Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s9564" xml:space="preserve">
              <pb o="212" file="242" n="242" rhead="GEOMETR. PRACT."/>
            hoc pacto. </s>
            <s xml:id="echoid-s9565" xml:space="preserve">Ex ſuperiori plano producto, hoc eſt, ex inferiori ſuperficie alicu-
              <lb/>
            ius plani, quod corporis ſupremæ baſi imponeretur, ad planum baſis oppoſitæ
              <lb/>
            perpendicularis demittatur. </s>
            <s xml:id="echoid-s9566" xml:space="preserve">Hæc enim accuratè dimenſa altitudinem Icoſaedri
              <lb/>
            dabit, eiuſque ſemiſsis altitudinem pyramidis, quæ quęritur. </s>
            <s xml:id="echoid-s9567" xml:space="preserve">Quam Geome-
              <lb/>
            tricè ita etiam explorabimus. </s>
            <s xml:id="echoid-s9568" xml:space="preserve">Fiat pentagonum ex 5. </s>
            <s xml:id="echoid-s9569" xml:space="preserve">lateribus Icoſaedri, inue-
              <lb/>
            ſtigetur que eius ſemidiameter, & </s>
            <s xml:id="echoid-s9570" xml:space="preserve">latus Decagoniin circulo illud pentagonum
              <lb/>
            circumſcribente, in partibus, in quibus latus Icoſaedri datum eſt, hac ſcilicet ra-
              <lb/>
            tione. </s>
            <s xml:id="echoid-s9571" xml:space="preserve">Concipiatur triangulum rectangulum, cuius baſis ſemidiameter dicti cir-
              <lb/>
            culi, latera verò ſemiſsis lateris pentagoni, hoc eſt, Icoſaedri, & </s>
            <s xml:id="echoid-s9572" xml:space="preserve">perpendicularis
              <lb/>
            è centro ad punctum medium dictilateris demiſſa. </s>
            <s xml:id="echoid-s9573" xml:space="preserve">Ita namque cognoſcetur ſe-
              <lb/>
            midiameter, ex iis, quæ lib. </s>
            <s xml:id="echoid-s9574" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9575" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9576" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9577" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s9578" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9579" xml:space="preserve">tradita ſunt. </s>
            <s xml:id="echoid-s9580" xml:space="preserve">Latus verò Decagoni
              <lb/>
            in eodem circulo deſcripti reperietur, vt paulo ante circa finem Num. </s>
            <s xml:id="echoid-s9581" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9582" xml:space="preserve">di-
              <lb/>
            ctum eſt. </s>
            <s xml:id="echoid-s9583" xml:space="preserve"> Et quia diameter ſphærę, ſiue Ico ſaedri potentia eſt
              <note symbol="a" position="left" xlink:label="note-242-01" xlink:href="note-242-01a" xml:space="preserve">1. corol. 16.
                <lb/>
              tertiidec.</note>
            tæ ſemidiametri; </s>
            <s xml:id="echoid-s9584" xml:space="preserve">ſi quadratum ſemidiametri inuentę quintupletur, procreabi-
              <lb/>
            tur quadratum diametri Icoſaedri, cuius radix quadrata diametrum offeret,
              <lb/>
            ideoque & </s>
            <s xml:id="echoid-s9585" xml:space="preserve">ſemidiameter Icoſaedri nota erit. </s>
            <s xml:id="echoid-s9586" xml:space="preserve">Vel aliter. </s>
            <s xml:id="echoid-s9587" xml:space="preserve"> Quoniam
              <note symbol="b" position="left" xlink:label="note-242-02" xlink:href="note-242-02a" xml:space="preserve">2. corol. 16.
                <lb/>
              tertiidec.</note>
            ſphærę, id eſt, Icoſaedri, componitur ex latere Hexagoni, & </s>
            <s xml:id="echoid-s9588" xml:space="preserve">duobus lateribus
              <lb/>
            decagoni in circulo pentagonum ex quinque lateribus Icoſaedri compoſitum
              <lb/>
            circumſcribente: </s>
            <s xml:id="echoid-s9589" xml:space="preserve">erit ſumma collecta ex ſemidiametro illius circuli, & </s>
            <s xml:id="echoid-s9590" xml:space="preserve">duobus
              <lb/>
            lateribus decagoni, diametro Icoſaedriæqualis: </s>
            <s xml:id="echoid-s9591" xml:space="preserve">ideo que rurſus ſemidiameter
              <lb/>
            Icoſaedrinota erit.</s>
            <s xml:id="echoid-s9592" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9593" xml:space="preserve">
              <emph style="sc">Hinc</emph>
            patet, Orontium cum illis, qui ipſum ſequuntur, decipi, qui putat,
              <lb/>
              <note position="left" xlink:label="note-242-03" xlink:href="note-242-03a" xml:space="preserve">Error Oron-
                <lb/>
              tii.</note>
            ex ſemiſſe ſemidiametri illius circuli, & </s>
            <s xml:id="echoid-s9594" xml:space="preserve">ex latere decagoni componi ſemiaxem
              <lb/>
            Icoſaedri, hoceſt, axem, vel altitudinem pyramidis, cuius baſis triangulum Ico-
              <lb/>
            ſaedri, & </s>
            <s xml:id="echoid-s9595" xml:space="preserve">vertex centrum ſp hæræ. </s>
            <s xml:id="echoid-s9596" xml:space="preserve">Nam vt ex iis conſtat, quę proximè ſcripſi-
              <lb/>
            mus, eo modo componitur ſemidiameter ſphærę, vel Icoſaedri, quæ maior
              <lb/>
            eſt prędicto axe. </s>
            <s xml:id="echoid-s9597" xml:space="preserve">Semidiameter porro circuli prædictum pentagonum circum-
              <lb/>
            ſcribentis reperiri quo que poterit, vt ad finem Num. </s>
            <s xml:id="echoid-s9598" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9599" xml:space="preserve">diximus, ſi nimirum
              <lb/>
            quadratum lateris decagoni ex quadrato lateris dicti pentagoni, quod à late-
              <lb/>
            re Icoſaedrinon differt, tollatur, & </s>
            <s xml:id="echoid-s9600" xml:space="preserve">reliqui numeri radix quadrata extrahatur:
              <lb/>
            </s>
            <s xml:id="echoid-s9601" xml:space="preserve"> propterea quodlatus pentagoni poteſt latera decagoni, & </s>
            <s xml:id="echoid-s9602" xml:space="preserve">hexagoni
              <note symbol="c" position="left" xlink:label="note-242-04" xlink:href="note-242-04a" xml:space="preserve">10. tertiidec.</note>
            circuli.</s>
            <s xml:id="echoid-s9603" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9604" xml:space="preserve">
              <emph style="sc">Iam</emph>
            verò cognita ſemidiametro Icoſaedri, inueniemus altitudinem pyra-
              <lb/>
              <note position="left" xlink:label="note-242-05" xlink:href="note-242-05a" xml:space="preserve">Perpendicu-
                <lb/>
              laris è centro
                <lb/>
              ſphæræ ad ba-
                <lb/>
              ſem Icoſaedri.</note>
            midis, cuius baſis eſt triangulum Icoſaedri, & </s>
            <s xml:id="echoid-s9605" xml:space="preserve">vertex eiuſdem centrum, hoc mo-
              <lb/>
            do. </s>
            <s xml:id="echoid-s9606" xml:space="preserve">Quoniam diameter Icoſaedri, eiuſdemque altitudo ſeſein centro ſecant bi-
              <lb/>
            fariam, concipiatur triangulum rectangulum, cuius baſis eſt diameter Icoſaedri
              <lb/>
            proximè cognita, latera verò circa angulum rectum, altitudo pyramidis, & </s>
            <s xml:id="echoid-s9607" xml:space="preserve">ſe-
              <lb/>
            midiameter circuli baſem Icoſaedri circumſcribentis. </s>
            <s xml:id="echoid-s9608" xml:space="preserve">Cum ergo hęc ſemidia-
              <lb/>
            meter cognoſci poſsit ex iis, quæ lib. </s>
            <s xml:id="echoid-s9609" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9610" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9611" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9612" xml:space="preserve">docuimus, cognoſcetur
              <note symbol="d" position="left" xlink:label="note-242-06" xlink:href="note-242-06a" xml:space="preserve">3. triang. re-
                <lb/>
              ctil.</note>
            latus reliquum pyramidis, videlicet altitudo, quæ inquiritur. </s>
            <s xml:id="echoid-s9613" xml:space="preserve">Semidiameter
              <lb/>
            porro circuli baſem triangularem Icoſaedri circumſcribentis effi cietur hoc et-
              <lb/>
            iam pacto cognita. </s>
            <s xml:id="echoid-s9614" xml:space="preserve"> Quoniam trianguli æquilaterilatus potentia triplum
              <note symbol="e" position="left" xlink:label="note-242-07" xlink:href="note-242-07a" xml:space="preserve">12. tertiidec.
                <lb/>
              Semidiame-
                <lb/>
              ter circuli tri-
                <lb/>
              angulum Ico-
                <lb/>
              ſaedricircum
                <lb/>
              ſcribentis.</note>
            ſemidiametri illius circuli; </s>
            <s xml:id="echoid-s9615" xml:space="preserve">ſi quadratum lateris Icoſaedri diuidatur per 3. </s>
            <s xml:id="echoid-s9616" xml:space="preserve">erit
              <lb/>
            Quotientis radix quadrata ſemidiameter quæſita.</s>
            <s xml:id="echoid-s9617" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9618" xml:space="preserve">6. </s>
            <s xml:id="echoid-s9619" xml:space="preserve">
              <emph style="sc">Eadem</emph>
            hacarte, quæ in Dodecaedro, & </s>
            <s xml:id="echoid-s9620" xml:space="preserve">Icoſaedro expoſita eſt, areas
              <lb/>
            Tetraedri, cubi, & </s>
            <s xml:id="echoid-s9621" xml:space="preserve">Octaedriinueſtigare licebit, ſi, lineis ex eorum centris ad o-
              <lb/>
            mnes angulos ductis, in pyramides æquales diſtribuantur. </s>
            <s xml:id="echoid-s9622" xml:space="preserve">Tetraedrum </s>
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