Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Table of Notes
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Secunde partis
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38
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denominata a maiori numero vſ ad dupluꝫ in-
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īcluſiue eſt maior quam ſubdupla ad illam ſuper-
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particularem datam. </
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<
s
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xml:space
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">Patet igitur correlarium.
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xml:space
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<
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N139C9
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xml:space
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preserve
">¶ Sequitur nono / in omni proportione ſuper-
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particulari proportio maximi extremi eiꝰ ad me-
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dium eſt maior quam ſubdupla ad proportioneꝫ
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medii ad minimū extremum: vt data proportione
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ſexquitertia que eſt .8. ad .6. proportio .8. ad .7. eſt
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maior quam ſubdupla ad proportionem .7. ad .6.
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<
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xml:space
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">Probatur / quia ꝓportio maximi extremi ad me-
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dium in proportione ſuperparticulari quecun
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fuerit illa denominatur a numero ſuperparticu-
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ri īmediate ſequenti numerum a quo denomina-
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tur proportio medii ad minimū extremum / vt patꝫ
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ex quarta parte ſexti correlarii: et ſic denomīatur
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a numero minori duplo ad numeruꝫ a quo deno-
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minatur proportio medii ad minimū extremum:
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igitur talis proportio maximi ad medium eſt ma
<
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ior quam ſubdupla ad proportionē medii ad mi-
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nimuꝫ extremum. </
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<
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correlario.
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xml:space
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<
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xml:space
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">¶ Sequitur decimo / in omni propor
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tione ſuperparticulari ꝓportio maximi extremi
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ad medium eſt maior quam ſubtripla ad illã pro-
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portionem ſuperparticularem. </
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<
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xml:space
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dato oppoſito puta / ſit ſubtripla aut mīor ſub-
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tripla: ſequeretur / ipſa eſſet ſubdupla adequate
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ad proportionem medii ad minimū extremū, vel
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minor quam ſubdupla: ſed conſequens eſt falſum /
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vt patet ex nono correlario: igitur illud ex quo ſe-
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quitur: et per conſequens correlarium verū / quod
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fuit probandum. </
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quando aliquid componitur ex duobus inequali
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bus adequate: et minus illornm eſt ſubtriplū eius
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puta vna tertia illud minus eſt ſubduplum ad re-
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ſiduū puta ad duas tertias: et ſi illud ſit minꝰ quã
<
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tertia illius totius illud eſt minus quã ſubdupluꝫ
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ad totū reſiduū: ſed ſic eſt in propoſito per te igi-
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tur intentum.
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xml:id
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xml:space
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<
s
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xml:space
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">¶ Sequitur vndecimo / data qua-
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cun proportione ſuperparticulari denominata
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ab aliquo numero: omnis proportio ſuperparti-
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cularis denominata a numero excedente illū per
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vnitatem adequate eſt maior quã medietas illius
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proportionis date. </
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<
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">Patet hoc correlariū ex octa
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uo correlario: quia omnis talis denoīatur nu-
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mero minori quam duplo ad numerū a quo deno
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minatur data ſuperparticularis.
<
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xml:space
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<
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xml:space
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">¶ Sequit̄̄ duo-
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decimo / data naturali ſerie proportionum ſu-
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perticulariū puta ſexquialtera, ſexquitertia, ſex-
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quiquarta, et ſic deinceps: quelibet proportio ſu-
<
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perparticularis que denomīatur ab altero duo-
<
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rum numerorum īmediate ſequentium numerū a
<
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quo denominatur ſexquialtera eſt maior quã me-
<
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dietas ſexquialtere: et quelibet denominata ab a-
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liquo trium numerorum īmediate ſequentium nu
<
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meruꝫ a quo denominatur ſexquitertia eſt maior
<
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quã medietas ſexquitertie: et quelibet denomina-
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ta ab aliquo quatuor numerorū īmediate ſequē-
<
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tium numerū a quo denomīatur ſexquiquarta eſt
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maior quam medietas eius: et ſic in infinitū ſemꝑ
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addendo vnū. </
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libet talis denominatur a numero duplo vel mi-
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nori duplo ad numerū a quo denominatur data
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proportio ſuꝑparticularis / vt patet intuenti: igi-
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tur quelibet talis eſt maior quam medietas date
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proportionis ſuperparticularis. </
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quentia ex octauo correlario.</
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">Quibuſcū dua
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bus ꝓportiõibus inequalibus propoſitis: maior
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Capitulū quintū.
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illarū minorem per proportionē que eſt inter de-
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nominationes earum excedit: vt captis quadru-
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pla et tripla: quadrupla que eſt maior excedit tri-
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plam per proportionem que eſt inter .4. et .3. que
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eſt ſexquitertia. </
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<
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">Et hoc ideo / quia tripla denomi-
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natur a ternario quadrupla vero a quaternario
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tum.</
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">Et hic aduerte / aliud eſt dicere proportio qua-
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drupla excedit triplam per proportionem ſexqui
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tertiam: et ſe habet ad triplam in proportione ſex
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quitertia. </
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<
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">Nam ſexdecupla excedit octuplam per
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proportionem duplam: et ſe habet ad illã in pro-
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portione ſexquitertia / vt poſtea patebit. </
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<
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">Et hoc do
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cumentum debes memorie cõmendare ſi vis calcu
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latorem intelligere in capitulo ſcḋo de medio nõ
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reſiſtēte / qḋ ego voco de medio vniformiter diffor
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miter reſiſtente. </
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<
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">Probatur concluſio ſupponēdo
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primū vnū manifeſtum / quod probatione non in-
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diget: videlicet quacun quantitate continua
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ſignata ad eã poteſt dari omnis proportio poſſi-
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bilis capiendo maiorē quantitatem: quo ſuppo-
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ſito capio duas proportiones f. maiorem et g. mi-
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norem: et vtriuſ illarum proportionum minimū
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extremum ſit c. quantitas continua: et aliud ex-
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tremū f. proportionis ſit a. et aliud g. proportiõis
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ſit b. / ita proportio f. ſit a. ad c. et proportio g. ſit
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b. ad c. / et ſint illi primi termini illarum proporti-
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onū gratia argumenti: et tunc dico / proportio f.
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maior excedit proportionem g. per proportioneꝫ /
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que eſt inter denominationes illaruꝫ / hoc eſt inter
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terminos a quibus ille proportiones denominã-
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tur puta inter a. et .b. </
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">Quod ſic probatur / q2 f. pro-
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portio a. ad .c. maior componitur adequate ex ꝓ-
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portione a. ad b. et ex proportione b. ad c. que eſt g /
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vt patet ex ſecunda concluſione huius: igitur pro-
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portio a. ad c. continet adequate proportioneꝫ b.
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ad c. et vltra proportionē que eſt a. ad b. / igitur ꝓ-
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portio f. que eſt a. ad c. excedit proportionē g. que
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eſt b. ad c. per ꝓportionē que eſt a. ad b. / quod fuit
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probandum. </
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<
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">Illa eni3 eſt proportio inter primos
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terminos illarum proportionū a quibus ille pro
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portiones f. et g. denominantur.
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">1. correĺ.</
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<
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xml:space
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">¶ Ex hac conclu-
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ſione ſequitur primo / capto vno termino habē-
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te duas proportiones maioris inequalitatis ad
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duos terminos minores inequales / vt oportet: ꝓ-
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portio inter illos duos minores terminos eſt illa
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per quam maior proportio excedit minorē: vt ca-
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pto octonario numero habente proportioneꝫ ad
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ternariū et quaternariū: dico / ꝓportio octona-
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rii ad ternariū que eſt maior excedit proportionē
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octonarii ad quaternariū minorē per ꝓportionē
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que eſt inter quaternariū et ternariū. </
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<
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N13B50
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">Probatur /
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ſint due ꝓportiones puta f. ꝓportio que ſit a. ad
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c. et g. ꝓportio minor que ſit a. ad b. / et tūc ego dico /
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ꝓportio b. ad c. eſt illa per quã ꝓportio f. exce-
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dit ꝓportionē g. </
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<
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">Probatur / q2 ꝓportio f. cõponi-
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tur adequate ex ꝓportione a. ad b. et ex ꝓportione
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b. ad c. / vt patet ex ſecūda concluſione: igitur ꝓpor
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tio f. que eſt a. ad c. addit adequate ſupra ꝓportio
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nē g. que eſt a. ad b. ꝓportionē b. ad c. / et per conſe-
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quens f. ꝓportio excedit ꝓportionē g. ꝑ ꝓportio-
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nē b. ad c. adequate cū illaꝫ adequate addat vltra
<
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alteraꝫ / et illa videlicet b. ad c. eſt proportio que eſt
<
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inter terminos minores illarum duarum propor
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tionum inequalium / igitur correlarium verum.</
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<
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xml:id
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xml:space
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">2. correĺ.</
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>
<
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<
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xml:space
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">¶ Sequitur ſecundo / ſi duo numeri ſiue quanti-
<
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tates ſe habent in proportione tripla ſubquadru
<
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plum maioris eſt ſubſexquitertium minoris: et ſi
<
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duo numeri ſe habēt in ꝓportiõe dupla ſubq̈dru-
<
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plū maioris eſt ſubduplū minoris: quēadmodum </
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