Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s9719" xml:space="preserve">
              <pb o="215" file="245" n="245" rhead="LIBER QVINTVS."/>
            adſit, ſed ſolum eius latus datum ſitac cognitum. </s>
            <s xml:id="echoid-s9720" xml:space="preserve">Sit ergo primo datum latus
              <lb/>
            Tetraedri A B, quotcunque palmorum,
              <lb/>
              <figure xlink:label="fig-245-01" xlink:href="fig-245-01a" number="157">
                <image file="245-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/245-01"/>
              </figure>
            conſtruaturque triangulum æquilaterum
              <lb/>
            A B C, pro baſe Tetraedri: </s>
            <s xml:id="echoid-s9721" xml:space="preserve">Diuiſo autem
              <lb/>
            latere A B, bifariam in D, iungatur recta
              <lb/>
              <note symbol="a" position="right" xlink:label="note-245-01" xlink:href="note-245-01a" xml:space="preserve">ſchol. 26.
                <lb/>
              primi.</note>
            C D, quæ ad AB, perpendicularis erit.</s>
            <s xml:id="echoid-s9722" xml:space="preserve"> Conſtructo quo que Iſoſcele ABE, cuius
              <lb/>
            vtrumque latus rectæ CD, æqualeſit, de-
              <lb/>
            mittatur ad AE, perpendicularis BF, cuius
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            quarta pars ſit F G. </s>
            <s xml:id="echoid-s9723" xml:space="preserve">Dico FG, altitudinem
              <lb/>
              <note position="right" xlink:label="note-245-02" xlink:href="note-245-02a" xml:space="preserve">Altitudo py-
                <lb/>
              ramidis Te-
                <lb/>
              traedri.</note>
            eſſe vnius pyramidis, hoc eſt, æqualem eſ-
              <lb/>
            ſe perpendiculari ex centro ſphæræ Tetra-
              <lb/>
            edro circumſcriptæ ad vnam baſem deductæ. </s>
            <s xml:id="echoid-s9724" xml:space="preserve">Quoniam enim, vt ad finem Eucli-
              <lb/>
            dis ex Hypſicle demonſtrauimus, E, angulus eſt inclinationis vnius baſis Tetra-
              <lb/>
            edriad alteram, eſt que EB, perpendiculari CD, æqualis: </s>
            <s xml:id="echoid-s9725" xml:space="preserve">ſi triangulum B E F,
              <lb/>
            concipiatur circa EF, moueri, donec rectum ſit ad baſem Tetraedri, cadet pun-
              <lb/>
            ctum B, in verticem Tetraedri; </s>
            <s xml:id="echoid-s9726" xml:space="preserve">ac proinde perpendicularis BF, altitudo erit Te-
              <lb/>
            traedri. </s>
            <s xml:id="echoid-s9727" xml:space="preserve"> Et quia altitudo Tetraedri duas partes tertias diamet@i ſphæræ
              <note symbol="b" position="right" xlink:label="note-245-03" xlink:href="note-245-03a" xml:space="preserve">2. corol. 13.
                <lb/>
              tertijdec.</note>
            net: </s>
            <s xml:id="echoid-s9728" xml:space="preserve">ſi ſemidiameter ponatur 6. </s>
            <s xml:id="echoid-s9729" xml:space="preserve">erit altitudo B F, 4. </s>
            <s xml:id="echoid-s9730" xml:space="preserve">& </s>
            <s xml:id="echoid-s9731" xml:space="preserve">ſemidiameter 3. </s>
            <s xml:id="echoid-s9732" xml:space="preserve"> Cum ergo altitudo vnius pyramidis ſit tertia pars ſemidiametri, erit BG, ſemidiame-
              <lb/>
              <note symbol="c" position="right" xlink:label="note-245-04" xlink:href="note-245-04a" xml:space="preserve">2. corol. 13.
                <lb/>
              tertijdec.</note>
            ter, & </s>
            <s xml:id="echoid-s9733" xml:space="preserve">G F, altitudo vnius pyramidis. </s>
            <s xml:id="echoid-s9734" xml:space="preserve">Quam etiam inueniemus, licet Iſoſceles
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            AEB, non extruatur, hoc modo. </s>
            <s xml:id="echoid-s9735" xml:space="preserve">Sumpta dH, tertia parte perpendicularis CD,
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            exciteturad CD, perpendicularis HK, quæ ex D, adinteruallum CD, ſecetur in
              <lb/>
            K. </s>
            <s xml:id="echoid-s9736" xml:space="preserve">Dico HI, quartam partem ipſius HK, eſſe altitudinem vnius pyramidis Ere-
              <lb/>
            cto enim triangulo DHK, ſupra baſem Tetraedri ABC, cadet punctũ K, in ver-
              <lb/>
            ticem Tetraedri, quod D K, ducta æqualis ſit perpendiculari ex medio latere ad
              <lb/>
            angulum baſis oppoſitum ductæ. </s>
            <s xml:id="echoid-s9737" xml:space="preserve">Ergo vt prius, HK, altitudo erit Tetraedri, & </s>
            <s xml:id="echoid-s9738" xml:space="preserve">
              <lb/>
              <note symbol="d" position="right" xlink:label="note-245-05" xlink:href="note-245-05a" xml:space="preserve">2. corol. 13.
                <lb/>
              tertijdec.</note>
            HI, perpendicularis ex centro ſphæræ in H, centrum baſis cadens. </s>
            <s xml:id="echoid-s9739" xml:space="preserve"> Nam D H, tertia pars perpendicularis CD, in centrum trianguli cadit.</s>
            <s xml:id="echoid-s9740" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9741" xml:space="preserve">
              <emph style="sc">Sit</emph>
            deinde datum latus Octaedri L M, ſupra quod conſtruatur triangulum
              <lb/>
            æquilaterum L M N, pro baſe Octaedri. </s>
            <s xml:id="echoid-s9742" xml:space="preserve">Diuiſo autem latere L M, bifariam in
              <lb/>
            O, iungaturrecta N O, quæ ad L M, erit perpendicularis. </s>
            <s xml:id="echoid-s9743" xml:space="preserve">Conſtructio
              <note symbol="e" position="right" xlink:label="note-245-06" xlink:href="note-245-06a" xml:space="preserve">ſchol. 26.
                <lb/>
              primi.</note>
            Iſoſcele QRS, ſupra baſem QR, æqualem diametro ſphæræ, vel quadrati ex la-
              <lb/>
            tere Octaedri deſcripti, (quæ habebitur, ſi educatur perpendicularis MP, lateri
              <lb/>
            L M, æqualis. </s>
            <s xml:id="echoid-s9744" xml:space="preserve">Iuncta enim recta L P, diameter erit illius quadrati, vel ſphæræ.)
              <lb/>
            </s>
            <s xml:id="echoid-s9745" xml:space="preserve">vtrum que laterum QS, RS, æquale habens perpendiculari N O; </s>
            <s xml:id="echoid-s9746" xml:space="preserve">ducatur ex R,
              <lb/>
            ad QS, perpendicularis RT, quæbifariam ſecetur in V. </s>
            <s xml:id="echoid-s9747" xml:space="preserve">Dico T V, eſſe altitudi-
              <lb/>
              <note position="right" xlink:label="note-245-07" xlink:href="note-245-07a" xml:space="preserve">Altitudo py-
                <lb/>
              ramidis Octa-
                <lb/>
              edri.</note>
            nem pyramidis quæſitam, hoc eſt, æqualem eſſe perpendiculari ex centro ſphę-
              <lb/>
            ræ ad vnam baſem Octaedri cadenti. </s>
            <s xml:id="echoid-s9748" xml:space="preserve">Quoniam enim, vt ad finem Euclidis ex
              <lb/>
            Hypſicle demonſtrauimus, augulus QSR, in clinationem vnius baſis ad alteram
              <lb/>
            indicat, eſt que obtuſus, erit perpendicularis R T, cadens ad partes anguli acuti
              <lb/>
            R S T, æqualis altitudini Octaed@i, id eſt, perpendicularibaſium Octaedri oppo-
              <lb/>
            ſitarum centra connectenti, vt ex Octaedro materiali perſpicuum eſt: </s>
            <s xml:id="echoid-s9749" xml:space="preserve">Ac pro-
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            pterea eius ſemiſsis T V, altitudo erit pyramidis q̃ſita, quod altitudo Octaedri
              <lb/>
            bifariam ſecetur in centro.</s>
            <s xml:id="echoid-s9750" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9751" xml:space="preserve">
              <emph style="sc">Si</emph>
            deturlatus cubi, ſiue hexaedri, erit eius ſemiſsis altitudo pyramidis quæ-
              <lb/>
              <note position="right" xlink:label="note-245-08" xlink:href="note-245-08a" xml:space="preserve">Altitudo py-
                <lb/>
              ramidis cubi.</note>
            fita: </s>
            <s xml:id="echoid-s9752" xml:space="preserve">propterea quod cubialtitudo eiuſdem lateriſit æqualis.</s>
            <s xml:id="echoid-s9753" xml:space="preserve"/>
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