Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secunde partis" file="0036" n="36"/>
                Probatur prima pars: quia ſemper vter extre-
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                morum acquirit equalē proportionē: igitur con-
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                tinuo inter ea manet eadem proportio. </s>
                <s xml:id="N136EE" xml:space="preserve">Secunda
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                pars probatur: quia continuo manet eadem pro-
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                portio inter medium et tertium continuo etiam
                  <lb/>
                manet eadem roportio que antea erat inter ſecun­
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                dum et tertium eadem ratione qua inter extrema
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                manet eadem proportio: igttur continuo illi ter-
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                mini manent proportionabiles arithmetice.</s>
              </p>
              <p xml:id="N13725">
                <s xml:id="N13726" xml:space="preserve">Patet conſequentia ex precedenti correlario.
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                </s>
                <s xml:id="N1372A" xml:space="preserve">Tertia autem ſic probatur: quia ſemper illi ex-
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                ceſſus cõtinuo manent partes aliquote cõſimilis
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                denominationis ſuorū numerorū: igitur in ea ꝓ-
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                portione qua numeri fiunt maiores et illi exceſſus
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                etiã fiūt maiores: quia ſunt partes aliquote illoꝝ
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                numerorū eiuſdē denominationis. </s>
                <s xml:id="N13737" xml:space="preserve">Et ſic patet cor­
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                relariū.
                  <note position="left" xlink:href="note-0036-01a" xlink:label="note-0036-01" xml:id="N13A78" xml:space="preserve">4. correĺ
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                  Calcu. in
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                  prīcipio
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                  de ītē. ele.</note>
                </s>
                <s xml:id="N13741" xml:space="preserve">¶ Sequitur quarto:  ſi ſint tres termini
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                arithmetice ꝓportionabiles: et ſtante maximo il-
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                lorū īuariato deſcreſcat minimus illoꝝ ſucceſſiue:
                  <lb/>
                ita  cõtinue illi tres maneant arithmetice ꝓpor-
                  <lb/>
                tionabiles: neceſſe eſt mediū in duplo tardius cõ-
                  <lb/>
                tinuo decreſcere minimo: neceſſe quo eſt ꝓporti-
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                onē extremi ad extremū continuo augeri: vt datis
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                his tribus terminis .12.8.4. et ſtantibus .12. decre­
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                ſcant .4. perdendo binariū: ſi illi tres termini de-
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                beant cõtinuo manere arithmetice ꝓportionabi-
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                les: neceſſe eſt numerū mediū perdere vnitatē: et ſic
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                manebunt arithmetice ꝓportiõabiles. </s>
                <s xml:id="N1375A" xml:space="preserve">Manebūt
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                em̄ .12.7.2. et manebit maior ꝓportio quã erat an­
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                tea inter extrema. </s>
                <s xml:id="N13761" xml:space="preserve">Probatur / et ſint a.b.c. tres ter-
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                mini arithmetice ꝓportionabiles a. maximus c.
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                vero minimus: et perdat c. vnã partē ſui que ſit d.
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                et medietas d. ſit e. / et tunc dico /  cum c. perdit d.b.
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                perdit e. adequate. </s>
                <s xml:id="N1376C" xml:space="preserve">Quod ſic ꝓbatur: quoniã illi
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                tres termini cõtinuo manēt ꝓportiõabiles arith-
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                metice: igitur medium inter extrema eſt medietas
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                aggregati et extremis vt ex ſuperioribus conſtat:
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                ſed facta tali diminutiõe aggregatū ex extremis
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                eſt minus per d. latitudinē quã antea: quia illam
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                perdit adequate: igitur medietas illius aggrega­
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                ti effecta eſt minor per medietatē illius quod per-
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                dit totū puta per medietatē ipſiꝰ d: ſed medietas
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                ipſius d. eſt e. / igitur medietas illius aggregati fa­
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                cta eſt minor per e. adeq̈te: et illa medietas eſt me-
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                diū inter illa extrema: igitur medietas inter illa
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                extrema perdidit e. / quod fuit probandū. </s>
                <s xml:id="N13787" xml:space="preserve">Secūda
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                vero pars patet ex priori parte decime ſuppoſiti-
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                onis ſecundi capitis huius: quoniã numerus mi-
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                nor creſcit ſtante maiore. </s>
                <s xml:id="N13790" xml:space="preserve">Et hec eſt quedã ſuppo-
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                ſitio quã ponit: et aliter probat calculator in prin­
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                cipio capituli de intenſione elementi.
                  <note position="left" xlink:href="note-0036-02a" xlink:label="note-0036-02" xml:id="N13A84" xml:space="preserve">5. correĺ.</note>
                </s>
                <s xml:id="N1379C" xml:space="preserve">¶ Sequitur
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                quinto /  oīs ꝓportio cõponitur ex duabus pro-
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                tionibus puta maximi termini ad mediū: et medii
                  <lb/>
                ad minimū: et proportio maximi ad mediū minor
                  <lb/>
                eſt quã ſubdupla ad ipſam que eſt extremi ad ex-
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                tremū: et proportio medii termini ad minimū ma­
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                ior eſt quam ſubdupla: vt proportio ſexquialtera
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                que eſt .6. ad .4. cõponitur ex proportione .6. ad .5
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                et .5. ad .4. et proportio .6. ad .5. minor eſt quã ſub-
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                dupla: et .5. ad .4. maior eſt quã ſubdupla ad ſex-
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                quialterã. </s>
                <s xml:id="N137B3" xml:space="preserve">Prima pars huius patet ex concluſiõe /
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                et ſecūda probatur: quia omne cõpoſitū adequate
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                ex duobus inequalibus eſt maius quam duplum
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                ad minus illorum: et minus quam duplum ad ma­
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                ius illorum / vt patet ex ſexta ſuppoſitione huius
                  <lb/>
                ſed omnis proportio componitur ex duabus pro­
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                portionibus inequalibus quarum minor eſt ma-
                  <cb chead="Capitulū quartū"/>
                oris extremi ad medium: et maior medii ad mini-
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                mum extremum: vt patet ex eadem cõcluſione: igi-
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                tur omnis proportio eſt maior quãdupla ad pro-
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                portionem que eſt maioris extremi ad medium: et
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                minor quam dupla ad proportionem quē eſt me-
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                dii termini ad minimum extremum. </s>
                <s xml:id="N137CF" xml:space="preserve">Patet conſe­
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                quentia in primo prime: et ſic patet correlarium.
                  <lb/>
                  <note position="right" xlink:href="note-0036-03a" xlink:label="note-0036-03" xml:id="N13A8A" xml:space="preserve">6. correĺ.</note>
                </s>
                <s xml:id="N137DB" xml:space="preserve">¶ Sequitur ſexto:  omnis proportio ſuperpar-
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                ticularis componitur ex duabus quarum vna eſt
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                maximi termini ad medium: et alia eſt medii ad mi­
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                nus extremum: et vtra illarum eſt ſuperparticu-
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                laris: et proportio medii ad minimum demonina-
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                tur a parte aliquota denominata a numero du-
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                plo ad numerū a quo denominatur pars aliquo-
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                ta a qua denoīatur ꝓportio maximi ad minimū:
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                et ꝓportio maximi termini ad medium denoīatur
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                a parte aliquota denominata a numero īmedia-
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                te ſequente numerum illum duplum: vt proportio
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                ſexquialtera que eſt .6. ad .4. cõponitur ex duabꝰ
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                inequalibus / vt dictum eſt: et vtra illarum eſt ſu-
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                perparticularis. </s>
                <s xml:id="N137F8" xml:space="preserve">Nam proportio .6. ad .5. eſt ſu-
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                perparticularis et .5. ad .4. ſimiliter: et proportio
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                que eſt .5. ad .4. denomīatur a quarta que eſt pars
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                aliquota denominata a numero in duplo maiore
                  <lb/>
                quam ſit numerus a quo denominatur medietas
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                a qua medietate denominatur ſexquialtera. </s>
                <s xml:id="N13805" xml:space="preserve">De-
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                nominatur enim medietas a binario, et quarta a
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                quaternario, et quinta denominatur a quinario
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                qui eſt numerus ſequens immediate quaternariū
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                </s>
                <s xml:id="N1380F" xml:space="preserve">Probatur prima pars huius ex correlario imme­
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                diate precedenti: et ſecunda probatur / et quia om-
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                nis proportio ſuperparticularis reperitur inter
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                duos numeros immediatos: vt patet ex eius gene­
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                ratione poſita in prima parte: capio igitur vnam
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                proportionem ſuperparticularem que ſit f. et du-
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                os terminos eius in numeris immediatos: puta
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                a. maiorem: et c. minorem: et tunc dico /  propor-
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                tio ſuperparticularis inter illos duos numeros
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                immediatos cõponitur adequate ex duabus pro-
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                portionibus ſuperparticularibus: ex vna videli-
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                cet que eſt maximi ad medium: et altera que eſt me­
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                dii ad extremum. </s>
                <s xml:id="N1382A" xml:space="preserve">Probatur quoniam cum a. et c.
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                ſunt nnmeri immediati: et a. maior: ſequitur /  a.
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                excedit c. per vnitatem: dupletur igitur tam c. quã
                  <lb/>
                a. / et manifeſtum eſt /  inter illos duos numeros
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                duplatos manet eadeꝫ proportio que erat antea
                  <lb/>
                puta f. / vt patet ex correlario decime ſuppoſitio-
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                nis ſecundi capitis huius: igitur exceſſus maioris
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                termini. </s>
                <s xml:id="N1383B" xml:space="preserve">ſic duplati ad minorem etiam ſit dupla-
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                tum erit in duplo maior: vt patet ex tertio corre-
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                lario huius concluſionis: et antea erat vnitas / er-
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                go modo eſt dualitas: et per conſequens inter nu­
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                merum maiorem ipſius proportionis f. et nume-
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                rum minorem mediat numerus excedens minimū
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                illorum per vnitatem: et qui exceditur maximo
                  <lb/>
                illorum per vnitatem. </s>
                <s xml:id="N1384C" xml:space="preserve">Patet hec conſequentia /
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                quia omnis numerus excedens alterum per dua-
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                litatem diſtat ab eo per vnum numerum tantum
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                in naturali ſerie numerorum / vt ſatis conſtat: ſit
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                igitur talis numerus medius b. / et ſequitur /  ma-
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                ximi termini illius proportionis f. ſuperparticu-
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                laris date ad ipſum b. eſt proportio ſuperparti-
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                cularis: et ipſius b. ad minimum extremum eiuſ-
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                dem proportionis f. eſt etiam proportio ſuper-
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                particularis: quia illi tres numeri ſunt imme-
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                diati / igitur illa proportio f. ſuperparticularis </s>
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