Vitruvius, De architectura libri decem, 1567

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          <div xml:id="echoid-div276" type="section" level="2" n="61">
            <p style="it">
              <s xml:id="echoid-s7573" xml:space="preserve">
                <pb o="81" file="514.01.113" n="113" rhead="TERTIVS."/>
              Vtriuſque generis proprium eſt, ut ſecundum collationem , comparationemq́; </s>
              <s xml:id="echoid-s7574" xml:space="preserve">res æquales, uel inæquales´ mutuo
                <lb/>
              dicantur, nam res omnis alteri comparata uel æqualis illi eſſe cernitur, uel inæqualis, nam uel tanta est, quan-
                <lb/>
              ta eſt alia, uel maior, uel minor. </s>
              <s xml:id="echoid-s7575" xml:space="preserve">Proportio igitur eſt ex earum numero, quæ per ſe nec ſunt, nec intelliguntur ,
                <lb/>
              ſed id, quod ſunt ad aliud referuntur. </s>
              <s xml:id="echoid-s7576" xml:space="preserve">& </s>
              <s xml:id="echoid-s7577" xml:space="preserve">ex proportionibus aliæ inter res æquales, aliæ inter inæquales cadent.
                <lb/>
              </s>
              <s xml:id="echoid-s7578" xml:space="preserve">Vnde proportio est determinata rerum duarum, quæ ſub eodem genere continentur, habitudo. </s>
              <s xml:id="echoid-s7579" xml:space="preserve">Determinata in-
                <lb/>
              quam non ad nos relata, neque in ſe certa , ſed huiuſmodi , ut alia eſſe non poßit. </s>
              <s xml:id="echoid-s7580" xml:space="preserve">Sub eodem autem genere,
                <lb/>
              quia nemo lineam maiorem, uel minorem, uel æqualem ſuperficiei dixerit, ſed alteri lineæ, nec ſuperficiem cor-
                <lb/>
              pori, ſed alteri plano comparabit. </s>
              <s xml:id="echoid-s7581" xml:space="preserve">cum ergo proportio in quantitate ſit , duplexq́; </s>
              <s xml:id="echoid-s7582" xml:space="preserve">ſit quantitatis genus ( ut di-
                <lb/>
              xi ) certe tam in menſuris, quàm in numeris, ac etiam in his rebus, quæ ex menſuris, & </s>
              <s xml:id="echoid-s7583" xml:space="preserve">numeris componun-
                <lb/>
                <note position="left" xlink:label="note-514.01.113-01" xlink:href="note-514.01.113-01a" xml:space="preserve">10</note>
              tur, inuenietur, Quæ igitur ad menſuras proportio pertinet, Geometrarum propria eſt; </s>
              <s xml:id="echoid-s7584" xml:space="preserve">quæ ad numeros, eorum
                <lb/>
              qui numer andi ſcientiam profitentur; </s>
              <s xml:id="echoid-s7585" xml:space="preserve">quæ ad utrumque genus Muſicis accommodatur, harmonicaq́; </s>
              <s xml:id="echoid-s7586" xml:space="preserve">nuncupa-
                <lb/>
              tur. </s>
              <s xml:id="echoid-s7587" xml:space="preserve">Nos modo de ſimplicibus illis dicemus , ſed libro quinto de permixta illa ex utriſque generibus , cum de
                <lb/>
              muſica ratione tractabimus copioſe diſſeremus. </s>
              <s xml:id="echoid-s7588" xml:space="preserve">Magnitudines, & </s>
              <s xml:id="echoid-s7589" xml:space="preserve">numeri inter ſe comparati, aut pares,
                <lb/>
              æqualesq́; </s>
              <s xml:id="echoid-s7590" xml:space="preserve">ſunt, aut ſe ſe mutuo ſuccedunt . </s>
              <s xml:id="echoid-s7591" xml:space="preserve">Pares, æqualesq́; </s>
              <s xml:id="echoid-s7592" xml:space="preserve">intelligo, cum nihil amplius uni ſit, quàm alteri.
                <lb/>
              </s>
              <s xml:id="echoid-s7593" xml:space="preserve">impares, & </s>
              <s xml:id="echoid-s7594" xml:space="preserve">ſe ſe eccedentes, cum quid maius, aut minus in eis eſſe dignoſcitur. </s>
              <s xml:id="echoid-s7595" xml:space="preserve">Aequalium, pariumq́; </s>
              <s xml:id="echoid-s7596" xml:space="preserve">nullum
                <lb/>
              aliud genus eſt, nam cum res inter ſe æquabili ratione conferuntur , unius formæ comparatio inter eas cadit. </s>
              <s xml:id="echoid-s7597" xml:space="preserve">
                <lb/>
              Inæqualium plura ſunt genera, ac totidem, quotres una altera maior eſſe potest . </s>
              <s xml:id="echoid-s7598" xml:space="preserve">maiorem intelligo contine-
                <lb/>
              re minorem integram, uel amplius aliquid. </s>
              <s xml:id="echoid-s7599" xml:space="preserve">Inæqualitatis igitur proportio, aut ſimplex est, aut compoſita. </s>
              <s xml:id="echoid-s7600" xml:space="preserve">ſim
                <lb/>
              plex es
                <unsure/>
              t, cum quod maius est, minus continet uel pluribus uicibus. </s>
              <s xml:id="echoid-s7601" xml:space="preserve">ſi pluribus uicibus ita , ut ex inte-
                <lb/>
                <note position="left" xlink:label="note-514.01.113-02" xlink:href="note-514.01.113-02a" xml:space="preserve">20</note>
              gro nihil ſuperſit, tunc proportio ea conficitur, quæ multiplex nominatur. </s>
              <s xml:id="echoid-s7602" xml:space="preserve">Ita quatuor continet duo bis . </s>
              <s xml:id="echoid-s7603" xml:space="preserve">Sex
                <lb/>
              continet duo ter. </s>
              <s xml:id="echoid-s7604" xml:space="preserve">Octo duo quater. </s>
              <s xml:id="echoid-s7605" xml:space="preserve">& </s>
              <s xml:id="echoid-s7606" xml:space="preserve">nihil amplius. </s>
              <s xml:id="echoid-s7607" xml:space="preserve">ſi una uice quod plus eſt continet minus, uel unã inſuper præ-
                <lb/>
              ter integram partem eius continet, uel plures. </s>
              <s xml:id="echoid-s7608" xml:space="preserve">ſi unam, uel illa est dimidium, uel tertia pars, uel quarta, & </s>
              <s xml:id="echoid-s7609" xml:space="preserve">ita
                <lb/>
              in infinitum. </s>
              <s xml:id="echoid-s7610" xml:space="preserve">Atque hæc proportio ſuperparticularis nominatur. </s>
              <s xml:id="echoid-s7611" xml:space="preserve">ſeſquialtera ſi dimidium, ſeſquitertia, ſi ter-
                <lb/>
              tiam partem, ſeſquiquarta ſi quartam continet, & </s>
              <s xml:id="echoid-s7612" xml:space="preserve">ita in infinitum. </s>
              <s xml:id="echoid-s7613" xml:space="preserve">Si vero maius continet minus integre , & </s>
              <s xml:id="echoid-s7614" xml:space="preserve">
                <lb/>
              aliquas eius partes, tunc ſuperpartiens nominatur illarum comparatio . </s>
              <s xml:id="echoid-s7615" xml:space="preserve">aliquas autem partes intelligo, non
                <lb/>
              ut aiunt numerantes , ſed ex illis conſtantes. </s>
              <s xml:id="echoid-s7616" xml:space="preserve">Pars enim numerans dicitur ea, quæ integram rem certis uicibus
                <lb/>
              deſumpta conſtituit. </s>
              <s xml:id="echoid-s7617" xml:space="preserve">Cuiuſmodi duo pars est numerans ſenarium, nam ter duo conſtituunt ſex . </s>
              <s xml:id="echoid-s7618" xml:space="preserve">quatuor item
                <lb/>
              numerat. </s>
              <s xml:id="echoid-s7619" xml:space="preserve">12. </s>
              <s xml:id="echoid-s7620" xml:space="preserve">nam ter quatuor duodecim efficiunt . </s>
              <s xml:id="echoid-s7621" xml:space="preserve">huiuſmodi partes quotas etiam appellare ſolent . </s>
              <s xml:id="echoid-s7622" xml:space="preserve">ſuper-
                <lb/>
              partiens igitur ratio eſt quotiens maius, minus ſemel continet , & </s>
              <s xml:id="echoid-s7623" xml:space="preserve">partem ipſius minoris non quotam, & </s>
              <s xml:id="echoid-s7624" xml:space="preserve">nu-
                <lb/>
                <note position="left" xlink:label="note-514.01.113-03" xlink:href="note-514.01.113-03a" xml:space="preserve">30</note>
              merantem, ſed ex illis quotis, & </s>
              <s xml:id="echoid-s7625" xml:space="preserve">numerantibus prouenientem. </s>
              <s xml:id="echoid-s7626" xml:space="preserve">ſic quinque continet tres ſemel , & </s>
              <s xml:id="echoid-s7627" xml:space="preserve">duas eius
                <lb/>
              partes, unde appellatur proportio bipartiens tertias, ſeptem continet quatuor, & </s>
              <s xml:id="echoid-s7628" xml:space="preserve">tres quartas, & </s>
              <s xml:id="echoid-s7629" xml:space="preserve">appellatur
                <lb/>
              proportio tripartiens quartas, & </s>
              <s xml:id="echoid-s7630" xml:space="preserve">nouem continet quinque, & </s>
              <s xml:id="echoid-s7631" xml:space="preserve">quatuor eius quintas, & </s>
              <s xml:id="echoid-s7632" xml:space="preserve">appellatur proportio
                <lb/>
              quadripartiens quintas, atque ita in infinitum. </s>
              <s xml:id="echoid-s7633" xml:space="preserve">Simplices igitur comparationes hæ ſunt, multiplex, ſupraparti-
                <lb/>
              cularis, ſuprapartiens. </s>
              <s xml:id="echoid-s7634" xml:space="preserve">Compoſitæ autem vim, & </s>
              <s xml:id="echoid-s7635" xml:space="preserve">naturam ſimplicium tenent , nam vel multiplex eſt, & </s>
              <s xml:id="echoid-s7636" xml:space="preserve">ſu-
                <lb/>
              praparticularis, vel multiplex, & </s>
              <s xml:id="echoid-s7637" xml:space="preserve">ſuprapartiens. </s>
              <s xml:id="echoid-s7638" xml:space="preserve">quatenus multiplex maius continebit rem integram pluri-
                <lb/>
              bus uicibus; </s>
              <s xml:id="echoid-s7639" xml:space="preserve">quatenus ſupra particularis, continebit amplius, aliquam eius partem, quotam ſcilicet , & </s>
              <s xml:id="echoid-s7640" xml:space="preserve">nu-
                <lb/>
              merantem; </s>
              <s xml:id="echoid-s7641" xml:space="preserve">quatenus ſupraparticularis continebit amplius partem non quotam , & </s>
              <s xml:id="echoid-s7642" xml:space="preserve">numerantem, ſed ex quo-
                <lb/>
              tis, & </s>
              <s xml:id="echoid-s7643" xml:space="preserve">numerantibus prouenientem. </s>
              <s xml:id="echoid-s7644" xml:space="preserve">multiplex ſupraparticularis eſt inter decem, & </s>
              <s xml:id="echoid-s7645" xml:space="preserve">quatuor, nam decem con-
                <lb/>
              tinet quatuor bis, unde multiplicis nomen ſortitur , & </s>
              <s xml:id="echoid-s7646" xml:space="preserve">dimidium , vnde ſeſquialtera nuncupatur, quare du-
                <lb/>
              plex ſeſquialtera dicetur. </s>
              <s xml:id="echoid-s7647" xml:space="preserve">multiplex ſuprapartiens est inter octo, & </s>
              <s xml:id="echoid-s7648" xml:space="preserve">tria, nam octo continet tria bis, unde mul-
                <lb/>
                <note position="left" xlink:label="note-514.01.113-04" xlink:href="note-514.01.113-04a" xml:space="preserve">40</note>
              tiplicis ratio naſcitur, & </s>
              <s xml:id="echoid-s7649" xml:space="preserve">duo, unde ſuprapartiens nominatur, atque ita inter octo & </s>
              <s xml:id="echoid-s7650" xml:space="preserve">tria erit proportio dupla
                <lb/>
              bipartiens tertias . </s>
              <s xml:id="echoid-s7651" xml:space="preserve">Oriuntur autem proportiones e numeris certo quodam ordine diſpoſitis , & </s>
              <s xml:id="echoid-s7652" xml:space="preserve">collocatis. </s>
              <s xml:id="echoid-s7653" xml:space="preserve">ac
                <lb/>
              primum ſupraparticulares rationes ita fient. </s>
              <s xml:id="echoid-s7654" xml:space="preserve">Diſpone a ternario numeros ſe ſe ut natura fert, conſequentes. </s>
              <s xml:id="echoid-s7655" xml:space="preserve">at-
                <lb/>
              que eo in primo , & </s>
              <s xml:id="echoid-s7656" xml:space="preserve">ſuperiori ordine collocabis , ſubſcribes eoſdem a binario natura ſe conſequentes quem-
                <lb/>
              admodum uides.</s>
              <s xml:id="echoid-s7657" xml:space="preserve"/>
            </p>
            <p style="it">
              <s xml:id="echoid-s7658" xml:space="preserve">3.</s>
              <s xml:id="echoid-s7659" xml:space="preserve">#4.</s>
              <s xml:id="echoid-s7660" xml:space="preserve">#5.</s>
              <s xml:id="echoid-s7661" xml:space="preserve">#6.</s>
              <s xml:id="echoid-s7662" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7663" xml:space="preserve">#8.</s>
              <s xml:id="echoid-s7664" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7665" xml:space="preserve">#10.</s>
              <s xml:id="echoid-s7666" xml:space="preserve">#11.</s>
              <s xml:id="echoid-s7667" xml:space="preserve">#12.</s>
              <s xml:id="echoid-s7668" xml:space="preserve">#hi ſunt numeri a quibus incipit comparatio.
                <lb/>
              </s>
              <s xml:id="echoid-s7669" xml:space="preserve">2.</s>
              <s xml:id="echoid-s7670" xml:space="preserve">#3.</s>
              <s xml:id="echoid-s7671" xml:space="preserve">#4.</s>
              <s xml:id="echoid-s7672" xml:space="preserve">#5.</s>
              <s xml:id="echoid-s7673" xml:space="preserve">#6.</s>
              <s xml:id="echoid-s7674" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7675" xml:space="preserve">#8.</s>
              <s xml:id="echoid-s7676" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7677" xml:space="preserve">#10.</s>
              <s xml:id="echoid-s7678" xml:space="preserve">#11.</s>
              <s xml:id="echoid-s7679" xml:space="preserve">#hi ſunt numeri in quos deſinit comparatio , & </s>
              <s xml:id="echoid-s7680" xml:space="preserve">ad
                <lb/>
              quos ſuperiores ſe habent in ſupraparticulari proportione, nam. </s>
              <s xml:id="echoid-s7681" xml:space="preserve">3. </s>
              <s xml:id="echoid-s7682" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7683" xml:space="preserve">2. </s>
              <s xml:id="echoid-s7684" xml:space="preserve">est ſeſquialtera. </s>
              <s xml:id="echoid-s7685" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7686" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7687" xml:space="preserve">3. </s>
              <s xml:id="echoid-s7688" xml:space="preserve">
                <lb/>
              ſeſquitertia. </s>
              <s xml:id="echoid-s7689" xml:space="preserve">5. </s>
              <s xml:id="echoid-s7690" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7691" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7692" xml:space="preserve">ſeſquiquarta, & </s>
              <s xml:id="echoid-s7693" xml:space="preserve">ita in infinitum. </s>
              <s xml:id="echoid-s7694" xml:space="preserve">Superpartiens oritur imparibus numeris ſua ſe-
                <lb/>
              rie ſuprapoſitis, modo a quinario incipias, & </s>
              <s xml:id="echoid-s7695" xml:space="preserve">a numeris naturali ordine, ſi conſequentibus a ternario in
                <lb/>
                <note position="left" xlink:label="note-514.01.113-05" xlink:href="note-514.01.113-05a" xml:space="preserve">50</note>
              hunc modum.</s>
              <s xml:id="echoid-s7696" xml:space="preserve"/>
            </p>
            <p style="it">
              <s xml:id="echoid-s7697" xml:space="preserve">5.</s>
              <s xml:id="echoid-s7698" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7699" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7700" xml:space="preserve">#11.</s>
              <s xml:id="echoid-s7701" xml:space="preserve">#13.</s>
              <s xml:id="echoid-s7702" xml:space="preserve">#15.</s>
              <s xml:id="echoid-s7703" xml:space="preserve">#17.</s>
              <s xml:id="echoid-s7704" xml:space="preserve">#19.</s>
              <s xml:id="echoid-s7705" xml:space="preserve">#21.</s>
              <s xml:id="echoid-s7706" xml:space="preserve">#numeri a quibus incipit comparatio.
                <lb/>
              </s>
              <s xml:id="echoid-s7707" xml:space="preserve">3.</s>
              <s xml:id="echoid-s7708" xml:space="preserve">#4.</s>
              <s xml:id="echoid-s7709" xml:space="preserve">#5.</s>
              <s xml:id="echoid-s7710" xml:space="preserve">#6.</s>
              <s xml:id="echoid-s7711" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7712" xml:space="preserve">#8.</s>
              <s xml:id="echoid-s7713" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7714" xml:space="preserve">#10.</s>
              <s xml:id="echoid-s7715" xml:space="preserve">#11.</s>
              <s xml:id="echoid-s7716" xml:space="preserve">#numeri in quos deſinit comparatio. </s>
              <s xml:id="echoid-s7717" xml:space="preserve">atque ita. </s>
              <s xml:id="echoid-s7718" xml:space="preserve">5. </s>
              <s xml:id="echoid-s7719" xml:space="preserve">
                <lb/>
              ad 3. </s>
              <s xml:id="echoid-s7720" xml:space="preserve">bipartiens tertias. </s>
              <s xml:id="echoid-s7721" xml:space="preserve">7. </s>
              <s xml:id="echoid-s7722" xml:space="preserve">ad 4. </s>
              <s xml:id="echoid-s7723" xml:space="preserve">tripartiens quartas. </s>
              <s xml:id="echoid-s7724" xml:space="preserve">9. </s>
              <s xml:id="echoid-s7725" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7726" xml:space="preserve">5, quadripartiens quintas, & </s>
              <s xml:id="echoid-s7727" xml:space="preserve">ſic de
                <lb/>
              ſingulis. </s>
              <s xml:id="echoid-s7728" xml:space="preserve">quod ſi ſuperiorem imparium numerorum ordinem a quinario orſus adſcribes, inferius autem a bina-
                <lb/>
              rio natur ali ordine præcedentes numeros ſubſcribes , differentias multiplicis ſupraparticularis , ſub ratione
                <lb/>
              dupla, content as conficies.</s>
              <s xml:id="echoid-s7729" xml:space="preserve"/>
            </p>
            <p>
              <s xml:id="echoid-s7730" xml:space="preserve">5.</s>
              <s xml:id="echoid-s7731" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7732" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7733" xml:space="preserve">#11.</s>
              <s xml:id="echoid-s7734" xml:space="preserve">#13.</s>
              <s xml:id="echoid-s7735" xml:space="preserve">#15.</s>
              <s xml:id="echoid-s7736" xml:space="preserve">#17.</s>
              <s xml:id="echoid-s7737" xml:space="preserve">#19.</s>
              <s xml:id="echoid-s7738" xml:space="preserve">#21.</s>
              <s xml:id="echoid-s7739" xml:space="preserve"/>
            </p>
            <p style="it">
              <s xml:id="echoid-s7740" xml:space="preserve">2.</s>
              <s xml:id="echoid-s7741" xml:space="preserve">#3.</s>
              <s xml:id="echoid-s7742" xml:space="preserve">#4.</s>
              <s xml:id="echoid-s7743" xml:space="preserve">#5.</s>
              <s xml:id="echoid-s7744" xml:space="preserve">#6.</s>
              <s xml:id="echoid-s7745" xml:space="preserve">#7.</s>
              <s xml:id="echoid-s7746" xml:space="preserve">#8.</s>
              <s xml:id="echoid-s7747" xml:space="preserve">#9.</s>
              <s xml:id="echoid-s7748" xml:space="preserve">#10. </s>
              <s xml:id="echoid-s7749" xml:space="preserve">Siaſeptenario incœperis per impares procedendo,
                <lb/>
              & </s>
              <s xml:id="echoid-s7750" xml:space="preserve">eandem in inferiori ſerie ſucceſſionem a binario ſeruaueris, multiplicis ſuperparticularis formas , ſub ratio-
                <lb/>
                <note position="left" xlink:label="note-514.01.113-06" xlink:href="note-514.01.113-06a" xml:space="preserve">60</note>
              ne tripla constitues.</s>
              <s xml:id="echoid-s7751" xml:space="preserve"/>
            </p>
          </div>
        </div>
      </text>
    </echo>
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