vnum velocitate vt quatuor exempli gra-
<
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/>
tia / deinde volo / potētia ſtante reſiſtentia: dimi-
<
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/>
nuatur vſ ad ſubduplū: et arguo ſic / ille motꝰ ſiue
<
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/>
velocitas vt quatuor diminuetur ad ſubduplum:
<
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/>
igitur perdit medietatē ſui. </
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>
<
s
xml:id
="
N13E54
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xml:space
="
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">Patet cõſequentia ex
<
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ſuppoſitione tertia: et per cõſequens nõ manebit
<
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/>
niſi velocitas vt duo: et deperdet̄̄ velocitas vt duo /
<
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igitur tanta ꝓportio deperdita eſt quanta manet
<
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</
s
>
<
s
xml:id
="
N13E5E
"
xml:space
="
preserve
">Patet hec cõſequētia / q2 ab equalibꝰ ꝓportioni-
<
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/>
bus equales latitudines motuū ꝓueniūt: ſed ma-
<
lb
/>
net quadrupla / ergo deperdita eſt ei equalis: ſed
<
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/>
deperdita eſt dūtaxat ꝓportio dupla: ergo dupla
<
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eſt equalis quadruple: quod fuit inferendum.</
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>
</
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<
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xml:id
="
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<
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xml:id
="
N13E86
"
xml:space
="
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">Secundo arguitur ſic / ſi illa poſitio
<
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eſſet vera / ſequeretur / quarta alicuiꝰ et ſua medie
<
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/>
tas eſſent equales ſed cõſequens eſt falſum: igitur
<
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/>
illud ex quo ſequitur. </
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>
<
s
xml:id
="
N13E8F
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xml:space
="
preserve
">Sequela ꝓbatur / q2 dupla
<
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eſt quarta pars octuple et medietas octuple ꝑ po-
<
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ſitionē: igitur propoſitū. </
s
>
<
s
xml:id
="
N13E96
"
xml:space
="
preserve
">Maior probatur / q2 du
<
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pla eſt quarta pars ipſius octuple cū octuple ad
<
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/>
duplam ſit proportio quadrupla / vt patet ex po-
<
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ſitiõe. </
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>
<
s
xml:id
="
N13E9F
"
xml:space
="
preserve
">Minor probatur: et volo / octupla perdat
<
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ꝓportionē duplã adequate: et manifeſtū eſt / effi-
<
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citur quadrupla: et per cõſequens ſubdupla ad id
<
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/>
quod erat antea / vt patet ex poſitione: igitur ꝑdit
<
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/>
medietatē ſui. </
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>
<
s
xml:id
="
N13EAA
"
xml:space
="
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">Patet cõſequentia ex tertia et ſexta
<
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ſuppoſitionibus: et non perdit niſi duplam: ergo
<
lb
/>
dupla eſt medietas octuple / quod fuit probandū.
<
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/>
<
note
position
="
right
"
xlink:href
="
note-0040-04a
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xlink:label
="
note-0040-04
"
xml:id
="
N13EDD
"
xml:space
="
preserve
">Cõfirma
<
lb
/>
tio ṗma.</
note
>
</
s
>
<
s
xml:id
="
N13EB8
"
xml:space
="
preserve
">¶ Et confirmatur / quia ſi poſitio eſſet vera / ſeque-
<
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/>
retur / aliquid contineret alterum bis adequate
<
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/>
et tamen non eſſet duplum ad illud: ſed minꝰ quaꝫ
<
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/>
duplum: conſequens eſt manifeſte falſuꝫ et contra
<
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/>
diffinitionem proportionis duple: igitur. </
s
>
<
s
xml:id
="
N13EC3
"
xml:space
="
preserve
">Seque
<
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/>
la probatur: quia proportio dupla ſexquiquar-
<
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/>
ta bis adequate continet ſexquialteram: patet
<
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/>
in his terminis .9.6.4. </
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>
<
s
xml:id
="
N13ECC
"
xml:space
="
preserve
">Nouem enim ad quatuor
<
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eſt proportio dupla ſexquiquarta: et componitur
<
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/>
adequate ex proportione .9. ad .6. et .6. ad .4. qua
<
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/>
rum vtra eſt ſexquialtera: et tamen ipſa propor
<
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/>
tio dupla ſexquiquarta eſt minor quam dupla
<
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/>
ad ſexquialteram: igitur propoſitum.</
s
>
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>
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