Vitruvius, De architectura libri decem, 1567

Table of Notes

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          <div xml:id="echoid-div276" type="section" level="2" n="61">
            <p style="it">
              <s xml:id="echoid-s7981" xml:space="preserve">
                <pb o="84" file="514.01.116" n="116" rhead="LIBER"/>
              reliquæ rationis antecedens, & </s>
              <s xml:id="echoid-s7982" xml:space="preserve">ita quaſi decuſſatim ducendo rationis diuidendæ numerum conſequentem, per
                <lb/>
              diuidentis antecedentem, naſcetur reliquæ ſeu ortæ rationis conſequens numerus. </s>
              <s xml:id="echoid-s7983" xml:space="preserve">Exempla id aperte oſten-
                <lb/>
              dunt. </s>
              <s xml:id="echoid-s7984" xml:space="preserve">In multiplicibus primum id experiemur. </s>
              <s xml:id="echoid-s7985" xml:space="preserve">Subtrahenda ſit dupla a tripla, partire denominatorem triplæ,
                <lb/>
              per duo ipſius duplæ rationis deominatorem , fiet 1. </s>
              <s xml:id="echoid-s7986" xml:space="preserve">cum dimidio; </s>
              <s xml:id="echoid-s7987" xml:space="preserve">orietur igitur ex hac partitione ſeſquial-
                <lb/>
              tera proportio. </s>
              <s xml:id="echoid-s7988" xml:space="preserve">Similiter ſubducenda quadrupla ab octupla , reliqua erit dupla . </s>
              <s xml:id="echoid-s7989" xml:space="preserve">ſint huius numeri in quadru-
                <lb/>
              pla proportione 16. </s>
              <s xml:id="echoid-s7990" xml:space="preserve">& </s>
              <s xml:id="echoid-s7991" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7992" xml:space="preserve">& </s>
              <s xml:id="echoid-s7993" xml:space="preserve">64. </s>
              <s xml:id="echoid-s7994" xml:space="preserve">& </s>
              <s xml:id="echoid-s7995" xml:space="preserve">8. </s>
              <s xml:id="echoid-s7996" xml:space="preserve">in octupla. </s>
              <s xml:id="echoid-s7997" xml:space="preserve">duc octo in ſexdecim fient centum uiginti octo, & </s>
              <s xml:id="echoid-s7998" xml:space="preserve">
                <lb/>
              4. </s>
              <s xml:id="echoid-s7999" xml:space="preserve">in 64. </s>
              <s xml:id="echoid-s8000" xml:space="preserve">fient. </s>
              <s xml:id="echoid-s8001" xml:space="preserve">256. </s>
              <s xml:id="echoid-s8002" xml:space="preserve">interigitur 128. </s>
              <s xml:id="echoid-s8003" xml:space="preserve">& </s>
              <s xml:id="echoid-s8004" xml:space="preserve">256. </s>
              <s xml:id="echoid-s8005" xml:space="preserve">erit proportio dupla, nam bis 128. </s>
              <s xml:id="echoid-s8006" xml:space="preserve">producunt. </s>
              <s xml:id="echoid-s8007" xml:space="preserve">256.
                <lb/>
              </s>
              <s xml:id="echoid-s8008" xml:space="preserve">
                <note style="it" position="right" xlink:label="note-514.01.116-01" xlink:href="note-514.01.116-01a" xml:space="preserve">
                  <lb/>
                64--8 # Octupla.
                  <lb/>
                16--4 # quadrupla.
                  <lb/>
                128--256 # dupla.
                  <lb/>
                </note>
              In ſuperparticularibus quoque rationibus ablatio huiuſmodi hoc modo fiet.
                <lb/>
              </s>
              <s xml:id="echoid-s8009" xml:space="preserve">Auferenda ſit a ſeſquialtera ſeſquitertia. </s>
              <s xml:id="echoid-s8010" xml:space="preserve">partiri 1. </s>
              <s xml:id="echoid-s8011" xml:space="preserve">cum dimidio denomina-
                <lb/>
                <note position="left" xlink:label="note-514.01.116-02" xlink:href="note-514.01.116-02a" xml:space="preserve">10</note>
              torem ſeſquialteræ per 1. </s>
              <s xml:id="echoid-s8012" xml:space="preserve">& </s>
              <s xml:id="echoid-s8013" xml:space="preserve">tertiam denominatorem ſeſquitertiæ, reliqua
                <lb/>
              ratio erit 1 octaua hoc eſtſeſquioctaua.</s>
              <s xml:id="echoid-s8014" xml:space="preserve"/>
            </p>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            6--4 # ſeſquialtera.
              <lb/>
            8--6 # ſeſquitertia.
              <lb/>
            36--32 # ſeſquioctaua.
              <lb/>
            </note>
            <p style="it">
              <s xml:id="echoid-s8015" xml:space="preserve">In ſuperpartientibus demum auferatur bipartiens tertias , a tripartiente
                <lb/>
              quartas , diuide 1. </s>
              <s xml:id="echoid-s8016" xml:space="preserve">& </s>
              <s xml:id="echoid-s8017" xml:space="preserve">tres quartæ per 1. </s>
              <s xml:id="echoid-s8018" xml:space="preserve">& </s>
              <s xml:id="echoid-s8019" xml:space="preserve">duas tertias fiet vnum & </s>
              <s xml:id="echoid-s8020" xml:space="preserve">
                <lb/>
              vigeſima pars, a quibus ſeſquigeſima ratio denominatur.</s>
              <s xml:id="echoid-s8021" xml:space="preserve"/>
            </p>
            <note position="right" xml:space="preserve">
              <lb/>
            7--4 # tripartiens quartas.
              <lb/>
            5--3 # bipartiens tertias.
              <lb/>
            21--20 # ſeſquigeſima.
              <lb/>
            </note>
            <p style="it">
              <s xml:id="echoid-s8022" xml:space="preserve">Partita igitur ratione maioris inæqualitatis per rationem maioris, cum diſ-
                <lb/>
              ſimiles fuerint, ratio fiet maioris inæqualitatis, & </s>
              <s xml:id="echoid-s8023" xml:space="preserve">utraque minor.
                <lb/>
              </s>
              <s xml:id="echoid-s8024" xml:space="preserve">Eodem reſponſu intelliges de diſſimilibus minoris inæqualitatis com-
                <lb/>
              parationibus, fiet enim comparatio minoris inæqualitatis , & </s>
              <s xml:id="echoid-s8025" xml:space="preserve">utra-
                <lb/>
              que ſimili modo minor. </s>
              <s xml:id="echoid-s8026" xml:space="preserve">At ſi utræque rationes, aut maioris aut
                <lb/>
                <note position="left" xlink:label="note-514.01.116-05" xlink:href="note-514.01.116-05a" xml:space="preserve">20</note>
              minoris inæqualitatis fuerint, & </s>
              <s xml:id="echoid-s8027" xml:space="preserve">ſimiles, quod æque eſt, ac ſi data
                <lb/>
              ratio per ſe ipſam diuidatur, quæ ueniet ratio erit æqualitatis. </s>
              <s xml:id="echoid-s8028" xml:space="preserve">Sed ſi altera maioris, alter a minoris inæquali-
                <lb/>
              tatis extiterit, ea quę prodibit ratio, in ipſa ratione diuidenda firmabitur, quæ ſcilicet per maiorem numerum fie
                <lb/>
              ri ſolet. </s>
              <s xml:id="echoid-s8029" xml:space="preserve">Cæterum ſi ordinem mutaueris, ita ut rationẽ ſubtrahendam, alteri ſupraponas, eademq́; </s>
              <s xml:id="echoid-s8030" xml:space="preserve">multiplicandi
                <lb/>
              forma ſeruata permutatam rationem ſenties, ita ut quemadmodum in priori exemplo ſublata tripla, una ex
                <lb/>
              dupla, fit ſeſquialtera, ita hic inuerſo ordine fiet ſubſeſquialtera, ſimilis ratio in reliquis.</s>
              <s xml:id="echoid-s8031" xml:space="preserve"/>
            </p>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            Dupla. # 8--4
              <lb/>
            Tripla. # 18--6
              <lb/>
            Subſeſquialtera. # 48--72
              <lb/>
            </note>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            ſeſquitertia # 8--6
              <lb/>
            ſeſquialtera # 6--4
              <lb/>
            ſubſeſquioctaua # 32--36
              <lb/>
            </note>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            bipartiens tertias. # 5--3
              <lb/>
            tripartiens quartas. # 7--4
              <lb/>
            ſubſeſquigeſima. # 20--21
              <lb/>
            </note>
            <p style="it">
              <s xml:id="echoid-s8032" xml:space="preserve">Poſſem hoc loco proportionum proprietates afferre, ac attendere quod ab æqualitate, inæqualitas omnis
                <lb/>
                <note position="left" xlink:label="note-514.01.116-09" xlink:href="note-514.01.116-09a" xml:space="preserve">30</note>
              prouenit, æqualitatemq; </s>
              <s xml:id="echoid-s8033" xml:space="preserve">eſſe inæqualitatis principium, ac demum ad æqualitatem omnem inæqualitatem redu-
                <lb/>
              ci. </s>
              <s xml:id="echoid-s8034" xml:space="preserve">quibus in rebus multa ſecretioris philoſophiæ arcana continentur, ſed hæc ſuo loco reſeruanda ſunt, altio-
                <lb/>
              ris enim ſunt indagationis, & </s>
              <s xml:id="echoid-s8035" xml:space="preserve">uſque ad diuinitatem pertingunt. </s>
              <s xml:id="echoid-s8036" xml:space="preserve">nunc de perquirendis ignotis numeris per eos,
                <lb/>
              qui noti ſunt, regulas aureas apponemus, ac primum id in minimis terminis exequemur. </s>
              <s xml:id="echoid-s8037" xml:space="preserve">Duo igitur ad minus
                <lb/>
              ſunt numerorum termini, quibus cognitis, tertium inueſtigamus, ſiue ille terminus extremus ſit , ſiue medius,
                <lb/>
              loquor autem nunc de ijs, qui ſe mutua ac continenti comparatione reſpiciunt. </s>
              <s xml:id="echoid-s8038" xml:space="preserve">Esto duo numeri præcedentes
                <lb/>
              inter ſe , aliqua ratione comparati. </s>
              <s xml:id="echoid-s8039" xml:space="preserve">Verbi gratia. </s>
              <s xml:id="echoid-s8040" xml:space="preserve">36. </s>
              <s xml:id="echoid-s8041" xml:space="preserve">12. </s>
              <s xml:id="echoid-s8042" xml:space="preserve">uolo tertium inuenire, ad quem posterior
                <lb/>
              ſcilicet 12. </s>
              <s xml:id="echoid-s8043" xml:space="preserve">ſe habeat, quemadmodũ. </s>
              <s xml:id="echoid-s8044" xml:space="preserve">36. </s>
              <s xml:id="echoid-s8045" xml:space="preserve">ad ipſum. </s>
              <s xml:id="echoid-s8046" xml:space="preserve">Multiplicetur ſeu diuidatur in ſe poſterior ille nume-
                <lb/>
              rus , qui ſecundum locum tenere debet idest. </s>
              <s xml:id="echoid-s8047" xml:space="preserve">12. </s>
              <s xml:id="echoid-s8048" xml:space="preserve">inſe, qui ab ea ductione prouenit numerus, id erit 144.
                <lb/>
              </s>
              <s xml:id="echoid-s8049" xml:space="preserve">
                <note position="left" xlink:label="note-514.01.116-10" xlink:href="note-514.01.116-10a" xml:space="preserve">40</note>
              per 144. </s>
              <s xml:id="echoid-s8050" xml:space="preserve">priorem hoc eſt 36. </s>
              <s xml:id="echoid-s8051" xml:space="preserve">partiare, certe prodibit numerus ille, quem uolebam, id eſt 4. </s>
              <s xml:id="echoid-s8052" xml:space="preserve">qui ſe ad
                <lb/>
              duodecim, uel ad quem 12. </s>
              <s xml:id="echoid-s8053" xml:space="preserve">ſe habebunt, quemadmodum 36. </s>
              <s xml:id="echoid-s8054" xml:space="preserve">ad 12. </s>
              <s xml:id="echoid-s8055" xml:space="preserve">in tripla enim proportione erit.
                <lb/>
              </s>
              <s xml:id="echoid-s8056" xml:space="preserve">Quod ſi duxeris 36. </s>
              <s xml:id="echoid-s8057" xml:space="preserve">inſe, emergent 1296. </s>
              <s xml:id="echoid-s8058" xml:space="preserve">quæ per 12. </s>
              <s xml:id="echoid-s8059" xml:space="preserve">partita reddent 108. </s>
              <s xml:id="echoid-s8060" xml:space="preserve">quare 108. </s>
              <s xml:id="echoid-s8061" xml:space="preserve">primus
                <lb/>
              erit trium proportione ſe conſequentium numerorum, propoſitis numeris præponendus; </s>
              <s xml:id="echoid-s8062" xml:space="preserve">nam numerus 108. </s>
              <s xml:id="echoid-s8063" xml:space="preserve">
                <lb/>
              comparatus ad 36. </s>
              <s xml:id="echoid-s8064" xml:space="preserve">eandem ſeruabit rationem ad 36. </s>
              <s xml:id="echoid-s8065" xml:space="preserve">quam idem 36. </s>
              <s xml:id="echoid-s8066" xml:space="preserve">ad 12. </s>
              <s xml:id="echoid-s8067" xml:space="preserve">triplam ſcilicet , quæ
                <lb/>
              est ex genere multip licium. </s>
              <s xml:id="echoid-s8068" xml:space="preserve">Datis igitur duobus numeris tertium ſeu poſtremum inuenimus. </s>
              <s xml:id="echoid-s8069" xml:space="preserve">quod ſi medium
                <lb/>
              uoluerimus inuenire inter duos propoſitos numeros proportione reſpondentiem, ducendi ſunt propoſiti illi nume-
                <lb/>
              ri inter ſe, & </s>
              <s xml:id="echoid-s8070" xml:space="preserve">quadrata ipſorum radix inuenienda, nempe ea erit medius ille numerus, ad quem prior ita ſe ha-
                <lb/>
              bebit, quemadmodum ille ad poſteriorem. </s>
              <s xml:id="echoid-s8071" xml:space="preserve">Hic ad Arithmeticos accedendum, qui de extrahendis numerorum
                <lb/>
              radicibus regulas ponunt. </s>
              <s xml:id="echoid-s8072" xml:space="preserve">Radices autem numerorum intelligo eos numeros, qui in ſe ducti efficiunt eam ſum
                <lb/>
              mam, de qua radicem trahimus, nam quatuor radix est ſexdecim, ducta enim in ſe quatuor efficiunt ſexde-
                <lb/>
                <note position="left" xlink:label="note-514.01.116-11" xlink:href="note-514.01.116-11a" xml:space="preserve">50</note>
              cim. </s>
              <s xml:id="echoid-s8073" xml:space="preserve">Exemplo ſint 25. </s>
              <s xml:id="echoid-s8074" xml:space="preserve">& </s>
              <s xml:id="echoid-s8075" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8076" xml:space="preserve">uolo numerum medium inuenire, ad quem 25. </s>
              <s xml:id="echoid-s8077" xml:space="preserve">ſe habeat ea ratione, qua
                <lb/>
              ille ſe ad 4. </s>
              <s xml:id="echoid-s8078" xml:space="preserve">habebit. </s>
              <s xml:id="echoid-s8079" xml:space="preserve">dueigitur 4. </s>
              <s xml:id="echoid-s8080" xml:space="preserve">in 25. </s>
              <s xml:id="echoid-s8081" xml:space="preserve">fient 100. </s>
              <s xml:id="echoid-s8082" xml:space="preserve">cuius radix eſt decem, ergo 25. </s>
              <s xml:id="echoid-s8083" xml:space="preserve">ad decem ſe ha-
                <lb/>
              bebit, ut decem ad 4. </s>
              <s xml:id="echoid-s8084" xml:space="preserve">nempe in proportione dupla ſexquialtera. </s>
              <s xml:id="echoid-s8085" xml:space="preserve">Atque hæc ſatis dicta ſint in minimo nu-
                <lb/>
              merorum ordine. </s>
              <s xml:id="echoid-s8086" xml:space="preserve">Nunc ad plures ordines tranſeundum, & </s>
              <s xml:id="echoid-s8087" xml:space="preserve">quærendum, qua ratione tribus terminis nume-
                <lb/>
              rorum propoſitis, & </s>
              <s xml:id="echoid-s8088" xml:space="preserve">notis alius inueniatur . </s>
              <s xml:id="echoid-s8089" xml:space="preserve">Fieri autem potest, ut uel primus, uel ſecundus, uel tertius, uel
                <lb/>
              quartus ignotus ſit, reliquis tribus perſpectis. </s>
              <s xml:id="echoid-s8090" xml:space="preserve">debemus tamen in experiendo quartum locum ignoto numero
                <lb/>
              reſeruare. </s>
              <s xml:id="echoid-s8091" xml:space="preserve">ita ut primus tertio re, & </s>
              <s xml:id="echoid-s8092" xml:space="preserve">ratione conueniat, nam ita fiet, ut ſecundus quarto ignoto reſpondeat.
                <lb/>
              </s>
              <s xml:id="echoid-s8093" xml:space="preserve">Eſto exempli gratia 30. </s>
              <s xml:id="echoid-s8094" xml:space="preserve">20. </s>
              <s xml:id="echoid-s8095" xml:space="preserve">24. </s>
              <s xml:id="echoid-s8096" xml:space="preserve">16. </s>
              <s xml:id="echoid-s8097" xml:space="preserve">quiſeſquialte ratione reſpiciant, eſto etiam ignotus numerus 16. </s>
              <s xml:id="echoid-s8098" xml:space="preserve">
                <lb/>
              duc 24. </s>
              <s xml:id="echoid-s8099" xml:space="preserve">per 20. </s>
              <s xml:id="echoid-s8100" xml:space="preserve">efficies 480. </s>
              <s xml:id="echoid-s8101" xml:space="preserve">partire 480. </s>
              <s xml:id="echoid-s8102" xml:space="preserve">per 30. </s>
              <s xml:id="echoid-s8103" xml:space="preserve">reſultabit ignotus ille, & </s>
              <s xml:id="echoid-s8104" xml:space="preserve">quæſitus numerus ſex
                <lb/>
              decim. </s>
              <s xml:id="echoid-s8105" xml:space="preserve">At ſi primus numerus quæratur uidelicet 30. </s>
              <s xml:id="echoid-s8106" xml:space="preserve">eum quarto loco ponito, nam cum ſit 30. </s>
              <s xml:id="echoid-s8107" xml:space="preserve">ad 20. </s>
              <s xml:id="echoid-s8108" xml:space="preserve">
                <lb/>
                <note position="left" xlink:label="note-514.01.116-12" xlink:href="note-514.01.116-12a" xml:space="preserve">60</note>
              </s>
            </p>
          </div>
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