Alvarus, Thomas, Liber de triplici motu, 1509

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              <p xml:id="N11F9F">
                <s xml:id="N11FCB" xml:space="preserve">
                  <pb chead="Prime partis" file="0023" n="23"/>
                uatiõis puta partes aliquote .b. et partes aliquo­
                  <lb/>
                te .d. / vt patet ex ſexta ſuppoſitione: et qualibet illa­
                  <lb/>
                rum in .b. equalis erit ipſi .a. quia antea erat equa­
                  <lb/>
                lis: ēt quelibet in .d. et equalis ipſi .c. eadē ratione /
                  <lb/>
                igitur .c. eſt pars aliquota .d. illius denominatio-
                  <lb/>
                nis cuius .a. eſt pars aliquota .b. / quod fuit probã-
                  <lb/>
                dum. </s>
                <s xml:id="N11FE7" xml:space="preserve">Et ſic patet: ſecunda pars ſuppoſitionis: et
                  <lb/>
                prima patet de ſe: quia vter talium numerorum
                  <lb/>
                habet ad talem partem aliquotam ſui ꝓportionē
                  <lb/>
                duplam q2 eſt ſua medietas </s>
                <s xml:id="N11FF0" xml:space="preserve">Continet eteuim eam
                  <lb/>
                bis: igitur ad eam habet proportionem duplam.
                  <lb/>
                </s>
                <s xml:id="N11FF6" xml:space="preserve">¶ Ex iſta ſuppoſitione ſequitur:  ſi vtra illaruꝫ
                  <lb/>
                quantitatum ſiue numerorum ſit diuiſorum in ꝑ-
                  <lb/>
                tes aliquotas eiuſdem denominationis ꝑdat vnã
                  <lb/>
                talē partē aliquotã adequate: eq̈le proportionem
                  <lb/>
                deꝑdit </s>
                <s xml:id="N12001" xml:space="preserve">Patꝫ / q2 eq̈lē ꝓportionē vter hab3 ad ag­
                  <lb/>
                gregatū ex oībꝰ dēpta vna / vt ptꝫ ex .8. ſuppoſitiõe
                  <lb/>
                et illam deperdit / vt conſtat igitur. </s>
                <s xml:id="N12008" xml:space="preserve">¶ Sequitur ſe­
                  <lb/>
                cundo /  ſi vter duorum numerorum ſit diuiſus
                  <lb/>
                in ꝑtes aliquotas eiuſdē denominatiõs: et acq̇rat
                  <lb/>
                vnã illarum partiū ſupra ſe p̄ciſe eq̈le ꝓportionē
                  <lb/>
                acquirit vter </s>
                <s xml:id="N12013" xml:space="preserve">Patet ex priori correlario. </s>
                <s xml:id="N12016" xml:space="preserve">q2 quã­
                  <lb/>
                do vter illorum illam partem deperdit equalem
                  <lb/>
                ꝓportionē deperdit / ergo quando acquirit equa-
                  <lb/>
                lem acquirit: igitur.</s>
              </p>
              <p xml:id="N1201F">
                <s xml:id="N12020" xml:space="preserve">Nona ſuppoſitio </s>
                <s xml:id="N12023" xml:space="preserve">Si duo numeri in­
                  <lb/>
                equales ſiue quantitates ſe habeant in aliqua ꝓ-
                  <lb/>
                portione: et maior illorum deperdat aliquam pro­
                  <lb/>
                portionem ſtante minori inuariato: tunc ꝓportio
                  <lb/>
                inter maiorē et minorē deꝑdit illã ꝓportionē quã
                  <lb/>
                deꝑdit maior adeq̈te. </s>
                <s xml:id="N12030" xml:space="preserve">dūmõ minor ſēꝑ maneat mi­
                  <lb/>
                nor. </s>
                <s xml:id="N12035" xml:space="preserve">vt ſi ꝓportionis q̄ eſt inter .8. et .4. maior nūe­
                  <lb/>
                rus puta octonariꝰ ꝑdat ꝓportionē ſexquitertiaꝫ
                  <lb/>
                que eſt octo ad ſex illam ꝓportionem deperdit ꝓ-
                  <lb/>
                portio que eſt inter octo et quattuor. </s>
                <s xml:id="N1203E" xml:space="preserve">Probatur / et
                  <lb/>
                ſint .a.b. numerus maior et .c. numerus minor in-
                  <lb/>
                ter quos ſit proportio .g. ſit .b. numerus maior
                  <lb/>
                c. / et manifeſtum eſt /  ꝓportio .a.b. ad .c. componi-
                  <lb/>
                tur ex proportione .a.b. ad .b et .b. ad .c. vt poſtea vi­
                  <lb/>
                debitur. </s>
                <s xml:id="N1204B" xml:space="preserve">Deperdat igitur numerus maior ꝓpor-
                  <lb/>
                tioneꝫ que eſt .a.b. ad .b. / et arguitur ſic / ꝓportio .g.
                  <lb/>
                componebatur antea ex proportione .a.b. ad .b.
                  <lb/>
                et .b. ad .c. modo non manet niſi proportio .b. ad .c. /
                  <lb/>
                igitur proportio .g. ꝑdit ꝓportionē ab. ad .b. et illã
                  <lb/>
                deperdat numerus maior / igitur.</s>
              </p>
              <p xml:id="N12058">
                <s xml:id="N12059" xml:space="preserve">Decima ſuppoſitio </s>
                <s xml:id="N1205C" xml:space="preserve">Si duo numeri
                  <lb/>
                ſiue quantitates inequales ſe habeant in aliqua
                  <lb/>
                proportione: et minor deperdat aliquam propor­
                  <lb/>
                tionem ſtante moiore: illam proportionem acqui­
                  <lb/>
                rit proportio que eſt inter maiorem quantitatem
                  <lb/>
                et minorem. </s>
                <s xml:id="N12069" xml:space="preserve">et ſi tantam proportionem deperdat
                  <lb/>
                quantitas maior ſicut minor: tunc proportio in-
                  <lb/>
                ter maiorem et minorem nec augetur nec diminui­
                  <lb/>
                tur: ſed ſemper manet equalis extremis manenti-
                  <lb/>
                bus quãtitatis. </s>
                <s xml:id="N12074" xml:space="preserve">vt ſi proportionis que eſt inter .8.
                  <lb/>
                et .quattuor. minor numerus perdat proportionē
                  <lb/>
                duplam ſtante maiore proportio inter maiorem
                  <lb/>
                et minorem acquirit proportionem duplaꝫ: et ſi qñ
                  <lb/>
                numerus minor perdit duplã etiaꝫ maior perdat
                  <lb/>
                duplã: illi numeri manebūt in eadem proportiõe
                  <lb/>
                in qua antea ſe habebant. </s>
                <s xml:id="N12083" xml:space="preserve">Erunt enim tn fine
                  <lb/>
                4. et .2. </s>
                <s xml:id="N12088" xml:space="preserve">Probatur prima pars ſuppoſitionis. </s>
                <s xml:id="N1208B" xml:space="preserve">et
                  <lb/>
                ſint .a. numerus maior et .b.c. numerus minor īter
                  <lb/>
                quos ſit proportio .g. et īuariato .a. perdat nume-
                  <lb/>
                rus minor proportionē que eſt .b.c. ad .c. / et manife­
                  <cb chead="Capitulum ſecundum"/>
                ſtum eſt /  in fine proportio īter illos numeros cõ-
                  <lb/>
                ponetur ex proportione .a. ad .b.c. et .b.c. ad .c. et an­
                  <lb/>
                tea proportio illa inter illos numeros puta .g. e-
                  <lb/>
                rat preciſe proportio .a. ad .b.c: et modo ꝓportio
                  <lb/>
                inter illos numeros cõponitur ex illa ꝓportione
                  <lb/>
                g. que eſt .a. ad .b.c. et ex proportione .b.c. ad .c. / ergo
                  <lb/>
                acquiſiuit ꝓportionē que eſt .b.c. ad .c. et illam de-
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                perdit quantitas minor .b.c. / igitur ꝓpoſitū. </s>
                <s xml:id="N120A5" xml:space="preserve">Se-
                  <lb/>
                cunda pars facile deducitur ex prima et penultīa
                  <lb/>
                ſuppoſitione: quoniam quantam ꝓportionem de­
                  <lb/>
                perdit quantitas minor tantam acquirit ꝓportio
                  <lb/>
                inter maiorem et minorem ſtante maiore: vt patet
                  <lb/>
                ex priori parte iſtius ſuppoſitionis: et quantam ꝓ­
                  <lb/>
                portionem deperdit quantitas maior tantam de­
                  <lb/>
                perdit proportio inter ipſam et minorē quantita­
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                tem ſtante minore: vt patet ex penultima: igitur ſi
                  <lb/>
                tantam ꝓportionem deꝑdat maior quantitas ſi-
                  <lb/>
                cut deperdit minor quantitas: proportio illa in­
                  <lb/>
                ter maiorē et minorem nullã proportionē acquirit
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                nec deperdit: et ſic in illas quantitates manet
                  <lb/>
                eadem proportio. </s>
                <s xml:id="N120C2" xml:space="preserve">¶ Ex quo ſequitur /  ſi tantam
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                proportioneꝫ adequate acquirat quãtitas minor
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                quantam acquirit quãtitas maior: ſemper mane­
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                bit eadem proportio. </s>
                <s xml:id="N120CB" xml:space="preserve">Probatur / quia ſi ille quan­
                  <lb/>
                titates illas proportiones equales quas acquiſi-
                  <lb/>
                uerunt deperdant manebunt in eadem proportio­
                  <lb/>
                ne in qua modo ſe habent: et illa eſt proportio in
                  <lb/>
                qua ſe habebant ante acquiſitionem illarum pro­
                  <lb/>
                portionum equaliū: igitur quando quãtitates ac­
                  <lb/>
                quirunt ꝓpportiones equales ipſe manet in eadē
                  <lb/>
                proportione in qua ſe habebant antea.</s>
              </p>
              <p xml:id="N120DC">
                <s xml:id="N120DD" xml:space="preserve">Undecima ſuppoſitio. </s>
                <s xml:id="N120E0" xml:space="preserve">Quecū pro­
                  <lb/>
                portio eſt inter aliquos numeros ſiue quãtitates
                  <lb/>
                talis eſt inter partes aliquotas conſimilis deno-
                  <lb/>
                minationis. </s>
                <s xml:id="N120E9" xml:space="preserve">vt qualis eſt proportio inter .8. et .4.
                  <lb/>
                talis eſt intermedietatē .8. et medietateꝫ .4. et quar­
                  <lb/>
                tam .8. et quartam .4. </s>
                <s xml:id="N120F0" xml:space="preserve">Probatur / ſint duo numeri
                  <lb/>
                primus .a.b.c. ſecundus .d.e.f. diuiſi in partes ali-
                  <lb/>
                quotas eiuſedem denominationis puta primus in
                  <lb/>
                a.b.c. et ſecundus in .d.e. et .f. / tunc dico /  qualis eſt
                  <lb/>
                proportio .a.b.c. ad .d.e.f. talis eſt .c. ad .f. </s>
                <s xml:id="N120FB" xml:space="preserve">Quod ꝓ­
                  <lb/>
                batur ſic. </s>
                <s xml:id="N12100" xml:space="preserve">et ſit inter illos numeros ſiue quantita­
                  <lb/>
                tes .g. ꝓportio: et deperdat numerus maior .a. per­
                  <lb/>
                tem aliquotam et minor .d. partem aliquotam cõ­
                  <lb/>
                ſimilis denominationis: et manifeſtum eſt /  quã­
                  <lb/>
                tam proportionem deperdit numerus maior tan­
                  <lb/>
                tam deperdit numerus minor / vt patet ex prīo cor­
                  <lb/>
                relario octaue ſuppoſitionis / ergo reſidui numeri
                  <lb/>
                adhuc manent in eadē proportione puta .g. </s>
                <s xml:id="N12111" xml:space="preserve">Pa-
                  <lb/>
                tet conſequentia ex ſeunda parte decime ſuppoſi­
                  <lb/>
                tionis: et reſidui numeri puta .b.c. et .e.f. adhuc ma­
                  <lb/>
                nent diuiſi in partes aliquotas eiuſdem denomi-
                  <lb/>
                nationis / vt patet ex ſexta ſuppoſitiõe: perdat igi­
                  <lb/>
                tur numerus maior .b. partem aliquotam et nume­
                  <lb/>
                rus minor .e. partem aliquotam: et ſequitur /  eq̈­
                  <lb/>
                lē ꝓportionē deperdit nūerꝰ maior et nūerꝰ minor /
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                vt iã argutū eſt: ergo reſidui numeri manent in ca­
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                dem proportione in qua antea ſe habebant puta
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                g. / vt patet ex ſecunda parte decime ſuppoſitionis
                  <lb/>
                et reſidui numeri ſunt .c. et .f. / ergo c. et .f. ſe habent
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                in .g. proportione et .c. et .f. ſunt ꝑtes aliquote eiuſ­
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                dem denominationis datorum numerorum ſe ha­
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                bentium in .g. proportione: igitur in quacun por­
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                portione ſe habent alique quantitaters in eadem
                  <lb/>
                ſe habent ſue partes aliquote eiuſdem denomina­
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                tionis / quod fuit probandum. </s>
                <s xml:id="N12136" xml:space="preserve">¶ Et hac ſuppoſi- </s>
              </p>
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