Alvarus, Thomas, Liber de triplici motu, 1509

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            <pb chead="Secunde partis" file="0020" n="20"/>
            <p xml:id="N11956">
              <s xml:id="N11957" xml:space="preserve">¶ Sequitur ſecunda pars de pro-
                <lb/>
              portionalitatibus et de quibuſdam
                <lb/>
              proportionum et proportionalita­
                <lb/>
              tum proprietatibus et accidentiis.</s>
            </p>
            <div xml:id="N11960" level="3" n="1" type="chapter" type-free="capitulum">
              <head xml:id="N11965" xml:space="preserve">Capitulum primum in quo a:
                <lb/>
              gitur de diffinitione et diuiſione
                <lb/>
              proportionalitatum.</head>
              <note position="left" xml:id="N1196C" xml:space="preserve">Nicho-
                <lb/>
              machus.</note>
              <p xml:id="N11972">
                <s xml:id="N11973" xml:space="preserve">pRoportionalitas iux­
                  <lb/>
                ta nichomachi ſententiam
                  <lb/>
                plurimum ad aſtrologiam
                  <lb/>
                muſicam, veterum lectio-
                  <lb/>
                nes intelligendas confert.
                  <lb/>
                </s>
                <s xml:id="N1197F" xml:space="preserve">Sed profecto ad phiſicam
                  <lb/>
                phiſicaſ calculatões nõ mi­
                  <lb/>
                nꝰ cõducit </s>
                <s xml:id="N11986" xml:space="preserve">Ad cuiꝰ ītelligēti­
                  <lb/>
                am aduertēdū eſt differētiã eſſe inter ꝓportionē et
                  <lb/>
                ꝓportionalitatē.
                  <note position="left" xlink:href="note-0020-01a" xlink:label="note-0020-01" xml:id="N11A73" xml:space="preserve">ꝓportio.</note>
                </s>
                <s xml:id="N11992" xml:space="preserve">¶ Proportio em̄ / vt dictum eſt
                  <lb/>
                habitudo eſt duarū quantitatū ad inuicē cõpara-
                  <lb/>
                tarū. </s>
                <s xml:id="N11999" xml:space="preserve">De qua ſuperius dictū eſt.
                  <note position="left" xlink:href="note-0020-02a" xlink:label="note-0020-02" xml:id="N11A79" xml:space="preserve">Propor­
                    <lb/>
                  tiõalitaſ</note>
                </s>
                <s xml:id="N119A1" xml:space="preserve">¶ Sed ꝓportiõa­
                  <lb/>
                litas eſt duarū ꝓportionū vel pluriū vnius ad al­
                  <lb/>
                teram certa habitudo. </s>
                <s xml:id="N119A8" xml:space="preserve">Ita vt ꝓportio: habitudo
                  <lb/>
                ſit numerorū ſiue quantitatū: ꝓportionalitas ve­
                  <lb/>
                ro proportionū collatio exiſtat. </s>
                <s xml:id="N119AF" xml:space="preserve">Sicut em̄ numeri
                  <lb/>
                ad inuicē cõparãtur in maioritate et in minoritate
                  <lb/>
                ita ꝓportiones ad inuiceꝫ in maioritate et minori­
                  <lb/>
                tate referūtur. </s>
                <s xml:id="N119B8" xml:space="preserve">¶ Naſcitur hinc oēm ꝓportionali­
                  <lb/>
                tatem ꝓportionē eſſe: quãuis nõ omīs ꝓportio ꝓ-
                  <lb/>
                portionalitas exiſtat.
                  <note position="left" xlink:href="note-0020-03a" xlink:label="note-0020-03" xml:id="N11A81" xml:space="preserve">Correla­
                    <lb/>
                  riū ṗmū</note>
                </s>
                <s xml:id="N119C4" xml:space="preserve">Patet hoc correlariū ex ſe
                  <lb/>
                </s>
                <s xml:id="N119C8" xml:space="preserve">Nam ꝓportio, aut genus, aut loco generis ſe ha-
                  <lb/>
                bet cū huic termino ꝓportionalitas comparatur
                  <lb/>
                </s>
                <s xml:id="N119CE" xml:space="preserve">Et aduerte /  in ꝓpoſito idem eſt medietas equa-
                  <lb/>
                litas et ꝓportionalitas: et eodē modo diffiniūtur.
                  <lb/>
                  <note position="left" xlink:href="note-0020-04a" xlink:label="note-0020-04" xml:id="N11A89" xml:space="preserve">medietaſ</note>
                </s>
                <s xml:id="N119DA" xml:space="preserve">Medietas em̄ eſt duarum vel pluriū ꝓportionum
                  <lb/>
                vnius ad alterã certa habitudo: vt habitudo que
                  <lb/>
                eſt inter ꝓportionē duplã et quadrupã.
                  <note position="left" xlink:href="note-0020-05a" xlink:label="note-0020-05" xml:id="N11A8F" xml:space="preserve">Diuiſio
                    <lb/>
                  ꝓportio­
                    <lb/>
                  nalitate.</note>
                </s>
                <s xml:id="N119E6" xml:space="preserve">¶ Poſita
                  <lb/>
                diffintione ꝓportionalitatis ponēda eſt diuiſio.
                  <lb/>
                  <note position="left" xlink:href="note-0020-06a" xlink:label="note-0020-06" xml:id="N11A99" xml:space="preserve">Undecim
                    <lb/>
                  medieta­
                    <lb/>
                  tes.</note>
                </s>
                <s xml:id="N119F2" xml:space="preserve">Apud recentiores mathematicos vndecim ſunt
                  <lb/>
                ꝓportionalitates ſiue medietates: quarū vltima
                  <lb/>
                perfectiſſima eſt: qm̄ in ea oēs conſonãtie muſica­
                  <lb/>
                les ſimplices reperiūtur. </s>
                <s xml:id="N119FB" xml:space="preserve">Sed apud ãtiquos tres
                  <lb/>
                ꝓportionalitates famate reperiūtur: videlicet a-
                  <lb/>
                rithmetica, geometrica, et muſica ſiue harmonica
                  <lb/>
                  <note position="left" xlink:href="note-0020-07a" xlink:label="note-0020-07" xml:id="N11AA3" xml:space="preserve">ꝓportio­
                    <lb/>
                  nalitas
                    <lb/>
                  arithme­
                    <lb/>
                  tica.</note>
                </s>
                <s xml:id="N11A09" xml:space="preserve">¶ Unde ꝓportionalitas arithmetica eſt quando
                  <lb/>
                diſpoſitis tribus quattuor vel pluribus terminis
                  <lb/>
                inter eos eedem differētie: ſed nõ eedem ꝓportio-
                  <lb/>
                nes reperiūtur. </s>
                <s xml:id="N11A12" xml:space="preserve">Exemplū / vt diſpoſitis his tribus
                  <lb/>
                terminis ſine numeris .1.3.5. inter quos nõ eadem
                  <lb/>
                ꝓportio reperitur: ſed bene eadē differētia. </s>
                <s xml:id="N11A19" xml:space="preserve">Uniꝰ
                  <lb/>
                em̄ ad .3. eſt ꝓpotio ſubtripla: et triū ad .5. eſt pro-
                  <lb/>
                portio ſubſuꝑbipartiēs tertias. </s>
                <s xml:id="N11A20" xml:space="preserve">Modo ille pro-
                  <lb/>
                portiones nõ ſunt ſimiles. </s>
                <s xml:id="N11A25" xml:space="preserve">Differentia tamen. </s>
                <s xml:id="N11A28" xml:space="preserve">i ex­
                  <lb/>
                ceſſus quo ſecūdus numerꝰ excedit primū eſt equa­
                  <lb/>
                lis differentie qua tertius excedit ſecundum: quia
                  <lb/>
                vtra dr̄a eſt binarius. </s>
                <s xml:id="N11A31" xml:space="preserve">In ꝓpoſito em̄ / hoc eſt in
                  <lb/>
                data diffinitione per terminos intelligas nume-
                  <lb/>
                ros ſereatim poſitos vel ea que ſe habēt vt nume­
                  <lb/>
                ri ſereatim poſiti:
                  <note position="left" xlink:href="note-0020-08a" xlink:label="note-0020-08" xml:id="N11AAF" xml:space="preserve">Differen­
                    <lb/>
                  tia.</note>
                et ꝑ differētias ītelligas exceſſū
                  <lb/>
                quo vnus numerus excedit alterū. </s>
                <s xml:id="N11A41" xml:space="preserve">Reperies autē /
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                hanc ꝓportionalitatē in naturali ſerie numerorū
                  <lb/>
                capiendo .6.7.8. comperies inter illos terminos
                  <lb/>
                diuerſas ꝓportiones: quoniã primi ad ſecundum
                  <lb/>
                eſt ꝓportio ſubſexquitertia / et ſecundi ad tertiū eſt
                  <lb/>
                ꝓportio ſubſexq̇ſeptīa et eſt equalis differētia in-
                  <cb chead="Capitulum primū."/>
                tes illos terminos. </s>
                <s xml:id="N11A51" xml:space="preserve">Quare in illis terminis repe­
                  <lb/>
                ritur ꝓportionalitas arithmetica. </s>
                <s xml:id="N11A56" xml:space="preserve">Sunt enim illi
                  <lb/>
                termini continuo proportionabiles arithmetice.
                  <lb/>
                  <note position="right" xlink:href="note-0020-09a" xlink:label="note-0020-09" xml:id="N11AB7">
                    <s xml:id="N11ABB" xml:space="preserve">Tertimini
                      <lb/>
                    ↄ̨tinuo ꝓ-
                      <lb/>
                    portiõa-
                      <lb/>
                    les ꝓpor­
                      <lb/>
                    tõalitate
                      <lb/>
                    aritithme
                      <lb/>
                    tica.
                      <lb/>
                    </s>
                    <s xml:id="N11ACB" xml:space="preserve">Corrrela­
                      <lb/>
                    riū ſcḋm</s>
                  </note>
                </s>
                <s xml:id="N11A62" xml:space="preserve">¶ Unde termini continuo proportionabiles pro-
                  <lb/>
                portionalitate arithmetica ſunt illi inter quos cõ-
                  <lb/>
                tinuo eſt equalis exceſſus ita  ſicut primus exce-
                  <lb/>
                dit ſecundum aliquo exceſſu: ita ſecundus excedat
                  <lb/>
                tertium equali exceſſu: et tertius quartum / et ſic con­
                  <lb/>
                ſequenter: vel econtra ſi incipias a minoribus.</s>
              </p>
              <p xml:id="N11AD0">
                <s xml:id="N11AD1" xml:space="preserve">¶ Ex quo elicitur omēs numeros in naturali ſerie
                  <lb/>
                numerorum eſſe terminos continuo proportiona­
                  <lb/>
                biles proportionalitate arithmetica: quoniã con­
                  <lb/>
                tinuo ſe excedunt equali exceſſu puta vnitate</s>
              </p>
              <note position="right" xml:id="N11ADA" xml:space="preserve">Correla-
                <lb/>
              riū ṫciū.</note>
              <p xml:id="N11AE0">
                <s xml:id="N11AE1" xml:space="preserve">¶ Sequitur vlterius proportiones duplam qua-
                  <lb/>
                druplam, octuplam, ſexdecuplam, trigecuplam
                  <lb/>
                ſecundam / et ſic conſequenter aſcēdendo per nume­
                  <lb/>
                ros pariter pares: eſſe terminos continuo propor­
                  <lb/>
                tionabiles arithmetice. </s>
                <s xml:id="N11AEC" xml:space="preserve">quoniã continuo ille pro-
                  <lb/>
                portiones ſe excedūt per equalem proportionem:
                  <lb/>
                puta duplam </s>
                <s xml:id="N11AF3" xml:space="preserve">Nam quadrupla excedit duplã per
                  <lb/>
                duplam: et octupla excedit quadruplam etiam per
                  <lb/>
                duplam: et ſimiliter ſexdecupla excedit octuplam
                  <lb/>
                per duplã: igitur ille proportiones continuo ſūt
                  <lb/>
                proportionabiles arithmetice. </s>
                <s xml:id="N11AFE" xml:space="preserve">Antecedens patet /
                  <lb/>
                quia addendo duplam ſupraduplã efficitur qua-
                  <lb/>
                drupla: et addendo duplam ſupraquadruplã effi­
                  <lb/>
                citur octupla: et ſic conſequenter. </s>
                <s xml:id="N11B07" xml:space="preserve">Et ille proporti-
                  <lb/>
                ones continuo per illa additamenta ſe excedūt: et
                  <lb/>
                illa additamenta cõtinuo ſunt proportiones du-
                  <lb/>
                ple / igitur cõtinuo ſe excedunt per proportionem
                  <lb/>
                dulam: quod fuit probandum. </s>
                <s xml:id="N11B12" xml:space="preserve">Huius medietatis
                  <lb/>
                proprietates in ſequenti capite patebunt.
                  <note position="right" xlink:href="note-0020-10a" xlink:label="note-0020-10" xml:id="N11B6D" xml:space="preserve">Geome-
                    <lb/>
                  trica me-
                    <lb/>
                  dietas.</note>
                </s>
                <s xml:id="N11B1C" xml:space="preserve">
                  <gap/>
                Geo-
                  <lb/>
                metrica autem medietas ſiue ꝓportionalitas eſt
                  <lb/>
                quotienſcun tribus diſpoſitis terminis: aut plu­
                  <lb/>
                ribus inter eos eedem proportiones reperiuntur
                  <lb/>
                eedeꝫ vero differētie nequa̄. </s>
                <s xml:id="N11B28" xml:space="preserve">Et per eaſdē ꝓpor-
                  <lb/>
                tiones in propoſitio ītelligas proportiones equa­
                  <lb/>
                les. </s>
                <s xml:id="N11B2F" xml:space="preserve">Et per equales proportiones intelligas pro-
                  <lb/>
                portiones eiuſdem denominationis. </s>
                <s xml:id="N11B34" xml:space="preserve">Cuiuſmodi
                  <lb/>
                ſunt proportio .4. ad .2. et 12. ad .6. </s>
                <s xml:id="N11B39" xml:space="preserve">Sunt em̄ eiuſ-
                  <lb/>
                dem denominationis: eſt enim vtra illarum du-
                  <lb/>
                pla: vt conſtat ex priori parte. </s>
                <s xml:id="N11B40" xml:space="preserve">Unde omnes duple
                  <lb/>
                ſunt equales: oēs ſexquialtere, et oēs ſuprabipar-
                  <lb/>
                tientes tertias. </s>
                <s xml:id="N11B47" xml:space="preserve">Exemplū / huius medietatis in his
                  <lb/>
                terminis .2:4.8. reperitur: quoniã qualis eſt pro-
                  <lb/>
                portio primi ad ſecūdum talis eſt proportio ſecū­
                  <lb/>
                di ad tertium: vtrobi enim ſubdupla proportio
                  <lb/>
                inuenitur: ſed non ſunt eedem differentie: quoniã
                  <lb/>
                tertius terminus ſecundum numero quaternario
                  <lb/>
                excedit: ſecūdus vero primum binario dumtaxat
                  <lb/>
                  <note position="right" xlink:href="note-0020-11a" xlink:label="note-0020-11" xml:id="N11B77" xml:space="preserve">Correla­
                    <lb/>
                  riū q̈rtū.</note>
                </s>
                <s xml:id="N11B5D" xml:space="preserve">¶ Educitur ex dictis omnes numeros pariter pa-
                  <lb/>
                res cõtinuo geometrice proportionari. </s>
                <s xml:id="N11B62" xml:space="preserve">Inter eas
                  <lb/>
                enim cõtinuo proportio dupla eſt: vt patet in his
                  <lb/>
                terminis. 2 4 8 16</s>
              </p>
              <note position="right" xml:id="N11B7F" xml:space="preserve">Correla­
                <lb/>
              riū quītã</note>
              <p xml:id="N11B85">
                <s xml:id="N11B86" xml:space="preserve">¶ Sequitur ſecundo omnes numeros impares cõ­
                  <lb/>
                tinuo ſe triplantes incipiendo a ternario conti-
                  <lb/>
                nuo proportionari geometrice. </s>
                <s xml:id="N11B8D" xml:space="preserve">Nam ſi continuo
                  <lb/>
                ſe triplant: continuo ſe habent in proportione tri­
                  <lb/>
                pla: ex quo quilibet ſequens immediate preceden­
                  <lb/>
                tem ter continet: vt patet in his terminis .3.9.2.7.
                  <lb/>
                  <note position="right" xlink:href="note-0020-12a" xlink:label="note-0020-12" xml:id="N11BB7" xml:space="preserve">Correla­
                    <lb/>
                  riū ſextã</note>
                </s>
                <s xml:id="N11B9D" xml:space="preserve">¶ Elicitur tertio omnes proportiones denomi-
                  <lb/>
                natas a numeris pariter paribus relinquendo
                  <lb/>
                poſt ſecundum numerum pariter parem vnum nu­
                  <lb/>
                merum: poſt quartum duos poſt ſeptimum quat­
                  <lb/>
                tuor: et ſic conſequenter duplando continuo nu-
                  <lb/>
                meros intermiſſos: eſſe terminos </s>
              </p>
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