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

Table of handwritten notes

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          <pb o="437" file="0155" n="164" rhead="ET HYPERBOLÆ QUADRATURA."/>
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        <div xml:id="echoid-div180" type="section" level="1" n="85">
          <head xml:id="echoid-head121" xml:space="preserve">PROP. XVII. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3401" xml:space="preserve">IIsdem poſitis: </s>
            <s xml:id="echoid-s3402" xml:space="preserve">dico exceſſum A ſupra C majorem eſſe
              <lb/>
            quadruplo exceſſus C ſupra E. </s>
            <s xml:id="echoid-s3403" xml:space="preserve">ex prædictis manifeſtæ
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            ſunt ſequentes tres analogiæ; </s>
            <s xml:id="echoid-s3404" xml:space="preserve">prima, quoniam A, C, B, ſunt
              <lb/>
            continuè proportionales; </s>
            <s xml:id="echoid-s3405" xml:space="preserve">ſecunda, quoniam C, D, B, ſunt
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            harmonicè proportionales; </s>
            <s xml:id="echoid-s3406" xml:space="preserve">& </s>
            <s xml:id="echoid-s3407" xml:space="preserve">tertia, quoniam C, E, D, ſunt
              <lb/>
            continuè proportionales; </s>
            <s xml:id="echoid-s3408" xml:space="preserve">& </s>
            <s xml:id="echoid-s3409" xml:space="preserve">ideo exceſſus A ſupra C, hoc eſt
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            A - C, eſt ad exceſſum C ſupra E, hoc eſt C - E in ratio-
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            ne compoſita ex proportionibus
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              <note position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve">
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              A - C:C: - B::A:C
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              C - B:C - D::A + :CA
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              C - D:C - E::E + C:C
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              </note>
            A ad C, A + C ad A & </s>
            <s xml:id="echoid-s3410" xml:space="preserve">E + C ad C;
              <lb/>
            </s>
            <s xml:id="echoid-s3411" xml:space="preserve">& </s>
            <s xml:id="echoid-s3412" xml:space="preserve">ideo A - C eſt ad C - E,
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            ut A C + E C + A E + CC ad
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            CC; </s>
            <s xml:id="echoid-s3413" xml:space="preserve">at B minor eſt quam E,
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            & </s>
            <s xml:id="echoid-s3414" xml:space="preserve">ideo AB, ſeu CC minor eſt quam A E; </s>
            <s xml:id="echoid-s3415" xml:space="preserve">& </s>
            <s xml:id="echoid-s3416" xml:space="preserve">igitur AE +
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            CC major eſt quam 2 CC. </s>
            <s xml:id="echoid-s3417" xml:space="preserve">atque A C + E C eſt ad 2 CC ut
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            A + E ad 2 C; </s>
            <s xml:id="echoid-s3418" xml:space="preserve">ſed A + E major eſt quam duo C, & </s>
            <s xml:id="echoid-s3419" xml:space="preserve">ideo
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            A C + E C major eſt quam 2 CC; </s>
            <s xml:id="echoid-s3420" xml:space="preserve">& </s>
            <s xml:id="echoid-s3421" xml:space="preserve">proinde A C + E C + A E
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            + C C major eſt quam 4 CC; </s>
            <s xml:id="echoid-s3422" xml:space="preserve">& </s>
            <s xml:id="echoid-s3423" xml:space="preserve">igitur A - C major eſt qua-
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            druplo ipſius C - E, quod demonſtrare oportuit.</s>
            <s xml:id="echoid-s3424" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div182" type="section" level="1" n="86">
          <head xml:id="echoid-head122" xml:space="preserve">PROP. XVIII. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3425" xml:space="preserve">SInt duæ quantitates inæquales; </s>
            <s xml:id="echoid-s3426" xml:space="preserve">A
              <lb/>
              <note position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve">
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              # E
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              A # C # B
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              # D
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              </note>
            minor, B major, C media geome-
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            trica, D media arithmetica. </s>
            <s xml:id="echoid-s3427" xml:space="preserve">dico D
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            majorem eſſe quam C, quoniam B, C,
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            A, ſunt continuè proportionales; </s>
            <s xml:id="echoid-s3428" xml:space="preserve">erit divi-
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            dendo, permutando, & </s>
            <s xml:id="echoid-s3429" xml:space="preserve">componendo; </s>
            <s xml:id="echoid-s3430" xml:space="preserve">ut exceſſus B ſupra A
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            ad exceſſum C ſupra A, ita A + C ad A, atque A + C major eſt
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            duplo ipſius A; </s>
            <s xml:id="echoid-s3431" xml:space="preserve">& </s>
            <s xml:id="echoid-s3432" xml:space="preserve">proinde exceſſus B ſupra A major eſt duplo
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            exceſſus C ſupra; </s>
            <s xml:id="echoid-s3433" xml:space="preserve">ſed exceſſus B ſupra A duplus eſt exceſſus
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            D ſupra A, & </s>
            <s xml:id="echoid-s3434" xml:space="preserve">ideo exceſſus D ſupra A major eſt exceſſu </s>
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