Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s9426" xml:space="preserve">
              <pb o="210" file="240" n="240" rhead="GEOMETR. PRACT."/>
            tienda erit vtraque figura tuminterior, tum exterior. </s>
            <s xml:id="echoid-s9427" xml:space="preserve">Si namque illa ex hac de-
              <lb/>
              <note position="left" xlink:label="note-240-01" xlink:href="note-240-01a" xml:space="preserve">Solidit{as} vaſis
                <lb/>
              excauati.</note>
            trahetur, reliqua fiet ſoliditas vaſis excauati.</s>
            <s xml:id="echoid-s9428" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div582" type="section" level="1" n="208">
          <head xml:id="echoid-head224" xml:space="preserve">DE AREA QVINQVE COR-
            <lb/>
          porum regularium.</head>
          <head xml:id="echoid-head225" xml:space="preserve">
            <emph style="sc">Capvt</emph>
          IV.</head>
          <p>
            <s xml:id="echoid-s9429" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9430" xml:space="preserve">
              <emph style="sc">QVinqve</emph>
            tantum ſunt corpora regularia, Tetraedrum, Hexaedrum,
              <lb/>
            Octaedrum, Dodecaedrum, & </s>
            <s xml:id="echoid-s9431" xml:space="preserve">Ico ſaedrum, vt in ſcholio propoſ. </s>
            <s xml:id="echoid-s9432" xml:space="preserve">18.
              <lb/>
            </s>
            <s xml:id="echoid-s9433" xml:space="preserve">libr. </s>
            <s xml:id="echoid-s9434" xml:space="preserve">13. </s>
            <s xml:id="echoid-s9435" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s9436" xml:space="preserve">demonſtrauimus: </s>
            <s xml:id="echoid-s9437" xml:space="preserve">quæ ſic ab Euclide libr. </s>
            <s xml:id="echoid-s9438" xml:space="preserve">11. </s>
            <s xml:id="echoid-s9439" xml:space="preserve">defi-
              <lb/>
            niuntur.</s>
            <s xml:id="echoid-s9440" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9441" xml:space="preserve">
              <emph style="sc">Tetraedrvm</emph>
            eſt figura ſolida ſub quatuor triangulis æqualibus, & </s>
            <s xml:id="echoid-s9442" xml:space="preserve">æ-
              <lb/>
            quilateris contenta. </s>
            <s xml:id="echoid-s9443" xml:space="preserve">qualem figuram exprimit pyramis triangularis æquila-
              <lb/>
            tera.</s>
            <s xml:id="echoid-s9444" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9445" xml:space="preserve">
              <emph style="sc">Hexaedrvm</emph>
            eſt figura ſolida ſub ſex quadratis æqualibus contenta. </s>
            <s xml:id="echoid-s9446" xml:space="preserve">qua-
              <lb/>
            lem refert cubus, ſeu parallelepipedum baſium quadratarum, in quo omnes tres
              <lb/>
            dimenſiones ſunt æquales.</s>
            <s xml:id="echoid-s9447" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9448" xml:space="preserve">
              <emph style="sc">Octaedrvm</emph>
            eſt figura ſolida ſub octo triangulis æqualibus, & </s>
            <s xml:id="echoid-s9449" xml:space="preserve">æquilate-
              <lb/>
            ris contenta.</s>
            <s xml:id="echoid-s9450" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9451" xml:space="preserve">
              <emph style="sc">Dodecaedrvm</emph>
            eſt figura ſolida ſub duodecim pentagonis æqualibus,
              <lb/>
            & </s>
            <s xml:id="echoid-s9452" xml:space="preserve">æquilateris, & </s>
            <s xml:id="echoid-s9453" xml:space="preserve">æquiangulis contenta.</s>
            <s xml:id="echoid-s9454" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9455" xml:space="preserve">
              <emph style="sc">Icosaedrvm</emph>
            eſt figura ſolida ſub 20. </s>
            <s xml:id="echoid-s9456" xml:space="preserve">triangulis æqualibus, & </s>
            <s xml:id="echoid-s9457" xml:space="preserve">æquilate-
              <lb/>
            ris contenta.</s>
            <s xml:id="echoid-s9458" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9459" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9460" xml:space="preserve">
              <emph style="sc">Cvbi</emph>
            ſiue Hexaedri aream gigni ex multiplicatione lateris in ſe, & </s>
            <s xml:id="echoid-s9461" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-240-02" xlink:href="note-240-02a" xml:space="preserve">Area cubi, &
                <lb/>
              Tetraedri.</note>
            iterum in productum, cap. </s>
            <s xml:id="echoid-s9462" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9463" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s9464" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9465" xml:space="preserve">docuimus. </s>
            <s xml:id="echoid-s9466" xml:space="preserve">Item pyramidem, ſeu Tetrae-
              <lb/>
            drum produci ex eius altitudine (quæ mechanicè cognoſcetur, vt c. </s>
            <s xml:id="echoid-s9467" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9468" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s9469" xml:space="preserve">2.
              <lb/>
            </s>
            <s xml:id="echoid-s9470" xml:space="preserve">traditum eſt) in tertiam baſis partem: </s>
            <s xml:id="echoid-s9471" xml:space="preserve">vel ex eius baſe in tertiam partem altitu-
              <lb/>
            dinis, declara uimus cap. </s>
            <s xml:id="echoid-s9472" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9473" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s9474" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9475" xml:space="preserve">Quod ſi Geometricè inuenire lubeat altitu-
              <lb/>
              <note position="left" xlink:label="note-240-03" xlink:href="note-240-03a" xml:space="preserve">Altitudo Te-
                <lb/>
              traedri.</note>
            dinem Tetraedri, ita faciemus. </s>
            <s xml:id="echoid-s9476" xml:space="preserve">Quoniam quadratum diametri ſphæræ Tetrae-
              <lb/>
            drum ambientis eſt, vt 2. </s>
            <s xml:id="echoid-s9477" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s9478" xml:space="preserve"> quod diameter ſit potentia ſeſquialtera
              <note symbol="a" position="left" xlink:label="note-240-04" xlink:href="note-240-04a" xml:space="preserve">13 tertiide-
                <lb/>
              cimi.</note>
            pyramidis: </s>
            <s xml:id="echoid-s9479" xml:space="preserve">Sifiat, vt 2. </s>
            <s xml:id="echoid-s9480" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s9481" xml:space="preserve">ita quadratũ lateris Tetraedri ad aliud, pro dibit qua-
              <lb/>
            dratum diametri ſphærę, eiuſque quadrati quadrata radix diametrum ipſam ex-
              <lb/>
              <note symbol="b" position="left" xlink:label="note-240-05" xlink:href="note-240-05a" xml:space="preserve">2. coroll. 13.
                <lb/>
              tertiidec.</note>
            hibebit, cuius duæ tertiæ partes altitudinem Tetraedri offerent.</s>
            <s xml:id="echoid-s9482" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9483" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9484" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            verò Octaedrum diuiditur in duas pyramides ſimiles, &</s>
            <s xml:id="echoid-s9485" xml:space="preserve">
              <note symbol="c" position="left" xlink:label="note-240-06" xlink:href="note-240-06a" xml:space="preserve">2. coroll. 14.
                <lb/>
              tertiidec.</note>
            æquales, quarum baſis communis eſt quadratum à latere deſcriptum: </s>
            <s xml:id="echoid-s9486" xml:space="preserve">ſi vtriuſ-
              <lb/>
            que pyramidis inueſtigetur area, ignorari non poterit area Octaedri, cum ex
              <lb/>
              <note position="left" xlink:label="note-240-07" xlink:href="note-240-07a" xml:space="preserve">Area Octae-
                <lb/>
              dri.</note>
            areis illarum pyramidum conflata ſit. </s>
            <s xml:id="echoid-s9487" xml:space="preserve">Producetur autem area illarum duarum
              <lb/>
            pyramidum, ſi quadratum lateris Octaedri ducatur in diametrum Octaedri, & </s>
            <s xml:id="echoid-s9488" xml:space="preserve">
              <lb/>
            producti numeri tertia pars capiatur. </s>
            <s xml:id="echoid-s9489" xml:space="preserve">quia pro ductus ille numerus ex quadrato
              <lb/>
            lateris Octaedri in eiuſdem diametrum, eſt parallelepipedum duarum illarum
              <lb/>
            pyramidum triplum: </s>
            <s xml:id="echoid-s9490" xml:space="preserve"> propterea quod ſemiſsis illius parallelepipedi
              <note symbol="d" position="left" xlink:label="note-240-08" xlink:href="note-240-08a" xml:space="preserve">coroll. 7.
                <lb/>
              duodec.</note>
            habens baſem, & </s>
            <s xml:id="echoid-s9491" xml:space="preserve">altitudinem, cum vtralibet pyramidum, tripla eſt vnius pyra-
              <lb/>
            midis. </s>
            <s xml:id="echoid-s9492" xml:space="preserve">Diameter porro Octaedri, quę à diametro ſphærę, vel quadrati lateris. </s>
            <s xml:id="echoid-s9493" xml:space="preserve">O-
              <lb/>
              <note position="left" xlink:label="note-240-09" xlink:href="note-240-09a" xml:space="preserve">Diameter
                <lb/>
              Octaedri.</note>
            ctaedri non differt, inuenietur, ſi ex duplo quadrati lateris radix quadrata </s>
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