Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secunde partis" file="0029" n="29"/>
                lia demas etc. ſed modo manet proportio a. ad b.
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                et c. ad .d. / ergo ille ſunt equales / quod fuit ꝓbãduꝫ
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                </s>
                <s xml:id="N12A07" xml:space="preserve">Sꝫ vniuerſaliter probatur /  ſi ſicut ſe hꝫ a.b. ad b.
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                ita .cd. ad d. / tūc ſic̈ ſe hꝫ a. ad b. ita c. ad .d. </s>
                <s xml:id="N12A0C" xml:space="preserve">Qḋ ſic
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                probatur / qr. ſicut ſe hꝫ a.b. ad b. ita c.d. ad d. / ergo
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                ſicut ſe habet a.b. ad c.d. ita b. ad .d. / vt patet ex cõ­
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                cluſione. </s>
                <s xml:id="N12A15" xml:space="preserve">Uolo igit̄̄ /  a.b. ꝑdat .b. et c.d. ꝑdat d. ita­
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                 maneãt a. et c. / et tūc arguo ſic / a.b. et c.d. ſe habēt
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                in ea proportione in qua ſe habent b. et d. q̄ ſit f.
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                gr̄a argumenti: et a.b. terminus maior deperdit
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                d. et c.d. terminꝰ minor deperdit d. / ergo inter de-
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                perditum a maiori termino et deꝑditū a minori ē
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                ꝓportio f. puta īter b. et d. et talis ꝓportio puta f.
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                eſt īter a.b. et c.d. / vt ꝓbatū eſt: igit̄̄ facta tali deꝑ-
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                ditione vel diminutione inter reſiduū ex a.b. et re-
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                ſiduū ex c.d. manet ꝓportio f. / vt ptꝫ ex ſeptīo cor-
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                relario quarte cõcluſionis octaui capitis huiꝰ par­
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                tis: et reſiduū ex a.b. ē a: et reſiduū ex c.d. eſt c. / igit̄̄
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                īter a. et c. eſt f. ꝓportio ſicut inter .b. et d. et ꝑ ↄ̨ñs ſi­
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                cut ſe hꝫ a. ad c. ita b. ad d. puta in f. ꝓportione: et
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                ex cõſequēti ſeq̇tur ex cõcluſione /  ſicut ſe habet a
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                ad b. ita c. ad d. / qḋ fuit probandū. </s>
                <s xml:id="N12A36" xml:space="preserve">Et eodē mõ pro­
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                bares ſi a. eēt terminꝰ minor et b. maior. </s>
                <s xml:id="N12A3B" xml:space="preserve">et ēt c. mi­
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                nor et d. maior.
                  <note position="left" xlink:href="note-0029-01a" xlink:label="note-0029-01" xml:id="N12AF8" xml:space="preserve">5. correĺ.</note>
                </s>
                <s xml:id="N12A45" xml:space="preserve">¶ Seq̇tur quīto /  diſpoſitis ī hac
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                medietate q̈tuor terminis: ſicut aggregatū ex q̈r-
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                to et tertio ad tertiū ita aggregatum ex ſecūdo et
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                prīo ad primū vt diſpoſitis his termīs .8.4.6.3. ſi­
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                cut ſe hñt 3. et .6. ad .6 ita .4. et .8. ad .8. </s>
                <s xml:id="N12A50" xml:space="preserve">Probat̄̄ /
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                ſint 4 ṫmini ī hac medietate ↄ̨ſtituti a.b.c.d. / tūc
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                ſicut ſe habet d.c. ad c. ita b.a. ad a. </s>
                <s xml:id="N12A57" xml:space="preserve">Qḋ ſic probat̄̄ /
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                q2 bñ ſeq̇tur ſicut ſe habet a. ad b. ita c. ad d. / igitur
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                ſicut ſe habet a.b. ad b. ita ſe habet c.d. ad d. / vt ptꝫ
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                ex ṗmo correlario huiꝰ concluſionis: et vltra ſicut
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                ſe habet a.b. ad b. ita c.d. ad d. / igit̄̄ ſicut ſe hꝫ d. ad
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                d.c. ita b. ad b.a. / quod fuit ꝓbandū. </s>
                <s xml:id="N12A64" xml:space="preserve">Ptꝫ hec cõſe-
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                quētia ex ꝓbatione tertii correlarii huius ↄ̨cluſio­
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                nis. </s>
                <s xml:id="N12A6B" xml:space="preserve">Et ſic patet correlariū.
                  <note position="left" xlink:href="note-0029-02a" xlink:label="note-0029-02" xml:id="N12AFE" xml:space="preserve">6. correl.</note>
                </s>
                <s xml:id="N12A73" xml:space="preserve">¶ Sequitur ſexto /  diſ­
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                poſitis .3. terminis cõtinuo ꝓportionabilibus hac
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                medietate: et aliis tribus etiã cõtinuo ꝓportiona-
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                bilibꝰ eadē medietate: et eadē ꝓportione qua tres
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                priores cõtinuo proportionant̄̄: ſicut ſe habēt ex-
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                trema ṗmi ternarii: ita ſe habēt extrema ſecundi.
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                </s>
                <s xml:id="N12A81" xml:space="preserve">vt cõſtitutis .4. et .1.21.6.3. ſicut ſe habēt .4. ad .1. ita
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                21. ad .3. </s>
                <s xml:id="N12A86" xml:space="preserve">Sint ſex termini a.b.c.d.e.f. et continuo
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                ꝓportionētur tres primi termini ꝓportione g. / et
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                eadē ꝓportione cõtinuo ꝓportionent̄̄ alii tres pu­
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                ta d.e.f. / et ſit ꝓportio cõpoſita adequate ex dupli­
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                ci g.h. / tūc dico /  eadē eſt ꝓportio a. ad c. q̄ eſt d. ad
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                f. </s>
                <s xml:id="N12A93" xml:space="preserve">Quod ſic oñditur. </s>
                <s xml:id="N12A96" xml:space="preserve">q2 ꝓportio a. ad .c. eſt .h. et ea-
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                dē eſt d. ad .f. / igit̄̄ eadē eſt proportio a. ad c. q̄ eſt d.
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                ad f. / qḋ fuit ꝓbãdū. </s>
                <s xml:id="N12A9D" xml:space="preserve">q2 vtrobi h. proportio </s>
                <s xml:id="N12AA0" xml:space="preserve">Pro­
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                batur maior: quia proportio a. ad c. cõponitur ex
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                duplici g. proportione adeq̈te puta ex proportio-
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                ne que eſt a. ad b. q̄ eſt g. et b. ad .c. q̄ etiã eſt g. / igitur
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                illa proportio a. ad c. eſt h. </s>
                <s xml:id="N12AAB" xml:space="preserve">Patet conſequētia / q2
                  <lb/>
                proportio h. vt ponit̄̄ cõponitur ex duplici g. ade-
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                quate. </s>
                <s xml:id="N12AB2" xml:space="preserve">Et iſto mõ probabis minorē: qm̄ proportio
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                d. ad f. componitur ex duplici g. puta ex proportio­
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                ne g. q̄ eſt d. ad e. et ex proportione g. que eſt e. ad f.
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                adequate. </s>
                <s xml:id="N12ABB" xml:space="preserve">Et ſic patet correlariū. </s>
                <s xml:id="N12ABE" xml:space="preserve">Et pari demon-
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                ſtratione oſtendes:  conſtitutis tribus quaterna­
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                riis continuo proportionabilibus eadem propor­
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                tione: et quin quinariis: et in quo volueris nūe-
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                ro: in quacun proportione ſe habent extrema
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                vniꝰ in eadē ſe habent extrema cuiuſuis alterius.</s>
              </p>
              <note position="left" xml:id="N12B04" xml:space="preserve">5. ꝓp̄etaſ
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              medieta­
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              tis geo-
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              metrice.</note>
              <p xml:id="N12B0E">
                <s xml:id="N12B0F" xml:space="preserve">Quinta concluſio </s>
                <s xml:id="N12B12" xml:space="preserve">Quotlibet in hac
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                medietate geometrica terminis conſtitutis conti­
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                nuo proportionabilibus:: qualis eſt illoruꝫ termi­
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                norū cõtinuo ꝓportio: talis eſt inter eorū differen­
                  <cb chead="Capitulum tertiū."/>
                tias ſiue exceſſns. </s>
                <s xml:id="N12B1E" xml:space="preserve">vt conſtitutis his terminis .16.8
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                4.2.1. qualis eſt ꝓportio .6. ad .8. talis eſt exceſſus
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                quo .16. excedunt .8. ad exceſſum quo .8. excedūt .4.
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                et exceſſus quo .4. excedunt .2. ad exceſſum quo duo
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                excedunt vnum / vt patet. </s>
                <s xml:id="N12B29" xml:space="preserve">Eſt enim inter illos exceſ-
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                ſus proportio dupla quēadmodū īter terīos </s>
                <s xml:id="N12B2E" xml:space="preserve">Pro­
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                bat̄̄ / ſint .3. ṫmini cõtinuo ꝓportionabiles .f. ꝓpor-
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                tione puta .ab.cd.e. et exceſſus quo primus excedit
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                ſecunduꝫ ſit a: et exceſſus quo ſecundus excedit ter­
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                tium ſit c. / tūc dico /  ſicut f. ꝓportio eſt inter illos
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                terminos: vcꝫ īter primum et ſecundum et inter ſe-
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                cundum et tertium. </s>
                <s xml:id="N12B3D" xml:space="preserve">ita etiã eſt f. proportio inter a.
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                et c. exceſſus ita  a. ad c. eſt proportio f. </s>
                <s xml:id="N12B42" xml:space="preserve">Quod ſic
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                oſtendit̄̄ / q2 b. ad d. eſt ꝓportio f. et a. ad .c. eſt eadeꝫ
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                proportio / igitur a. ad c. eſt f. proportio / quod fuit
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                ꝓbandū. </s>
                <s xml:id="N12B4B" xml:space="preserve">Probatur maior / quia b. eſt equale c.d.
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                q2 a.b: excedebat preciſe per a. ipſum .cd. et ſic re-
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                moto exceſſu .b. manebit equale c.d. et d. eſt equale
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                e. eadem rõne: et inter .cd. et e. eſt f. proportio / vt po­
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                nitur: ergo inter b: et d. eſt eadem f. proportio </s>
                <s xml:id="N12B56" xml:space="preserve">Pa­
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                tet conſequentia / q2 oīm equaliū eſt eadem ꝓpor-
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                tio </s>
                <s xml:id="N12B5D" xml:space="preserve">Minor ꝓbatur / et capio vnū terminū ad quem
                  <lb/>
                a. habeat ꝓportionē f. qui ſit g. / et arguo ſic / ſicut ſe
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                habet b. ad d. ita ſe habet a. ad g. puta in f. propor­
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                tione: ergo ſicut ſe habet b. ad d: puta in f. ꝓportio­
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                ne ita ſe habet a.b. ad g.d. puta in f. proportione.
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                </s>
                <s xml:id="N12B69" xml:space="preserve">Patet hec conſequentia ex ſecundo correlario q̈r­
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                te concluſionis: et .ab. etiam ad .cd. eſt proportio
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                f. / vt ponitur igitur g.d. et c.d. ſunt equalia. </s>
                <s xml:id="N12B70" xml:space="preserve">Patet
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                conſequentia / quia idem tertium eandē ꝓportiõeꝫ
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                hꝫ ad vtrum illoꝝ: et vltra .gd. et c.d. ſūt eq̈lia: g̊
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                eodē cõi dēpto puta d. r̄ſidua manebūt eq̈lia / ſꝫ re­
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                ſidua ſunt g. et c. / g̊ g. et c. ſunt eq̈lia et a. ad g. eſt f.
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                proportio / vt poſitū eſt / ergo a. ad c. eſt f. proportio /
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                quod fuit ꝓbandum </s>
                <s xml:id="N12B7F" xml:space="preserve">Patet hec conſequētia / quia
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                eiuſdē tertii ad vtrū duorū equaliū eſt eadem ꝓ­
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                portio. </s>
                <s xml:id="N12B86" xml:space="preserve">Et ſic ptꝫ concluſio </s>
                <s xml:id="N12B89" xml:space="preserve">Qm̄ eo modo quo pro­
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                batū eſt in illis tribus terminis probabitur quot­
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                cnn diſpoſitis cõtinuo proportionabilibus hac
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                medietate. </s>
                <s xml:id="N12B92" xml:space="preserve">Et hec ſit quinta proprietas medieta-
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                tis geometrice.
                  <note position="right" xlink:href="note-0029-03a" xlink:label="note-0029-03" xml:id="N12BDF" xml:space="preserve">1. correĺ.</note>
                </s>
                <s xml:id="N12B9C" xml:space="preserve">¶ Ex hac concluſione ſequitur pri-
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                mo /  ſi duo numeri inequales continuo diminuã­
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                tur continuo in eadem ꝓportione manentes: con-
                  <lb/>
                tinno deperditū maiori numero ſe habet in eadeꝫ
                  <lb/>
                ꝓportione ad deperditū minori numero in qua cõ­
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                tinuo ſe habent illi numeri qui diminuuntur. </s>
                <s xml:id="N12BA9" xml:space="preserve">vt ſi
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                numerus octonarius et quaternarius continuo di­
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                minuantur continuo manētes in ꝓportiõe dnpla:
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                continuo deperditum ab octonario ſe habebit in
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                ꝓportione dupla ad deperditum a quaternario.
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                </s>
                <s xml:id="N12BB5" xml:space="preserve">Hoc correlarium facile ex demonſtratione conclu­
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                ſionis probatur.
                  <note position="right" xlink:href="note-0029-04a" xlink:label="note-0029-04" xml:id="N12BE5" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N12BBF" xml:space="preserve">¶ Sequit̄̄ ſecundo /  ſi nõ conti-
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                nuo deperditum maiori numero ſe habeat ad de­
                  <lb/>
                perditum a minori numero in eadem proportiõe:
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                in qua continuo ſe habent illi numeri q̇ diminuun­
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                tur: illi duo numeri inequales q̇ cõtinuo diminuū­
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                tur non ſe habent in eadem ꝓportione etc. </s>
                <s xml:id="N12BCC" xml:space="preserve">Patet
                  <lb/>
                hoc correlarium ex priori / qm̄ p̄cedens correlariū
                  <lb/>
                eſt vna conditionalis a: igitur ex oppoſito ↄ̨ñtis
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                eius ſequit̄̄ oppoſitum añcedentis: et ꝑ conſequēs
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                conditionalis in qua arguitur ex oppoſito conſe­
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                quentis illius ad oppoſitum añtis eſt vera: et ta-
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                lis eſt correlarium / igitur correlarium verum.</s>
              </p>
              <note position="right" xml:id="N12BEB" xml:space="preserve">3. correĺ.</note>
              <p xml:id="N12BEF">
                <s xml:id="N12BF0" xml:space="preserve">¶ Sequitur tertio /  ſi continuo deperdita a duo­
                  <lb/>
                bus numeris inequalibus manent in eadem pro-
                  <lb/>
                portione in qua ſe habent illi numeri in principio
                  <lb/>
                deperditionis: numeri remanētes cõtinuo manēt
                  <lb/>
                in eadem ꝓportione. </s>
                <s xml:id="N12BFB" xml:space="preserve">vt ſi numerus duodenarius
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                et ſenarius diminuantur: et continuo deperdituꝫ </s>
              </p>
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