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

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              <pb o="445" file="0163" n="172" rhead="ET HYPERBOLÆ QUADRATURA."/>
            F D M K, P L M D. </s>
            <s xml:id="echoid-s3685" xml:space="preserve">dico triangulum A F L eſſe medium arith-
              <lb/>
            meticum inter parallelogramma F D M K, P L M D. </s>
            <s xml:id="echoid-s3686" xml:space="preserve">Grego-
              <lb/>
            rius à S. </s>
            <s xml:id="echoid-s3687" xml:space="preserve">Vincentio in Lib. </s>
            <s xml:id="echoid-s3688" xml:space="preserve">de Hyperbola demonſtrat triangu-
              <lb/>
            lum A F L eſſe æquale trapezio D F L M, ſed manifeſtum eſt
              <lb/>
            trapezium D F L M eſſe medium arithmeticum inter paral-
              <lb/>
            lelogramma F D M K, P L M D; </s>
            <s xml:id="echoid-s3689" xml:space="preserve">& </s>
            <s xml:id="echoid-s3690" xml:space="preserve">ideo patet propo-
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            ſitum.</s>
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        <div xml:id="echoid-div201" type="section" level="1" n="97">
          <head xml:id="echoid-head133" xml:space="preserve">PROP. XXVII. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3692" xml:space="preserve">Iisdem poſitis: </s>
            <s xml:id="echoid-s3693" xml:space="preserve">ducatur A I rectam F L bifariam dividens in
              <lb/>
              <note position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">TAB.
                <lb/>
              XLVIII.
                <lb/>
              fig. 4.</note>
            I & </s>
            <s xml:id="echoid-s3694" xml:space="preserve">hyperbolam interſecans in puncto G, fiatque trape-
              <lb/>
            zium ſectori circumſcriptum A F G L, quod dico eſſe me-
              <lb/>
            dium geometricum inter parallellogramma F D M K, P L M D.
              <lb/>
            </s>
            <s xml:id="echoid-s3695" xml:space="preserve">ex demonſtratis Gregorii à S. </s>
            <s xml:id="echoid-s3696" xml:space="preserve">Vincentio evidens eſt trape-
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            zium A F G L æquale eſſe rectilineo D F G L M. </s>
            <s xml:id="echoid-s3697" xml:space="preserve">& </s>
            <s xml:id="echoid-s3698" xml:space="preserve">quoniam
              <lb/>
            A G I recta ſecat rectam F L bifariam in I, ex ejuſdem Gre-
              <lb/>
            gorii à S. </s>
            <s xml:id="echoid-s3699" xml:space="preserve">Vincentio Lib. </s>
            <s xml:id="echoid-s3700" xml:space="preserve">de hyperbola, manifeſtum eſt rectas
              <lb/>
            L M, G H, FD, eſſe continuè proportionales in eadem ratione
              <lb/>
            cum tribus continuè proportionalibus A D, A H, A M. </s>
            <s xml:id="echoid-s3701" xml:space="preserve">aſym-
              <lb/>
            ptoto A O per punctum G ducatur parallela recta R G S
              <lb/>
            rectis F D, M K, occurrens in punctis R, S. </s>
            <s xml:id="echoid-s3702" xml:space="preserve">quoniam rectæ
              <lb/>
            F D, G H, L M, ſunt continuè proportionales, erit dividen-
              <lb/>
            do & </s>
            <s xml:id="echoid-s3703" xml:space="preserve">permutando F R ad S L ut G H ad L M: </s>
            <s xml:id="echoid-s3704" xml:space="preserve">& </s>
            <s xml:id="echoid-s3705" xml:space="preserve">quoniam
              <lb/>
            rectæ M A, H A, D A, ſunt continuè proportionales, erit
              <lb/>
            etiam dividendo & </s>
            <s xml:id="echoid-s3706" xml:space="preserve">permutando M H ad H D hoc eſt S G ad
              <lb/>
            G R ut H A ad D A, vel ut G H ad L M; </s>
            <s xml:id="echoid-s3707" xml:space="preserve">& </s>
            <s xml:id="echoid-s3708" xml:space="preserve">proinde F R
              <lb/>
            eſt ad S L ut S G ad G R, cumque anguli F R G, G S L, ſint
              <lb/>
            æquales ob parallelas F R, S L, erunt triangula F R G, G L S,
              <lb/>
            æqualia; </s>
            <s xml:id="echoid-s3709" xml:space="preserve">& </s>
            <s xml:id="echoid-s3710" xml:space="preserve">proinde parallelogrammum R D M S æquale eſt
              <lb/>
            rectilineo D F G L M ſeu trapezio A F G L; </s>
            <s xml:id="echoid-s3711" xml:space="preserve">ſed parallelo-
              <lb/>
            grammum R D M S eſt medium geometricum inter parale-
              <lb/>
            logramma P D M L, F D M K, quoniam eandem habentia alti-
              <lb/>
            tudinem eorum baſes nempe L M, S M, K M, ſunt continuè
              <lb/>
            proportionales; </s>
            <s xml:id="echoid-s3712" xml:space="preserve">& </s>
            <s xml:id="echoid-s3713" xml:space="preserve">ideo trapezium A F G L eſt medium </s>
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