Vitruvius, De architectura libri decem, 1567

List of thumbnails

< >
61
61 (29) 		62
62 (30)
63
63 (31) 		64
64 (32)
65
65 (33) 		66
66 (34)
67
67 (35) 		68
68 (36)
69
69 (37) 		70
70 (38)
< >
page |< < (83) of 412 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div1" type="section" level="1" n="1">
          <div xml:id="echoid-div276" type="section" level="2" n="61">
            <p>
              <s xml:id="echoid-s7840" xml:space="preserve">
                <pb o="83" file="514.01.115" n="115" rhead="TERTIVS."/>
              tatis ad alteram, ſed unius proportionis ad alteram, uerbi gratia, ſi dixeris proportionem illam, quæ inter qua-
                <lb/>
              tuor cadit, & </s>
              <s xml:id="echoid-s7841" xml:space="preserve">duo, ſimilem eſſe illi proportioni , quæ cadit inter octo , & </s>
              <s xml:id="echoid-s7842" xml:space="preserve">quatuor, nam utraque dupla eſt,
                <lb/>
              ideo duplæ omnes, triplæ item, & </s>
              <s xml:id="echoid-s7843" xml:space="preserve">quadruplæ, reliquæq́; </s>
              <s xml:id="echoid-s7844" xml:space="preserve">ſiue ſint unius generis, cuiuſmodi ſunt , quæ cadunt
                <lb/>
              inter lineam, & </s>
              <s xml:id="echoid-s7845" xml:space="preserve">lineam, inter planum, & </s>
              <s xml:id="echoid-s7846" xml:space="preserve">planum, inter corpus, & </s>
              <s xml:id="echoid-s7847" xml:space="preserve">corpus, ſiue ſint generum diuerſorum, qua
                <lb/>
              les ſunt quæ cadunt inter lineam, & </s>
              <s xml:id="echoid-s7848" xml:space="preserve">planum, inter planum, & </s>
              <s xml:id="echoid-s7849" xml:space="preserve">corpus; </s>
              <s xml:id="echoid-s7850" xml:space="preserve">proportionales ſunt, hoc eſt poſſunt
                <lb/>
              proportionum ſimilitudine inter ſe conferri, & </s>
              <s xml:id="echoid-s7851" xml:space="preserve">conſequenter ſimiles habentur, & </s>
              <s xml:id="echoid-s7852" xml:space="preserve">ubi eſt proportionum col-
                <lb/>
              latio, quam proportionalitatem uocant, ibi neceſſario eſt proportio. </s>
              <s xml:id="echoid-s7853" xml:space="preserve">quoniam proportionalitas, ut ita dicam,
                <lb/>
              nil aliud eſt, quàm proportionum reſponſus , ſed none contrario, nam inter quatuor, & </s>
              <s xml:id="echoid-s7854" xml:space="preserve">duo proportio cadit,
                <lb/>
              ſed non proportionalitas. </s>
              <s xml:id="echoid-s7855" xml:space="preserve">In his igitur proportionum comparationibus, omne artis ſecretum ponitur. </s>
              <s xml:id="echoid-s7856" xml:space="preserve">In com-
                <lb/>
                <note position="left" xlink:label="note-514.01.115-01" xlink:href="note-514.01.115-01a" xml:space="preserve">10</note>
              ponendis igitur proportionibus, illud obſeruandum est, an ambæ rationes ſint ſimiles, an diſſimiles, hoc est an
                <lb/>
              ſint ambæ ex his, quæ a maiori in minus deſinunt, an ex his, quæ a minori in maius terminantur . </s>
              <s xml:id="echoid-s7857" xml:space="preserve">An altera
                <lb/>
              unius, altera ſit alterius generis, nam hoc plurimum intereſt. </s>
              <s xml:id="echoid-s7858" xml:space="preserve">ut ex regulis dignoſcemus. </s>
              <s xml:id="echoid-s7859" xml:space="preserve">In componendis
                <lb/>
              etiam proportionibus duo ſe ſe offerunt conſideranda, primum est denominatio compoſitæ proportionis. </s>
              <s xml:id="echoid-s7860" xml:space="preserve">Al-
                <lb/>
              terum est numerorum collectio ſub eadem producta ratione conſtitutorum. </s>
              <s xml:id="echoid-s7861" xml:space="preserve">Primum abſoluitur hoc modo in
                <lb/>
              his, quæ ſunt ſimilium generum, ac in multiplicibus. </s>
              <s xml:id="echoid-s7862" xml:space="preserve">Ducas unius in alterius denominatorem, orietur compo-
                <lb/>
              ſitæ rationis denominatio. </s>
              <s xml:id="echoid-s7863" xml:space="preserve">Verbigratia. </s>
              <s xml:id="echoid-s7864" xml:space="preserve">inter 12. </s>
              <s xml:id="echoid-s7865" xml:space="preserve">& </s>
              <s xml:id="echoid-s7866" xml:space="preserve">quatuor tripla eſt proportio, inter quatuor, & </s>
              <s xml:id="echoid-s7867" xml:space="preserve">octo
                <lb/>
              dupla. </s>
              <s xml:id="echoid-s7868" xml:space="preserve">duc tria in duo, fiet ſex. </s>
              <s xml:id="echoid-s7869" xml:space="preserve">ex tripla igitur, & </s>
              <s xml:id="echoid-s7870" xml:space="preserve">dupla fiet ſextupla proportio. </s>
              <s xml:id="echoid-s7871" xml:space="preserve">Secundum abſoluitur, & </s>
              <s xml:id="echoid-s7872" xml:space="preserve">
                <lb/>
              confirmatur his numeris , duc 12. </s>
              <s xml:id="echoid-s7873" xml:space="preserve">in octo fiet 96. </s>
              <s xml:id="echoid-s7874" xml:space="preserve">& </s>
              <s xml:id="echoid-s7875" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7876" xml:space="preserve">in. </s>
              <s xml:id="echoid-s7877" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7878" xml:space="preserve">fient 16. </s>
              <s xml:id="echoid-s7879" xml:space="preserve">atque ſi 96. </s>
              <s xml:id="echoid-s7880" xml:space="preserve">ad ſex-
                <lb/>
              decim comparaueris, cernes ſextuplam oriri proportionem, quã ex proępoſitis denominationibus collectã cernis.
                <lb/>
              </s>
              <s xml:id="echoid-s7881" xml:space="preserve">
                <note position="left" xlink:label="note-514.01.115-02" xlink:href="note-514.01.115-02a" xml:space="preserve">20</note>
                <note style="it" position="right" xlink:label="note-514.01.115-03" xlink:href="note-514.01.115-03a" xml:space="preserve">
                  <lb/>
                Tripla # 12--4
                  <lb/>
                Dupla # 8--4
                  <lb/>
                Sextupla # 96--16
                  <lb/>
                </note>
              Similis ratio in ſupraparticulari proportione obſeruatur. </s>
              <s xml:id="echoid-s7882" xml:space="preserve">Eſto 6. </s>
              <s xml:id="echoid-s7883" xml:space="preserve">ad 4. </s>
              <s xml:id="echoid-s7884" xml:space="preserve">ſeſ-
                <lb/>
              quialtera, &</s>
              <s xml:id="echoid-s7885" xml:space="preserve">. 8. </s>
              <s xml:id="echoid-s7886" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7887" xml:space="preserve">6. </s>
              <s xml:id="echoid-s7888" xml:space="preserve">ſeſquitertia. </s>
              <s xml:id="echoid-s7889" xml:space="preserve">duc unum ſemis denominatorem ſeſqui-
                <lb/>
              alteræ in unum & </s>
              <s xml:id="echoid-s7890" xml:space="preserve">tertiam denominatorem ſeſquitertiæ. </s>
              <s xml:id="echoid-s7891" xml:space="preserve">fient 2. </s>
              <s xml:id="echoid-s7892" xml:space="preserve">a quibus pro-
                <lb/>
              portio dupla nominatur. </s>
              <s xml:id="echoid-s7893" xml:space="preserve">Ex ſeſquialtera igitur, & </s>
              <s xml:id="echoid-s7894" xml:space="preserve">ſeſquitertia oritur dupla, quod
                <lb/>
              & </s>
              <s xml:id="echoid-s7895" xml:space="preserve">in numeris cernes hoc modo. </s>
              <s xml:id="echoid-s7896" xml:space="preserve">Duc 6 in 8 fient 48. </s>
              <s xml:id="echoid-s7897" xml:space="preserve">& </s>
              <s xml:id="echoid-s7898" xml:space="preserve">4 in 6 fient 24. </s>
              <s xml:id="echoid-s7899" xml:space="preserve">quo facto cernes duplam
                <lb/>
              inter eos numeros cadere comparationem. </s>
              <s xml:id="echoid-s7900" xml:space="preserve">Exemplum quoque in ſuprapartientibus ponam. </s>
              <s xml:id="echoid-s7901" xml:space="preserve">eſto bipartiens
                <lb/>
              tertias ratio, qualis eſt 5 ad 3 addenda trip artienti quartas, qualis eſt 7 ad 4. </s>
              <s xml:id="echoid-s7902" xml:space="preserve">denominator bipartiẽs
                <lb/>
              tertias eſt 1. </s>
              <s xml:id="echoid-s7903" xml:space="preserve">& </s>
              <s xml:id="echoid-s7904" xml:space="preserve">duæ tertiæ. </s>
              <s xml:id="echoid-s7905" xml:space="preserve">& </s>
              <s xml:id="echoid-s7906" xml:space="preserve">tripartiens quartas eſt unum & </s>
              <s xml:id="echoid-s7907" xml:space="preserve">tres quartæ. </s>
              <s xml:id="echoid-s7908" xml:space="preserve">ducito inuicem huiuſmodi de-
                <lb/>
              nomin. </s>
              <s xml:id="echoid-s7909" xml:space="preserve">itores, fient 2 & </s>
              <s xml:id="echoid-s7910" xml:space="preserve">undecim duodecimæ, a quibus dupla undecupartiens duodecimas nominatur. </s>
              <s xml:id="echoid-s7911" xml:space="preserve">quod
                <lb/>
              & </s>
              <s xml:id="echoid-s7912" xml:space="preserve">numeri hinc inde prouenientes oſtendunt. </s>
              <s xml:id="echoid-s7913" xml:space="preserve">ducendo 5 in 7. </s>
              <s xml:id="echoid-s7914" xml:space="preserve">qui ſunt antecedentes, & </s>
              <s xml:id="echoid-s7915" xml:space="preserve">3. </s>
              <s xml:id="echoid-s7916" xml:space="preserve">in 4. </s>
              <s xml:id="echoid-s7917" xml:space="preserve">qui
                <lb/>
                <note position="left" xlink:label="note-514.01.115-04" xlink:href="note-514.01.115-04a" xml:space="preserve">30</note>
              ſunt ſubſequentes, nam ex illis fient 35. </s>
              <s xml:id="echoid-s7918" xml:space="preserve">ex his 12. </s>
              <s xml:id="echoid-s7919" xml:space="preserve">inter quos ſupradicta proportio cadit.</s>
              <s xml:id="echoid-s7920" xml:space="preserve"/>
            </p>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            Bipartiens tertias # 5--3
              <lb/>
            Tripartiens quartas # 7--4
              <lb/>
            Dupla undecupartiens duodecimas. # 35--12
              <lb/>
            </note>
            <p style="it">
              <s xml:id="echoid-s7921" xml:space="preserve">Quod ſi diuerſorum generum proportiones interceßerint, ita
                <lb/>
              quod in una fiat comparatio maioris ad minus, in altera mi-
                <lb/>
              noris ad maius: </s>
              <s xml:id="echoid-s7922" xml:space="preserve">Tunc id obſeruandum eſt, vt partitione uta-
                <lb/>
              mur. </s>
              <s xml:id="echoid-s7923" xml:space="preserve">ita enim compoſitio duarum emerget . </s>
              <s xml:id="echoid-s7924" xml:space="preserve">partitur enim
                <lb/>
              maior per minorem hunc in modum. </s>
              <s xml:id="echoid-s7925" xml:space="preserve">ſubdupla ratio a binario nomen capit, quemadmodum & </s>
              <s xml:id="echoid-s7926" xml:space="preserve">dupla; </s>
              <s xml:id="echoid-s7927" xml:space="preserve">ſeſquial-
                <lb/>
              tera autem ab 1 & </s>
              <s xml:id="echoid-s7928" xml:space="preserve">duodecima, unum igitur & </s>
              <s xml:id="echoid-s7929" xml:space="preserve">dimidium minus es
                <unsure/>
              t quàm duo. </s>
              <s xml:id="echoid-s7930" xml:space="preserve">partire igitur duo per unum
                <lb/>
              & </s>
              <s xml:id="echoid-s7931" xml:space="preserve">dimidium. </s>
              <s xml:id="echoid-s7932" xml:space="preserve">edetur 1. </s>
              <s xml:id="echoid-s7933" xml:space="preserve">& </s>
              <s xml:id="echoid-s7934" xml:space="preserve">tertia. </s>
              <s xml:id="echoid-s7935" xml:space="preserve">Ex prædictis ergo rationibus, ratio emerget ſubſeſquitertia , nam diui-
                <lb/>
              denda est ea, quæ eſt minoris inæqualitatis, & </s>
              <s xml:id="echoid-s7936" xml:space="preserve">quæ prouenit inde ratio diuidendam rationem ſequi ſolet, & </s>
              <s xml:id="echoid-s7937" xml:space="preserve">
                <lb/>
              experire hoc per numeros 2 4. </s>
              <s xml:id="echoid-s7938" xml:space="preserve">inter quos eſt ſubdupla proportio, & </s>
              <s xml:id="echoid-s7939" xml:space="preserve">6. </s>
              <s xml:id="echoid-s7940" xml:space="preserve">& </s>
              <s xml:id="echoid-s7941" xml:space="preserve">4. </s>
              <s xml:id="echoid-s7942" xml:space="preserve">inter quos eſt ſeſquialte-
                <lb/>
              ra proportio. </s>
              <s xml:id="echoid-s7943" xml:space="preserve">duc 2. </s>
              <s xml:id="echoid-s7944" xml:space="preserve">in 6. </s>
              <s xml:id="echoid-s7945" xml:space="preserve">fient 12. </s>
              <s xml:id="echoid-s7946" xml:space="preserve">& </s>
              <s xml:id="echoid-s7947" xml:space="preserve">4 in 4. </s>
              <s xml:id="echoid-s7948" xml:space="preserve">fient 16. </s>
              <s xml:id="echoid-s7949" xml:space="preserve">comparato 12 ad 16. </s>
              <s xml:id="echoid-s7950" xml:space="preserve">uidebis ſubſeſ-
                <lb/>
                <note position="left" xlink:label="note-514.01.115-06" xlink:href="note-514.01.115-06a" xml:space="preserve">40</note>
              quitertiam proporportione procreari.</s>
              <s xml:id="echoid-s7951" xml:space="preserve"/>
            </p>
            <note style="it" position="right" xml:space="preserve">
              <lb/>
            ſubdupla. # 2--4
              <lb/>
            ſeſquialtera. # 6--4
              <lb/>
            ſubſeſquitertia. # 12--16
              <lb/>
            </note>
            <p style="it">
              <s xml:id="echoid-s7952" xml:space="preserve">Exemplum hoc ſatis eße potest pro omnibus diuerſorum generum comparatio-
                <lb/>
              nibus. </s>
              <s xml:id="echoid-s7953" xml:space="preserve">Cæterum & </s>
              <s xml:id="echoid-s7954" xml:space="preserve">illud innoteſcet, qua nam ratione plures quàm duæ rationes
                <lb/>
              inuicem componantur, nam quæ ex duabus prioribus compoſitis effecta fue-
                <lb/>
              rit, ea cum tertia eodem modo componenda es
                <unsure/>
              t, quo ſupra diximus. </s>
              <s xml:id="echoid-s7955" xml:space="preserve">Ad me-
                <lb/>
              moriam uero reuocanda eſt fractorum, & </s>
              <s xml:id="echoid-s7956" xml:space="preserve">integrorum ductio, partitio, & </s>
              <s xml:id="echoid-s7957" xml:space="preserve">collectio, ut facile exerceri poſſi-
                <lb/>
              mus in hoc genere uniuerſo. </s>
              <s xml:id="echoid-s7958" xml:space="preserve">Ex prædictis illud colligere poßumus, quod cum ſimiles proportiones componuntur.
                <lb/>
              </s>
              <s xml:id="echoid-s7959" xml:space="preserve">verbigratia. </s>
              <s xml:id="echoid-s7960" xml:space="preserve">maioris inæqualitatis ratio, ſimiliter minoris inæqualitatis, & </s>
              <s xml:id="echoid-s7961" xml:space="preserve">utraq; </s>
              <s xml:id="echoid-s7962" xml:space="preserve">maior generatur, quod ex
                <lb/>
              ſuperioribus exemplis innotuit. </s>
              <s xml:id="echoid-s7963" xml:space="preserve">Ex duabus quoque minoris inæqualitatis rationibus, ratio prouenit minoris
                <lb/>
              inæqualitatis, & </s>
              <s xml:id="echoid-s7964" xml:space="preserve">utraque minor erit. </s>
              <s xml:id="echoid-s7965" xml:space="preserve">Sed ex una maioris, & </s>
              <s xml:id="echoid-s7966" xml:space="preserve">altera minoris inæqualitatis comparatione, ra-
                <lb/>
                <note position="left" xlink:label="note-514.01.115-08" xlink:href="note-514.01.115-08a" xml:space="preserve">50</note>
              tio huiuſmodi gignitur, cuiuſmodi ea eſt, quæ ab ampliori numero nomen capit. </s>
              <s xml:id="echoid-s7967" xml:space="preserve">Sola uero æqualitatis ratio
                <lb/>
              in ſe ipſam ducta, rationem procreat æqualitatis. </s>
              <s xml:id="echoid-s7968" xml:space="preserve">Atque hæc de componendis proportionibus dicta ſint. </s>
              <s xml:id="echoid-s7969" xml:space="preserve">Nunc
                <lb/>
              quemadmodum ratio a ratione ſubtrahatur, & </s>
              <s xml:id="echoid-s7970" xml:space="preserve">quæ reliqua ſit dignoſcatur, dicendum. </s>
              <s xml:id="echoid-s7971" xml:space="preserve">Si prius obſeruabimus
                <lb/>
              id partitione quadam effici, & </s>
              <s xml:id="echoid-s7972" xml:space="preserve">nunquam maiorem a minori, ſed minorem tantum a maiori demi poſſe. </s>
              <s xml:id="echoid-s7973" xml:space="preserve">quis
                <lb/>
              enim maius a minori demet? </s>
              <s xml:id="echoid-s7974" xml:space="preserve">cum nunquam maius in minori reperiri poſſit? </s>
              <s xml:id="echoid-s7975" xml:space="preserve">Eſficitur demptio, ac ſubtractio
                <lb/>
              duobus modis, primo ſi partiaris maioris rationis denominatorem numerum per denominatorem minoris, id
                <lb/>
              quod reliquum erit, procreatus proportionis deominator erit. </s>
              <s xml:id="echoid-s7976" xml:space="preserve">Secundo in numeris experitur, qui ex ſupra
                <lb/>
              datis rationibus proueniunt, verbigratia. </s>
              <s xml:id="echoid-s7977" xml:space="preserve">Constituantur numeri rationis maioris, quæ & </s>
              <s xml:id="echoid-s7978" xml:space="preserve">est ipſa diuidenda,
                <lb/>
              & </s>
              <s xml:id="echoid-s7979" xml:space="preserve">ponatur ſupra numeros ipſius minoris, per quam ratio maior partiri debet, inde ducatur numerus antece-
                <lb/>
              dens diuidendæ rationis, quæ & </s>
              <s xml:id="echoid-s7980" xml:space="preserve">maior est, per numerum conſequentem minoris, & </s>
              <s xml:id="echoid-s7981" xml:space="preserve">diuidentis, certe orietur
                <lb/>
                <note position="left" xlink:label="note-514.01.115-09" xlink:href="note-514.01.115-09a" xml:space="preserve">60</note>
              </s>
            </p>
          </div>
        </div>
      </text>
    </echo>
Impressum