Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secūde partis
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43
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portionis multiplicis vel multiplicis ſuperparti
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cularis, vel multiplicis ſuprapartientis. </
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<
s
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N14262
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">Falſitas
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conſequentis probatur: quoniam ſi c. eſt pars ali
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quota multiplicis ꝓportionis: capio talem ꝓpor
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tionem multiplicem inter primos terminos eius:
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et arguo ſic: c: proportio multiplex ſuperparticu-
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laris, aut multiplex ſnpraꝑtiens, eſt pars aliquo
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ta alicuius ꝓportionis multiplicis: igitur ex ali-
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quot c. illa proportio multiplex componitur. </
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<
s
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N14273
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tur ex conſequenti ſequitur / alicuius termini in-
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termedii ad minimum extremū ipſius proportio-
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nis mĺtiplicis / qḋ minimū externū ē vnitas ē ꝓpor
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tio c. / vt patet ex tertia ſuppoſitione: et illa ꝓpor-
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tio c. eſt multiplex ſuꝑparticularis, aut multiplex
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ſuꝑperpartiens: igitur alicuius numeri ad vnita-
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tem eſt ꝓportio multiplex ſuprapartiens aut mul
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tiplex ſuperparticularis quod eſt oppoſitum ſex-
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te ſuppoſitionis: et per conſequens falſum: et ex
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conſequenti illud ex quo ſequitur videlicet / c. eſt
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ꝓportio multiplex ſuperparticularis, aut multi-
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plex ſuprapartiens. </
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<
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xml:space
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">Et ſic patet concluſio.</
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<
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">Quarta concluſio. </
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<
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N14295
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">Nulla proportio
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multiplex eſt commenſurabilis alicui proportio-
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ni rationali non multiplici. </
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<
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N1429C
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xml:space
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">Probatur: quia nul
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la ꝓportio multiplex eſt commenſurabilis alicui
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ſuperparticulari, aut ſuprapartienti / vt patet ex
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ſecunda, nec alicui multiplici ſuꝑparticulari, aut
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/>
multiplici ſuprapartienti / vt ptꝫ ex tertia, igit̄̄ nul
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la ꝓportio multiplex ↄ̨menſurabilis eſt alicui ꝓ-
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portioni rationali non multiplici. </
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<
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N142AB
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xml:space
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">Et ſic patet cõ
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cluſio.</
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">Nulla proportio
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ſuperparticularis eſt commenſurabilis alicui ꝓ-
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portioni ſuperparticulari. </
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<
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">Probatur ſupponen
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do / inter cuiuſlibet ꝓportionis ſuperparticula
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ris primos numeros nullus numerus mediat vt
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viſum eſt in prima parte vbi agebatur de genera
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tione ꝓportionum ſuperparticularium. </
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<
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poſito arguitur ſic: inter cuiuſlibet ꝓportionis ſu
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perparticularis primos numeros nullus mediat
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numerus: igitur nulla talis ex aliquot ītermediis
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ꝓportionibus adequate componitur. </
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<
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N142D1
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ſequentia / quia nulla eſt ꝓportio intermedia niſi
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ſit numerus intermedius: et vltra ex nullis ꝓpor-
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tionibus componitur. </
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<
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N142DA
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">igitur nulla ꝓportio ē pars
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aliquota eius: et per conſequens ipſa non eſt com
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menſurabilis alicui proportioni ſuperparticula
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ri. </
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<
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">Patet conſequentia / quia alias aliquid eſſet
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pars aliquota vtriuſ. </
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<
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N142E8
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xml:space
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">Et ſic patet concluſio.</
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<
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<
s
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xml:space
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">¶ Sed tu dices / hec ꝓbatio eſt inefficax: quoniã
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concedit / aliqua proportio ex nullis ꝓportioni
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bus componitur quod eſt contra ea que dicta ſūt
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capite quarto huius partis. </
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<
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N142F9
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">imo ꝓbatio nihil ali
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ud probat niſi / ex nullis ꝓportionibus equalibꝰ
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rationalibus componitur que ſint partes aliquo
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te illius: cum hoc tamen ſtat / aliqua ꝓportio ir-
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rationalis eſt pars aliquota duarum ꝓportionū
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ſuperparticularium: et ſic erunt commenſurabi-
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les.
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obiectio.</
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">¶ Sed hoc non obſtat / quia nulla ꝓportio ſuꝑ
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particularis componitur ex alia ſuperparticula
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ri et vna irrationali: ſicut nec aliq̄ rationalis cõ-
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ponitur ex vna rationali et altera irrationali a-
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dequate / vt probãt mathemathici. </
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<
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N14318
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">igitur nulla ſu
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ꝑparticularis continet alteram ſuꝑparticularem
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ſemel aut aliquoties et vnam partem aliquotam
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eius que ſit ꝓportio irrationalis: quia tunc com-
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poneretur ex rationali et irrationali adequate:
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nec aliqua ſuꝑparticularis continet alteram ſe-
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Capitulum ſextum
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mel vel aliquoties et aliquot partes eius aliquo-
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tas que ſint proportiones irrationales: quia tunc
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iam ille proportiones irrationales componerent
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vnam rationalem: quia alias componeretur illa
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ſuperparticularis ex rationali et irrationali: et ſi
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ille partes aliquote faciant vnam rationalem iaꝫ
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inter terminos illius ꝓportionis ſuꝑparticularis
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reperientur aliquot ꝓportiones rationales equa
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les / vt patet intuenti: quod tamen eſt falſum cum
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non reperiantur inter primos numeros alicuius
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ꝓportionis ſuꝑparticularis.</
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<
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<
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</
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<
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">tantum ꝓportio multiplex commenſuratur ꝓpor
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tioni multiplici. </
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">Probatur / quia proportio multi
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plex eſt commenſurabilis ꝓportioni multiplici / vt
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patet de quadrupla reſpectu duple: et inter ratio-
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nales nulla non multiplex eſt cõmenſurabilis ali
<
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cui ꝓportioni multiplici / vt patet ex quarta cõclu
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ſione / igitur propoſitum. </
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<
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dialectica.</
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<
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">Septima concluſio </
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">Omēs propor-
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tiones multiplices quarum denominationes ſunt
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de numero numerorum ſunt inter ſe cõmenſurabi
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les.
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">nicholaꝰ
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horen.</
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<
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">Hanc concluſionem ponit Nicholaus horen
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ſub forma dicta: ſed pono eam ſub alia forma cla
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riori. </
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<
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">Omnes ꝓportiones multiplices ꝓcedentes
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ſemper ſecundum dendminationem prime illarū
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ſunt cõmenſurabiles: ita ſi prima illarum ſit du
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pla. </
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<
s
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">ſecunda immediate ſequens ſit etiam dupla:
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et ſic conſequenter tales ſunt cõmenſurabiles. </
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">Et
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vt paucis abſoluam omnes ꝓportiones quarum
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quelibet īmediate ſequētes ſunt eiuſdem denomi
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nationis cum prima ſunt commenſurabiles </
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<
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">Pa-
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tet hec concluſio / quoniam omnes tales ita ſe ha-
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bent aliquid eſt pars aliquota vtriuſ / igitur.
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</
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<
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">Et ad hoc videndum diſponatur vna ſeries nūe-
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rorum incipiendo ab vnitate ſemper duplando et
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vna alia ſemper triplando, et alia quadruplan-
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do, et alia quintuplando, et ſic in infinitum. </
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<
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">et tunc
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dico / omnes ꝓportiones primi ordinis ſunt cõ-
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menſurabiles inter ſe. </
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<
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">et quelibet cuilibet alteri il
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lius ordines: </
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<
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">Et ſic etiam dicendum eſt de ꝓportio
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nibus alioruꝫ ordinum. </
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<
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<
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">Et ſic etiam conſtitues ordines multarum ſuper-
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particularium et ſuprapartientium etc. </
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<
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">Quod au
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tem iſte ſunt commenſurabiles probatur / quoniã
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quelibet illius ordinis eſt equalis prime aut com
<
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ponitur ex aliquot equalibus illi: igitur. </
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<
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">¶ Iſte
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concluſiones dempta prima et ſexta ſunt Nicho-
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lai horen cum ſuis probationibus ſaltem virtu-
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tes probationum et fundamenta ſunt ex ipſo.</
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cholauꝫ
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horen.</
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</
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">Fundatur enim principaliter probatio ſecūde ter
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tie et quarte in hac ſuppoſitione cuiuſlibet ꝓpor
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tionis multiplicis vnitas eſt minimum extremum
<
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</
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<
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">Modo illa ſuppoſitio falſa eſt / quoniam octo ad
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quatuor eſt proportio multiplex: tamen neutrum
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extremorum eius eſt vnitas: </
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<
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">Sed diceret Nicho-
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laus horen et bene / illa ſuppoſitio et ſi nõ ſit ve-
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ra diſtribuendo pro ſingulis generum. </
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<
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vera diſtribuendo pro generibus ſingulorum: et ī </
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