Vitruvius, De architectura libri decem, 1567

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              <s xml:id="echoid-s8108" xml:space="preserve">
                <pb o="85" file="514.01.117" n="117" rhead="TEBTIVS."/>
              ſic 24. </s>
              <s xml:id="echoid-s8109" xml:space="preserve">ad 16. </s>
              <s xml:id="echoid-s8110" xml:space="preserve">erit conuerſa ratione, ut 16. </s>
              <s xml:id="echoid-s8111" xml:space="preserve">ad 24. </s>
              <s xml:id="echoid-s8112" xml:space="preserve">ita 20. </s>
              <s xml:id="echoid-s8113" xml:space="preserve">ad 30. </s>
              <s xml:id="echoid-s8114" xml:space="preserve">ducantur igitur 20. </s>
              <s xml:id="echoid-s8115" xml:space="preserve">in 24.
                <lb/>
              </s>
              <s xml:id="echoid-s8116" xml:space="preserve">reddentur 480. </s>
              <s xml:id="echoid-s8117" xml:space="preserve">quæ partita per 16. </s>
              <s xml:id="echoid-s8118" xml:space="preserve">efficient 30. </s>
              <s xml:id="echoid-s8119" xml:space="preserve">Pari ratione ignoretur ſecundus numerus ſcilicet 20. </s>
              <s xml:id="echoid-s8120" xml:space="preserve">
                <lb/>
              Collocato ante primum duos poſteriores terminos hoc modo 24. </s>
              <s xml:id="echoid-s8121" xml:space="preserve">16. </s>
              <s xml:id="echoid-s8122" xml:space="preserve">30. </s>
              <s xml:id="echoid-s8123" xml:space="preserve">cum igitur ita ſe habeat 24. </s>
              <s xml:id="echoid-s8124" xml:space="preserve">ad
                <lb/>
              16. </s>
              <s xml:id="echoid-s8125" xml:space="preserve">ut 30. </s>
              <s xml:id="echoid-s8126" xml:space="preserve">ad 20. </s>
              <s xml:id="echoid-s8127" xml:space="preserve">Sequetur quodita ſe habebit 30. </s>
              <s xml:id="echoid-s8128" xml:space="preserve">ad 20. </s>
              <s xml:id="echoid-s8129" xml:space="preserve">ſic ut 24. </s>
              <s xml:id="echoid-s8130" xml:space="preserve">ad 16. </s>
              <s xml:id="echoid-s8131" xml:space="preserve">ducito ergo 30. </s>
              <s xml:id="echoid-s8132" xml:space="preserve">
                <lb/>
              per 16. </s>
              <s xml:id="echoid-s8133" xml:space="preserve">conficies 480. </s>
              <s xml:id="echoid-s8134" xml:space="preserve">quæ partita per 24. </s>
              <s xml:id="echoid-s8135" xml:space="preserve">efficient 20. </s>
              <s xml:id="echoid-s8136" xml:space="preserve">Cæterum ſi tertius ignoratur numerus, ſcili-
                <lb/>
              cet 24. </s>
              <s xml:id="echoid-s8137" xml:space="preserve">conuerte antecedentes numeros in conſequentes, & </s>
              <s xml:id="echoid-s8138" xml:space="preserve">conſequentes in antecedentes, erit igitur ut 20. </s>
              <s xml:id="echoid-s8139" xml:space="preserve">
                <lb/>
              ad 30. </s>
              <s xml:id="echoid-s8140" xml:space="preserve">ita 16. </s>
              <s xml:id="echoid-s8141" xml:space="preserve">ad 24. </s>
              <s xml:id="echoid-s8142" xml:space="preserve">duc igitur 16. </s>
              <s xml:id="echoid-s8143" xml:space="preserve">per 30. </s>
              <s xml:id="echoid-s8144" xml:space="preserve">fient 480. </s>
              <s xml:id="echoid-s8145" xml:space="preserve">quæ partita per 20. </s>
              <s xml:id="echoid-s8146" xml:space="preserve">oſtendent 24. </s>
              <s xml:id="echoid-s8147" xml:space="preserve">
                <lb/>
              Sed poſtquàm partitione abſoluta ipſius producti per primum, & </s>
              <s xml:id="echoid-s8148" xml:space="preserve">partiens eſt numerus, reliquus erit numerus
                <lb/>
              aliquis diuidens minor, obſeruabis eum in ſubtiliores partes distribuendum ea ratione, quam tertius numerus
                <lb/>
              ſeu terminus obtinet, atque productus inde numerus iterum per primum eſt partiendus, idq́; </s>
              <s xml:id="echoid-s8149" xml:space="preserve">hoc modo facien-
                <lb/>
                <note position="left" xlink:label="note-514.01.117-01" xlink:href="note-514.01.117-01a" xml:space="preserve">10</note>
              dum, quatenus ex continenti partitione nil amplius relinquatur, quare & </s>
              <s xml:id="echoid-s8150" xml:space="preserve">illud memoria mandandum primum
                <lb/>
              numerum, cum ipſo tertio, re ac nomine ſeu qualitate ſemper conuenire. </s>
              <s xml:id="echoid-s8151" xml:space="preserve">Vnde ſi primus trium numerorum
                <lb/>
              ſolis integris numeris cons
                <unsure/>
              titerit, tertius uero fractis, uel contra, illud prius conaberis , quàm prædictam re-
                <lb/>
              gulam experiaris, ut alterum propoſitorum numerorum , ad alterius denominationem reuoces. </s>
              <s xml:id="echoid-s8152" xml:space="preserve">De ijs rebus
                <lb/>
              exempla plurima ex Arithmeticorum abacis deſumere potes. </s>
              <s xml:id="echoid-s8153" xml:space="preserve">Probabis autem an recte ipſam regulam ex
                <lb/>
              pertus fueris, ſi primum in quartum duxeris, nam idem reddetur terminus, qui ex ductu ſecundi in tertium
                <lb/>
              redditur. </s>
              <s xml:id="echoid-s8154" xml:space="preserve">nam ſi inæquales prædictos numeros inueneris , ſcito te erraſſe , ad calculum igitur redeundum eſt .
                <lb/>
              </s>
              <s xml:id="echoid-s8155" xml:space="preserve">Ac ut abſolutam reddamus hanc pulcherrimam quæſtionem, ex Alchindo addemus admir abilem illam regu-
                <lb/>
              lam , quæ quinque terminis propoſitis, ſextum ignotum dignoſcere, quemcunque locum teneat, facile poſſimus. </s>
              <s xml:id="echoid-s8156" xml:space="preserve">
                <lb/>
              nam dari poſſunt ſex numerorum termini, ſic ſe mutua ratione reſpondentes , ut ratio primi ad ſecundum com-
                <lb/>
                <note position="left" xlink:label="note-514.01.117-02" xlink:href="note-514.01.117-02a" xml:space="preserve">20</note>
              ponatur, ex rationibus tertij ad quartum, & </s>
              <s xml:id="echoid-s8157" xml:space="preserve">ipſius quintĩ ad ſextum. </s>
              <s xml:id="echoid-s8158" xml:space="preserve"># Dentur igitur ſex termini ſibi ratio-
                <lb/>
              ne aliqua comparati. </s>
              <s xml:id="echoid-s8159" xml:space="preserve">1 2 3 4 6 9. </s>
              <s xml:id="echoid-s8160" xml:space="preserve">Dico rationem unius ad duo, quæ est ſubdupla, conſtare ex ratione
                <lb/>
              3. </s>
              <s xml:id="echoid-s8161" xml:space="preserve">ad 4. </s>
              <s xml:id="echoid-s8162" xml:space="preserve">quæ eſt ſubſeſquitertia, & </s>
              <s xml:id="echoid-s8163" xml:space="preserve">ratione 6. </s>
              <s xml:id="echoid-s8164" xml:space="preserve">ad 9. </s>
              <s xml:id="echoid-s8165" xml:space="preserve">quæ eſt ſubſeſquialtera. </s>
              <s xml:id="echoid-s8166" xml:space="preserve">Pone numeros eo quo ſu-
                <lb/>
              pra diximus ordine, fiet enim ratio 18. </s>
              <s xml:id="echoid-s8167" xml:space="preserve">ad 36. </s>
              <s xml:id="echoid-s8168" xml:space="preserve">ſimilis ei quæ est 1. </s>
              <s xml:id="echoid-s8169" xml:space="preserve">ad 2. </s>
              <s xml:id="echoid-s8170" xml:space="preserve">id eſt ſubdupla . </s>
              <s xml:id="echoid-s8171" xml:space="preserve">Compoſitio
                <lb/>
              huiuſmodi primorum terminorum caput, & </s>
              <s xml:id="echoid-s8172" xml:space="preserve">radix eſt omnium aliarum, quæ numero ſunt decem & </s>
              <s xml:id="echoid-s8173" xml:space="preserve">ſeptem.
                <lb/>
              </s>
              <s xml:id="echoid-s8174" xml:space="preserve">Multi uero ſunt modi , quibus hi termini permiſceri poſſunt , ſed uera compoſitio decem & </s>
              <s xml:id="echoid-s8175" xml:space="preserve">ſeptem, præter pri-
                <lb/>
              mum, constare dignoſcitur . </s>
              <s xml:id="echoid-s8176" xml:space="preserve">quorum ordines ſubſcripta tabella oſtendit. </s>
              <s xml:id="echoid-s8177" xml:space="preserve">Vtar autem numerorum notis , li-
                <lb/>
              cet uſus ſum in materna lingua literis a, b, c, d, e, f, ita enim Alchindus poſuerat. </s>
              <s xml:id="echoid-s8178" xml:space="preserve">ſed ad faciliorem intelli-
                <lb/>
              gentiam numerorum notas appingam. </s>
              <s xml:id="echoid-s8179" xml:space="preserve">Sex ordines notantur per primum, ſecundum , tertium , quartum,
                <lb/>
              quintum, & </s>
              <s xml:id="echoid-s8180" xml:space="preserve">ſextum, primus a læua , ſextus ad dexteram, cæteri ſunt intermedij. </s>
              <s xml:id="echoid-s8181" xml:space="preserve">Primorum uero termino-
                <lb/>
                <note position="left" xlink:label="note-514.01.117-03" xlink:href="note-514.01.117-03a" xml:space="preserve">30</note>
              rum ratione cum ſecundis in omnibus ordinibus, intelliguntur eſſe compoſitæ, ex rationibus tertij ad quartum,
                <lb/>
              & </s>
              <s xml:id="echoid-s8182" xml:space="preserve">quinti ad ſextum .</s>
              <s xml:id="echoid-s8183" xml:space="preserve"/>
            </p>
          </div>
          <div xml:id="echoid-div286" type="section" level="2" n="62">
            <head xml:id="echoid-head63" xml:space="preserve">TABVLASEX QVANTITATVM
              <lb/>
              <emph style="sc">Proportione respondentivm.</emph>
            </head>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            # ###### Ordo numerorum.
              <lb/>
            Modi compa- \\ rabiles. # Trimus. # Secundus. # Tertius. # Quartus. # Quintus. # Sextus.
              <lb/>
            Trimus modus. # 1 # 2 # 3 # 4 # 6 # 9
              <lb/>
            Secundus modus. # 1 # 2 # 3 # 9 # 6 # 4
              <lb/>
            Tertius modus. # 1 # 3 # 2 # 4 # 6 # 9
              <lb/>
            Quartus. # 1 # 3 # 2 # 9 # 6 # 4
              <lb/>
            Quintus. # 1 # 6 # 2 # 9 # 3 # 4
              <lb/>
            Sextus. # 1 # 6 # 2 # 4 # 3 # 9
              <lb/>
            Septimus. # 2 # 4 # 1 # 3 # 9 # 6
              <lb/>
            Octauus. # 2 # 4 # 1 # 6 # 9 # 3
              <lb/>
            Nonus. # 2 # 9 # 1 # 3 # 4 # 6
              <lb/>
            </note>
            <note position="left" xml:space="preserve">40</note>
            <note position="left" xml:space="preserve">50</note>
            <p>
              <s xml:id="echoid-s8184" xml:space="preserve">Trimus igitur modus , a quo cæteri uim ſumunt,
                <lb/>
              eſt in quo ratio primi numeri ad ſecundum con-
                <lb/>
              ſtat ex ratione tertij ad quartum, & </s>
              <s xml:id="echoid-s8185" xml:space="preserve">ratione quin-
                <lb/>
              ti ad ſextum.</s>
              <s xml:id="echoid-s8186" xml:space="preserve"/>
            </p>
            <note position="left" xml:space="preserve">60</note>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            # #### Ordo numerorum.
              <lb/>
            Modi compa- \\ rabiles. # Trimus. # Secundus. # Tertius. # Quartus. # Quintus. # Sextus.
              <lb/>
            Decimus. # 2 # 9 # 1 # 6 # 4 # 3
              <lb/>
            Vndecimus. # 3 # 4 # 1 # 2 # 9 # 6
              <lb/>
            Duodecimus. # 3 # 4 # 1 # 6 # 9 # 2
              <lb/>
            Decimus tertius. # 3 # 9 # 1 # 2 # 4 # 6
              <lb/>
            Decimus quartus. # 3 # 9 # 1 # 6 # 4 # 2
              <lb/>
            Decimus quinturs. # 4 # 6 # 2 # 1 # 3 # 9
              <lb/>
            Decimus ſextus. # 4 # 6 # 2 # 9 # 3 # 1
              <lb/>
            Decimus ſeptimus. # 6 # 9 # 1 # 2 # 4 # 3
              <lb/>
            Decimus octauus. # 6 # 9 # 1 # 3# 4 # 2
              <lb/>
            </note>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            Subſeſquitertia. # 3--4
              <lb/>
            Subſeſquialtera. # 6--9
              <lb/>
            Subdupla, ut 1. ad 2 # 18--36
              <lb/>
            </note>
          </div>
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