Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N11960" level="3" n="1" type="chapter" type-free="capitulum">
              <p xml:id="N11BC3">
                <s xml:id="N11DA7" xml:space="preserve">
                  <pb chead="Prime partis" file="0022" n="22"/>
                tertio adiecimus merito perfectiſſimam vocitari
                  <lb/>
                </s>
                <s xml:id="N11DB0" xml:space="preserve">Cuiꝰ probatio eſt / qm̄ in dicta medietate tres fa-
                  <lb/>
                mate ꝓportionalitates reperiuntur arithmetica
                  <lb/>
                geometrica, et harmonica. </s>
                <s xml:id="N11DB7" xml:space="preserve">In iſta etiã medietate
                  <lb/>
                oēs ſimplices harmonice cõſonantie reperiuntur
                  <lb/>
                  <note position="left" xlink:href="note-0022-01a" xlink:label="note-0022-01" xml:id="N11E6C" xml:space="preserve">tertium.
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N11DC3" xml:space="preserve">¶ Ex his omnibus demū infero oēm ſcientiã aliã
                  <lb/>
                oēm artem: philoſophie inſeruire. </s>
                <s xml:id="N11DC8" xml:space="preserve">ei ancillari
                  <lb/>
                at famulari. </s>
                <s xml:id="N11DCD" xml:space="preserve">vt facile ex his que dicta ſunt ꝑſpi­
                  <lb/>
                ci poteſt: et ſignanter inſeruirent iſta philoſophie.
                  <lb/>
                  <note position="left" xlink:href="note-0022-02a" xlink:label="note-0022-02" xml:id="N11E76">
                    <s xml:id="N11E7A" xml:space="preserve">pythago­
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                    ras.
                      <lb/>
                    </s>
                    <s xml:id="N11E80" xml:space="preserve">phūs
                      <lb/>
                    plinius.</s>
                  </note>
                </s>
                <s xml:id="N11DD9" xml:space="preserve">Pythagore qui aſtruxit celos corpora illa ſempi
                  <lb/>
                terna perpetuo harmonicis cõſonantiis circūuo-
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                lui teſte philoſopho ſecūdo celi et mundi: et plinio
                  <lb/>
                ſecundo naturalis hiſtorie.</s>
              </p>
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            <div xml:id="N11E85" level="3" n="2" type="chapter" type-free="capitulum">
              <head xml:id="N11E8A" xml:space="preserve">Capitulum ſecundum / in quo ꝓbantur
                <lb/>
              alique proprietates predictarum ꝓpor-
                <lb/>
              tionalitatem ſiue medietatum.</head>
              <p xml:id="N11E91">
                <s xml:id="N11E92" xml:space="preserve">AD inducendas mathemathi­
                  <lb/>
                co ordine aliquas ꝓprietates predicta­
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                rum medietatum: ponende ſunt alique
                  <lb/>
                ſuppoſitiones: quarū alique erunt diffinitiones:
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                et alique petentur ꝓpter earuꝫ euidentē noticiam:
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                alique vero probabuntur ſit igitur.</s>
              </p>
              <p xml:id="N11E9F">
                <s xml:id="N11EA0" xml:space="preserve">Prima ſuppoſitio / que et difinitio.
                  <lb/>
                </s>
                <s xml:id="N11EA4" xml:space="preserve">Medium eſt quod equali inter capidine diſtat ab
                  <lb/>
                vtro extemorum. </s>
                <s xml:id="N11EA9" xml:space="preserve">vt numerus ternarius eſt medi­
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                um inter quaternarium et binarium. </s>
                <s xml:id="N11EAE" xml:space="preserve">quia equali
                  <lb/>
                exceſſu ſiue equali differentia ab vtro illoruꝫ di­
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                ſtat: puta vnitate.</s>
              </p>
              <p xml:id="N11EB5">
                <s xml:id="N11EB6" xml:space="preserve">Secunda ſuppoſitio / que et difinitio
                  <lb/>
                </s>
                <s xml:id="N11EBA" xml:space="preserve">Partes aliquote eiuſdem denominationis ſunt
                  <lb/>
                ille q̄ ab eodē numero denominãtur vt medietates
                  <lb/>
                a binario: tertie. a ternario. </s>
                <s xml:id="N11EC1" xml:space="preserve">q̈rte a q̈ternario .etc̈.</s>
              </p>
              <p xml:id="N11EC4">
                <s xml:id="N11EC5" xml:space="preserve">Tertia ſuppoſitio / que etiam difini-
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                tio eſt </s>
                <s xml:id="N11ECA" xml:space="preserve">Aliquã quãtitatē continere aliquod equa-
                  <lb/>
                le in aliqua ꝓportione pluries adequate quã alia
                  <lb/>
                quantitas idem equale contineat: eſt illam quãti­
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                tatem in eadem ꝓportione ſe habere ad alteram
                  <lb/>
                vt ſi aliqua quantitas contineat in ꝓportione ſex­
                  <lb/>
                quialtera adequate plura pedalia quã vna altera
                  <lb/>
                minor talis quantitas ſe habet ad minorem in ꝓ-
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                portione ſexquialtera.</s>
              </p>
              <p xml:id="N11EDB">
                <s xml:id="N11EDC" xml:space="preserve">Quarta ſuppoſitio </s>
                <s xml:id="N11EDF" xml:space="preserve">Si aliqua quan­
                  <lb/>
                titas vel numerus contineat tota vice ſecūdum nu­
                  <lb/>
                merum: quota vice tertius numerus cõtinet quar­
                  <lb/>
                tum vel tota vice et aliquã vel aliquot partes ali-
                  <lb/>
                quotas eiuſdem denominationis quota tertiꝰ cõ­
                  <lb/>
                tinet quartum et aliquam partem vel aliquot par­
                  <lb/>
                tes aliquotas eius adequate: qualis ē proportio
                  <lb/>
                inter primū et ſecundum talis eſt inter tertiū et q̈r­
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                tum. </s>
                <s xml:id="N11EF2" xml:space="preserve">Patet hec ſuppoſitio ex diffinitione nume-
                  <lb/>
                rorum habentium ad reliquos eandeꝫ proportio-
                  <lb/>
                nem. </s>
                <s xml:id="N11EF9" xml:space="preserve">Sic eī tales numeri debent definiri vt cõſtat.</s>
              </p>
              <p xml:id="N11EFC">
                <s xml:id="N11EFD" xml:space="preserve">Quinta ſuppoſitio </s>
                <s xml:id="N11F00" xml:space="preserve">Si duo numeri
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                vel quantitates diuidantur in partes aliquotas
                  <lb/>
                eiuſdem denominationis: quot partes illiꝰ deno­
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                minationis ſunt in vno tot ſunt in altero. </s>
                <s xml:id="N11F09" xml:space="preserve">Patet /
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                quia ſi ſunt eiuſdem denominationis: ab eodē nu-
                  <lb/>
                mero denominantur: vt patet ex ſecunda ſuppoſi­
                  <lb/>
                tione / et per conſequēs ſunt equales numero. </s>
                <s xml:id="N11F12" xml:space="preserve">Tūc
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                enim alique partes aliquote alicuius quantitatis
                  <lb/>
                denominantur ab aliquo numero: quando talis
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                quãtitas diuiditur in tot partes equales quot ſūt
                  <lb/>
                vnitates in tali numero:</s>
              </p>
              <cb chead="Capitulum ſecundum"/>
              <p xml:id="N11F1F">
                <s xml:id="N11F20" xml:space="preserve">Sexta ſuppoſitio </s>
                <s xml:id="N11F23" xml:space="preserve">Si duo numeri
                  <lb/>
                vel quantitates diuidantur in partes aliquotas
                  <lb/>
                eiuſdem denominationis: et perdit aliquam vel
                  <lb/>
                aliquod partes aliquotas ex illa vter illorū re-
                  <lb/>
                manentibus aliquibus: reſidue erunt eiuſdē deno­
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                minationis. </s>
                <s xml:id="N11F30" xml:space="preserve">vt ſi bipedale diuidatur in .5. quin-
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                tas et pedale ſimiliter: et perdit bipedale duas q̇n­
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                tas ex eis: et pedale ſimiliter: reſidue partes erunt
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                eiuſdē denominatiõis: puta tertie: vt patet </s>
                <s xml:id="N11F39" xml:space="preserve">Pro­
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                batur / quia in principio decremēti ille partes ali-
                  <lb/>
                quote illarum quantitatum ſunt equales numero
                  <lb/>
                et equales numero deperdentur ab vtra illaruꝫ
                  <lb/>
                quantitatum / vt ponitur remanentibus aliquibus
                  <lb/>
                ex illis: ergo remantes manebunt equales nu-
                  <lb/>
                mero. </s>
                <s xml:id="N11F48" xml:space="preserve">Patet conſequentia / q2 ſi ab equalibus nu-
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                meris equales demas .etc̈. / et ꝑ conſequens ſemper
                  <lb/>
                denominabuntur ab equali numero: quare ſemꝑ
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                erunt eiuſdem denominationis / vt patet ex diffini­
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                tione.</s>
              </p>
              <p xml:id="N11F53">
                <s xml:id="N11F54" xml:space="preserve">Septima ſuppoſitio </s>
                <s xml:id="N11F57" xml:space="preserve">Qualis eſt pro­
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                portio alicuius ad aliquam eius partem aliquo-
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                tam: talis eſt cuiuſlibet alteriꝰ ad partē aliquotã
                  <lb/>
                eius conſiĺis denominationis. </s>
                <s xml:id="N11F60" xml:space="preserve">vt qualis eſt ꝓpor­
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                tio alicuius quãtitatis ad ſuã medietatē tertiam
                  <lb/>
                quartam .etc̈. talis eſt cuiuſlibet alterius ad ſuã me­
                  <lb/>
                dietatem tertiã quartã .etc̈. </s>
                <s xml:id="N11F69" xml:space="preserve">Patet hec ex q̈rta ſup­
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                poſitõe / hoc adito /  q̊ties aliq̈ quãtitas ↄ̨tinet ali­
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                quam ſui partem aliquotaꝫ: toties quelibet alia
                  <lb/>
                quantitas continet partem ſui aliquotam cõſimi­
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                lis denominationis: cum ſemper partes aliquote
                  <lb/>
                eiuſdem denominationis ſint equales numero / vt
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                patet ex quinta ſuppoſitione:</s>
              </p>
              <p xml:id="N11F78">
                <s xml:id="N11F79" xml:space="preserve">Octaua ſuppoſitio </s>
                <s xml:id="N11F7C" xml:space="preserve">Si aliqui duo nu­
                  <lb/>
                meri ſiue quantitates diuidantur in duas partes
                  <lb/>
                equales: cuiuſlibet illorum numerorum ad alterã
                  <lb/>
                illarum ſuarum partium eſt eadem ꝓportio. </s>
                <s xml:id="N11F85" xml:space="preserve">Et ſi
                  <lb/>
                vter duorum numerorum diuidatur in plures ꝑ­
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                tes aliquotas eiuſdem denominationis quaꝫ ſint
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                due: talis eſt ꝓportio vnius illorum numerorū ad
                  <lb/>
                aggregatū ex omnibus talibus partibus aliquo­
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                tis dempta vna: qualis eſt alterius ad aggrega-
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                tum ex omnibus dempta ſimiliter vna. / vt diuiſo
                  <lb/>
                ſenario in tres partes aliquotas: et ſimiliter ter­
                  <lb/>
                nario: talis eſt ꝓportio ipſius ſenarii ad aggre-
                  <lb/>
                gatum ex duabus tertiis eius qualis ē ternarii ad
                  <lb/>
                aggregatum ex duabus tertiis eius. </s>
                <s xml:id="N11F9C" xml:space="preserve">vt conſtat.</s>
              </p>
              <p xml:id="N11F9F">
                <s xml:id="N11FA0" xml:space="preserve">Probatur ſuppoſitio. </s>
                <s xml:id="N11FA3" xml:space="preserve">ſint duo numeri ſiue equa­
                  <lb/>
                les ſiue inequales. </s>
                <s xml:id="N11FA8" xml:space="preserve">primus .a.b. ſecundus .c.d. diui­
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                ſi in partes aliquotas eiuſdem denominationis
                  <lb/>
                et ſit primi numeri vna illarum partium .a. et reſi­
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                due .b. ſecundi vero numeri ſit conſimilis pars ali­
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                quota .c. et reſidue partes eiuſdem numeri .d. / et di­
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                co /  talis ē proportio a.b. ad .b. qualis eſt .c.d. ad
                  <lb/>
                d. </s>
                <s xml:id="N11FB7" xml:space="preserve">Quod probatur ſic / quia quota vice .a.b. conti-
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                net .b. et aliquam partem aliquotam ipſius .b. to-
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                ta vice .c.d. continet .d. quia ſemel / vt conſtat et vnã
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                partem eius aliquotam euſdem denominationis
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                cum parte aliquota ipſius .b. quam coutinet .a.b /
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                igitur qualis eſt proportio .a.b. ad b. talis eſt pro­
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                portio .c.d. ad .d. / quod fuit probãdū </s>
                <s xml:id="N11FC6" xml:space="preserve">Patet hec cõ­
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                ſequentia clare ex quarta ſuppoſitione. </s>
                <s xml:id="N11FCB" xml:space="preserve"> autem .c.
                  <lb/>
                ſit pars aliquota ipſius .d. eiuſdem denominatio­
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                nis cuius .a. eſt pars aliquota ipſius .b. / probatur /
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                quia ſi .a.b. numerus perdat .a. et .c.d. ꝑdat .c. / tunc
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                reſidue partes manebunt partes eiuſdem denomi­ </s>
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