Vitruvius, De architectura libri decem, 1567

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        <div xml:id="echoid-div1" type="section" level="1" n="1">
          <div xml:id="echoid-div286" type="section" level="2" n="62">
            <p style="it">
              <s xml:id="echoid-s8578" xml:space="preserve">
                <pb o="91" file="514.01.123" n="123" rhead="TERTIVS."/>
              dans uocabitur. </s>
              <s xml:id="echoid-s8579" xml:space="preserve">Perfectum uero eum numerum dicunt, qui conſtat ex ſumma omanium ſuarum partìum, quem-
                <lb/>
              admodum ſenarius, cuius partes 1. </s>
              <s xml:id="echoid-s8580" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8581" xml:space="preserve">3. </s>
              <s xml:id="echoid-s8582" xml:space="preserve">quæ ſimul ſumptæ, ſenarium reddunt, pari ſorma. </s>
              <s xml:id="echoid-s8583" xml:space="preserve">28. </s>
              <s xml:id="echoid-s8584" xml:space="preserve">numerus
                <lb/>
              perfectus eſt; </s>
              <s xml:id="echoid-s8585" xml:space="preserve">nam eius partes ſunt 1. </s>
              <s xml:id="echoid-s8586" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8587" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8588" xml:space="preserve">7. </s>
              <s xml:id="echoid-s8589" xml:space="preserve">14. </s>
              <s xml:id="echoid-s8590" xml:space="preserve">quarum ſumma 28. </s>
              <s xml:id="echoid-s8591" xml:space="preserve">reddunt. </s>
              <s xml:id="echoid-s8592" xml:space="preserve">Sed poſtquam de.
                <lb/>
              </s>
              <s xml:id="echoid-s8593" xml:space="preserve">numeris perfectis ſermo eſt inſtitus, dicam eorum ortus, uires, proprietatesq́;</s>
              <s xml:id="echoid-s8594" xml:space="preserve">. & </s>
              <s xml:id="echoid-s8595" xml:space="preserve">ad faciliorem intelligen-
                <lb/>
              tiam diffinitiones aliquas numer or um adducam. </s>
              <s xml:id="echoid-s8596" xml:space="preserve">Numeri quidam ſunt, qui pariter pares nominantur, quidam
                <lb/>
              uero, qui primi, & </s>
              <s xml:id="echoid-s8597" xml:space="preserve">incompoſiti dicuntur. </s>
              <s xml:id="echoid-s8598" xml:space="preserve">pariter pares numeri ſunt, qui cum eorum ſumma par ſit, uſque ad
                <lb/>
              unitatem in paria diuiduntur. </s>
              <s xml:id="echoid-s8599" xml:space="preserve">Exempligratia 64. </s>
              <s xml:id="echoid-s8600" xml:space="preserve">eſt numerus pariter par, & </s>
              <s xml:id="echoid-s8601" xml:space="preserve">ipſa ſmma 64. </s>
              <s xml:id="echoid-s8602" xml:space="preserve">par eſt, & </s>
              <s xml:id="echoid-s8603" xml:space="preserve">
                <lb/>
              eius prima partitio par eſt. </s>
              <s xml:id="echoid-s8604" xml:space="preserve">videlicet 32. </s>
              <s xml:id="echoid-s8605" xml:space="preserve">Cuius item diuiſio par eſt. </s>
              <s xml:id="echoid-s8606" xml:space="preserve">16. </s>
              <s xml:id="echoid-s8607" xml:space="preserve">enim eſt numerus par, dimidium
                <lb/>
              quoque 16. </s>
              <s xml:id="echoid-s8608" xml:space="preserve">par eſt, ut pote octo, patiter quoque octonarij dimidium par 4. </s>
              <s xml:id="echoid-s8609" xml:space="preserve">& </s>
              <s xml:id="echoid-s8610" xml:space="preserve">quaternarij duo, cuius di-
                <lb/>
                <note position="left" xlink:label="note-514.01.123-01" xlink:href="note-514.01.123-01a" xml:space="preserve">10</note>
              midium unitas. </s>
              <s xml:id="echoid-s8611" xml:space="preserve">Ecce qua ratione ſexagintaquatuor est numerus pariter par. </s>
              <s xml:id="echoid-s8612" xml:space="preserve">Sed numeri primi, & </s>
              <s xml:id="echoid-s8613" xml:space="preserve">incompo-
                <lb/>
              ſiti ſunt, qui a ſola unitate numer antur, neque alium babent numerum, qui eos integre partiatur, cuiuſmodi
                <lb/>
              eſt ternarius numerus, quinarius item, & </s>
              <s xml:id="echoid-s8614" xml:space="preserve">ſeptenarius, & </s>
              <s xml:id="echoid-s8615" xml:space="preserve">buinſmodi. </s>
              <s xml:id="echoid-s8616" xml:space="preserve">His poſitis perfectorum productio ponen-
                <lb/>
              da eſt. </s>
              <s xml:id="echoid-s8617" xml:space="preserve">Collocato ordine ſuo pariter pares numeros, & </s>
              <s xml:id="echoid-s8618" xml:space="preserve">eos collige; </s>
              <s xml:id="echoid-s8619" xml:space="preserve">ſi perueneris in eam numeri ſummam, quæ
                <lb/>
              ſit numeri primi, & </s>
              <s xml:id="echoid-s8620" xml:space="preserve">incompoſiti, & </s>
              <s xml:id="echoid-s8621" xml:space="preserve">eam duxeris per præcedentem numerum, facies perfectum numerum. </s>
              <s xml:id="echoid-s8622" xml:space="preserve">col-
                <lb/>
              locato igitur 1. </s>
              <s xml:id="echoid-s8623" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8624" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8625" xml:space="preserve">8. </s>
              <s xml:id="echoid-s8626" xml:space="preserve">16. </s>
              <s xml:id="echoid-s8627" xml:space="preserve">32. </s>
              <s xml:id="echoid-s8628" xml:space="preserve">64. </s>
              <s xml:id="echoid-s8629" xml:space="preserve">Collige 1. </s>
              <s xml:id="echoid-s8630" xml:space="preserve">& </s>
              <s xml:id="echoid-s8631" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8632" xml:space="preserve">reddentur tres, ducito tres numerum pri-
                <lb/>
              mum, & </s>
              <s xml:id="echoid-s8633" xml:space="preserve">incompoſitum per duo numerum præcedentem, efficientur ſex. </s>
              <s xml:id="echoid-s8634" xml:space="preserve">Senarius igitur erit perfectus nume-
                <lb/>
              rus. </s>
              <s xml:id="echoid-s8635" xml:space="preserve">pari forma collige 1. </s>
              <s xml:id="echoid-s8636" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8637" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8638" xml:space="preserve">reddentur ſeptem, qui numerus eſt primus, & </s>
              <s xml:id="echoid-s8639" xml:space="preserve">incompoſitus, ducito ſeptem
                <lb/>
              per quatuor, reddes. </s>
              <s xml:id="echoid-s8640" xml:space="preserve">28. </s>
              <s xml:id="echoid-s8641" xml:space="preserve">qui eſt ſecundus perfectus numerus. </s>
              <s xml:id="echoid-s8642" xml:space="preserve">Collige item 1. </s>
              <s xml:id="echoid-s8643" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8644" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8645" xml:space="preserve">8. </s>
              <s xml:id="echoid-s8646" xml:space="preserve">efficies quinde-
                <lb/>
              cim, is numerus non eſt primus, & </s>
              <s xml:id="echoid-s8647" xml:space="preserve">incompoſitus, immo babet ſuas partes numerantes ſcilicet ternarium, vel
                <lb/>
              quinarium. </s>
              <s xml:id="echoid-s8648" xml:space="preserve">quare ſequere ulterius 1. </s>
              <s xml:id="echoid-s8649" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8650" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8651" xml:space="preserve">8. </s>
              <s xml:id="echoid-s8652" xml:space="preserve">16. </s>
              <s xml:id="echoid-s8653" xml:space="preserve">collige efficies 31. </s>
              <s xml:id="echoid-s8654" xml:space="preserve">qui numerus primus, & </s>
              <s xml:id="echoid-s8655" xml:space="preserve">incom-
                <lb/>
                <note position="left" xlink:label="note-514.01.123-02" xlink:href="note-514.01.123-02a" xml:space="preserve">20</note>
              poſitus est. </s>
              <s xml:id="echoid-s8656" xml:space="preserve">duc igitur 16. </s>
              <s xml:id="echoid-s8657" xml:space="preserve">in 31. </s>
              <s xml:id="echoid-s8658" xml:space="preserve">reddentur 496. </s>
              <s xml:id="echoid-s8659" xml:space="preserve">numerus perfectus, & </s>
              <s xml:id="echoid-s8660" xml:space="preserve">ita in alijs. </s>
              <s xml:id="echoid-s8661" xml:space="preserve">nec mir ere in dena-
                <lb/>
              rio ſex. </s>
              <s xml:id="echoid-s8662" xml:space="preserve">in centenario 28. </s>
              <s xml:id="echoid-s8663" xml:space="preserve">in millenario 496. </s>
              <s xml:id="echoid-s8664" xml:space="preserve">perfectos numeros, boc eſt tam paucos reperiri, quia rara
                <lb/>
              ſunt, quæ perfecta ſunt. </s>
              <s xml:id="echoid-s8665" xml:space="preserve">Simili ratione in denis millibus perfectus eſt tantum 8128. </s>
              <s xml:id="echoid-s8666" xml:space="preserve">perfectorum numerorum
                <lb/>
              proprietas eſt, ut terminante primo m ſenarium, ſecundus deſinit in octonarium, tertius in ſenarium, quartus
                <lb/>
              in octonarium, & </s>
              <s xml:id="echoid-s8667" xml:space="preserve">nulla alia eſt perfectorum numerorum terminatio. </s>
              <s xml:id="echoid-s8668" xml:space="preserve">Sed qua ratione ternarius, & </s>
              <s xml:id="echoid-s8669" xml:space="preserve">denarius
                <lb/>
              numerus perfecti ſint appellati, nunc dicam. </s>
              <s xml:id="echoid-s8670" xml:space="preserve">Ternarius numerus complectitur primo primas numerorum diſ-
                <lb/>
              ferentias, paritatem ſcilicet, & </s>
              <s xml:id="echoid-s8671" xml:space="preserve">imparitatem, ſolusq́; </s>
              <s xml:id="echoid-s8672" xml:space="preserve">conficitur ex ſumma præcedentium numerorum, nam
                <lb/>
              1. </s>
              <s xml:id="echoid-s8673" xml:space="preserve">& </s>
              <s xml:id="echoid-s8674" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8675" xml:space="preserve">reddunt 3. </s>
              <s xml:id="echoid-s8676" xml:space="preserve">concluditq́; </s>
              <s xml:id="echoid-s8677" xml:space="preserve">principium, medium, & </s>
              <s xml:id="echoid-s8678" xml:space="preserve">finem. </s>
              <s xml:id="echoid-s8679" xml:space="preserve">bac de cauſa perfectus est numerus iudi-
                <lb/>
              catus. </s>
              <s xml:id="echoid-s8680" xml:space="preserve">Denarius uero perfectus est dictus, quoniam tanquam forma omnes alios numeros includit, ultra de-
                <lb/>
              cem enim progrediendo ad eandem unitatem, ex qua conficitur is numerus, redire oportet, ac denuo incipere.
                <lb/>
              </s>
              <s xml:id="echoid-s8681" xml:space="preserve">
                <note position="left" xlink:label="note-514.01.123-03" xlink:href="note-514.01.123-03a" xml:space="preserve">30</note>
              Atque boc dixit Vitr. </s>
              <s xml:id="echoid-s8682" xml:space="preserve">ex Platonis ſententia, qui denarium pefectum numerum nuncupauit.</s>
              <s xml:id="echoid-s8683" xml:space="preserve"/>
            </p>
            <p style="it">
              <s xml:id="echoid-s8684" xml:space="preserve">
                <emph style="sc">Mathematici</emph>
              uero contra diſputantes, ea re perfectum eſſe dixerunt numerum, qui ſex
                <lb/>
              dicitur, quod is numerus habet partitiones eorum rationibus ſex numero conuenientes; </s>
              <s xml:id="echoid-s8685" xml:space="preserve">ſic ſextantẽ,
                <lb/>
              unum@: </s>
              <s xml:id="echoid-s8686" xml:space="preserve">trientem, duo: </s>
              <s xml:id="echoid-s8687" xml:space="preserve">ſemiſſem, tria: </s>
              <s xml:id="echoid-s8688" xml:space="preserve">beſſem, quem dimiron dicunt quatuor: </s>
              <s xml:id="echoid-s8689" xml:space="preserve">quintarium, quem pen-
                <lb/>
              tamiron dicunt, quinque: </s>
              <s xml:id="echoid-s8690" xml:space="preserve">perfectum ſex. </s>
              <s xml:id="echoid-s8691" xml:space="preserve">{Eorum rationibus} idest ſecundum eorum Matbematicorum
                <lb/>
              rationes, qui uolunt eum eſſe perfectum numerum, qui ex ſumma eius partium redditur. </s>
              <s xml:id="echoid-s8692" xml:space="preserve">ideo dixit Vitr. </s>
              <s xml:id="echoid-s8693" xml:space="preserve">quod
                <lb/>
              is numerus babet partitiones corum rationibus, ſex numero conuenientes. </s>
              <s xml:id="echoid-s8694" xml:space="preserve">cum igitur perfectum ſenarium di-
                <lb/>
              xerint, idem illi euenire uoluerunt, quod denario. </s>
              <s xml:id="echoid-s8695" xml:space="preserve">nam cum unitatem ultra ſex adderent, ad eandem rationem
                <lb/>
              reuertebantur. </s>
              <s xml:id="echoid-s8696" xml:space="preserve">Solebant antiqui rem integram omnem Aſſem nominare. </s>
              <s xml:id="echoid-s8697" xml:space="preserve">eamq́; </s>
              <s xml:id="echoid-s8698" xml:space="preserve">in ſuas partes diuidere, & </s>
              <s xml:id="echoid-s8699" xml:space="preserve">
                <lb/>
              tanquàm felices Græcorum interpretes, maximè proprie res declarabant. </s>
              <s xml:id="echoid-s8700" xml:space="preserve">Voluere igitur ſenarium perfectum
                <lb/>
                <note position="left" xlink:label="note-514.01.123-04" xlink:href="note-514.01.123-04a" xml:space="preserve">40</note>
              eſſe, & </s>
              <s xml:id="echoid-s8701" xml:space="preserve">quia integrum, & </s>
              <s xml:id="echoid-s8702" xml:space="preserve">totum est quoddam, ideo eum Aſſem uocauere. </s>
              <s xml:id="echoid-s8703" xml:space="preserve">qui cum partes; </s>
              <s xml:id="echoid-s8704" xml:space="preserve">ſuas baberet, ex
                <lb/>
              eorum uocabulis, quales partes eſſent, declarabant. </s>
              <s xml:id="echoid-s8705" xml:space="preserve">Vnitatem, quæ ſexta eius eſt pars, ſextantem nomina-
                <lb/>
              runt. </s>
              <s xml:id="echoid-s8706" xml:space="preserve">2. </s>
              <s xml:id="echoid-s8707" xml:space="preserve">trientem, nam 2. </s>
              <s xml:id="echoid-s8708" xml:space="preserve">tertia ſenarij, boc est Aßis pars eſt. </s>
              <s xml:id="echoid-s8709" xml:space="preserve">Tria ſemiſſem, quaſi Aſſem dimidium uo
                <lb/>
              cauere. </s>
              <s xml:id="echoid-s8710" xml:space="preserve">4. </s>
              <s xml:id="echoid-s8711" xml:space="preserve">beſſem, quoniam à toto duas aufert partes, græcè dimiron eadem ratione dicitur. </s>
              <s xml:id="echoid-s8712" xml:space="preserve">5. </s>
              <s xml:id="echoid-s8713" xml:space="preserve">quinta-
                <lb/>
              rium latine, græce autẽ pentamiron nominatur. </s>
              <s xml:id="echoid-s8714" xml:space="preserve">Sed ubi ad ſenarium peruentum est, boc eſt ad Aſſem, ſi quis
                <lb/>
              ultra progredi uelit, iam alterum aſſem duplicare incipit. </s>
              <s xml:id="echoid-s8715" xml:space="preserve">ideo 12. </s>
              <s xml:id="echoid-s8716" xml:space="preserve">diplaſiona Græci dicunt. </s>
              <s xml:id="echoid-s8717" xml:space="preserve">Cum uero Vi-
                <lb/>
              tr. </s>
              <s xml:id="echoid-s8718" xml:space="preserve">dixit, ſic ſextantem unum, non uult rationem reddere cur ſenarius numerus perfectus a Matbematicis di-
                <lb/>
              ctus eſt. </s>
              <s xml:id="echoid-s8719" xml:space="preserve">nam ſupra eam point, dum dixit quod is numerus habet partitiones eorum rationibus ſex numero con
                <lb/>
              uenientes, nam ſumma partium eius numeri, totum integrum reddit. </s>
              <s xml:id="echoid-s8720" xml:space="preserve">Sed oſtendere uult ſenarium ob prædi-
                <lb/>
              ctam rationem cum perfectus ſit, ſuas quoque partes habere a Matbematicis nominatas. </s>
              <s xml:id="echoid-s8721" xml:space="preserve">Quod ſi quis ulterius
                <lb/>
                <note position="left" xlink:label="note-514.01.123-05" xlink:href="note-514.01.123-05a" xml:space="preserve">50</note>
              progrediuelit, id illi eueniat, quod denario, qui ab alij perfectus est dictus; </s>
              <s xml:id="echoid-s8722" xml:space="preserve">euenire ostenſum est. </s>
              <s xml:id="echoid-s8723" xml:space="preserve">Alioquin
                <lb/>
              uana fuiſſet Matbematicorum obiectio in eos, qui denarium perfectum statuere, ſi ijdem Mathematici ſena-
                <lb/>
              rium ob eandem rationem perfectum eſſe uoluiſſent, qua denarius est ab alijs perfectus nominatus. </s>
              <s xml:id="echoid-s8724" xml:space="preserve">ideo
                <lb/>
              ſequitur.</s>
              <s xml:id="echoid-s8725" xml:space="preserve"/>
            </p>
            <p>
              <s xml:id="echoid-s8726" xml:space="preserve">
                <emph style="sc">Cvm</emph>
              ad ſupputationcm creſcat ſupra ſex: </s>
              <s xml:id="echoid-s8727" xml:space="preserve">adiecto aſſe ephecton cum fact ſunt octo, quod eſt
                <lb/>
              tertia adiecta tertiarium, qui epitritos dicitur, dimidia adiecta cum facta ſunt nouem, ſeſquialterum
                <lb/>
              qui hemiolios appellatur, duabus partibus additis, & </s>
              <s xml:id="echoid-s8728" xml:space="preserve">decuſsi facto, beſalterum, quem epidimiron uo-
                <lb/>
              citant, in undeeim numero, quod adiecti ſunt quinque, quintarium, quod epipentamiron. </s>
              <s xml:id="echoid-s8729" xml:space="preserve">Duodecim
                <lb/>
              autem, quiex duobus ſimplicibus numeris eſt effectus diplaſrona.</s>
              <s xml:id="echoid-s8730" xml:space="preserve"/>
            </p>
            <p style="it">
              <s xml:id="echoid-s8731" xml:space="preserve">{Adiecto Aſſe.</s>
              <s xml:id="echoid-s8732" xml:space="preserve">} Non referuntur bæc uerba ad adiectionem unitatis ſupra ſenarium, nã ſextans adijcitur,
                <lb/>
                <note position="left" xlink:label="note-514.01.123-06" xlink:href="note-514.01.123-06a" xml:space="preserve">60</note>
              </s>
            </p>
          </div>
        </div>
      </text>
    </echo>
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