Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secunde partis
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48
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nūeris reperirent̄̄ irratiõales ꝓportiões: vt ſatis
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cõſtat ītelligēti. </
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<
s
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N14A80
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xml:space
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preserve
">Et ſic ptꝫ correlariū.
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xml:space
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">5. correĺ.</
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<
s
xml:id
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N14A88
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xml:space
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">¶ Sequit̄̄ q̇n
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to: ꝓpoſita q̈uis ꝓportiõe ratiõali: nõ difficile ē
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īueſtigare et ſcire an habeat ꝓportionē rõnalē ſub
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multiplicē: an aliquã aliã rationalē minoris ineq̈
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litatꝪ: vt ꝓpoſita ꝓportiõe dupla īueſtigare et ſci
<
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re poterimꝰ an habeat ſubduplã: ſubtriplã: ſubq̈-
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druplã rationalē .etc̈. nec ne: cõſiderando primū ex
<
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doctrina vndecime ↄ̨cluſiõis: an habeat medieta-
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tem: tertiã: quartã: quintã rationales: et cõperien-
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tes nõ: dicemus ipſam nõ habere ſubtriplam:
<
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ſubquadruplã .etc̈. rationales. </
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>
<
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xml:space
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">Et eadem ratione
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dicemꝰ ipſam nõ habere ſubſexq̇tertiã rationalē:
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q2 nõ habet ꝓportionē cõpoſitã ex tribus quartis
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eius rationalibus: nec ſubſexquialterã rationalē:
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q2 nõ habet ꝓportionē compoſitã ex duabus ter-
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tiis eius rationalibus. </
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<
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xml:space
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">Et ſic in omnibus aliis di
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ces. </
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<
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">Demonſtratio huius correlarii innititur huic
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baſi et fundamento / nun̄ aliqua ꝓportio ratio
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nalis cõponitur adequate ex vna rationali et vna
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irrationali. </
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<
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">Applica tu demonſtrationē. </
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<
s
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xml:space
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">Iſto mo
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do inquirere debes an habet ſubſuprapartientē
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rationalē aut ſub multiplicē ſubſuprapartientem
<
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rationalē: aut ſub multiplicē ſubſuꝑparticularē:
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/>
īueſtigando et inquirendo ex cõcluſione vndecima
<
lb
/>
an talis ꝓportio rationalis ꝓpoſita habeat par
<
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tem aliquotã rationalē vel partes a qua vel a qui
<
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bus denominatur dicta ꝓportio minoris inequa
<
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litatis: et ſi ſic aſcribenda eſt ei talis ꝓportio mi-
<
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noris inequalitatis rationalis: ſin minus: aſſeren
<
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dum eſt ipſam nõ habere talē ꝓportionē minoris
<
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inequalitatis rationalē. </
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<
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</
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xml:space
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tenebris īuoluere.
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xml:id
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xml:space
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">6. correĺ.</
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<
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xml:space
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">¶ Sequitur ſexto per modum
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epilopi oīm eoꝝ / que preſenti capite digeſta ſunt:
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quauis ꝓportione rationali ꝓpoſita: ſcire po-
<
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terimus an habeat aliquã ꝓportionē rationalem
<
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maioris inequalitatis ad ſeipſam et minoris ine-
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qualitatis: et quas habeat: et quas nõ. </
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<
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xml:id
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xml:space
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">Et hoc ca-
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put diligenter conſidera quoniã ex eo pendet fer-
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me vniuerſalis huiꝰ materie īquiſitio: et ſuprema
<
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eius difficultas. </
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<
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xml:id
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N14AFA
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xml:space
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">¶ His adde / doctrina huius ca-
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pitis habita: ꝓpoſita aliqua certa velocitate ꝓ-
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ueniente ab aliqua ꝓportione rationali nota: iu-
<
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dicare poterꝪ de quacū alia velocitate a quauis
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alia ꝓportiõe ꝓueniente cõmenſurabiles ſint. </
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<
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xml:space
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">nec
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ne. </
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<
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xml:space
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">Item ꝓpoſita quauis velocitate ꝓueniente ab
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aliqua ꝓportione ratiõali nota: ſcire de quacū
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alia velocitate date velocitati cõmenſurabili a q̈
<
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ꝓportiõe ꝓueniat: ratiõali vcꝫ vĺ irrationali / q̊ ex
<
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his ſcito et ſequētibꝰ: particulariꝰ ſcire poteris ex
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qua rationali vel irrationali ꝓueniat ſpecifice.</
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">Capitum ſeptimū / in quo agitur de medie
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rei inuentione et proportione proportionuꝫ
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rationalis et irrationalis.</
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>
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<
s
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xml:space
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">AD habendam aliqualē noti-
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ciã de ꝓportiõe ꝓportiõis rationalis et
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irrationalis et duarū irrationaliū ſit.</
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</
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<
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<
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">Prima ſuppoſitio. </
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">Oīs numerus ha
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bet numerū ad ſe duplū, triplū, quadruplū, et ſic
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in infinitū: aſcēdendo per ſpecies ꝓportionis mul
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tiplicis. </
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<
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N14B60
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">Iſta ſuppoſitio patet ex ſe / qm̄ dato vno
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numero ex duabus vnitatibus adequate cõpoſito
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dabitur vnus alter compoſitus ex quatuor: et ille
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erit duplus: et alter ex ſex: et erit triplus: et alter ex
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octo: et erit quadrupus: et ſic ſine termino.</
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</
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<
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">Secunda ſuppoſitio. </
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<
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">Omnis nume
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rus rerum diuiſibiliū ſiue quantitas habet cuius
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Capitulū ſeptimū.
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cū denominationis aliquam partem aliquotaꝫ
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cum fractione vel ſine fractione. </
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<
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N14B79
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xml:space
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">Uolo dicere / ſi-
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gnato quocun numero rerū diuiſibiliū talis nu
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merus habet medietatē tertiam, quartam, quin-
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tam, ſextam, ſeptimam, et ſic in infinitū. </
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<
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tur: quia capto numero duodenario ille habet me
<
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dietatem, puta numerum ſenariū: habet numerū
<
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quaternariū pro tertia, ternariū pro quarta, pro
<
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quinta vero habet numerū cū fractione, ad quam
<
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fractionē inueniendã oportet duodecim per quī
<
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diuidere: et exibit binariꝰ cū duabꝰ q̇ntis iuxta do-
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ctrinã ſuperiꝰ poſitã octauo capite ṗme partꝪ. </
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<
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xml:space
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ſic operãdū eſt in cuiꝰ vis alteriꝰ ꝑtꝪ aliq̊te īuētiõe.</
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</
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<
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">Supra quēcū
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numerū rerum diuiſibiliū contingit dare numeꝝ
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continentē ipſum et medietatē: et alium continentē
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ipſum et vnam tertiam, et duas tertias: aut tres
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quartas: et ſic de qnibuſcun aliis partibus ali-
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quotis. </
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ipſum et medietatē ſufficit addere illi medietatem
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ſui: et ad dandum numerū continentē ipſum et du-
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as tertias ſufficit ei addere illas duas tertias: vt
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patet ex ſe aſpicienti in numeris. </
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<
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tales partes īueniant̄̄ p̄cedēs ſuppoſitio declarat</
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</
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<
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">Quarta ſuppoſitio. </
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<
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">Quodlibet con-
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tinuū eſt duplū ad ſuã medietatē: triplū ad tertiã:
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quadruplū ad quartã: ſexquialterū ad duas ter-
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tias: et ſic de qualibet alia ſpecie ꝓportionis. </
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>
<
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tet hec ſuppoſitio ex diffinitionibus terminorum.</
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<
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">Quinta ſuppoſitio. </
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<
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xml:space
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">Omnis ꝓportio
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habet medietatē: tertiam: quartã: et ſic in infinitū.
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</
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<
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xml:space
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">Probatur hec ſuppoſitio / q2 oīs quantitas cõti-
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nua: et quodlibet cõtinuo ſucceſſiue diminuibile eſt
<
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huiuſmodi et oīs ꝓportio eſt quantitas continua
<
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aut cõtinuo partibiliter diminuibilis (et diſtribu-
<
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at ly omnis pro generibus ſingulorum more ma-
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themathicorum) / igitur propoſitum.</
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</
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<
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<
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xml:space
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">Sexta ſuppoſitio. </
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<
s
xml:id
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xml:space
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">Si aliq̄ due quã-
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titates cõtinue ſe habeant in aliqua proportione
<
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ratiõali vel irratiõali: dabilis eſt vna tertia qua-
<
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libet illarū maior que ſe habeat in eadē ꝓportiõe
<
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ad maiorē illaꝝ. </
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<
s
xml:id
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xml:space
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">vt ſi .4. et .2. ſe habeãt in aliqua ꝓ
<
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portione dabilis eſt alter numerus puta .8. qui in
<
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eadem ꝓportione ſe habeat ad .4. et ſi diameter a.
<
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ſe habeat in aliqua ꝓportione ad coſtã b. dabilis
<
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eſt vna alia quãtitas puta c. que ſe habet in eadeꝫ
<
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ꝓportione ad b. </
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<
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xml:id
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N14BFE
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xml:space
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preserve
">Patet hec ſuppoſitio ex ſe.</
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</
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<
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<
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xml:id
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xml:space
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">His poſitis ſit prima cõcluſio. </
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>
<
s
xml:id
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N14C05
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xml:space
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">Que-
<
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libet ꝓportio ratiõalis in q̈libet ꝓportiõe multi-
<
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plici ab aliq̈ ratiõali excedit̄̄. </
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>
<
s
xml:id
="
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xml:space
="
preserve
">Hoc eſt q̈libet ꝓpor-
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tio ratiõalis hꝫ ꝓportionē duplã: triplã: q̈druplã
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et ſic in īfinitū rõnales. </
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>
<
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="
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xml:space
="
preserve
">Probat̄̄ hec ↄ̨cĺo / qm̄ ſi illa
<
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ꝓportio fuerit mĺtiplex manifeſtū ē / ad nūeꝝ eiꝰ
<
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/>
maiorē dabit̄̄ aliq̇s nūerꝰ ſe hñs in eadē ꝓportiõe /
<
lb
/>
ad illū ſicut ille partes hꝫ ad minorē / vt ptꝫ ex ṗma ſup
<
lb
/>
poſitiõe: et tūc illiꝰ ad minimū erit ꝓportio dupla
<
lb
/>
ad ꝓportionē medii ad minimū: qm̄ illa cõponit̄̄
<
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/>
ex duabꝰ eq̈libꝰ illi: et ſi addat̄̄ q̈rtꝰ nūerꝰ ſe hñs in
<
lb
/>
eadē ꝓportione ad tertiū in qua tertius ſe habet
<
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ad ſecundū: ſicut poteſt fieri ex prima ſuppoſitiõe:
<
lb
/>
iã ꝓportio illius ad minimū erit tripla ad ꝓpor-
<
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tionē ſcḋi ad minimū: et cū poſſint ſic addi infiniti
<
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ṫmini ↄ̨tinuo ꝓportiõabiles illa ꝓportiõe mĺtipli
<
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ci / vt ptꝫ ex ṗma ſuppõe: ſequit̄̄ / ad illã ꝓportionē
<
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/>
dabit̄̄ ꝓportio dupla, tripla, q̈drupla, et ſic ī īfini
<
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tū. </
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>
<
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xml:space
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">Ptꝫ ↄ̨ña ex octaua ↄ̨cĺiõe p̄cedētꝪ capitꝪ </
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>
<
s
xml:id
="
N14C35
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xml:space
="
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">Si o
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illa ſit ſuꝑparticĺarꝪ ad maximū extremū eiꝰ adde </
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