Clavius, Christoph, Geometria practica

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            <s xml:id="echoid-s14670" xml:space="preserve">
              <pb o="313" file="343" n="343" rhead="LIBER SEPTIMVS."/>
            N, æqualis. </s>
            <s xml:id="echoid-s14671" xml:space="preserve">Eodem pacto, quia baſis coni O P Q. </s>
            <s xml:id="echoid-s14672" xml:space="preserve">æqualis eſt ambitui corporis
              <lb/>
            EFGHIKLM; </s>
            <s xml:id="echoid-s14673" xml:space="preserve">quia & </s>
            <s xml:id="echoid-s14674" xml:space="preserve">æqualis ſuperficiei ſphæræ N, quæ corpori illi Iſoperi-
              <lb/>
            metra eſt: </s>
            <s xml:id="echoid-s14675" xml:space="preserve">altitudo vero æqualis ſemidiametro ſphęræ ABCD; </s>
            <s xml:id="echoid-s14676" xml:space="preserve">erit ſolido EFG-
              <lb/>
            HIKLM, æqualis conus O P Q, per ea, quæ Archimedes lib 1. </s>
            <s xml:id="echoid-s14677" xml:space="preserve">de ſphæra & </s>
            <s xml:id="echoid-s14678" xml:space="preserve">cy-
              <lb/>
            lindro propoſ. </s>
            <s xml:id="echoid-s14679" xml:space="preserve">29. </s>
            <s xml:id="echoid-s14680" xml:space="preserve">demonſtrauit. </s>
            <s xml:id="echoid-s14681" xml:space="preserve">Quamobrem & </s>
            <s xml:id="echoid-s14682" xml:space="preserve">ſphæra N, maiorerit ſolido
              <lb/>
            EFGHIKLM, conicis ſuperficiebus contento. </s>
            <s xml:id="echoid-s14683" xml:space="preserve">Sphæra igitur omnibus cor-
              <lb/>
            poribus ſibi Iſoperimetris, & </s>
            <s xml:id="echoid-s14684" xml:space="preserve">circa alias ſphæras circumſcrip tibilibus, &</s>
            <s xml:id="echoid-s14685" xml:space="preserve">c. </s>
            <s xml:id="echoid-s14686" xml:space="preserve">maior
              <lb/>
            eſt. </s>
            <s xml:id="echoid-s14687" xml:space="preserve">quod demonſtrandum erat.</s>
            <s xml:id="echoid-s14688" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div895" type="section" level="1" n="313">
          <head xml:id="echoid-head340" xml:space="preserve">THEOR. 17. PROPOS. 19.</head>
          <note position="right" xml:space="preserve">Sphæramaior
            <lb/>
          eſt quolibet
            <lb/>
          cono & cy-
            <lb/>
          lindro ſibi Iſo-
            <lb/>
          perimetro.</note>
          <p>
            <s xml:id="echoid-s14689" xml:space="preserve">SPHÆRA quolibet cono, & </s>
            <s xml:id="echoid-s14690" xml:space="preserve">cylindro ſibi Iſoperimetro maior eſt.</s>
            <s xml:id="echoid-s14691" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14692" xml:space="preserve">
              <emph style="sc">Proposita</emph>
            enim quacunque ſphæra, ſi fiat conus baſem habens æqua-
              <lb/>
            lem ſuperficiei ſphærę, id eſt, quadruplam maximi in ſphæra circuli, altitudinem
              <lb/>
            verò ſemidiametro ſphæræ æqualem: </s>
            <s xml:id="echoid-s14693" xml:space="preserve"> erit ſphæra huic cono æqualis; </s>
            <s xml:id="echoid-s14694" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-343-02" xlink:href="note-343-02a" xml:space="preserve">9. quinti.</note>
            rea quod ad conum, cuius baſis eſt maximus in ſphæra circulus, & </s>
            <s xml:id="echoid-s14695" xml:space="preserve">altitudo ſe-
              <lb/>
            midiameter ſphæræ, tam ſphæra, ex propoſ. </s>
            <s xml:id="echoid-s14696" xml:space="preserve">32. </s>
            <s xml:id="echoid-s14697" xml:space="preserve">libri 1. </s>
            <s xml:id="echoid-s14698" xml:space="preserve">Archimedis de ſphæra & </s>
            <s xml:id="echoid-s14699" xml:space="preserve">
              <lb/>
            cylindro, quam prior conus baſem habens quadruplã maximi circuli in
              <note symbol="b" position="right" xlink:label="note-343-03" xlink:href="note-343-03a" xml:space="preserve">11. duodec.</note>
            ra, hoc eſt, ſuperficiei ſphærę æqualem, & </s>
            <s xml:id="echoid-s14700" xml:space="preserve">altitudinem ſemidiametrum ſphæræ,
              <lb/>
            proportionem habet quadruplam. </s>
            <s xml:id="echoid-s14701" xml:space="preserve">Cum ergo ambitus conibaſem habentis ſu-
              <lb/>
            perficiei ſphæræ æqualem maior ſit ambitu ſphæræ, quippe cumille hunc exce-
              <lb/>
            dattota ſuperficie coni, ſecluſa baſi, quæ ambitui ſphæræ ponitur æqualis, li-
              <lb/>
            quido conſtat, ſi fiat conus ſphærę Iſoperimeter, hunc eſſe illo cono, ac proin-
              <lb/>
            de & </s>
            <s xml:id="echoid-s14702" xml:space="preserve">ſphęra minorem.</s>
            <s xml:id="echoid-s14703" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14704" xml:space="preserve">
              <emph style="sc">Rvrsvs</emph>
            ſi fiat cylindrus baſem habens æqualem ſuperficiei ſphęræ, & </s>
            <s xml:id="echoid-s14705" xml:space="preserve">al-
              <lb/>
            titudinem ſemidiametrum ſphærę; </s>
            <s xml:id="echoid-s14706" xml:space="preserve"> erit hic cylindrus triplus illius coni
              <note symbol="c" position="right" xlink:label="note-343-04" xlink:href="note-343-04a" xml:space="preserve">10. duodec.</note>
            habentis æqualem eidem ſuperficiei ſphęræ, & </s>
            <s xml:id="echoid-s14707" xml:space="preserve">altitudinem ſemidiametrum ean-
              <lb/>
            dem ſphærę, quem ſphęræ æqualem eſſe proximè oſtendimus: </s>
            <s xml:id="echoid-s14708" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s14709" xml:space="preserve">tri-
              <lb/>
            plusipſius ſphæræ. </s>
            <s xml:id="echoid-s14710" xml:space="preserve">Tertia ergo pars illius cylindri (cylindrus videlicet eandem
              <lb/>
            habens baſem, altitudinem vero tertiam partem altitudinis cylindri illius: </s>
            <s xml:id="echoid-s14711" xml:space="preserve">
              <note symbol="d" position="right" xlink:label="note-343-05" xlink:href="note-343-05a" xml:space="preserve">14. duode.</note>
            ille cylindrus ſit huius triplus) æqualis erit ſphæræ. </s>
            <s xml:id="echoid-s14712" xml:space="preserve">Cum ergo poſterior hic
              <lb/>
            cylindrus habeat ambitum maiorẽ ambitu ſphęræ, quod ille hunc excedat am-
              <lb/>
            bitu totius cylindri, ſecluſa vna baſe; </s>
            <s xml:id="echoid-s14713" xml:space="preserve">quis non videt, ſi fiat cylindrus ſphęræ I-
              <lb/>
            ſoperimeter, hunc eſſe priore illo cylindro, acproinde & </s>
            <s xml:id="echoid-s14714" xml:space="preserve">ſp hæra maiorẽ? </s>
            <s xml:id="echoid-s14715" xml:space="preserve">Sphę-
              <lb/>
            ra ergo quolibet cono, & </s>
            <s xml:id="echoid-s14716" xml:space="preserve">cylindro ſibi Iſoperimetro maioreſt. </s>
            <s xml:id="echoid-s14717" xml:space="preserve">quod demon-
              <lb/>
            ſtrandum erat.</s>
            <s xml:id="echoid-s14718" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div898" type="section" level="1" n="314">
          <head xml:id="echoid-head341" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s14719" xml:space="preserve">
              <emph style="sc">Hæc</emph>
            omnia ferè ex Theone Alexandrino in commentarijs in Almageſtũ
              <lb/>
            Ptolemaei, & </s>
            <s xml:id="echoid-s14720" xml:space="preserve">ex Pappo Alexandrino in Mathematicis collectionibus, licet ple-
              <lb/>
            raque eorum clarius & </s>
            <s xml:id="echoid-s14721" xml:space="preserve">facilius demonſtrauerimus, excerpta ſunt: </s>
            <s xml:id="echoid-s14722" xml:space="preserve">quæ verò ſe-
              <lb/>
            quntur, à nobis inuenta ſunt, ac demonſtrata.</s>
            <s xml:id="echoid-s14723" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div899" type="section" level="1" n="315">
          <head xml:id="echoid-head342" xml:space="preserve">PROBL. 3. PROPOS. 20.</head>
          <p>
            <s xml:id="echoid-s14724" xml:space="preserve">DATO ſemicirculo vel quadranti, vel octauæ parti circuli, aut </s>
          </p>
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