Vitruvius, De architectura libri decem, 1567

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          <div xml:id="echoid-div276" type="section" level="2" n="61">
            <pb o="82" file="514.01.114" n="114" rhead="LIBER"/>
            <p>
              <s xml:id="echoid-s7752" xml:space="preserve">7#9#11#13#15#17#19#21</s>
            </p>
            <p style="it">
              <s xml:id="echoid-s7753" xml:space="preserve">2#3#4#5#6#7#8#9.</s>
              <s xml:id="echoid-s7754" xml:space="preserve">#Atque ita ſubſequentes rationes additione conficies. </s>
              <s xml:id="echoid-s7755" xml:space="preserve">Sed diſcri-
                <lb/>
              mina ſuperpartientis muliiplicis rationis ſub ratione dupla efficies, ſi ab octonario per tres unitates diſcedenles
                <lb/>
              numeros ordinatim ſuprapones , & </s>
              <s xml:id="echoid-s7756" xml:space="preserve">a ternario ſeriem naturæ ſeruaueris in inferiori loco, hoc modo.</s>
              <s xml:id="echoid-s7757" xml:space="preserve"/>
            </p>
            <p>
              <s xml:id="echoid-s7758" xml:space="preserve">8#11#14#17#20#23#26#29</s>
            </p>
            <p style="it">
              <s xml:id="echoid-s7759" xml:space="preserve">3#4#5#6#7#8#9#10.</s>
              <s xml:id="echoid-s7760" xml:space="preserve">#quod ſi multiplicem ſub ratione tripla, ac ſuprapartiente na-
                <lb/>
              ſci videre cupis, ſubſcribe ſeriem numerorum ab undecim, ſe ſe quatuor unitatibus excedentium, & </s>
              <s xml:id="echoid-s7761" xml:space="preserve">natur alem
                <lb/>
              ordinem a ternario ſubſcribe, hoc modo.</s>
              <s xml:id="echoid-s7762" xml:space="preserve"/>
            </p>
            <p>
              <s xml:id="echoid-s7763" xml:space="preserve">11#15#19#23#27#31#35#39</s>
            </p>
            <p>
              <s xml:id="echoid-s7764" xml:space="preserve">3#4#5#6#7#8#9#10.</s>
              <s xml:id="echoid-s7765" xml:space="preserve">#Atque hæc ſatis pro exemplo ſint poſita , Cætera per quàm
                <lb/>
              facilia studioſis eſſe poſſunt, & </s>
              <s xml:id="echoid-s7766" xml:space="preserve">quia comparationum termini immutarri poßunt, ita ut a minori incipere , & </s>
              <s xml:id="echoid-s7767" xml:space="preserve">
                <lb/>
                <note position="left" xlink:label="note-514.01.114-01" xlink:href="note-514.01.114-01a" xml:space="preserve">10</note>
              in maius terminare poßumus: </s>
              <s xml:id="echoid-s7768" xml:space="preserve">ideo non ignorandum eandem eße comparationem mutatis terminis , eo ſolo di-
                <lb/>
              ſcrimine, ut ſi comparamus quod maius eſt, ad minus , ut 4. </s>
              <s xml:id="echoid-s7769" xml:space="preserve">ad. </s>
              <s xml:id="echoid-s7770" xml:space="preserve">2. </s>
              <s xml:id="echoid-s7771" xml:space="preserve">tunc utimur ea particula, ſupra.
                <lb/>
              </s>
              <s xml:id="echoid-s7772" xml:space="preserve">ut cum dicimus ſupraparticularis. </s>
              <s xml:id="echoid-s7773" xml:space="preserve">ſuprapartiens. </s>
              <s xml:id="echoid-s7774" xml:space="preserve">cum uero comparamus quod minus eſt ad maius, ut 2. </s>
              <s xml:id="echoid-s7775" xml:space="preserve">ad
                <lb/>
              4. </s>
              <s xml:id="echoid-s7776" xml:space="preserve">tunc utimur ea particula, ſub. </s>
              <s xml:id="echoid-s7777" xml:space="preserve">atque in ſubmultiplicem, ſubparticularem, ſubpartientem nominamus. </s>
              <s xml:id="echoid-s7778" xml:space="preserve">
                <lb/>
              Totidem igitur differentiæ, formæq́; </s>
              <s xml:id="echoid-s7779" xml:space="preserve">comparationum inueniuntur , cum minus ad maius comparabitur, quod
                <lb/>
              ſunt eæ, quibus maius minori confertur. </s>
              <s xml:id="echoid-s7780" xml:space="preserve">Cæterum utſciamus & </s>
              <s xml:id="echoid-s7781" xml:space="preserve">uno obtutu cernamus a quibus numeris pro-
                <lb/>
              portiones nomina ſua ſortiantur, hanc regulam obſeruabimus . </s>
              <s xml:id="echoid-s7782" xml:space="preserve">Erit enim hoc utile in ædificijs , cæterisq́; </s>
              <s xml:id="echoid-s7783" xml:space="preserve">re-
                <lb/>
              bus arte, & </s>
              <s xml:id="echoid-s7784" xml:space="preserve">ratione perfectis, ad dignoſcendum quæ nam proportio maior , quæ minor, quæ amplior, quæue
                <lb/>
              ſtrictior habeatur, magni enim refert, qua quis utatur proportione. </s>
              <s xml:id="echoid-s7785" xml:space="preserve">Cum alia ædes magnificæ, alia priuatorũ
                <lb/>
                <note position="left" xlink:label="note-514.01.114-02" xlink:href="note-514.01.114-02a" xml:space="preserve">20</note>
              domus, alia templa proportione perficiantur. </s>
              <s xml:id="echoid-s7786" xml:space="preserve">Quare hoc diligentiſſime obſeruandum Architectis cenſemus.
                <lb/>
              </s>
              <s xml:id="echoid-s7787" xml:space="preserve">Denominatio proportionum partitione dignoſcitur. </s>
              <s xml:id="echoid-s7788" xml:space="preserve">Partitio eſt experientia quædam, qua ſcitur , quot uicibus
                <lb/>
              numerus minor , in maiori inſit, & </s>
              <s xml:id="echoid-s7789" xml:space="preserve">quid inde ſuperſit. </s>
              <s xml:id="echoid-s7790" xml:space="preserve">ex partitione igitur, & </s>
              <s xml:id="echoid-s7791" xml:space="preserve">reliquo nomen proportio quæq; </s>
              <s xml:id="echoid-s7792" xml:space="preserve">
                <lb/>
              ſuum capit, Verbi gratia. </s>
              <s xml:id="echoid-s7793" xml:space="preserve">Quæro quæ nam proportio ſit inter octo, & </s>
              <s xml:id="echoid-s7794" xml:space="preserve">quatuor, ſtatim partior octo per qua-
                <lb/>
              tuor, id eſt experior quot uicibus quatuor constituant octo, & </s>
              <s xml:id="echoid-s7795" xml:space="preserve">inuenio duabus uicibus quatuor in octo ineſſe,
                <lb/>
              & </s>
              <s xml:id="echoid-s7796" xml:space="preserve">nihil inde ſupereſſe, ab hoc igitur numero, duo, proportionis nomen deſumo, aioq́; </s>
              <s xml:id="echoid-s7797" xml:space="preserve">inter octo , & </s>
              <s xml:id="echoid-s7798" xml:space="preserve">quatuor
                <lb/>
              duplam cadere proportionem. </s>
              <s xml:id="echoid-s7799" xml:space="preserve">Quæro iter um quæ proportio inter quinque, & </s>
              <s xml:id="echoid-s7800" xml:space="preserve">ſexdecim intercidat , partior
                <lb/>
              ſexdecim per quinarium, inuenio quinarium ter in ſexdecim, & </s>
              <s xml:id="echoid-s7801" xml:space="preserve">amplius unum eſſe. </s>
              <s xml:id="echoid-s7802" xml:space="preserve">Triplam igitur a ternis ui-
                <lb/>
              cibus proportionem illam nomino, & </s>
              <s xml:id="echoid-s7803" xml:space="preserve">quia ſupereſt unitas, quæ pars quinta eſt quinarij, ideo addo ſeſqui quin-
                <lb/>
              tam, & </s>
              <s xml:id="echoid-s7804" xml:space="preserve">triplam eam ſeſquiquintam nomino proportionem. </s>
              <s xml:id="echoid-s7805" xml:space="preserve">Similis ratio in reliquis reperitur. </s>
              <s xml:id="echoid-s7806" xml:space="preserve">Quod cum
                <lb/>
                <note position="left" xlink:label="note-514.01.114-03" xlink:href="note-514.01.114-03a" xml:space="preserve">30</note>
              ita ſit, facile reperiemus, quæ nam æqualis, quæ maior, aut minor proportio intelligatur. </s>
              <s xml:id="echoid-s7807" xml:space="preserve">nam æquales & </s>
              <s xml:id="echoid-s7808" xml:space="preserve">ſi-
                <lb/>
              miles ſunt, quæ eaſdem babent nominationes. </s>
              <s xml:id="echoid-s7809" xml:space="preserve">Inæquales, & </s>
              <s xml:id="echoid-s7810" xml:space="preserve">dißimiles, quarum non eſt ſimilis denominatio.
                <lb/>
              </s>
              <s xml:id="echoid-s7811" xml:space="preserve">primum id oſtendam in multiplicibus, deinde in reliquis. </s>
              <s xml:id="echoid-s7812" xml:space="preserve">Ex multiplicibus illæ ſunt maiores , quarum nomina-
                <lb/>
              tio a maiori numero prouenit. </s>
              <s xml:id="echoid-s7813" xml:space="preserve">Quadrupla igitur maior est proportio, quàm dupla, quoniam illa a quater-
                <lb/>
              nario, hæc a binario ſumitur. </s>
              <s xml:id="echoid-s7814" xml:space="preserve">In ſuperparticularibus ratio contraria eſt, nam illa maior eſt, quæ a minori nu-
                <lb/>
              mero conſtituitur, & </s>
              <s xml:id="echoid-s7815" xml:space="preserve">hoc eſt ratione minutorũ, quas fractiones nominant, ne de uerbis contendamus. </s>
              <s xml:id="echoid-s7816" xml:space="preserve">nam quo
                <lb/>
              plures ſunt fractiones, eo minor eſt proportio, ideo ſeſquitertia minor ſeſquialtera est, nam ſeſquitertia a terna-
                <lb/>
              rio, ſeſquialtera a dimidio nominatur, maior igitur pars eſt dimidium, quàm tertia pars, atque ita de ſingulis,
                <lb/>
              ſed in ſuprapartientibus intelligendum hoc est, quod ſupra non attuli. </s>
              <s xml:id="echoid-s7817" xml:space="preserve">Dixi ſuprapartientem proportionem
                <lb/>
              fieri cum id, quod amplius, & </s>
              <s xml:id="echoid-s7818" xml:space="preserve">maius eſt continet id, quod minus eſt, & </s>
              <s xml:id="echoid-s7819" xml:space="preserve">plures eius partes , ſed id hoc modo ca-
                <lb/>
                <note position="left" xlink:label="note-514.01.114-04" xlink:href="note-514.01.114-04a" xml:space="preserve">40</note>
              piendum est, ut tam de numero partium, quàm de ipſa denominatione, & </s>
              <s xml:id="echoid-s7820" xml:space="preserve">de utroque ſimul intelligatur. </s>
              <s xml:id="echoid-s7821" xml:space="preserve">De nu-
                <lb/>
              mero quidem partium quando plus continet, quod minus eſt ſemel, & </s>
              <s xml:id="echoid-s7822" xml:space="preserve">duas uerbi gratia eius partes ſic ſuper-
                <lb/>
              bipartiens nominatur. </s>
              <s xml:id="echoid-s7823" xml:space="preserve">ſi tres ſupratripartiens, atque ita in infinitum. </s>
              <s xml:id="echoid-s7824" xml:space="preserve">De denominatione uero partium, cum
                <lb/>
              plus continet minus ſemel, & </s>
              <s xml:id="echoid-s7825" xml:space="preserve">partes eius, quæ ſunt uel terliæ, uel quartæ, uel huiuſmodi partes totius, unde
                <lb/>
              proportio dicitur ſuprapartiens tertias, uel quartas, uel huiuſmodi. </s>
              <s xml:id="echoid-s7826" xml:space="preserve">De utroque ſimul, ut cum dicimus pro-
                <lb/>
              portionem ſuprabipartientem tertias, uel quartas, atque ita in infinitum. </s>
              <s xml:id="echoid-s7827" xml:space="preserve">Sed in his maior ea proportio eſt,
                <lb/>
              quæ a maiori numero nomen capit, in his dico, quæ ſecundum priorem modum, quo numerus partium expri-
                <lb/>
              mitur, continetur. </s>
              <s xml:id="echoid-s7828" xml:space="preserve">nam ubi exprimuntur quales ſint minoris partes, ibi eadem ratio, ut ſi dicas proportionem
                <lb/>
              ſupra octo partientem undecimas eſſe maiorem proportioni, ſupr atripartientem undecimas, quoniam hæc a
                <lb/>
              numero minori, qui ternarius eſt, illa ab octonario, qui maior eſt denominatus, cum ſecunda denominatio , ut-
                <lb/>
                <note position="left" xlink:label="note-514.01.114-05" xlink:href="note-514.01.114-05a" xml:space="preserve">50</note>
              pote ab undecima deducta, ſit eadem in utroque, atque it a facillime comparamus eam cognitionem, qua ma-
                <lb/>
              iores, & </s>
              <s xml:id="echoid-s7829" xml:space="preserve">minores proportiones habemus, unde alia, ut di
                <unsure/>
              xi, in templis, alia in publicis , alia in priuatis ædifi-
                <lb/>
              cijs utemur. </s>
              <s xml:id="echoid-s7830" xml:space="preserve">Similis quoque ratio in atrijs, tablinis, œcis , exedris, reliquiſque conclauibus habetur. </s>
              <s xml:id="echoid-s7831" xml:space="preserve">nam quæq;
                <lb/>
              </s>
              <s xml:id="echoid-s7832" xml:space="preserve">babitatio ſuis gaudet proportionibus. </s>
              <s xml:id="echoid-s7833" xml:space="preserve">Cæterum quoniam ſæpe locorum anguſtiæ, proportiones mutare nos co-
                <lb/>
              gunt, ſæpe etiam ſtudio alios ædificiorum proſpectus, alias formas oſtendi uolumus, ideo non ignorandum eſt, qua
                <lb/>
              nam via proportiones inter ſe permutentur, ſubtrahantur, partiantur, componantur, & </s>
              <s xml:id="echoid-s7834" xml:space="preserve">amplientur. </s>
              <s xml:id="echoid-s7835" xml:space="preserve">Sic enim,
                <lb/>
              & </s>
              <s xml:id="echoid-s7836" xml:space="preserve">medium inter extrema, & </s>
              <s xml:id="echoid-s7837" xml:space="preserve">extrema ex medijs inueniemus, tam in rebus, quas efficiemus , quàm in muſi-
                <lb/>
              cis conſonantijs, de quibus quinto libro dicemus. </s>
              <s xml:id="echoid-s7838" xml:space="preserve">Et in reliquis diſciplinis, quarum cognitione homines dele-
                <lb/>
              ctantur. </s>
              <s xml:id="echoid-s7839" xml:space="preserve">Quemadmodum igitur eſt quædam comparatio rerum ſub uno genere comprehenſarum, quæ propor-
                <lb/>
              tio nominatur, ita & </s>
              <s xml:id="echoid-s7840" xml:space="preserve">quædam proportionum inter ſe comparatio, quæ reſpectus eſt quidam non unius quanti-
                <lb/>
                <note position="left" xlink:label="note-514.01.114-06" xlink:href="note-514.01.114-06a" xml:space="preserve">60</note>
              </s>
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