Clavius, Christoph, In Sphaeram Ioannis de Sacro Bosco commentarius

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          <p>
            <s xml:id="echoid-s1599" xml:space="preserve">
              <pb o="34" file="070" n="71" rhead="Comment. in I. Cap. Sphæræ"/>
            dus, remiſſe autem calidus exiſtit: </s>
            <s xml:id="echoid-s1600" xml:space="preserve">frigiditatis cum humiditate, ex qua philoſo-
              <lb/>
            phi aquam colligunt, quam frigidam dicunt in ſummo, humidam vero remiſ-
              <lb/>
            ſe: </s>
            <s xml:id="echoid-s1601" xml:space="preserve">ſiccitatis cum frigiditate, ex qua terra conficitur, quæ in ſummo ſicca, frigi-
              <lb/>
            da uero remiſſe eſſe prædicatur: </s>
            <s xml:id="echoid-s1602" xml:space="preserve">caliditatis cum frigiditate: </s>
            <s xml:id="echoid-s1603" xml:space="preserve">& </s>
            <s xml:id="echoid-s1604" xml:space="preserve">humiditatis cum
              <lb/>
            ſiccitate. </s>
            <s xml:id="echoid-s1605" xml:space="preserve">Sed quoniam duæ hæ poſtremæ combinationes impoſſibiles ſunt,
              <lb/>
            cum ſint contrariorum; </s>
            <s xml:id="echoid-s1606" xml:space="preserve">quorum ea eſt natura, vt vnum alterum ſemper expel-
              <lb/>
            lat: </s>
            <s xml:id="echoid-s1607" xml:space="preserve">Neque enim una, eademq́ue res numero calida, & </s>
            <s xml:id="echoid-s1608" xml:space="preserve">frigida; </s>
            <s xml:id="echoid-s1609" xml:space="preserve">neque humida
              <lb/>
            ſimul, & </s>
            <s xml:id="echoid-s1610" xml:space="preserve">ſicca eſſe poteſt; </s>
            <s xml:id="echoid-s1611" xml:space="preserve">idcirco inutiles cenſentur, neque quicquam ex eis cõ
              <lb/>
            ſtitui poteſt. </s>
            <s xml:id="echoid-s1612" xml:space="preserve">Hæ autem omnes combinationes luce clarius in figura propoſi-
              <lb/>
            ta conſpiciuntur. </s>
            <s xml:id="echoid-s1613" xml:space="preserve">Quod autem diximus, unam qualitatem in quolibet elemẽ
              <lb/>
            to eſſe in ſummo gradu, & </s>
            <s xml:id="echoid-s1614" xml:space="preserve">in remiſſo alteram, intelligendum eſt ex ſententia
              <lb/>
            @uorunda
              <unsure/>
            m philoſophorum. </s>
            <s xml:id="echoid-s1615" xml:space="preserve">Multi enim arbitrantur, utramque qualitatem in
              <lb/>
            quouis elemento eſſe in ſummo grad@</s>
          </p>
          <note position="left" xml:space="preserve">Digreſſio
            <lb/>
          pulcherri-
            <lb/>
          ma de rerũ
            <lb/>
          cõbinatio--
            <lb/>
          nibus, fiue
            <lb/>
          cõparatio--
            <lb/>
          nibus.</note>
          <p>
            <s xml:id="echoid-s1616" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            vero diximus, inter quatuor res non poſſe fieri plures com-
              <lb/>
            binationes, quàm ſex, ſi binæ tantum ſemper ſumantur, uiſum mihi eſt, paulo
              <lb/>
            uberius explicare, quotnam combinationes huiuſmodi fieri poſsint inter quot-
              <lb/>
            cunque res propoſitas; </s>
            <s xml:id="echoid-s1617" xml:space="preserve">Ad multa enim conducit huiuſce rei notitia, eſtq́ue per
              <lb/>
            ſe iucundiſſima. </s>
            <s xml:id="echoid-s1618" xml:space="preserve">Propoſito ergo numero aliquarum rerum, multiplicetur is
              <lb/>
            per numerum proxime minorem. </s>
            <s xml:id="echoid-s1619" xml:space="preserve">Nam producti numeri medietas indicabit
              <lb/>
              <note position="left" xlink:label="note-070-02" xlink:href="note-070-02a" xml:space="preserve">Quot com-
                <lb/>
              binationes
                <lb/>
              fieri poſ-
                <lb/>
              f
                <unsure/>
              int inter
                <lb/>
              quotcunq;
                <lb/>
              res, ſi binæ
                <lb/>
              ſ
                <unsure/>
              umantur.</note>
            numerum combinationum, quæ fieri poſſunt inter res propoſitas. </s>
            <s xml:id="echoid-s1620" xml:space="preserve">Vt in propo-
              <lb/>
            ſito exemplo, quoniam ſunt quatuor qualitates primæ, ſi multiplicentur 4. </s>
            <s xml:id="echoid-s1621" xml:space="preserve">per
              <lb/>
            3. </s>
            <s xml:id="echoid-s1622" xml:space="preserve">officientur 12. </s>
            <s xml:id="echoid-s1623" xml:space="preserve">quare ſex combinationes inter ipſas fieri poſſunt. </s>
            <s xml:id="echoid-s1624" xml:space="preserve">Quòd ſi fue-
              <lb/>
            rint quinque res combinandæ, multiplicanda ſunt 5. </s>
            <s xml:id="echoid-s1625" xml:space="preserve">per 4. </s>
            <s xml:id="echoid-s1626" xml:space="preserve">Nam producti me-
              <lb/>
            dietas, nempe 10. </s>
            <s xml:id="echoid-s1627" xml:space="preserve">oſtendet numerum combinationum: </s>
            <s xml:id="echoid-s1628" xml:space="preserve">quot uidelicet Porphy
              <lb/>
            rius inter quinque prædicabilia inſtituit.</s>
            <s xml:id="echoid-s1629" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1630" xml:space="preserve">
              <emph style="sc">Potest</emph>
            hæc regula tradita in duas diſtrahi, prout ſcilicet numerus re-
              <lb/>
            rum par, uel impar fuerit. </s>
            <s xml:id="echoid-s1631" xml:space="preserve">Sienim numerus rerum fuerit par, multiplicandus
              <lb/>
            erit numerus proxime minor per medietatem numeri rerum: </s>
            <s xml:id="echoid-s1632" xml:space="preserve">Nam productus
              <lb/>
            numerus continuo oſtendet combinationum numerum. </s>
            <s xml:id="echoid-s1633" xml:space="preserve">Vt ſi ſcire lubet, quo@
              <lb/>
            fieri poſſint combinationes inter 10. </s>
            <s xml:id="echoid-s1634" xml:space="preserve">res, multiplicabuntur 9. </s>
            <s xml:id="echoid-s1635" xml:space="preserve">per 5. </s>
            <s xml:id="echoid-s1636" xml:space="preserve">ut fiant 45.
              <lb/>
            </s>
            <s xml:id="echoid-s1637" xml:space="preserve">quot nimirum combinationes fieri inter decem res poſſunt. </s>
            <s xml:id="echoid-s1638" xml:space="preserve">Si uero numerus
              <lb/>
            rerum extiterit impar, multiplicandus is erit per medietatem numeri proxime
              <unsure/>
              <lb/>
            minoris; </s>
            <s xml:id="echoid-s1639" xml:space="preserve">Hac enim ratione numerus procreatus indicabit, quot fieri poſſintcõ
              <lb/>
            binationes. </s>
            <s xml:id="echoid-s1640" xml:space="preserve">Vt ſi res fuerint 15. </s>
            <s xml:id="echoid-s1641" xml:space="preserve">Multiplicatis 15. </s>
            <s xml:id="echoid-s1642" xml:space="preserve">per 7. </s>
            <s xml:id="echoid-s1643" xml:space="preserve">efficietur numerus con
              <lb/>
            binationum inter ipſas, nempe 105. </s>
            <s xml:id="echoid-s1644" xml:space="preserve">Inter 9. </s>
            <s xml:id="echoid-s1645" xml:space="preserve">uero res fient combinationes 36. </s>
            <s xml:id="echoid-s1646" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s1647" xml:space="preserve">fic de cæteris.</s>
            <s xml:id="echoid-s1648" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1649" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi ſcire placuerit, quotcunque rebus propoſitis, quot fimpliciter
              <lb/>
              <note position="left" xlink:label="note-070-03" xlink:href="note-070-03a" xml:space="preserve">Quot com-
                <lb/>
              binationes
                <lb/>
              fieri poſ-
                <lb/>
              fint inter
                <lb/>
              quotcunq;
                <lb/>
              res abſolu-
                <lb/>
              te, ſi non ſo
                <lb/>
              l
                <unsure/>
              um binæ,
                <lb/>
              ſed etiam
                <lb/>
              @ernæ, qua-
                <lb/>
              ternæ, qui-
                <lb/>
              @æ, &c. ſu-
                <lb/>
              mantur
                <unsure/>
              .</note>
            coniunctiones ex ipſis poſſint fieri, non ſolum intelligendo, quando binæ ſu-
              <lb/>
            muntur, ut in præcedenti regula, ſed etiam quando ternæ, quaternæ, quinæ,
              <lb/>
            &</s>
            <s xml:id="echoid-s1650" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1651" xml:space="preserve">hoc eſt, quotnã modis diſtinctis inter ſeſe poſſint cõparari; </s>
            <s xml:id="echoid-s1652" xml:space="preserve">efficietur id hac
              <lb/>
            arte, & </s>
            <s xml:id="echoid-s1653" xml:space="preserve">regula. </s>
            <s xml:id="echoid-s1654" xml:space="preserve">Accipiantur tot numeri, incipiendo ab unitate, in dupla propo@
              <lb/>
            tione, quot res ſunt propoſitæ, & </s>
            <s xml:id="echoid-s1655" xml:space="preserve">à ſumma omnium illorum ſubtrahatur nume
              <lb/>
            rus rerum: </s>
            <s xml:id="echoid-s1656" xml:space="preserve">Reliquus enim numerus indicabit, quotnam comparationes diuerſę
              <lb/>
            effici poſſint.</s>
            <s xml:id="echoid-s1657" xml:space="preserve">|Facile aũt habebitur ſumma quotcunq. </s>
            <s xml:id="echoid-s1658" xml:space="preserve">numerorum duplæ pro-
              <lb/>
            portionis ab 1. </s>
            <s xml:id="echoid-s1659" xml:space="preserve">incipientis, ſi ultimus numerus duplicetur, & </s>
            <s xml:id="echoid-s1660" xml:space="preserve">ex producto unitas
              <lb/>
            abijciatur. </s>
            <s xml:id="echoid-s1661" xml:space="preserve">Vt ſi lubeat ſcire ſummam horum numerorum in dupla proportio-
              <lb/>
            ne, 1. </s>
            <s xml:id="echoid-s1662" xml:space="preserve">2. </s>
            <s xml:id="echoid-s1663" xml:space="preserve">4. </s>
            <s xml:id="echoid-s1664" xml:space="preserve">8. </s>
            <s xml:id="echoid-s1665" xml:space="preserve">16. </s>
            <s xml:id="echoid-s1666" xml:space="preserve">32. </s>
            <s xml:id="echoid-s1667" xml:space="preserve">64. </s>
            <s xml:id="echoid-s1668" xml:space="preserve">duplicandus erit numerus ultimus 64. </s>
            <s xml:id="echoid-s1669" xml:space="preserve">ut fiant 128. </s>
            <s xml:id="echoid-s1670" xml:space="preserve">@
              <lb/>
            quibus reiecta unitate, remanent 127. </s>
            <s xml:id="echoid-s1671" xml:space="preserve">pro ſumma omnium illorum </s>
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