Clavius, Christoph, Geometria practica
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          <p style="it">
            <s xml:id="echoid-s10213" xml:space="preserve">3. </s>
            <s xml:id="echoid-s10214" xml:space="preserve">VEL ex duab{us} tertiis partib{us} ar@æ circuli maximi in totam diametrum.</s>
            <s xml:id="echoid-s10215" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10216" xml:space="preserve">4. </s>
            <s xml:id="echoid-s10217" xml:space="preserve">VEL ex ſemidiametro in quatuor terti{as} part{es} areæ circuli maximi.</s>
            <s xml:id="echoid-s10218" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10219" xml:space="preserve">5. </s>
            <s xml:id="echoid-s10220" xml:space="preserve">VEL ex ſemiſſe areæ circuli maximi in quatuor terti{as} partes diametri.</s>
            <s xml:id="echoid-s10221" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10222" xml:space="preserve">6. </s>
            <s xml:id="echoid-s10223" xml:space="preserve">VEL ex dupla diametro in tertiam partem areæ circuli maximi.</s>
            <s xml:id="echoid-s10224" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10225" xml:space="preserve">7. </s>
            <s xml:id="echoid-s10226" xml:space="preserve">VEL ex diametro in 6. </s>
            <s xml:id="echoid-s10227" xml:space="preserve">partem ſuperficiei ſphæræ.</s>
            <s xml:id="echoid-s10228" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10229" xml:space="preserve">8. </s>
            <s xml:id="echoid-s10230" xml:space="preserve">VEL denique ex tertia parte diametriin ſemiſſem ſuperficiei conuexæ ſphæræ.</s>
            <s xml:id="echoid-s10231" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s10232" xml:space="preserve">
              <emph style="sc">Primvm</emph>
            demonſtatum à nobis eſt in commentariis in ſphęram, quam de-
              <lb/>
              <note position="left" xlink:label="note-254-01" xlink:href="note-254-01a" xml:space="preserve">Demonſtra-
                <lb/>
              tio primæ par-
                <lb/>
              tis.</note>
            monſtrationem repetemus lib. </s>
            <s xml:id="echoid-s10233" xml:space="preserve">7. </s>
            <s xml:id="echoid-s10234" xml:space="preserve">de Iſoperimetris. </s>
            <s xml:id="echoid-s10235" xml:space="preserve">Idem tamen aliter hac ratio-
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            ne demonſtrabimus. </s>
            <s xml:id="echoid-s10236" xml:space="preserve">Concipiatur conus, cuius baſis maximus circulus ſphę-
              <lb/>
            rę, & </s>
            <s xml:id="echoid-s10237" xml:space="preserve">altitudo ſemidiameter eiuſdem. </s>
            <s xml:id="echoid-s10238" xml:space="preserve">Item alius conus, cuius baſis quadrupla
              <lb/>
            ſit maximi circuli, & </s>
            <s xml:id="echoid-s10239" xml:space="preserve">altitudo ſemidiameter eadem. </s>
            <s xml:id="echoid-s10240" xml:space="preserve">Et quia prioris coni tam
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            ſphęra, per propoſ. </s>
            <s xml:id="echoid-s10241" xml:space="preserve">32. </s>
            <s xml:id="echoid-s10242" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10243" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10244" xml:space="preserve">Archimedis de ſphęra, & </s>
            <s xml:id="echoid-s10245" xml:space="preserve">cylindro, quadrupla
              <lb/>
            eſt, quam poſterior conus: </s>
            <s xml:id="echoid-s10246" xml:space="preserve"> erunt poſterior conus, & </s>
            <s xml:id="echoid-s10247" xml:space="preserve">ſphęra inter ſe
              <note symbol="a" position="left" xlink:label="note-254-02" xlink:href="note-254-02a" xml:space="preserve">11. duodec.</note>
            quales.</s>
            <s xml:id="echoid-s10248" xml:space="preserve"/>
          </p>
          <note symbol="b" position="left" xml:space="preserve">9. quinti.</note>
          <p>
            <s xml:id="echoid-s10249" xml:space="preserve">
              <emph style="sc">Rvrsvs</emph>
            quia circulus, cuius ſemidiameter ęqualis eſt toti diametro ſpę-
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            rę, quadruplus eſt circuli maximi. </s>
            <s xml:id="echoid-s10250" xml:space="preserve"> (cum enim ſit circulus ad circulum, vt
              <note symbol="c" position="left" xlink:label="note-254-04" xlink:href="note-254-04a" xml:space="preserve">2. duodec.</note>
            dratum diametri ad quadratum diametri: </s>
            <s xml:id="echoid-s10251" xml:space="preserve">quadratum autem prioris diametri
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            quadruplum ſit quadrati diametri poſterioris, ex ſcholio propoſ. </s>
            <s xml:id="echoid-s10252" xml:space="preserve">4. </s>
            <s xml:id="echoid-s10253" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10254" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10255" xml:space="preserve">Eucl.
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            </s>
            <s xml:id="echoid-s10256" xml:space="preserve">quòd illa diameter ſit huius dupla; </s>
            <s xml:id="echoid-s10257" xml:space="preserve">quando quidem ſemiſsis prioris diametri
              <lb/>
            ſumpta eſt poſteriori diametro ęqualis; </s>
            <s xml:id="echoid-s10258" xml:space="preserve">erit quoque circulus circuli quadru-
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            plus.) </s>
            <s xml:id="echoid-s10259" xml:space="preserve">eritidem circulus, cuius ſemidiameter diametro ſphęrę ęqualis eſt, ęqua-
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            lis baſi poſterioris coni, cum huius baſis quadrupla etiam poſita ſit maximi circu-
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            li. </s>
            <s xml:id="echoid-s10260" xml:space="preserve">Quia verò etiam ſuperficies ſphæræ quadrupla eſt circuli maximi, ex propoſ. </s>
            <s xml:id="echoid-s10261" xml:space="preserve">
              <lb/>
            31. </s>
            <s xml:id="echoid-s10262" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10263" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10264" xml:space="preserve">Archimedis de ſphæra, & </s>
            <s xml:id="echoid-s10265" xml:space="preserve">cylindro: </s>
            <s xml:id="echoid-s10266" xml:space="preserve"> Erunt ſuperficies ſphæræ, baſis
              <note symbol="d" position="left" xlink:label="note-254-05" xlink:href="note-254-05a" xml:space="preserve">9. quinti.</note>
            ſterioris coni, & </s>
            <s xml:id="echoid-s10267" xml:space="preserve">circulus ſemidiametrum habens æqualem diametro ſphæræ, inter
              <lb/>
            ſe æquales.</s>
            <s xml:id="echoid-s10268" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10269" xml:space="preserve">
              <emph style="sc">Postremo</emph>
            concipiatur cylindrus, cuius baſis ſit prædictus circulus ſemidia-
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            metrum diametro ſphæræ habens æqualem, altitudo verò ſemidiameter ſphæræ.
              <lb/>
            </s>
            <s xml:id="echoid-s10270" xml:space="preserve"> Erit hic cylindrus triplus poſterioris coni prædicti: </s>
            <s xml:id="echoid-s10271" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s10272" xml:space="preserve">ſphæræ,
              <note symbol="e" position="left" xlink:label="note-254-06" xlink:href="note-254-06a" xml:space="preserve">10. duodec.</note>
            ei cono eſt oſtenſa æqualis. </s>
            <s xml:id="echoid-s10273" xml:space="preserve"> Idem autem cylindrus triplus quoque eſt
              <note symbol="f" position="left" xlink:label="note-254-07" xlink:href="note-254-07a" xml:space="preserve">11. duodec.</note>
            qui eandem habeat altitudinem, & </s>
            <s xml:id="echoid-s10274" xml:space="preserve">baſem terriæ parti illius cylindri, hoc
              <lb/>
            eſt, tertiæ parti ſuperficiei ſphæræ, æqualem. </s>
            <s xml:id="echoid-s10275" xml:space="preserve"> Ergo poſterior cylindrus,
              <note symbol="g" position="left" xlink:label="note-254-08" xlink:href="note-254-08a" xml:space="preserve">9. quinti.</note>
            ſem habenstertiæ parti ſuperficiei ſphæræ æqualem, altitudinem verò ſemidia-
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            metro eiuſdem ſphæræ æqualem,) & </s>
            <s xml:id="echoid-s10276" xml:space="preserve">ſphæra æquales ſunt. </s>
            <s xml:id="echoid-s10277" xml:space="preserve">Cum ergo cylindrus
              <lb/>
            hic poſterior contineatur ſub ſemidiametro ſphæræ, & </s>
            <s xml:id="echoid-s10278" xml:space="preserve">tertia parte ſuperficiei
              <lb/>
            ſphæricæ: </s>
            <s xml:id="echoid-s10279" xml:space="preserve">liquidò conſtat, ſphæræ ſoliditatem gigni ex ſemidiametro in partem
              <lb/>
            tertiam ſuperficiei ſphæræ. </s>
            <s xml:id="echoid-s10280" xml:space="preserve">Velex @. </s>
            <s xml:id="echoid-s10281" xml:space="preserve">totius diametri in {2/@}. </s>
            <s xml:id="echoid-s10282" xml:space="preserve">ſuperficiei ſphæræ: </s>
            <s xml:id="echoid-s10283" xml:space="preserve">cum
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            hic numerus illi ſit æqualis. </s>
            <s xml:id="echoid-s10284" xml:space="preserve">quod eſt primum.</s>
            <s xml:id="echoid-s10285" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10286" xml:space="preserve">
              <emph style="sc">Concipiatvr</emph>
            rurſum cylindrus, cuius baſis maximus circulus ſphæræ, & </s>
            <s xml:id="echoid-s10287" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-254-09" xlink:href="note-254-09a" xml:space="preserve">Demonſtra-
                <lb/>
              tio ſecundæ
                <lb/>
              partis.</note>
            altitudo diameter ſphæræ. </s>
            <s xml:id="echoid-s10288" xml:space="preserve">Erit hic cylindrus ſeſquialter ſphæræ, ex coroll. </s>
            <s xml:id="echoid-s10289" xml:space="preserve">pro-
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            poſ. </s>
            <s xml:id="echoid-s10290" xml:space="preserve">32. </s>
            <s xml:id="echoid-s10291" xml:space="preserve">libr. </s>
            <s xml:id="echoid-s10292" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10293" xml:space="preserve">Archimedis de ſphæra, & </s>
            <s xml:id="echoid-s10294" xml:space="preserve">cylindro. </s>
            <s xml:id="echoid-s10295" xml:space="preserve">Quod ſi ex parte ſuperiori per
              <lb/>
            tettiam par
              <unsure/>
            tem diametri ſphæræ, vel axis cylindri, ducatur baſibus cylindri pla-
              <lb/>
            num parallelum: </s>
            <s xml:id="echoid-s10296" xml:space="preserve"> erit totus cylindrus ad cylindrum abſciſſum, cums axis
              <note symbol="h" position="left" xlink:label="note-254-10" xlink:href="note-254-10a" xml:space="preserve">13. duodec.</note>
            tertiæ partes ſunt totius axis, ſeſquialter; </s>
            <s xml:id="echoid-s10297" xml:space="preserve">Ac proinde poſterior hic cylindrus
              <lb/>
              <note symbol="i" position="left" xlink:label="note-254-11" xlink:href="note-254-11a" xml:space="preserve">9. quinti.</note>
            abſciſſus, qui quidem continetur ſub maximo circulo, nempe ſub ſua baſi, & </s>
            <s xml:id="echoid-s10298" xml:space="preserve">dua-
              <lb/>
            bustertiis partibus diametriſphæræ, ſphæræ æqualis erit. </s>
            <s xml:id="echoid-s10299" xml:space="preserve">Pater igitur etiam ſe-
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            cundum.</s>
            <s xml:id="echoid-s10300" xml:space="preserve"/>
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