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

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          <pb o="435" file="0153" n="162" rhead="ET HYPERBOLÆ QUADRATURA."/>
        </div>
        <div xml:id="echoid-div174" type="section" level="1" n="82">
          <head xml:id="echoid-head118" xml:space="preserve">PROP. XIV. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3342" xml:space="preserve">Sint duo polygona complicata A, B, nem-
              <lb/>
              <note position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve">
                <lb/>
              A # B
                <lb/>
              C # D
                <lb/>
              E # F
                <lb/>
              </note>
            pe A intra circuli vel ellipſeos ſectorem & </s>
            <s xml:id="echoid-s3343" xml:space="preserve">
              <lb/>
            B extra: </s>
            <s xml:id="echoid-s3344" xml:space="preserve">continuetur ſeries convergens horum
              <lb/>
            polygonorum complicatorum ſecundum no-
              <lb/>
            ſtram methodum ſubduplam deſcriptorum, ita
              <lb/>
            ut polygona intra circulum ſint A, C, E, &</s>
            <s xml:id="echoid-s3345" xml:space="preserve">c, & </s>
            <s xml:id="echoid-s3346" xml:space="preserve">extra cir-
              <lb/>
            culum B, D, F, &</s>
            <s xml:id="echoid-s3347" xml:space="preserve">c; </s>
            <s xml:id="echoid-s3348" xml:space="preserve">dico A + E minorem eſſe quam 2 C:
              <lb/>
            </s>
            <s xml:id="echoid-s3349" xml:space="preserve">ex prædictis manifeſtæ ſunt ſequentes analogiæ; </s>
            <s xml:id="echoid-s3350" xml:space="preserve">prima quo-
              <lb/>
            niam A, C, B, ſunt continue pro-
              <lb/>
              <note position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve">
                <lb/>
              C - A:B - C::A:C
                <lb/>
              B - C:D - C::A + C:A
                <lb/>
              </note>
            portionales; </s>
            <s xml:id="echoid-s3351" xml:space="preserve">& </s>
            <s xml:id="echoid-s3352" xml:space="preserve">ſecunda quoniam
              <lb/>
            C, D, B, ſunt harmonice pro-
              <lb/>
            portionales: </s>
            <s xml:id="echoid-s3353" xml:space="preserve">& </s>
            <s xml:id="echoid-s3354" xml:space="preserve">proinde exceſſus
              <lb/>
            C ſupra A, hoc eſt C — A, eſt ad exceſſum D ſupra C ſeu
              <lb/>
            D - C in ratione compoſita ex proportione A ad C & </s>
            <s xml:id="echoid-s3355" xml:space="preserve">ex
              <lb/>
            proportione A + C ad A, hoc eſt in ratione A + C ad C;
              <lb/>
            </s>
            <s xml:id="echoid-s3356" xml:space="preserve">at A + C eſt major quam C, & </s>
            <s xml:id="echoid-s3357" xml:space="preserve">ideo exceſſus C ſupra A eſt
              <lb/>
            major quam exceſſus D ſupra C, eſt autem D major quam
              <lb/>
            E, & </s>
            <s xml:id="echoid-s3358" xml:space="preserve">proinde exceſſus C ſupra A multò major eſt quam
              <lb/>
            exceſſus E ſupra C; </s>
            <s xml:id="echoid-s3359" xml:space="preserve">eſt igitur A + E minor quam 2 C; </s>
            <s xml:id="echoid-s3360" xml:space="preserve">
              <lb/>
            quod demonſtrare oportuit.</s>
            <s xml:id="echoid-s3361" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div176" type="section" level="1" n="83">
          <head xml:id="echoid-head119" xml:space="preserve">PROP. XV. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3362" xml:space="preserve">Iiſdem poſitis: </s>
            <s xml:id="echoid-s3363" xml:space="preserve">dico exceſſum C ſupra A minorem eſſe qua-
              <lb/>
            druplo exceſſus E ſupra C. </s>
            <s xml:id="echoid-s3364" xml:space="preserve">ex prædictis manifeſtæ ſunt
              <lb/>
            ſequentes tres analogiæ, prima quoniam A, C, B, ſunt con-
              <lb/>
            tinuè proportionales; </s>
            <s xml:id="echoid-s3365" xml:space="preserve">ſecunda, quoniam C, D, B, ſunt har-
              <lb/>
            monicè proportionales; </s>
            <s xml:id="echoid-s3366" xml:space="preserve">& </s>
            <s xml:id="echoid-s3367" xml:space="preserve">tertia, quoniam C, E, D, ſunt con-
              <lb/>
            tinuè proportionales; </s>
            <s xml:id="echoid-s3368" xml:space="preserve">& </s>
            <s xml:id="echoid-s3369" xml:space="preserve">ideo
              <lb/>
              <note position="right" xlink:label="note-0153-03" xlink:href="note-0153-03a" xml:space="preserve">
                <lb/>
              C - A:B - C::A:C
                <lb/>
              B - C:D - C::A + C:A
                <lb/>
              D - C:E - C::E + C:C
                <lb/>
              </note>
            exceſſus C ſupra A (hoc eſt)
              <lb/>
            C - A eſt ad exceſſum E ſu-
              <lb/>
            pra C ſeu E - C, ut A C + E C
              <lb/>
            + AE + CC ad CC; </s>
            <s xml:id="echoid-s3370" xml:space="preserve">at B </s>
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