Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N1082A" level="3" n="3" type="chapter" type-free="capitulum">
              <p xml:id="N109ED">
                <s xml:id="N109F8" xml:space="preserve">
                  <pb chead="Prime partis" file="0010" n="10"/>
                laris: eſt ſexquialtera: vel ſexquitertia: vel minor
                  <lb/>
                ſexquitertia: et nulla proportio diametri ad coſtã
                  <lb/>
                eſt ſexquialtera: vel ſexquitertia vel minor ſexq̇ter­
                  <lb/>
                tia. / ergo nulla proportio diametri ad coſtã: eſt ſu-
                  <lb/>
                perparticularis. </s>
                <s xml:id="N10A08" xml:space="preserve">Cõſequētia ptꝫ cū maiore mani-
                  <lb/>
                feſte: et probatur minor. </s>
                <s xml:id="N10A0D" xml:space="preserve">qm̄ oīs proportio ſexqui-
                  <lb/>
                altera: vel ſexquitertia: vel minor ſexquitertia. eſt
                  <lb/>
                maior vel minor: medietate duple. et nulla propor­
                  <lb/>
                tio diametri ad coſtã: eſt maior vel minor medieta­
                  <lb/>
                te duple. q2 eſt equalis medietati duple. / vt patꝫ ex
                  <lb/>
                tertia ſuppoſitiõe. </s>
                <s xml:id="N10A1A" xml:space="preserve">igitur nulla ꝓportio diametri
                  <lb/>
                ad coſtã: eſt ſexquialtera. vel ſexq̇tertia: vel minor
                  <lb/>
                ſexquitertia. </s>
                <s xml:id="N10A21" xml:space="preserve">Cõſequētia patet cū minore: et maior
                  <lb/>
                probatur: q2 ſexquialtera eſt maior quã medietas
                  <lb/>
                duple. et ſexquitertia minor quã medietas duple / et
                  <lb/>
                ex cõſequēti: ꝑ locū a maiori: quelibet minor ſexq̇-
                  <lb/>
                tertia: eſt minor quã medietas duple. / ergo oīs pro­
                  <lb/>
                portio ſexquialtera. vel ſexquitertia: vĺ minor ſex­
                  <lb/>
                quitertia: eſt maior: vel minor: medietate duple.
                  <lb/>
                </s>
                <s xml:id="N10A31" xml:space="preserve">Probatur tamē ãtecedēs. </s>
                <s xml:id="N10A34" xml:space="preserve">q2 dupla. cõponit̄̄ ade-
                  <lb/>
                quate ex ſexquialtera: et ſexquitertia. / vt patet ex
                  <lb/>
                ſecūda parte. </s>
                <s xml:id="N10A3B" xml:space="preserve">et ſexquialtera eſt maior. </s>
                <s xml:id="N10A3E" xml:space="preserve">et ſexquiter­
                  <lb/>
                tia mīor. </s>
                <s xml:id="N10A43" xml:space="preserve">igitur ſexquialtera eſt maior quã medie­
                  <lb/>
                tas duple. et ſexquitertia minor quã medietas du­
                  <lb/>
                ple. </s>
                <s xml:id="N10A4A" xml:space="preserve">Patet conſequētia ex ſexta ſuppoſitione q̈rti
                  <lb/>
                capitis ſecūde partis.</s>
              </p>
              <p xml:id="N10A4F">
                <s xml:id="N10A50" xml:space="preserve">Tertia concluſio. </s>
                <s xml:id="N10A53" xml:space="preserve">Nulla proportio
                  <lb/>
                diametri ad coſtã eſt aliqua proportio ſuprapar-
                  <lb/>
                tiens. </s>
                <s xml:id="N10A5A" xml:space="preserve">Probatur. </s>
                <s xml:id="N10A5D" xml:space="preserve">q2 oīs proportio ſuprapartiēs:
                  <lb/>
                reperibilis eſt inter duos numeros: quoꝝ alter eſt
                  <lb/>
                impar. </s>
                <s xml:id="N10A64" xml:space="preserve">et nulla proportio diametri ad coſtã: repe­
                  <lb/>
                ribilis eſt inter duos numeros: quoꝝ alter eſt īpar /
                  <lb/>
                ergo nulla proporito diametri ad coſtã: eſt aliqua
                  <lb/>
                proportio ſuprapartiens </s>
                <s xml:id="N10A6D" xml:space="preserve">Patet conſequentia in
                  <lb/>
                ſcḋo ſcḋe vt prius. </s>
                <s xml:id="N10A72" xml:space="preserve">et maior ex quarta ſuppoſitiõe
                  <lb/>
                et minor probat̄̄. </s>
                <s xml:id="N10A77" xml:space="preserve">q2 ſi nõ detur oppoſitū. </s>
                <s xml:id="N10A7A" xml:space="preserve">videlicet /
                  <lb/>
                 proportio diametri ad coſtã: reperitur inter du­
                  <lb/>
                os numeros: quoꝝ alter eſt impar. </s>
                <s xml:id="N10A81" xml:space="preserve">ita  diameter
                  <lb/>
                et coſta: ſe habere poſſūt vt duo nūeri: quoꝝ alter
                  <lb/>
                eſt impar. </s>
                <s xml:id="N10A88" xml:space="preserve">vel igitur diameter erit numerꝰ impar:
                  <lb/>
                vel coſta ſi diameter: ſequitur /  quadratū ipſius
                  <lb/>
                diametri: erit numerꝰ impar. </s>
                <s xml:id="N10A8F" xml:space="preserve">Patet cõſequētia ex
                  <lb/>
                quinta ſuppoſitione. </s>
                <s xml:id="N10A94" xml:space="preserve">et vltra quadratū diametri:
                  <lb/>
                eſt numerꝰ impar. </s>
                <s xml:id="N10A99" xml:space="preserve">ergo quadratū diametri: nõ eſt
                  <lb/>
                duplū ad quadratū coſte. </s>
                <s xml:id="N10A9E" xml:space="preserve">Patet conſequentia ex
                  <lb/>
                ſexta ſuppoſitione. </s>
                <s xml:id="N10AA3" xml:space="preserve">et cõſequēs eſt falſum: vt patet
                  <lb/>
                ex ſecūda ſuppoſitione. </s>
                <s xml:id="N10AA8" xml:space="preserve">igitur et antecedens: </s>
                <s xml:id="N10AAB" xml:space="preserve">Non
                  <lb/>
                eſt igitur dicendū /  diameter eſt numerus impar
                  <lb/>
                reſpectu coſte. </s>
                <s xml:id="N10AB2" xml:space="preserve">ſi vero coſta ſit nūerꝰ īpar reſpectu
                  <lb/>
                diametri: ſequit̄̄ /  quadratū eiꝰ erit numerꝰ īpar
                  <lb/>
                ſed quadratū eiꝰ: eſt etiã quadratū diametri. </s>
                <s xml:id="N10AB9" xml:space="preserve">qm̄
                  <lb/>
                ipſa coſta: eſt diameter mīoris quadrati. </s>
                <s xml:id="N10ABE" xml:space="preserve">vt patet
                  <lb/>
                in ſuperiori figura. </s>
                <s xml:id="N10AC3" xml:space="preserve">Igit̄̄ quadratū diametri: eſt
                  <lb/>
                numerꝰ impar. </s>
                <s xml:id="N10AC8" xml:space="preserve">Patet cõſequētia ex quinta ſuppo­
                  <lb/>
                ſitione. </s>
                <s xml:id="N10ACD" xml:space="preserve">et per cõſequēs quadratū diametri: nõ eſt
                  <lb/>
                duplū ad q̈dratū coſte. </s>
                <s xml:id="N10AD2" xml:space="preserve">Patet cõſequētia ex ſexta
                  <lb/>
                ſuppoſitione. </s>
                <s xml:id="N10AD7" xml:space="preserve">et cõſequēs eſt falſum. </s>
                <s xml:id="N10ADA" xml:space="preserve">vt patet ex ſe­
                  <lb/>
                cūda ſuppoſitione: igitur et ãtecedēs. </s>
                <s xml:id="N10ADF" xml:space="preserve">Et ſic patet:
                  <lb/>
                 nec diameter ſe habet ſicut nūerꝰ īpar: nec coſta
                  <lb/>
                  <note position="left" xlink:href="note-0010-01a" xlink:label="note-0010-01" xml:id="N10B21" xml:space="preserve">Quid ſit
                    <lb/>
                  quãtita-
                    <lb/>
                  tē ſe hr̄e
                    <lb/>
                  vt nūerꝰ.</note>
                </s>
                <s xml:id="N10AEB" xml:space="preserve">¶ Aliquam autem quantitatem: ſe habere vt nu-
                  <lb/>
                merus impar reſpectu alterius: eſt ipſam diuidi
                  <lb/>
                ſaltē ad ymaginationē: in partes equales denoīa­
                  <lb/>
                tas a numero impari. </s>
                <s xml:id="N10AF4" xml:space="preserve">vt in tres tertias: in quin
                  <lb/>
                quītas in ſeptem ſeptimas / et ſic cõſequēter. </s>
                <s xml:id="N10AF9" xml:space="preserve">et hoc
                  <lb/>
                reſpectu alterius quãtitatis: diuiſe in partes illis
                  <cb chead="Capitulū quartū."/>
                equales. </s>
                <s xml:id="N10B01" xml:space="preserve">vt ſi pedale diuidatur in tres tertias et bi­
                  <lb/>
                pedale in ſexſexas quarum ſextarum quelibet eſt
                  <lb/>
                equalis vni tertie pedalis: tūc dico:  pedale ſe hꝫ
                  <lb/>
                vt nūerꝰ impar: reſpectu bipedalis. </s>
                <s xml:id="N10B0A" xml:space="preserve">Tu tamē ad-
                  <lb/>
                uerte  etiã poteſt ſe habere vt nūerꝰ par: reſpectu
                  <lb/>
                bipedalis. </s>
                <s xml:id="N10B11" xml:space="preserve">tamē ſemꝑ īter pedale et bipedale erit
                  <lb/>
                ꝓportio dupla. </s>
                <s xml:id="N10B16" xml:space="preserve">Diameter autē et coſta: nū̄ ſic ſe
                  <lb/>
                poſſunt habere:  diameter ſe habeat vt numerus
                  <lb/>
                impar reſpectu coſte: vel econtra / vt ꝓbatū eſt.</s>
              </p>
              <p xml:id="N10B2D">
                <s xml:id="N10B2E" xml:space="preserve">Quarta cõcluſio. </s>
                <s xml:id="N10B31" xml:space="preserve">Omnis proportio
                  <lb/>
                diametri ad coſtã: eſt irrationalis </s>
                <s xml:id="N10B36" xml:space="preserve">Probatur hec
                  <lb/>
                cõcluſio. </s>
                <s xml:id="N10B3B" xml:space="preserve">q2 oīs ꝓportio rationalis: eſt multiplex:
                  <lb/>
                aut multiplex ſuꝑparticularis, aut multiplex ſu-
                  <lb/>
                prapartiens, aut ſuꝑparticularis, aut ſuprapar­
                  <lb/>
                tiens, et nulla ꝓportio diametri ad coſtã: eſt mul-
                  <lb/>
                tiplex, aut multiplex ſuperparticularis, aut mul-
                  <lb/>
                tiplex ſuprapartiēs. </s>
                <s xml:id="N10B48" xml:space="preserve">vt patet ex prima cõcluſione
                  <lb/>
                aut ſuꝑparticularis. </s>
                <s xml:id="N10B4D" xml:space="preserve">vt ptꝫ ex ſcḋa: aut ſuprapar-
                  <lb/>
                tiens: vt patet ex tertia. / igitur nulla ꝓportio dia­
                  <lb/>
                metri ad coſtã: eſt rationalis. </s>
                <s xml:id="N10B54" xml:space="preserve">Cõſequētia patet vt
                  <lb/>
                ſupra: et maior ex fine primi capitis. </s>
                <s xml:id="N10B59" xml:space="preserve">Illa enim eſt
                  <lb/>
                ſūma diuiſio ꝓportiõis rationalis: et vltra nulla
                  <lb/>
                ꝓportio diametri ad coſtã: eſt ratiõalis. </s>
                <s xml:id="N10B60" xml:space="preserve">et eſt pro­
                  <lb/>
                portio: igitur eſt proportio irrationalis. </s>
                <s xml:id="N10B65" xml:space="preserve">Patet
                  <lb/>
                cõſequentia a ſufficienti diuiſione.</s>
              </p>
            </div>
            <div xml:id="N10B6A" level="3" n="4" type="chapter" type-free="capitulum">
              <head xml:id="N10B6F" xml:space="preserve">Capitulum quartum / in quo agitur de
                <lb/>
              infinitis ſpeciebus proportionis irratio­
                <lb/>
              nalis: et de earum procreatione.</head>
              <p xml:id="N10B76">
                <s xml:id="N10B77" xml:space="preserve">PRoportio irrationalis: per-
                  <lb/>
                inde at rationalis: in infinitas di-
                  <lb/>
                ſtribuitur ſpecies </s>
                <s xml:id="N10B7E" xml:space="preserve">Ad quod mathema­
                  <lb/>
                tica induſtria inferendū: ponūtur alique ſuppões</s>
              </p>
              <p xml:id="N10B83">
                <s xml:id="N10B84" xml:space="preserve">Prima ſuppoſitio. </s>
                <s xml:id="N10B87" xml:space="preserve">Si due quantita­
                  <lb/>
                tes: ſe habent vt duo numeri: aggregatū ex eis: ſe
                  <lb/>
                habebit vt vnꝰ numerꝰ. </s>
                <s xml:id="N10B8E" xml:space="preserve">Probatur. </s>
                <s xml:id="N10B91" xml:space="preserve">q2 ſemꝑ ex ad­
                  <lb/>
                ditiõe numeri ad numerū: reſultat numerꝰ maior</s>
              </p>
              <p xml:id="N10B96">
                <s xml:id="N10B97" xml:space="preserve">Secūda ſuppoſitio </s>
                <s xml:id="N10B9A" xml:space="preserve">Si alique quan­
                  <lb/>
                titates. </s>
                <s xml:id="N10B9F" xml:space="preserve">ſe habeant in ꝓportione rationali: ille ſe
                  <lb/>
                habebunt: vt duo numeri: et econtra. </s>
                <s xml:id="N10BA4" xml:space="preserve">Patet ſup-
                  <lb/>
                poſitio hec ex diffinitione ꝓportiõis ratioalis: cū
                  <lb/>
                ſuo correlario: primo capite poſita.</s>
              </p>
              <p xml:id="N10BAB">
                <s xml:id="N10BAC" xml:space="preserve">Tertia ſuppoſitio. </s>
                <s xml:id="N10BAF" xml:space="preserve">Si due quantita­
                  <lb/>
                tes ſe habeant in ꝓportione ratiõali: aggregatū
                  <lb/>
                ex eis: ſe habet in ꝓportione ratiõali: ad quãlibet
                  <lb/>
                illaꝝ quantitatū. </s>
                <s xml:id="N10BB8" xml:space="preserve">Probatur hec ſuppoſitio. </s>
                <s xml:id="N10BBB" xml:space="preserve">qm̄ ſi
                  <lb/>
                ſe habent in ꝓportione rationali: iã quelib3 illaꝝ
                  <lb/>
                ſe habet vt numerꝰ: vt patet ex ſecūda ſuppoſitõe
                  <lb/>
                et ſi quelibet illaꝝ ſe habet vt uūerꝰ: ſe aggregatū
                  <lb/>
                ex eis: ſe habet vt nūerꝰ. </s>
                <s xml:id="N10BC6" xml:space="preserve">vt patet ex prima ſuppo­
                  <lb/>
                ſitiõe. </s>
                <s xml:id="N10BCB" xml:space="preserve">et ꝑ cõſequens illiꝰ agggregati: quod ſe ha­
                  <lb/>
                bet vt numerꝰ: ad vtrã illarū quantitatū: que ſe
                  <lb/>
                habent vt numeri: erit ꝓportio rationalis. </s>
                <s xml:id="N10BD2" xml:space="preserve">vt ptꝫ
                  <lb/>
                ex ſecūda ſuppoſitione: quod fuit ꝓbandum.</s>
              </p>
              <p xml:id="N10BD7">
                <s xml:id="N10BD8" xml:space="preserve">Qurata ſuppoſitio. </s>
                <s xml:id="N10BDB" xml:space="preserve">Coſte: ad exceſſū
                  <lb/>
                quo diameter excedit coſtã: ꝓportio irrationalis
                  <lb/>
                </s>
                <s xml:id="N10BE1" xml:space="preserve">Probatur. </s>
                <s xml:id="N10BE4" xml:space="preserve">q2 ſi eſſet rationalis: iã ſe haberent vt
                  <lb/>
                duo numeri. </s>
                <s xml:id="N10BE9" xml:space="preserve">vt patet ex ſecūda ſuppoſitiõe. </s>
                <s xml:id="N10BEC" xml:space="preserve">et ſi ſe
                  <lb/>
                haberēt vt duo numeri: aggregatū ex eis: qḋ ade­
                  <lb/>
                q̈te eſt diameter haberet ſe in ꝓportione ratiõali
                  <lb/>
                ad vtrū illoꝝ. </s>
                <s xml:id="N10BF5" xml:space="preserve">et ꝑ cõſequēs ad coſtam. </s>
                <s xml:id="N10BF8" xml:space="preserve">vt patet ex
                  <lb/>
                tertia ſuppoſitione: et ſic diametri ad coſtam: eſſet
                  <lb/>
                rationalis proportio. </s>
                <s xml:id="N10BFF" xml:space="preserve">quod eſt contra quratã cõ­
                  <lb/>
                cluſionem precedentis capitis.</s>
              </p>
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