Alvarus, Thomas
,
Liber de triplici motu
,
1509
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dratus: inter tales numeros reperitur medium ꝓ
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portionabile ꝓportione rationali ita primi ad
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ipſum ſit ea proportio rationalis que eſt ipſiꝰ ad
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tertium. </
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<
s
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xml:space
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">et illius numeri quadrati tale medium eſt
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vnum latus. </
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<
s
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xml:space
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">Probatur prima pars huius corre-
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larii / quia illa pars eſt vna cõditionalis ex cuiꝰ op
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poſito conſequentis / ſequitur oppoſitum antece-
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dentis: vt patet ex ſecundo correlario: igitur illa
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pars vera. </
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<
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xml:space
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">Secunda probatur ex correlario īme-
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diate precendenti. </
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<
s
xml:id
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xml:space
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">¶ Sequitur quīto / inter ṗmos
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numeros ꝓportionis duple: triple: octuple: ſexq̇-
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altere etc̈. non inuenitur medium ꝓportionabile ꝓ
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portione rationali </
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<
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">Probatur primo de dupla / q̄
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eſt inter iſtos terminos .4.2. quoniam numerus q̇
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fit ex ductu vnius extremi in alterum puta .4. in .2.
<
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non eſt quadratus / igitur inter illa extrema non ī
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uenitur medium ꝓportionabile proportione ra-
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tionali </
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<
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xml:space
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">Añs patet intelligenti diffinitionem nu-
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meri quadrati. </
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<
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xml:space
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">et conſequentia patet ex ſecundo
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correlario. </
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<
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xml:space
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">Et eodē modo ꝓbabis reliquas ꝑtes.
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</
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<
s
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xml:space
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">¶ Et ex hoc habes pulchrū documentuꝫ ab cogno
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ſcendū quãdo aliqua ꝓportio īeq̈litatꝪ habet ſub
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duplam proportionem ad eam rationalem. </
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<
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">Quã
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do enim numerus reſultans ex ductu vnius extre-
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mi in alterum non eſt quadratus / tunc talis ꝓpor
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tio non habet ꝓportionem rationalem ſubduplã
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ad illam cum non habeat medium ꝓportionabile
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ꝓportione rationali. </
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<
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xml:id
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xml:space
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">et ſic tale medium inter ter-
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minos illius ꝓportionis non ſe habet vt numerꝰ
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reſpectu alicuius extremi illius ꝓportionis. </
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<
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xml:space
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">Si eī
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ſe haberet vt numerus: maioris extremi ad ipſum
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eſſet aliqua ꝓportio rationalis: et ipſius ad mini
<
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mum extremum eſſet eadem ꝓportio rationalis: et
<
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ſic iam ibi eſſent tres numeri continuo ꝓportiona
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biles in hac medietate geometrica: et ſic numerus
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qui fit ex ductu extremi in extremū eſſet quadratꝰ /
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vt patet ex primo correlario / quod eſt oppoſitū da
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ti.
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ꝓportio
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alio mõ
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ponenda
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oñditur.</
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>
</
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<
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xml:space
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">Et ex hoc facile elicitur ꝓportionem irrationa-
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lem neceſſario ponendã eſſe: quod nota.</
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</
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<
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">Gratia ordinis obſeruandi medieta
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tis harmonice aliquas proprietates ponã quas
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non intendo demonſtrare: quia huic operi paruꝫ
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conducunt.
<
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xml:id
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xml:space
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">ṗma ꝓṗe
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tas medi
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etatꝪ har
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monice.</
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</
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<
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xml:space
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">¶ Prima proprietas </
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monica in maioribus terminis maiorem ſeruat ꝓ
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portionē quam in minoribus. </
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<
s
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xml:space
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">Hoc eſt dicere / ca
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ptis tribus terminis hac medietate ꝓportionabi
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libus: maior eſt proportio maximi ad mediū: quã
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medii ad minimū. </
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<
s
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">vt conſtitutis his terminis .12.8
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6. maior eſt proportio .12. ad .8. que eſt ſexquialte
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ra quã .8. ad .6. que eſt ſexquitertia.
<
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xml:space
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">ſcḋa ꝓṗe
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tas medi
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etatꝪ har
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monice.</
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</
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<
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xml:space
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">¶ Secunda ꝓ-
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prietas. </
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<
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">tribus terminis in hac medietate conſtitu
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tis medius terminus in collectas extremitates du
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ctus dupluꝫ numero qui fit ex extremo in extremū
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ꝓducit. </
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<
s
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">vt conſtitutis predictis terminis .12.8.6. et
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collectis extremis puta .6. et .12. que .18. conſtituūt
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numerus qui fit ex ductu medii puta octonarii in
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collectas extremitates puta ī .18. eſt duplus ad nu
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merum qui fit ex ductu extremorum .12. ſcilicet ī .6
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</
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<
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">Quod patet / quia ille eſt .144. hic vero .72. mõ con
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ſtat illū eſſe dupluꝫ ad hunc.
<
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xml:id
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xml:space
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">3. ꝓṗetas
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medieta
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tis har-
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monice.</
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<
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">¶ Tertia proprietas
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in hac medietate determinatis extremis medius
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terminus reperitur ſi per extremorum coniuncto-
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rum numerum: numerus qui ex differentia extre-
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morum in minimū conſurgit diuiditur. </
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<
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">iſ qui
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ex diuiſiõe relinquit̄̄ accipiat̄̄: at minimo extre-
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mo aggregatur. </
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<
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">vt determinatis his terminis .6.
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et .3. / ſi vis inuenire medium harmonicum inter il-
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los addas extremū extrēo puta .3. ip̄is .6 et erūt 9. /
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deiñ ducas dnr̄aꝫ inter .6. et .3. in .3. mīmū extremū:
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et quia illa differentia eſt .3. ex ductu eius in .3. fi-
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unt .9. diuidas / igitur .9. per .9. et relictū ex diuiſio
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ne erit vnitas: addas igitur vnitatem ternario: et
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aggregatum ex illa vnitate et ternario eſt mediuꝫ
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harmonicum inter ſex. et tria: eſt enim aggregatū
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illud quaternarius numerus. </
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">Modo .6.4.3: ꝓpor
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tionantur harmonice. </
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<
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">¶ Et hic aduerte / quibuſ-
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cū duobus numeris inequalibus cõſtitutis hac
<
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doctrina mediante reperies medium terminū in-
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ter eos: et hoc cum fractione aut ſine inter .4. enim
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et .3. medium harmonicū eſt .3. cuꝫ tribus ſeptimis
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</
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<
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cum partim ex his / que dicta ſunt / patet et comple
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te in poſterum dicetur.</
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agitur de quibuſdam propor
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tionalitatibus et modis argu
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endi in eis.</
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<
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portionabiliter ſiue in ꝓportionalitati-
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bus quibus nonun̄. </
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<
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">et philoſophi et cal
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culatores phiſici vtūtur ponit Euclides ſexto ele-
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mentorum et recentiores mathematici poſt eum.
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</
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<
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tur conuerſa: ſecunda permutata: tertia coniun-
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cta. </
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<
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</
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<
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xml:space
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">¶ Pro intelligentia primi modi arguendi aduer
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tendum eſt / in propoſito antecedens alicuius ꝓ
<
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portionis dicitur terminus / qui ad alterum com-
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paratur et conſequens terminus cui aliquis com
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paratur / vt cum dicitur quatuor ad duo ille termi
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nus quatuor eſt antecedens et duo conſequens / et
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ſi dicamus duo ad quatuor duo dicuntur antece-
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dens et quatuor conſequens
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litas con
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uerſa</
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<
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xml:space
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">¶ Iſto ſuppoſito pro
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portionalitas conuerſa eſt quando ex anteceden-
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tibus fiunt conſequētia: et eocontra. </
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<
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">Uel aliter eſt
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proportionalis illatio in qua ex proportionibus
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maioris inequalitatis concluduntur proportio-
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nes minoris ineq̈litatis eis correſpondentes. </
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<
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arguendo ſicut ſe habet octo ad quatuor ita duo a
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d vnum / igitur ſicut ſe habet vnum ad duo ita qua
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tuor ad octo. </
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<
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portionibus minoris inequalitatis ꝓportiones
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maioris īeq̈litatꝪ eis correſpõdētes.
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<
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xml:space
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ta ꝓportiõalitas dicit̄̄ / cū ex ãtecedēte ſcḋe ꝓporti-
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onis ſit ↄ̨ñs prime et ex ↄ̨ñti prime ſit añs ſcḋe. </
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>
<
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xml:space
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aliter eſt diſpoſitis quatuor terminis geometri-
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ce proportionalibus primi ad tertium. </
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<
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xml:space
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">et ſecundi
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ad quartum proportionalis illatio ſic arguendo
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ſicut ſe habet .8. ad .4. ita .2. ad .1. / igitur ſicut ſe ha
<
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bent .8. ad .2. ita .4. ad vnū. </
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<
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xml:space
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">Et iſto modo arguen-
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endi vtitur philoſophus in pleriſ locis vt in fi-
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ne ſecundi perihermenias: in tertio topi. </
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>
<
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xml:space
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">et in pri
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mo celi et mundi in tractatu de infinito.
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">Cõiūcta.</
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</
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<
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xml:space
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">¶ Coniun
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cta proportionalitas eſt a diſiunctis terminis geo
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meteice proportionabilibus ad coniunctos pro-
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portionalis illatio. </
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<
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xml:id
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xml:space
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">tali modo arguendo: ſicut ſe
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habent .8. ad .4. ita .2. ad .1. / igitur ſicut ſe habent.
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</
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<
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">octo et quatuor ad quatuor ita duo et vnū ad vnū
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<
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xml:id
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xml:space
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">diſiūcta.</
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>
</
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>
<
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xml:space
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">¶ Diſiuncta proportionalitas eſt a cõiunctis ter-
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minis geometrice proportionabilibus ad diſiun
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ctos proportionalis illatio. </
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<
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xml:space
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">tali modo arguendo /
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ſicut ſe habent 8. et .4. ad .4. ita duo et vnū ad vnū /
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igitur ſicut ſe habent octo ad quatuor ita duo ad
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vnum.
<
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xml:id
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xml:space
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">Euerſa.</
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>
</
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>
<
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xml:id
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xml:space
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">¶ Euerſa ꝓportionalitas eſt a diuiſis ter-
<
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minis geometrice proportionabilibus ad coniun
<
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ctos ordine conuerſo ad coniunctam proportio- </
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