Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secunde partis
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file
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0049
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49
"/>
tur aliquis numeris cū fractione vel ſine habens
<
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ſe in eadem proportione ad illud maius extremū:
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vt patet ex tertia ſuppoſitione: et tūc illius nume-
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ri ad minimū numerū erit ꝓportio dupla ad illaꝫ
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ſuperparticularē: q2 ibi erūt tres termini cõtinuo
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ꝓportionabiles .etc̈. </
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<
s
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N14C49
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xml:space
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preserve
">Et iſto modo poteris cõſttue-
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re .5. terminos .6.7. continuo ꝓportionabiles: illa
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ꝓportione ſuperparticulari data: et ſic in infinitū /
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igit̄̄ dabitur ad eam quadrupla, quītupla, ſextu-
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pla rationalis: et ſic in infinitū. </
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<
s
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N14C54
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xml:space
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preserve
">Et eodē modo pro
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babis de quocū genere ꝓportionū rationaliuꝫ
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</
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<
s
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N14C5A
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xml:space
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">Et ſic patet concluſio.</
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<
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<
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N14C61
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xml:space
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">Quãuis quelibet
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ꝓportio rationalis in qualibet ꝓportione multi-
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plici ab aliqua ꝓportione ratiõali excedatur: ita
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quelibet ꝓportio rationalis habeat duplã, tri-
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plam, quadruplã, rationales / et ſic in infinitū: ni-
<
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chilominus nõ quelibet ꝓportio ratiõalis habet
<
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ſubduplã, ſubtriplã, ſubquadruplã, rationales.
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etc̈. </
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<
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xml:space
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">Prima pars huiꝰ concluſionis patet ex priori
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concluſione: et ſecunda ꝓbatur: quia ꝓportio du-
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pla non habet ſubduplã rationalē, nec ſubtriplã,
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nec ſubquadruplã .etc̈. / vt patet ex doctrina vnde-
<
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cime concluſionis precedentis capitis: igitur non
<
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quelibet ꝓportio rationalis habet ſubduplã ſub
<
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triplã, ſubq̈druplã ratiõales .etc̈. </
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<
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xml:space
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">Ptꝫ igit̄̄ ↄ̨cluſio</
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<
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<
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xml:space
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">Aliqua ꝓportio ra-
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tionalis eſt dupla, tripla, quadrupla, et ſic in infi
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nitū alicui ꝓportioni irratiõali. </
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<
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">Probatur / quia
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ꝓportio dupla eſt huiuſmodi / igitur. </
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<
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xml:space
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">Antecedens
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ꝓbatur / quia ꝓportio dupla habet medietatē ter
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tiam, quartã, quintã .etc̈. / vt patet ex quinta ſuppo
<
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ſitione: et ad medietatē ſui eſt dupla, et ad tertiaꝫ
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tripla, et ſic in infinitū / vt patet ex quarta ſuppo-
<
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ſitione: et nec eius medietas, nec eius tertia, et ſic
<
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in infinitū ſunt ꝓportiones rationales / vt patet ex
<
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ꝓbatione precedentis cõcluſionis: igit̄̄ ſunt ꝓpor
<
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tiões irratiõales: igit̄̄ ipſa ꝓportio dupla eſt du-
<
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pla, tripla, quadrupla, et ſic in infinitū alicui pro
<
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portioni irrationali / quod fuit probandum.</
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>
</
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<
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="
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">Quarta cõcluſio. </
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<
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="
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xml:space
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">Quelibet ꝓportio
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rationalis eſt cõmenſurabilis alicui proportioni
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irrationali. </
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<
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="
N14CB6
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xml:space
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">Probatur hec concluſio / qm̄ nulla ꝓ-
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portio ratiõalis habet quãlibet ſui partē aliquo-
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tam rationalē ꝓportionē: igitur quelibet eſt com
<
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menſurabilis alicui rationali. </
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>
<
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="
N14CBF
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xml:space
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preserve
">Patet cõſequētia
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ſuppoſita cõſtantia: qm̄ quelibet quãlibet aliquo
<
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tam habet) vt ly quãlibet diſtribuat pro generibꝰ
<
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ſingulorū (et nõ quãlibet habet rationalē ꝓporti-
<
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onē: igitur aliquam habet que eſt irrationalis ꝓ-
<
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portio: et illi eſt cõmenſurabilis / vt patet ex quarta
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ſuppoſitione: igitur ꝓpropoſitū. </
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<
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">Probat̄̄ antecedēs /
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qm̄ inter nulliꝰ ꝓportionis terminos inueniūtur
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tot numeri cõtinuo ꝓportionabiles quot poſſunt
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ſignari partes aliquote: igitur aliqua pars ali-
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quota erit ꝓportio irratiõalis. </
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<
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">Et ſic ptꝫ ↄ̨cluſio:</
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">Non oīs proportio
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irrationalis eſt ſubdupla, aut ſubtripla, et ſic con
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ſequēter ad aliquã irrationalē: īmo multe irrati-
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onales ſunt ſubduple aut ſubtriple .etc̈. ad ratio-
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nales. </
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<
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">Probatur hec ↄ̨cluſio facile: qm̄ medietas
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duple, quintuple, triple, octuple .etc̈. nõ eſt ſubdu-
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pla ad aliquã irrationalē: et tñ eſt irrationalis / vt
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ſatis patet ex decima ↄ̨cluſione cū ſuo primo cor-
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relario precedentis capitis / igitur concluſio vera.</
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</
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<
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<
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">Sexta concluſio. </
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<
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">Quelibet ꝓportio
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Capitulum ſeptimū.
"/>
in qualibet proportione rationali ab aliqua pro
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portione rationali vel irratiõali exceditur. </
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<
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batur hec concluſio: quoniã data quacū propor
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tione ad illam poteſt dari dupla, tripla, quadru
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pla, et ſic cõſequenter procedendo per oēs ſpecies
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ꝓportionis multiplicis: quoniã poſſunt dari tres
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termini continuo ꝓportionabiles tali ꝓportione
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data: et quatuor, et quin, et ſex, et ſic conſequēter
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vt docet ſexta ſuppoſitio: et etiam data quacun
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dabitur vna que contineat ipſam et medietatē eiꝰ
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et alia que continet ipſam et vnã tertiã eius, et vnã
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quartam, et ſic in infinituꝫ. </
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<
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cõtinet ipſam et duas tertias eius, vel tres quar-
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tas: et ſic in infinītum ſecundū omnē ſpeciem pro-
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portionis rationalis tam ſimplicis quam cõpo-
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ſite: et quelibet talis proportio erit rationalis vel
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irrationalis / vt patet ex primo capite prime par-
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tis: igitur quelibet proportio in qualibet propor
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tione rationali ab aliqua proportione rationali
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vel irrationali exceditur. </
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<
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<
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tio in qualibet proportione rationali aliquã ra-
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tionalem vel irratiõalem excedit. </
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quelibet proportio poteſt diuidi in duas equales
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ratiõales vel non rationales: in .3. in .4. in .5. in .6. /
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et ſic in infinitū. </
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<
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">vt patet ex quinta ſuppoſitione / et
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ſui medietatē in proportione dupla excedit: et ter-
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tiã in tripla: et quartã in q̈drupla: et ſic in infinitū /
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vt patet ex prima ſuppoſitione: et duas tertias in
<
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ſexquialtera: et tres quartas ī ſexquitertia: et tres
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quintas in ſuprabipartiente tertias: et ſic in infi-
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nitum diſcurrendo per ſingulas ſpecies propor-
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tionuꝫ rationalium: igitur quelibet proportio in
<
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qualibet proportione rationali aliquam ratio-
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nalem vel irrationalem excedit.</
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</
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<
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irrationales inter terminos proportionis ratio
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nalis mediantes ſit.</
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<
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medie rei inuentionis. </
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<
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xml:space
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">Si datis duabus rectis li-
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neis proportionabilibus proportione rationali
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vel irrationali in directum protractis coniūctis
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at ligatis: deſcribatur ſemicirculus: et a cõmuni
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medio ſiue puncto in quo vniuntur eleuetur linea
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directe orthogonaliter ad peripheriam vſ ſemi
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circuli. </
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<
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">talis linea ſcḋm cõtinuã ꝓportionalitatē
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inter datas lineas mediabit. </
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">Huiꝰ cõcluſionis ſen
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ſus talis eſt. </
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<
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xml:space
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">Si velis inter duas lineas ꝓportiõa-
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biles ꝓportione dupla aut quacun alia īuenire
<
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vnã que ſe habeat in eadē ꝓportione ad minorē in
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qua ſe habet maior ad ipſam: ↄ̨iūge illas duas li
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neas et ſuꝑ illas deſcribas ſemicirculū: et a pūcto
<
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in quo iūgunt̄̄ ille due linee oriat̄̄ directe et ortho-
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gonaliter vna alia linea vſ ad circūferentiã cir-
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culi: et illa eſt linea q̄ querit̄̄: et ꝓportio maioris li-
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nee ad illã mediã eſt medietas ꝓportiõis q̄ eſt īter
<
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illã lineã maiorē et minimã ſic ↄ̨iunctas. </
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huius concluſionis patet in hac figura.</
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