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

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          <pb o="500" file="0222" n="233" rhead="CHRIST. HUGENII"/>
          <p>
            <s xml:id="echoid-s4785" xml:space="preserve">In curvâ propoſita cujus æquatio x
              <emph style="super">3</emph>
            + y
              <emph style="super">3</emph>
            - axy = 0,
              <lb/>
            fiet ſecundum hanc regulam dividenda quantitas 3y
              <emph style="super">3</emph>
            - axy,
              <lb/>
            diviſor verò 3xx - ay; </s>
            <s xml:id="echoid-s4786" xml:space="preserve">ideoque z = {3y
              <emph style="super">3</emph>
            - axy/3xx - ay}, quæ
              <lb/>
            eſt longitudo cognita, cum dentur x, y & </s>
            <s xml:id="echoid-s4787" xml:space="preserve">a.</s>
            <s xml:id="echoid-s4788" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4789" xml:space="preserve">Eſto item alia curva A B H, cujus æquatio axx - x
              <emph style="super">3</emph>
            -
              <lb/>
              <note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve">TAB. XLV.
                <lb/>
              fig. 3.</note>
            qqy = 0, poſito ſcilicet a & </s>
            <s xml:id="echoid-s4790" xml:space="preserve">q eſſe lineas datas, A F vero
              <lb/>
            = x, F B = y. </s>
            <s xml:id="echoid-s4791" xml:space="preserve">Sit B E tangens, & </s>
            <s xml:id="echoid-s4792" xml:space="preserve">F E dicatur ut ante,
              <lb/>
            z. </s>
            <s xml:id="echoid-s4793" xml:space="preserve">Hîc fiet ſecundum regulam, dividenda quantitas - qqy;
              <lb/>
            </s>
            <s xml:id="echoid-s4794" xml:space="preserve">diviſor autem 2ax - 3xx; </s>
            <s xml:id="echoid-s4795" xml:space="preserve">unde z = {- qqy/2ax - 3xx} Ubi cum
              <lb/>
            dividenda quantitas ſit negata, ſi fuerit etiam diviſor minor
              <lb/>
            nihilo, hoc eſt ſi 2a minor quam 3x, erit z ſive fe ſumen-
              <lb/>
            da in partem ab A averſam. </s>
            <s xml:id="echoid-s4796" xml:space="preserve">Si vero 2a major quam 3x, ſu-
              <lb/>
            menda erit F E verſus A, ex præcepto regulæ.</s>
            <s xml:id="echoid-s4797" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4798" xml:space="preserve">Horum vero rationem, ipſiuſque regulæ & </s>
            <s xml:id="echoid-s4799" xml:space="preserve">compendii quò
              <lb/>
              <note position="left" xlink:label="note-0222-02" xlink:href="note-0222-02a" xml:space="preserve">TAB. XLV.
                <lb/>
              fig. 4.</note>
            reducta eſt, originem ut explicemus, proponatur ut ante
              <lb/>
            curva B C, ad cujus punctum B tangens ducenda ſit.</s>
            <s xml:id="echoid-s4800" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4801" xml:space="preserve">Intelligatur primum recta E B D, quæ non tangat curvam
              <lb/>
            ſed eam ſecet in B, atque item in alio puncto D, ipſi B
              <lb/>
            proximo; </s>
            <s xml:id="echoid-s4802" xml:space="preserve">rectæ autem A G occurrat in E; </s>
            <s xml:id="echoid-s4803" xml:space="preserve">& </s>
            <s xml:id="echoid-s4804" xml:space="preserve">ab utriſque
              <lb/>
            punctis B, D ducantur ad rectam A G, iiſdem angulis in-
              <lb/>
            clinatæ B F, D G; </s>
            <s xml:id="echoid-s4805" xml:space="preserve">& </s>
            <s xml:id="echoid-s4806" xml:space="preserve">ſit A F = x, F B = y, ſicut antea;
              <lb/>
            </s>
            <s xml:id="echoid-s4807" xml:space="preserve">ponatur que etiam F G data eſſe, quæ ſit e, quæraturque F E
              <lb/>
            = z.</s>
            <s xml:id="echoid-s4808" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4809" xml:space="preserve">Eſt itaque ſicut E F ad F B, hoc eſt, ſicut z ad y, ita E G,
              <lb/>
            hoc eſt, z + e ad G D; </s>
            <s xml:id="echoid-s4810" xml:space="preserve">quæ erit y + {ey/z}; </s>
            <s xml:id="echoid-s4811" xml:space="preserve">& </s>
            <s xml:id="echoid-s4812" xml:space="preserve">hoc quidem
              <lb/>
            in qualibet curva ita ſe habere manifeſtum eſt.</s>
            <s xml:id="echoid-s4813" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4814" xml:space="preserve">Nunc porrò conſideretur æquatio naturam curvæ conti-
              <lb/>
            nens, ex. </s>
            <s xml:id="echoid-s4815" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s4816" xml:space="preserve">illa ſuperius propoſita x
              <emph style="super">3</emph>
            + y
              <emph style="super">3</emph>
            - xya = 0,
              <lb/>
            ubi a rectam longitudine datam, velut A H ſignificabat; </s>
            <s xml:id="echoid-s4817" xml:space="preserve">& </s>
            <s xml:id="echoid-s4818" xml:space="preserve">
              <lb/>
            patet, cum punctum D in curva ponatur, debere eodem mo-
              <lb/>
            do duas A G, G D, hoc eſt x + e & </s>
            <s xml:id="echoid-s4819" xml:space="preserve">y + {ey/z} ad ſe mutuo </s>
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