Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N1082A" level="3" n="3" type="chapter" type-free="capitulum">
              <pb chead="Prime partis" file="0009" n="9"/>
              <head xml:id="N10833" xml:space="preserve">Capitulū tertiū / in quo oſtenditur: et de­
                <lb/>
              mõſtratur: proportionem irrationalem
                <lb/>
              eſſe ponendam.</head>
              <p xml:id="N1083A">
                <s xml:id="N1083B" xml:space="preserve">AD demonſtrandum inter a-
                  <lb/>
                liquas magnitudines ꝓportionē irra­
                  <lb/>
                tionalem inueniri: que nullo pacto ſit
                  <lb/>
                ſicut numeri ad numerum.</s>
              </p>
              <p xml:id="N10844">
                <s xml:id="N10845" xml:space="preserve">Suppono primo /  proportio qua-
                  <lb/>
                dratorum ſuperficialium: eſt proportio coſtarum
                  <lb/>
                dublicata. </s>
                <s xml:id="N1084C" xml:space="preserve">Hoc eſt ſi inter coſtas duorum quadra­
                  <lb/>
                torum ſuperficialium: ſit aliqua proportio maio-
                  <lb/>
                ris inequalitatis: inter quadrata erit proportio
                  <lb/>
                dupla: ad illã: que eſt inter coſtas ſignatorū qua-
                  <lb/>
                dratorū: vt ſi inter coſtas duorū quadratorū ine-
                  <lb/>
                qualiū ſuperficialiū: fuerit proportio dupla: inter
                  <lb/>
                quadrata erit proportio q̈drupla </s>
                <s xml:id="N1085B" xml:space="preserve">Hec ſuppoſitio
                  <lb/>
                clare ꝓbatur: et demõſtratur: inferiꝰ. in tertia ꝑte
                  <lb/>
                tractatu ſecūdo: capitulo .2. </s>
                <s xml:id="N10862" xml:space="preserve">Uideas eã ibi.</s>
              </p>
              <p xml:id="N10865">
                <s xml:id="N10866" xml:space="preserve">Secunda ſuppoſitio. </s>
                <s xml:id="N10869" xml:space="preserve">Quadratum
                  <lb/>
                diametri: ſe hꝫ ad q̈dratū coſte in ꝓportiõe dupla
                  <lb/>
                </s>
                <s xml:id="N1086F" xml:space="preserve">Hoc eſt q̈dratū cuiꝰ q̈libet coſta. </s>
                <s xml:id="N10872" xml:space="preserve">eſt eq̈lis diametro
                  <lb/>
                alicuiꝰ q̈drati ſe hꝫ in ꝓportiõe dupla: ad illud q̈-
                  <lb/>
                dratū. </s>
                <s xml:id="N10879" xml:space="preserve">Probat̄̄ hec ſuppoſitio: et ſit vnū q̈dratum
                  <lb/>
                magnū: cuiꝰ latꝰ ſit .d.c. et diameṫ ſit a.c. ſit aliḋ
                  <lb/>
                paruū cū iſto cõicans cuiꝰ coſta ſit .c.f. et diameter
                  <lb/>
                ſit .d.c et diuidat̄̄ q̈dratū maiꝰ: ꝑ duos diametros
                  <lb/>
                in quatuor triãgulos equales: vt ptꝫ in hac figura /
                  <lb/>
                  <figure xlink:href="fig-0009-01a" xlink:label="fig-0009-01" xml:id="N108CE" number="3">
                    <image file="0009-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YHKVZ7B4/figures/0009-01"/>
                  </figure>
                quo poſito argr̄ ſic / ma-
                  <lb/>
                gnū q̈dratū ē duplū
                  <lb/>
                ad paruū q̈dratū et
                  <lb/>
                ipſū magnū q̈dratū
                  <lb/>
                eſt quadratū diame­
                  <lb/>
                tri ipſius parui qua­
                  <lb/>
                drati. </s>
                <s xml:id="N10897" xml:space="preserve">vt ptꝫ manife­
                  <lb/>
                ſte / igit̄̄ quadratū di­
                  <lb/>
                ametti: ſe hꝫ ad q̈dra­
                  <lb/>
                tū coſte: in ꝓportiõe
                  <lb/>
                dupla. </s>
                <s xml:id="N108A2" xml:space="preserve">Cõſeq̄ntia ptꝫ
                  <lb/>
                cū mīore. </s>
                <s xml:id="N108A7" xml:space="preserve">et argr̄ maior. </s>
                <s xml:id="N108AA" xml:space="preserve">q2 q̈dratū magnū: cõtinet
                  <lb/>
                q̈termedietatē parui q̈drati. </s>
                <s xml:id="N108AF" xml:space="preserve">adeq̈te igr̄ ipſū ma-
                  <lb/>
                gnū q̈dratū: cõtinet bis adeq̈te: paruū q̈dratū. </s>
                <s xml:id="N108B4" xml:space="preserve">Cõ­
                  <lb/>
                ſequentia ptꝫ ex ſe: et ꝓbat̄̄ añs. </s>
                <s xml:id="N108B9" xml:space="preserve">q2 q̈dratū magnū
                  <lb/>
                q̈ter ↄ̨tinet tm̄: ſicut ē triãgulꝰ .d.e.c. / vt ptꝫ. </s>
                <s xml:id="N108BE" xml:space="preserve">et ille tri­
                  <lb/>
                angulꝰ eſt medietas parui quadrati: vt manifeſte
                  <lb/>
                ptꝫ in figura. </s>
                <s xml:id="N108C5" xml:space="preserve">igit̄̄ magnū quadratū: quater conti-
                  <lb/>
                net adequate: mediante parui / qḋ fuit ꝓbandum.</s>
              </p>
              <p xml:id="N108D4">
                <s xml:id="N108D5" xml:space="preserve">Terita ſuppoſitio. </s>
                <s xml:id="N108D8" xml:space="preserve">diametri ad coſtã
                  <lb/>
                eſt ꝓportio: que eſt medietas duple. </s>
                <s xml:id="N108DD" xml:space="preserve">Probatur / q2
                  <lb/>
                quadrati diametri ad quadratū coſte eſt ꝓportio
                  <lb/>
                dupla: vt ptꝫ ex ſcḋa ſuppoſitione. </s>
                <s xml:id="N108E4" xml:space="preserve">ergo diametri
                  <lb/>
                ad coſtã: eſt ꝓportio ſubdupla ad duplã. </s>
                <s xml:id="N108E9" xml:space="preserve">et ꝑ conſe­
                  <lb/>
                quēs medietas duple. </s>
                <s xml:id="N108EE" xml:space="preserve">Patet cõſequētia ex prima
                  <lb/>
                ſuppoſitione. </s>
                <s xml:id="N108F3" xml:space="preserve">Qm̄ ſemꝑ ꝓportio quadratorū: eſt
                  <lb/>
                dupla ad ꝓportionē coſtaꝝ. </s>
                <s xml:id="N108F8" xml:space="preserve">Et ſic ꝓportio coſtaꝝ
                  <lb/>
                eſt medietas ꝓportionis quadratoꝝ. </s>
                <s xml:id="N108FD" xml:space="preserve">Cum igitur
                  <lb/>
                proportio quadratoruꝫ fuerit dupla: coſtaꝝ pro-
                  <lb/>
                portio erit medietas duple.</s>
              </p>
              <note position="left" xml:id="N10904" xml:space="preserve">Numeri
                <lb/>
              primi.</note>
              <p xml:id="N1090A">
                <s xml:id="N1090B" xml:space="preserve">Quarta ſuppoſitio cuinſlibet ꝓpor­
                  <lb/>
                tionis ſuprapartientis alter primorū numeroruꝫ
                  <lb/>
                eſt impar. </s>
                <s xml:id="N10912" xml:space="preserve">Sunt autē primi numeri alicuius ꝓpor­
                  <lb/>
                tionis: qui in ea ꝓportiõe ſunt numeri: vt tria et .2.
                  <lb/>
                ſunt primi numeri ꝓportionis ſexquialtere: quia
                  <lb/>
                in naturali ſerie numeroruꝫ: inter nullos minores
                  <cb chead="Capitulum tertiū."/>
                ꝓportio ſexquialtera inuenit̄̄: </s>
                <s xml:id="N1091E" xml:space="preserve">Probatur ſuppoſi­
                  <lb/>
                tio. </s>
                <s xml:id="N10923" xml:space="preserve">q2 ſi non: detur oppoſitū. </s>
                <s xml:id="N10926" xml:space="preserve">videlicet /  vter ſit
                  <lb/>
                numerus par. </s>
                <s xml:id="N1092B" xml:space="preserve">et arguitur ſic. </s>
                <s xml:id="N1092E" xml:space="preserve">vter iſtorꝝ eſt nume­
                  <lb/>
                rus par. </s>
                <s xml:id="N10933" xml:space="preserve">ergo ſequitur /  vter illoꝝ eſt medietas /
                  <lb/>
                vt patet ex diffinitione numeri paris: et proportio
                  <lb/>
                medietatū: eſt eadē cū ꝓportione totoꝝ: vt conſtat
                  <lb/>
                et inferius ꝓbabis: igitur illi non erant primi nu-
                  <lb/>
                meri talis ꝓportiõis. </s>
                <s xml:id="N1093E" xml:space="preserve">q2 nõ erant minimi illiꝰ pro­
                  <lb/>
                portionis: cū ſue medietates ſint numeri minores
                  <lb/>
                et ꝑ ↄ̨ñs: nõ dediſti ṗmos nūeros: talis ꝓpoſitiõis</s>
              </p>
              <p xml:id="N10945">
                <s xml:id="N10946" xml:space="preserve">Quīta ſuppoſitio. </s>
                <s xml:id="N10949" xml:space="preserve">Omne quadratū
                  <lb/>
                numeri īparis: eſt īpar. </s>
                <s xml:id="N1094E" xml:space="preserve">Probatur: q2 oē quadra-
                  <lb/>
                tum numeri īparis: eſt ille numerꝰ: qui reſultat ex
                  <lb/>
                ductu numeri īparis: in ſeipſum ſemel. </s>
                <s xml:id="N10955" xml:space="preserve">vt patet ex
                  <lb/>
                ſcḋo arithmetice nichomachi. </s>
                <s xml:id="N1095A" xml:space="preserve">ſed oīs numerꝰ: re-
                  <lb/>
                ſultãs ex ductu numeri īparis in ſeipſum: eſt īpar /
                  <lb/>
                igitur oē quadratū numeri īparis: eſt īpar. </s>
                <s xml:id="N10961" xml:space="preserve">Pro-
                  <lb/>
                batur minor: q2 ſi numerꝰ īpar: multiplicetur per
                  <lb/>
                numeꝝ parē immediate precedentē ipſum vt .5. per
                  <lb/>
                4. / tunc reſultaret numerꝰ par: ſed quãdo multipli­
                  <lb/>
                catur per ſeipſum: ſiue dicetur ī ſeipſum ſemel (qḋ­
                  <lb/>
                ideꝫ ē) adhuc illi nūero pari: qui reſultabat ex mul­
                  <lb/>
                tiplicatione numeri paris: immediate preceden-
                  <lb/>
                tis: additur numerꝰ īpar: vt patet intelligenti. </s>
                <s xml:id="N10972" xml:space="preserve">igr̄
                  <lb/>
                totū reſultans: erit nūerꝰ īpar. </s>
                <s xml:id="N10977" xml:space="preserve">Patet cõſequētia:
                  <lb/>
                qm̄ ſi numerꝰ īpar: addatur numero pari: reſulta­
                  <lb/>
                bit numerꝰ īpar. </s>
                <s xml:id="N1097E" xml:space="preserve">Exemplū / vt ſi ternariꝰ: multipli­
                  <lb/>
                cetur per numeꝝ parem: īmediate precedentē: puta
                  <lb/>
                binariū: reſultabit numerꝰ par: puta ſenariꝰ. </s>
                <s xml:id="N10985" xml:space="preserve">et ſi
                  <lb/>
                vlteriꝰ addatur numerꝰ teruariꝰ: ſupra ſenariū re­
                  <lb/>
                ſultabit nouenarius: qui eſt numerꝰ īpar reſultãs
                  <lb/>
                ex ductu ternarii in ſeipſum ſemel.</s>
              </p>
              <p xml:id="N1098E">
                <s xml:id="N1098F" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N10992" xml:space="preserve">nullus numerus
                  <lb/>
                impar: eſt duplas ad aliquē numerū. </s>
                <s xml:id="N10997" xml:space="preserve">Probatur:
                  <lb/>
                q2 ſi eſſet duplus ad aliquē numerū: iã ille numerꝰ
                  <lb/>
                eſſet ſua medietas adequate: et ſic diuideret̄̄ in du-
                  <lb/>
                as medietates: et ꝑ cõſequēs nõ eſſet impar.</s>
              </p>
              <p xml:id="N109A0">
                <s xml:id="N109A1" xml:space="preserve">Hīs iactis ſuppoſitiõibus: ſit prima
                  <lb/>
                cõcluſio. </s>
                <s xml:id="N109A6" xml:space="preserve">Nulla ꝓportio diametri ad coſtã: ē mĺti­
                  <lb/>
                plex, aut mĺtiplex ſuꝑparticularis: aut multiplex
                  <lb/>
                ſuprapartiēs. </s>
                <s xml:id="N109AD" xml:space="preserve">Probat̄̄ hec cõcluſio: oīs ꝓportio
                  <lb/>
                mĺtiplex, aut mĺtiplex ſuꝑparticĺaris, aut mĺti-
                  <lb/>
                plex ſuprapartiēs eſt dupla aut maior dupla: ſed
                  <lb/>
                nulla ꝓportio diametri ad coſtã: ē dupla aut ma-
                  <lb/>
                ior dupla: igit̄̄ nulla ꝓportio diametri ad coſtam
                  <lb/>
                eſt mĺtiplex: aut mĺtiplex ſuꝑparticularꝪ, aut mĺ-
                  <lb/>
                tiplex ſuprapartiēs. </s>
                <s xml:id="N109BC" xml:space="preserve">Ptꝫ ↄ̨ña in ſcḋo ſcḋe et maior
                  <lb/>
                ſimiliter: q2 oīs proportio multiplex: eſt dupla: vĺ
                  <lb/>
                mior: et oīs ꝓportio multiplex ſuperparticularis
                  <lb/>
                aut multiplex ſuprapartiens: eſt maior dupla: vt
                  <lb/>
                patebit ex cſḋa parte: igitur oīs proportio multi­
                  <lb/>
                plex: aut multiplex ſuꝑparticularis: aut mĺtiplex
                  <lb/>
                ſuprapartiens: eſt dupla: vel maior dupla. </s>
                <s xml:id="N109CB" xml:space="preserve">Iã ꝓ-
                  <lb/>
                batur minor. </s>
                <s xml:id="N109D0" xml:space="preserve">q2 oīs proportio diametri ad coſtã:
                  <lb/>
                eſt medietas duple: ſiue ſubdupla ad duplã (quod
                  <lb/>
                idē eſt) adequate: ergo nulla proportio diametri
                  <lb/>
                ad coſtã: eſt ipſa tota dupla: vel maior dupla </s>
                <s xml:id="N109D9" xml:space="preserve">Pa­
                  <lb/>
                tet antecedēs. </s>
                <s xml:id="N109DE" xml:space="preserve">ex tertia ſuppoſitione: et probat̄̄ cõ­
                  <lb/>
                ſequētia. </s>
                <s xml:id="N109E3" xml:space="preserve">q2 alias medietas eſſet equalis ſuo toti:
                  <lb/>
                vel maior. </s>
                <s xml:id="N109E8" xml:space="preserve">quod nõ eſt poſibile: deductis ſophiſta­
                  <lb/>
                rum quiſquiliis.</s>
              </p>
              <p xml:id="N109ED">
                <s xml:id="N109EE" xml:space="preserve">Secunda concluſio. </s>
                <s xml:id="N109F1" xml:space="preserve">nulla proportio
                  <lb/>
                diametri ad coſtã: eſt aliqua proportio ſuꝑparti-
                  <lb/>
                cularis. </s>
                <s xml:id="N109F8" xml:space="preserve">Probatur: q2 oīs proportio ſuꝑparticu- </s>
              </p>
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