Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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              <p xml:id="N12BEF">
                <s xml:id="N12BFB" xml:space="preserve">
                  <pb chead="Secūde partis" file="0030" n="30"/>
                a duodenario ſe habeat in proportio dupla a
                  <lb/>
                ſenario: continuo illud quod remanet ex duode-
                  <lb/>
                nario ſe habet in proportione dupla ad illud qḋ
                  <lb/>
                remanet a numero ſenario. </s>
                <s xml:id="N12C0B" xml:space="preserve">Et ſub tenore huiꝰ exē­
                  <lb/>
                pli ego intelligo correlarium </s>
                <s xml:id="N12C10" xml:space="preserve">Non enī in iſtis exa­
                  <lb/>
                ctꝰ ſenſus dialecticus ex expetendus ſed ipſa ma­
                  <lb/>
                thematica ſententia eſt efflagitanda. </s>
                <s xml:id="N12C17" xml:space="preserve">Hoc correla­
                  <lb/>
                rium perinde at primum demonſtrationem con­
                  <lb/>
                cluſionis exquirit. </s>
                <s xml:id="N12C1E" xml:space="preserve">Applica vt vales.</s>
              </p>
              <note position="left" xml:id="N12C21" xml:space="preserve">4. correĺ.</note>
              <p xml:id="N12C25">
                <s xml:id="N12C26" xml:space="preserve">¶ Sequitur quarto /  quandocun duo numeri ī­
                  <lb/>
                equales continuo creſcunt: et continuo ſe habent
                  <lb/>
                in eadem proportione: oportet /  continuo acqui­
                  <lb/>
                ſitū maiori numero ſe habeat in eadeꝫ proportio­
                  <lb/>
                ne ad acquiſitum minori in qua ſe habent illi nūe­
                  <lb/>
                ri creſcentes. </s>
                <s xml:id="N12C33" xml:space="preserve">vt ſi numerus quaternarius et ſena-
                  <lb/>
                rius continuo creſcant et continuo manent in pro­
                  <lb/>
                portione ſexquialtera: oportet /  continuo acqui­
                  <lb/>
                ſitum ſenario ſe habeat in proportione ſexquial-
                  <lb/>
                tera ad acquiſitum quaternario. </s>
                <s xml:id="N12C3E" xml:space="preserve">Hoc correlariuꝫ
                  <lb/>
                eadem cum precedentibus demonſtratione oſten-
                  <lb/>
                ditur. </s>
                <s xml:id="N12C45" xml:space="preserve">¶ Sequitur quinto /  datis quibuſcun
                  <lb/>
                duobus numeris inequalibus ſe habentibus ī ali­
                  <lb/>
                q̄ ꝓportiõe et ī ea ꝓportiõe ī q̄ mīor excedit̄̄ a maio­
                  <lb/>
                re ī eadē cõtinuo tardiꝰ creſcat maiore: cõtinuo ta­
                  <lb/>
                les numeri manent in eadem proportione. </s>
                <s xml:id="N12C50" xml:space="preserve">vt da-
                  <lb/>
                tis: 4. et .6. ſe habentibus in proportione ſexqui-
                  <lb/>
                altera: ſi quando ſex acquiſiuerint aliquod cremē­
                  <lb/>
                tum. </s>
                <s xml:id="N12C59" xml:space="preserve">quatuor acquirant in ſexquialtero minus: ip­
                  <lb/>
                ſi continuo manent in proportione ſexquialtera.
                  <lb/>
                </s>
                <s xml:id="N12C5F" xml:space="preserve">Probatur hoc correlarium / quoniam ſi in eadem
                  <lb/>
                proportione in qua numerꝰ maior ſe habet ad mi­
                  <lb/>
                norem velocius creſcat quaꝫ minor: ſequitur /  cõ­
                  <lb/>
                tinuo inter acquiſitum minori numero eſt eadem
                  <lb/>
                proportio que eſt inter illos numeros. </s>
                <s xml:id="N12C6A" xml:space="preserve">vt patet ex
                  <lb/>
                probatioue concluſionis: et per conſequens con-
                  <lb/>
                tinuo tales numeri manent in eadem proportione
                  <lb/>
                </s>
                <s xml:id="N12C72" xml:space="preserve">Et ſic patet correlarium</s>
              </p>
              <p xml:id="N12C75">
                <s xml:id="N12C76" xml:space="preserve">Sexta concluſio </s>
                <s xml:id="N12C79" xml:space="preserve">Datis tribus nu-
                  <lb/>
                meris in hac medietate conſtitutis: quod fit ex du­
                  <lb/>
                ctu extremi in extremum equale eſt quadrato me-
                  <lb/>
                dii: hoc eſt illi numero qui reſultat ex ductu medii
                  <lb/>
                termiui in ſeipſum. </s>
                <s xml:id="N12C84" xml:space="preserve">vt conſtitutis his tribus termi­
                  <lb/>
                nis .8.4.2. numerus ſexdenarius reſultans ex du-
                  <lb/>
                ctu octonarii in binarium eſt equalis numero qui
                  <lb/>
                fit ex ductu quaternarii in ſeipſū / vt conſtat. </s>
                <s xml:id="N12C8D" xml:space="preserve">Pro­
                  <lb/>
                batur hec concluſio / ſint tres numeri a.b.c. in hac
                  <lb/>
                medietate conſtituti continuo ꝓportionabiles .g.
                  <lb/>
                ꝓportione: et ſit d. numerus reſultãs ex ductu a. in
                  <lb/>
                b. et e. ſit numerus reſultãs ex ductu b. in idē b. et f.
                  <lb/>
                numerus reſultans ex ductu a. in c. / tunc dico /  e. et
                  <lb/>
                f. ſunt equales. </s>
                <s xml:id="N12C9C" xml:space="preserve">Qḋ ſic ꝓbatur: qm̄ d. ad e. eſt ꝓpor­
                  <lb/>
                tio g. et d. ad f. eſt eadē ꝓportio g. / ergo e. et f. ſunt
                  <lb/>
                equalia / quod fuit ꝓbandū. </s>
                <s xml:id="N12CA3" xml:space="preserve">Patet conſequētia et
                  <lb/>
                maior oſtenditur. </s>
                <s xml:id="N12CA8" xml:space="preserve">quia ſicut ſe hꝫ d. ad a. ita ſe ha-
                  <lb/>
                bet e. ad .b. q2 toties adeq̈te a. cõtinet̄̄ in d. quoties
                  <lb/>
                eſt vnitas in b. et toties continetur b. in e. quoties
                  <lb/>
                eſt vnitas in b. cum d. fiat ex ductu a in b. et e. ex du­
                  <lb/>
                ctu b. in b. / igit̄̄ ſicut ſe habet d. ad a. ita e. ad b. </s>
                <s xml:id="N12CB3" xml:space="preserve">Con­
                  <lb/>
                ſequētia claret ex tertia ſuppoſitione huius capi-
                  <lb/>
                tis: et ex conſequēti ſicut ſe hꝫ d. ad a. ita e. ad b: er­
                  <lb/>
                go ſicut ſe habet d. ad e. ita ſe habet a. ad b. ſed a.
                  <lb/>
                ad b. eſt g. proportio / ergo .d. ad e. eſt g. ꝓportio / qḋ
                  <lb/>
                fuit ꝓbandū. </s>
                <s xml:id="N12CC0" xml:space="preserve">Patet igitur maior. </s>
                <s xml:id="N12CC3" xml:space="preserve">Iã probat̄̄ mi-
                  <lb/>
                nor. </s>
                <s xml:id="N12CC8" xml:space="preserve">q2 d. in g. ꝓportione pluries cõtinet a. quaꝫ f.
                  <lb/>
                contineat idē a. adeq̈te: ergo d. ſe habet ad f. in g.
                  <lb/>
                ꝓportione </s>
                <s xml:id="N12CCF" xml:space="preserve">Patet conſequentia ex tertia ſuppoſi­
                  <lb/>
                ctione p̄allegata. </s>
                <s xml:id="N12CD4" xml:space="preserve">Probatur antexedens / q2 d. toti­
                  <lb/>
                es continet a. quoties eſt vnitas in b. cuꝫ a. in b. du­
                  <lb/>
                catur et inde reſultat d. et f. toties continet a quo
                  <cb chead="Capitulum ſecundum"/>
                ties eſt vnitas in c. eadē rõue: ſꝫ in g. ꝓportiõe plu­
                  <lb/>
                ries continetnr vnitas in b. quã in c. cū b. et c. ſe ha­
                  <lb/>
                beant in g. ꝓportione: ergo in g. ꝓportione pluri­
                  <lb/>
                es cõtinetur a. in d. quã in f. / qḋ fuerat oñdendū. </s>
                <s xml:id="N12CE4" xml:space="preserve">Et
                  <lb/>
                ſic patet cõclnſio q̄ ꝓfecto pulchra eſt et induſtria q̄
                  <lb/>
                ſit huius medietatis. </s>
                <s xml:id="N12CEB" xml:space="preserve">ſexta proprietas.
                  <note position="right" xlink:href="note-0030-01a" xlink:label="note-0030-01" xml:id="N12E33" xml:space="preserve">1. correĺ.</note>
                </s>
                <s xml:id="N12CF3" xml:space="preserve">¶ Ex hac
                  <lb/>
                concluſione ſeq̇tur ṗmo /  ī hac medietate id quod
                  <lb/>
                fit ex ductu vnius extremi ad triū termīorū alterū
                  <lb/>
                extremū eſt numerꝰ quadratꝰ: </s>
                <s xml:id="N12CFC" xml:space="preserve">Probatur / q2 talis
                  <lb/>
                numerus eſt equalis quadrato medii termini / g̊ eſt
                  <lb/>
                numerus quadratus </s>
                <s xml:id="N12D03" xml:space="preserve">Cõſequētia patet de ſe et añ­
                  <lb/>
                cedens ex concluſione.
                  <note position="right" xlink:href="note-0030-02a" xlink:label="note-0030-02" xml:id="N12E39" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N12D0D" xml:space="preserve">¶ Sequitur ſecundo /  ſi cõ­
                  <lb/>
                ſtitutis duobus numeris ſe habentibus in aliqua
                  <lb/>
                ꝓportione maioris ineqnalitatis rationali. </s>
                <s xml:id="N12D14" xml:space="preserve">nūe-
                  <lb/>
                rus q̇ fit ex ductu vnius extremi in alterū non eſt q̈­
                  <lb/>
                dratꝰ: inter tales termīos nõ eſt medium ꝓportio­
                  <lb/>
                nabile ꝓportione rationali: ita  ṗmi ad illḋ me-
                  <lb/>
                diū ſit eadē ꝓportio rationalis que eſt illius me-
                  <lb/>
                dii ad tertiuꝫ. </s>
                <s xml:id="N12D21" xml:space="preserve">Probatur hoc correlarium / qnia ſi
                  <lb/>
                inter tales numeros reperiatur mediū ꝓportiõa-
                  <lb/>
                bile ꝓportione rationali: puta aliquis numerus
                  <lb/>
                medio loco proportionabilis: iam ſequitur /  ibi­
                  <lb/>
                dē reperiuntur tres numeri cõtinuo ꝓportionabi­
                  <lb/>
                les hac medietate: et ꝑ cõſequēs numerꝰ q̇ fit ex du­
                  <lb/>
                ctu extremi in extremum eſt equalis quadrato me­
                  <lb/>
                dii / vt patet ex coucluſione: igitur talis numerus ē
                  <lb/>
                quadratus / vt patet ex primo correlario / quod eſt
                  <lb/>
                oppoſitū añcedetis correlari ꝓbãdi. </s>
                <s xml:id="N12D36" xml:space="preserve">īfert igir̄ cor­
                  <lb/>
                relarii oppoſitū conſequentis oppoſitū añceden-
                  <lb/>
                tis / et ꝑ conſequēs correlarium verum.
                  <note position="right" xlink:href="note-0030-03a" xlink:label="note-0030-03" xml:id="N12E3F" xml:space="preserve">3. correĺ.</note>
                </s>
                <s xml:id="N12D42" xml:space="preserve">¶ Sequitur
                  <lb/>
                tertio /  ſi medium proportionabile īter duos nu­
                  <lb/>
                meros ſe habētes in proportione maioris inequa­
                  <lb/>
                litatis nõ ſit latus numeri contenti ſub extremis:
                  <lb/>
                tunc numerus qui fit ex ductu vnius extremi in al-
                  <lb/>
                terū nõ eſt quadratus. </s>
                <s xml:id="N12D4F" xml:space="preserve">Probatur / ſint a.c. duo nu­
                  <lb/>
                meri ſe habentes in proportione maioris inequa­
                  <lb/>
                litatis a. maior c. minor: et numerus qui fit ex du­
                  <lb/>
                ctu a. in c. ſit d. et e. ſit medium ꝓpõrtionale inter a
                  <lb/>
                et c. / tunc dico /  ſi e. non ſit latus ipſius d: d. nõ eſt
                  <lb/>
                numerus quadratus. </s>
                <s xml:id="N12D5C" xml:space="preserve">Quod ſic oñditur: q2 ſi d: ſit
                  <lb/>
                numerus quadratus ſequitur /  eius latus eſt e. /
                  <lb/>
                igitur ex oppoſito ſequitur oppoſitum: et per con­
                  <lb/>
                ſequens correlarium verum. </s>
                <s xml:id="N12D65" xml:space="preserve">Probatur antece-
                  <lb/>
                dens / quia ſi d. eſt numerus quadratus cum nõ ſit
                  <lb/>
                quadratus a. nec quadratus ipſius c. / vt conſtat:
                  <lb/>
                qm̄ quando duo numeri inequales in ſeipſos du­
                  <lb/>
                cuntur quod inde ſit neutrius illoꝝ eſt quadratuꝫ:
                  <lb/>
                ſed eſt alicuius numeri minoris maiore illorum et
                  <lb/>
                maioris minore: ſit igitur talis numerus b. cuius
                  <lb/>
                d. eſt quadratum / et ſequitur /  a. ad b. eſt aliqua ꝓ­
                  <lb/>
                portio: conſtituo igitur tres terminos continuo ꝓ­
                  <lb/>
                portionabiles illa proportione a. ad b. que ſint a.
                  <lb/>
                b.h. / et ſequitur ex cõcluſione /  numerus qui fit ex
                  <lb/>
                ductu a. in h. eſt equalis ipſi d. et per te numerꝰ qui
                  <lb/>
                fit ex ductu a. in c. eſt equalis ipſi d. </s>
                <s xml:id="N12D80" xml:space="preserve">Imo eſt ipſum
                  <lb/>
                d. / igitur h. et c: ſunt numeri equales. </s>
                <s xml:id="N12D85" xml:space="preserve">Patet hec cõ­
                  <lb/>
                ſequentia / q2 ex ductu vuius tertii in vtrū illorū
                  <lb/>
                reſultat idem numerus. </s>
                <s xml:id="N12D8C" xml:space="preserve">et ſic tot vnitates continet
                  <lb/>
                c. ſicut h. / et per conſequens ſunt equales. </s>
                <s xml:id="N12D91" xml:space="preserve">ſed inter
                  <lb/>
                a. et h. eſt mediū ꝓportionale quod eſt latus qua-
                  <lb/>
                drati quod fit ex ductu a. in h. quod latus eſt b. / igi­
                  <lb/>
                tur inter a. et c. eſt mediū ꝓportionale quod eſt la-
                  <lb/>
                tus quadrati quod fit ex ductu a. in h. / et per conſe­
                  <lb/>
                quens medium e. inter a. et c. eſt latus numeri d. q̇
                  <lb/>
                fit ex ductu a. in c. / quod fuit probandum. </s>
                <s xml:id="N12DA0" xml:space="preserve">Et ſic pa­
                  <lb/>
                tet correlarium.
                  <note position="right" xlink:href="note-0030-04a" xlink:label="note-0030-04" xml:id="N12E45" xml:space="preserve">4. correĺ.</note>
                </s>
                <s xml:id="N12DAA" xml:space="preserve">¶ Sequitur quarto /  conſtitutis
                  <lb/>
                duobus terminis ſe habentibus in aliqua ꝓpor-
                  <lb/>
                tione maioris inequalitatis rationali ſi numerus
                  <lb/>
                qui fit ex ductu vnius extremi in alterum ſit qua- </s>
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