Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div885" type="section" level="1" n="309">
          <p>
            <s xml:id="echoid-s14558" xml:space="preserve">
              <pb o="310" file="340" n="340" rhead="GEOMETR. PRACT."/>
            ra erit, quæ vel ęqualis ſit ſolido E, atque adeo minor quam ſphęra ABC, vel
              <lb/>
            minor quidem quàm ſphæra ABC, maior verò quam magnitudo E, quæ minor
              <lb/>
            ponitur, quam ſphęra ABC. </s>
            <s xml:id="echoid-s14559" xml:space="preserve"> Deſcribatur deinde intra ſphæram ABC,
              <note symbol="a" position="left" xlink:label="note-340-01" xlink:href="note-340-01a" xml:space="preserve">17. duodec.</note>
            quod minimè tangat ſphæram L M N; </s>
            <s xml:id="echoid-s14560" xml:space="preserve">ita vt vnaquæque perpendicularium ex
              <lb/>
            centro D, ad baſes huius corporis inſcripti cadentium minor ſit ſemidiametro
              <lb/>
            AD. </s>
            <s xml:id="echoid-s14561" xml:space="preserve">Siigitur à centro D, ad omnes eius angulos lineæ extendantur, vt totum
              <lb/>
            corpus in pyramides reſoluatur, quarũ baſes ſunt eædem, quæ corporis ABC,
              <lb/>
            vertex autem communis centrum D; </s>
            <s xml:id="echoid-s14562" xml:space="preserve"> erit quęlibet pyramis ęqualis ſolido
              <note symbol="b" position="left" xlink:label="note-340-02" xlink:href="note-340-02a" xml:space="preserve">14. hui{us}.</note>
            ctangulo contento ſub eius perpendiculari, & </s>
            <s xml:id="echoid-s14563" xml:space="preserve">tertia parte baſis; </s>
            <s xml:id="echoid-s14564" xml:space="preserve">Et ideo ſoli-
              <lb/>
            dum rectangulum contentum ſub ſemidiametro A D, & </s>
            <s xml:id="echoid-s14565" xml:space="preserve">tertia parte baſis cu-
              <lb/>
            iuſuis pyramidis, maius erit pyramide ipſa. </s>
            <s xml:id="echoid-s14566" xml:space="preserve">Et quoniam omnia ſolida rectan-
              <lb/>
            gula contenta ſunt ſingulis perpendicularibus ex centro D, ad baſes corporis
              <lb/>
            dicti protractis, & </s>
            <s xml:id="echoid-s14567" xml:space="preserve">ſingulis tertijs partibus baſium, ſimul æqualia ſunt toti cor-
              <lb/>
            pori; </s>
            <s xml:id="echoid-s14568" xml:space="preserve">efficiunt autem omnes tertiæ partes baſium ſimul tertiam partem ambi-
              <lb/>
            tus corporis, erit ſolidum rectangulum contentum ſub ſemidiametro A D, & </s>
            <s xml:id="echoid-s14569" xml:space="preserve">
              <lb/>
            tertia parte ambitus dicti corporis ſphęræ ABC, inſcripti, maius corpore inſcri-
              <lb/>
            pto. </s>
            <s xml:id="echoid-s14570" xml:space="preserve">Cum igitur ambitus ſphęræ A B C, maior ſit ambitu corporis ſibi inſcri-
              <lb/>
            pti, atque adeo & </s>
            <s xml:id="echoid-s14571" xml:space="preserve">tertia pars ambitus ſphęræ maior tertia parte ambitus dicti
              <lb/>
            corporis; </s>
            <s xml:id="echoid-s14572" xml:space="preserve">erit ſolidum rectangulum contentum ſub AD, ſemidiametro, & </s>
            <s xml:id="echoid-s14573" xml:space="preserve">ter-
              <lb/>
            tia parte ambitus ſphęræ ABC, hoc eſt, ſolidum E, multo maius corpore inſcri-
              <lb/>
            pto intra ſphęram ABC: </s>
            <s xml:id="echoid-s14574" xml:space="preserve">Ponebatur autem ſphęra L M N, vel ęqualis ſolido
              <lb/>
            E, vel maior. </s>
            <s xml:id="echoid-s14575" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s14576" xml:space="preserve">ſphęra L M N, maior erit corpore intra ſphęram A B C,
              <lb/>
            deſcripto, pars toto, quod eſt abſurdum. </s>
            <s xml:id="echoid-s14577" xml:space="preserve">Non igitur ſolidum E, m@nu
              <unsure/>
            s erit ſphę-
              <lb/>
            ra ABC. </s>
            <s xml:id="echoid-s14578" xml:space="preserve">Cum ergo neque maius ſit oſtenſum, æquale omnino erit: </s>
            <s xml:id="echoid-s14579" xml:space="preserve">Ac propte-
              <lb/>
            rea area cuiuslibet ſphęræ ęqualis eſt ſolido rectangulo comprehenſo ſub ſemi-
              <lb/>
            diametro ſphęræ, & </s>
            <s xml:id="echoid-s14580" xml:space="preserve">tertia parte ambitus ſphęræ. </s>
            <s xml:id="echoid-s14581" xml:space="preserve">quod demonſtrandum erat.</s>
            <s xml:id="echoid-s14582" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div889" type="section" level="1" n="310">
          <head xml:id="echoid-head337" xml:space="preserve">THEOR. 15. PROPOS. 17.</head>
          <note position="left" xml:space="preserve">Sphæra ma-
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          ior eſt omni-
            <lb/>
          b{us} corpori-
            <lb/>
          b{us} ſibi Iſope-
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          rimetris, &
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          circa ali{as}
            <lb/>
          ſphær{as} cir-
            <lb/>
          cumſcriptibi-
            <lb/>
          lib{us}, quæ pla
            <lb/>
          nis ſuperficie-
            <lb/>
          b{us} continen-
            <lb/>
          tur.</note>
          <p>
            <s xml:id="echoid-s14583" xml:space="preserve">SPHÆRA omnibus corporibus ſibi Iſoperimetris, quæ planis
              <lb/>
            ſuperficiebus contineantur, circaque alias ſphæras circumſcriptibilia
              <lb/>
            ſint, hoc eſt, quorum omnes perpendiculares ad baſes productæ ab
              <lb/>
            aliquo puncto medio ſint æquales, maior eſt.</s>
            <s xml:id="echoid-s14584" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14585" xml:space="preserve">
              <emph style="sc">Esto</emph>
            ſphæra A, cuius centrum A, & </s>
            <s xml:id="echoid-s14586" xml:space="preserve">ſemidiameter AB: </s>
            <s xml:id="echoid-s14587" xml:space="preserve">Solidum autem cir-
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            ca aliquam ſphæram circumſcriptibile ſibi Iſoperimetrum C, cuius vna perpen-
              <lb/>
            dicularium C D. </s>
            <s xml:id="echoid-s14588" xml:space="preserve">Dico ſphæram A, maior
              <unsure/>
            em eſſe ſolido C. </s>
            <s xml:id="echoid-s14589" xml:space="preserve">Intelligatur enim
              <lb/>
            circa ſphęram A, corpus deſcriptum ſimile prorſus ſolido C, ita vt ſingula quo-
              <lb/>
            que latera contingant ſphęram A, hoc eſt, eius perpendiculares, quarum vna
              <lb/>
            ſit AB, ſint quo que æquales, nempe ſemidiametri ſphęræ A, exiſtentes. </s>
            <s xml:id="echoid-s14590" xml:space="preserve">Ita que
              <lb/>
            quoniam ambitus corporis circa ſphęram A, maior eſt ambitu ſphęræ A, (per ea,
              <lb/>
            quę ab Archimede ſunt demonſtrata lib 1. </s>
            <s xml:id="echoid-s14591" xml:space="preserve">de ſphæra & </s>
            <s xml:id="echoid-s14592" xml:space="preserve">cylindro, propoſ. </s>
            <s xml:id="echoid-s14593" xml:space="preserve">27.)
              <lb/>
            </s>
            <s xml:id="echoid-s14594" xml:space="preserve">erit quoque eiuſdem corporis ambitus maior ambitu corpori C. </s>
            <s xml:id="echoid-s14595" xml:space="preserve">Quare per-
              <lb/>
            pendicularis AB, hoc eſt ſemidiameter ſphærę A, maior erit perpendiculari CD.</s>
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