Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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        <div xml:id="echoid-div31" type="section" level="1" n="16">
          <pb o="326" file="0028" n="29" rhead="THEOR. DE QUADRAT."/>
        </div>
        <div xml:id="echoid-div36" type="section" level="1" n="17">
          <head xml:id="echoid-head29" xml:space="preserve">
            <emph style="sc">Theorema</emph>
          VIII.</head>
          <p style="it">
            <s xml:id="echoid-s354" xml:space="preserve">IN ſemicirculo & </s>
            <s xml:id="echoid-s355" xml:space="preserve">quolibet circuli ſectore, habet
              <lb/>
            arcus ad duas tertias rectæ ſibi ſubtenſæ hanc ra-
              <lb/>
            tionem, quam ſemidiameter ad eam, quæ ex centro
              <lb/>
            ducitur ad ſectoris centrum gravitatis.</s>
            <s xml:id="echoid-s356" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s357" xml:space="preserve">Eſto primùm ſemicirculus A B C, deſcriptus centro D,
              <lb/>
              <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">TAB. XXXVI.
                <lb/>
              Fig. 3.</note>
            ſectuſque bifariam rectâ B D, in qua centrum gravitatis
              <lb/>
            ſemicirculi ſit E . </s>
            <s xml:id="echoid-s358" xml:space="preserve">Dico arcum A B C eſſe ad duas
              <note symbol="1" position="left" xlink:label="note-0028-02" xlink:href="note-0028-02a" xml:space="preserve">Theor. 4. h.</note>
            A C, ſicut B D ad D E. </s>
            <s xml:id="echoid-s359" xml:space="preserve">Jungantur enim A B, B C. </s>
            <s xml:id="echoid-s360" xml:space="preserve">Igi-
              <lb/>
            tur, ut ſemicirculus ad triangulum A B C, ſic ſunt duæ ter-
              <lb/>
            tiæ B D ad D E , eſt enim B D æqualis diametro
              <note symbol="2" position="left" xlink:label="note-0028-03" xlink:href="note-0028-03a" xml:space="preserve">Theor. 7. h.</note>
            nis reliquæ. </s>
            <s xml:id="echoid-s361" xml:space="preserve">Verùm etiam ut ſemicirculus, id eſt, ut trian-
              <lb/>
            gulus habens baſin æqualem arcui A B C & </s>
            <s xml:id="echoid-s362" xml:space="preserve">altitudinem B D,
              <lb/>
            ad A B C triangulum, ita eſt arcus A B C ad A C re-
              <lb/>
            ctam; </s>
            <s xml:id="echoid-s363" xml:space="preserve">ergo ut arcus A B C ad A C, ita ſunt duæ tertiæ
              <lb/>
            B D ad D E, & </s>
            <s xml:id="echoid-s364" xml:space="preserve">permutando, ut arcus A B C ad duas tertias
              <lb/>
            B D, ita A C ad D E, ſive ita {2/3} A C ad {2/3} D E, unde rur-
              <lb/>
            ſus permutando, ut arcus A B C ad {2/3} A C, ita {2/3} B D ad {2/3}
              <lb/>
            D E, ſive ita, B D ad D E.</s>
            <s xml:id="echoid-s365" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">TAB. XXXVI.
            <lb/>
          Fig. 4.</note>
          <p>
            <s xml:id="echoid-s366" xml:space="preserve">Sit deinde ſector D A B C, ſemicirculo minor, bifariam
              <lb/>
            ſectus rectâ D B, in qua ſectoris centrum gravitatis ponatur
              <lb/>
            E punctum, & </s>
            <s xml:id="echoid-s367" xml:space="preserve">ducatur ſubtenſa A C. </s>
            <s xml:id="echoid-s368" xml:space="preserve">Dico rurſus, arcum
              <lb/>
            A B C ad duas tertias rectæ A C eam habere rationem,
              <lb/>
            quam B D ad D E. </s>
            <s xml:id="echoid-s369" xml:space="preserve">Jungantur enim A B, B C, & </s>
            <s xml:id="echoid-s370" xml:space="preserve">totius
              <lb/>
            circuli ſit diameter K D B, quæ producatur in Q, ut fiat
              <lb/>
            Q F, ad B F, ſicut portio A C B ad A B C triangulum,
              <lb/>
            & </s>
            <s xml:id="echoid-s371" xml:space="preserve">jungantur A Q, Q C; </s>
            <s xml:id="echoid-s372" xml:space="preserve">erit jam triangulus A Q C portio-
              <lb/>
            ni A C B æqualis. </s>
            <s xml:id="echoid-s373" xml:space="preserve">Ponantur deinde centra gravitatis, G tri-
              <lb/>
            anguli A C D, & </s>
            <s xml:id="echoid-s374" xml:space="preserve">H portionis A C B; </s>
            <s xml:id="echoid-s375" xml:space="preserve">& </s>
            <s xml:id="echoid-s376" xml:space="preserve">ſicut D Q ad
              <lb/>
            Q F, ita ſit H D ad D R.</s>
            <s xml:id="echoid-s377" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s378" xml:space="preserve">Quia igitur ſicut portio A C B ſive triangulus A Q C ad
              <lb/>
            triangulum A B C, id eſt, ut Q F ad B F, ita {2/3} K F ad
              <lb/>
            D H , erit rectangulum ſub Q F, D H, æquale
              <note symbol="3" position="left" xlink:label="note-0028-05" xlink:href="note-0028-05a" xml:space="preserve">Theor. 7. h.</note>
            </s>
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