Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secūde partis
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nalem intermediam: et per conſequens iam nõ ha
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bet ſubquadruplam rationalem. </
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<
s
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N145C9
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xml:space
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">Patet hec con-
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ſequentia / quia ex oppoſito ſequitur oppoſituꝫ / vt
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patet ex decima diffinitione quinti elementorum
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euclidis. </
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<
s
xml:id
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N145D2
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xml:space
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preserve
">Iam probo priorem conſequentiam vi-
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delicet / ſi inter terminos date proportionis non
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fuerit numerus qui ſit medium proportionabile:
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non reperiuntur ibi .5. numeri cõtinuo proportio
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nabiles. </
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<
s
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N145DD
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xml:space
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preserve
">Que probatur ſic: q2 ex oppoſito conſe-
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quentis ſequitur oppoſitum ãtecedentis: q2 ſi ſūt
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ibi quin numeri continuo ꝓportionabiles iam
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ibi tertius numerus eſt medio loco ꝓportionabi-
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lis: quia primi ad ipſum eſt ea proportio que ē ip
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ſius ad quintum / vt conſtat: quia ex equalibus cõ-
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ponuntur ille ꝓportiones adequate. </
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<
s
xml:id
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N145EC
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xml:space
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">Et ſic proba
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bis alias partes.
<
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xlink:href
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note-0045-01a
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note-0045-01
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xml:id
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xml:space
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">correĺm.</
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>
</
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<
s
xml:id
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N145F6
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xml:space
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preserve
">¶ Ex hac concluſione ſequitur /
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ſi inter terminos alicuius proportionis fuerit nu
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merus qui ſit medium proportionabile ipſa ha-
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bet ſubduplam rationalem et ſi ipſius numeri me
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dii proportio ad aliud extremuꝫ minus date pro-
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portionis haberit numerum qui ſit medium pro-
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portionabile: tunc tota proportio habet ſubqua
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druplam rationalem: et ſi iteruꝫ illius numeri me
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dii proportio ad minus extremum date ꝓportio-
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nis habuerit numerum qui ſit medium ꝓportio-
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nabile: iam data proportio habebit ſuboctuplaꝫ
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rationalem / et ſic in infinitum. </
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<
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xml:space
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rium ex concluſione et eius ꝓbatione: auxilianti-
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bus correlariis ſexte concluſionis ſecūdi capitis</
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>
</
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<
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<
s
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">Decima concluſio notanda. </
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<
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ſita quauis proportione rationali an habeat ſub
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duplam rationalem inueſtigare. </
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<
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xml:id
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N1462B
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xml:space
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">vt propoſita du
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pla aut tripla volo īueſtigare et ſcire ex predictis
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an habeat ſubduplã rationalem. </
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<
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N14632
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xml:space
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">Sit propoſita
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proportio rationalis f. inter a. numerū maiorem
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et b. numerum minoreꝫ. </
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<
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xml:space
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">et volo inueſtigare vtrum
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f. ꝓportio habeat ſubduplã rationalem: tunc du-
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cam maiorem numerum in minorem / hoc eſt multi
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plicabo a. per b. et ſi numerus inde ꝓueniens fue-
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rit quadratus: dico / habet ſubduplam rationa
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lem. </
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<
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">ſin minus non habet ſubduplam rationalem
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</
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<
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xml:space
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">Probatur prima pars videlicet / ſi numerus qui
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fit ex ductu ipſius a. in b. ſit quadratus: tunc ha-
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bet ſubduplam rationalem. </
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<
s
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N14651
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xml:space
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">quia ſit talis numerꝰ
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eſt quadratus: tunc inter a. et b. eſt medius nume-
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rus proportionabilis / vt patet ex quarto correla
<
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rio ſexte concluſionis ſecundi capitis huius par-
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tis: et ſi ſit numerus qui ſit medium ꝓportionabi
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le inter a. et b. / ſequitur / illa proportio habet ſub
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duplam rationalem. </
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<
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">Patet conſequentia ex cor-
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relario precedentis. </
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<
s
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N14665
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">Iam probatur ſecunda pars /
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quia ſi numerus qui fit ex ductu a. in b. non ſit qua
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dratus: iam inter a. et b. non eſt numerus qui ē me
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dio loco proportionabilis / vt patet ex ſecundo cor
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relario ſexte concluſionis ſecundi capitis huius:
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et ſi non eſt numerus qui eſt medio loco proportio
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nabilis inter a. et b. iam ille non habet ſubduplaꝫ
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rationalem / vt patet ex concluſione nona huius.</
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<
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xml:space
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">Patet igitur concluſio.
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">correĺm.</
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<
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xml:space
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">¶ Ex hac ſequitur / du-
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pla non habet ſubduplam rationalem, nec tripla
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nec octupla, nec aliqua ſuperparticularis. </
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<
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batur / quoniam ducendo quatuor per duo reſul-
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tat numerus octonarius qui non eſt quadratus / vt
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conſtat: et ducendo .6. per duo: reſultat numerus
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duodenarius qui etiam non eſt quadratus: et du
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cendo .16. per duo conſurgit numerus .32. qui non
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eſt quadratus vt apparet intelligenti. </
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<
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">Item ducē
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do .3: per duo producuntur .6. qui non ſunt nume-
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rus quadratus: et ſic probabis de qualibet alia ꝓ
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Capitulum ſextum
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portione ſuperparticulari.
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">2. correĺ.</
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xml:space
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">¶ Sequitur ſecundo /
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propoſita qua volueris ꝓportione rationali. </
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<
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">ī
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ueſtigare poterimus vtrum habeat ſubquadru-
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plam rationalē ſuboctuplaꝫ, ſubſexdecuplam, et
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ſic in infinitum procedendo per numeros pariter
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pares. </
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<
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xml:space
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">vt propoſita proportione ſexdecupla: vo-
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lo inueſtigare: vtrum habeat ſubquadruplam ra
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tionalem, ſuboctuplam, ſubſexdecuplam, et ſic in
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infinitum. </
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<
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xml:space
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">Ad quod inueſtigandum ſiue ſciendum
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ſit f. ꝓportio inter a. maiorem numerum et b. mi-
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norem: tunc aut inter a. et b. eſt numerus qui ſit me
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dium ꝓportionabile aut non. </
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<
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xml:space
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">ſi nõ: iam ſequitur /
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non habet ſubquadruplam rationalē nec ſub-
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octuplam etc. / vt patet ex nona concluuſione: ſi ſic
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ſignetur ille et ſit h. / et tunc videndum eſt an nume
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rus / qui fit ex ductu h. in b. ſit quadratus: et ſi ſic iã
<
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talis ꝓportio f. que eſt inter a. et b. habet ſubqua-
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druplam: ſi vero talis numerus non ſit quadratꝰ
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dico / talis proportio non habet ſubquadruplã
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rationalem. </
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<
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">Primum iſtorum probatur. </
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<
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N146DC
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">quia ſi
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talis numerus qui fit ex ductu h. in b. ſit quadra-
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tus: iam inter h. et b. eſt numerus medio loco pro-
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portionabilis qui ſit k. / vt patet ex quarto correla
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rio preallegato ſexte concluſionis ſecundi capitis
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huius: et ex conſequenti iam ꝓportio h. ad b. que
<
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eſt ſubdupla ad ꝓportionem f. habet ſubduplam
<
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proportionem rationalem / vt patet ex correlario
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none concluſionis: et ſi habet ſubduplam iam pro
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portio f. habet ſubquadruplam: quia omne ſub-
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duplum ſubdupli eſt ſubquadruplum dupli / vt pa
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tet ex ſecundo correlario quarte concluſionis q̈r-
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ti capitis huius / quod erat oſtendendum. </
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<
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batur ſecundum: quia ſi numerus qui fit ex ductu
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h. in b. non ſit quadratus iam proportio que eſt ī-
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ter h. et b. non habet numerū medio loco ꝓportio
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nabilem / vt patet ex ſecundo correlario ſexte con-
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cluſionis preallegate: et ſi non habet mediū nume
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rū ꝓportionabilem iã non habet ſubduplã ratio
<
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nalem: et ſic eius medietas non eſt proportio rõa-
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lis et eius medietas eſt ſubquadruplum ꝓportio
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nis f. que eſt a. ad b. / vt cõſtat: igitur proportio ſub
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quadrupla ad f. non eſt rationalis / quod fuit oſtē-
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dendum. </
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<
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">Alie particule correlarii ſimilem demon
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ſtrationem ſortiuntur. </
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<
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">Si eni3 non inueniatur ra
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tionalis ſubquadrupla: nec ſuboctuplã rõnalem
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inuenies. </
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<
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">Si vero ſubquadrupla reperta fuerit ra
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tionalis: conſidera an ex ductu vnius extremita-
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lis ſubquadrupli in alterum reſultat numerꝰ qua
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dratus: et ſi ſic concludas datam ꝓportionem ha
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bere ſuboctuplam rationalē: quia ſua quarta ha
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bet ſubduplam rationalem. </
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<
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eam non habere talem ſuboctuplam rationalem.
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</
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<
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xml:space
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">Et ſic in aliis operaberis.
<
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xml:id
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">3. correl.</
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<
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">¶ Sequitur tertio / ſi
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gnata quauis ꝓportione rationali: inueſtigare et
<
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ſcire poterimus an habeat ſexquialteram ratio-
<
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nalem, ſexquiquartaꝫ, ſexquioctauam, ſexquiſex
<
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decimã, ſexquitrigeſimã ſecundam, ſexquitrigeſi
<
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mã quartã, et ſic in infinituꝫ: ꝓcedendo per ſpecies
<
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ꝓportionis ſuperparticularis denominatas a ꝑ
<
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tibus aliquotis que partes aliquote a nūeris pa-
<
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riter paribus denominantur. </
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<
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">vt ꝓpoſita ꝓportio
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ne quadrupla: volo inueſtigare et ſcire an ip̄a ha
<
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beat ſexquialteram rationalem: tūc videbo an ha
<
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beat medietatem rationalem per doctrinam deci
<
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me concluſionis huius: et tunc ſi habeat medieta-
<
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tem rationalem: manifeſtum eſt habet ſexquial<lb/>teram rationalem: quia non oportet ad dandam
<
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ſexquialteram ipſius quadruple aliud quam ad-
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dere ipſi quadruple ſuã medietatem puta duplã: </
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