Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 1: Opera mechanica

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            <s xml:id="echoid-s2489" xml:space="preserve">
              <pb o="110" file="0162" n="176" rhead="CHRISTIANI HUGENII"/>
            æqualitatis; </s>
            <s xml:id="echoid-s2490" xml:space="preserve">liquet rationem B G ad G M fore eandem quæ N H
              <lb/>
              <note position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">
                <emph style="sc">De linea-</emph>
                <lb/>
                <emph style="sc">RUM CUR-</emph>
                <lb/>
                <emph style="sc">VARUM</emph>
                <lb/>
                <emph style="sc">EVOLUTIO-</emph>
                <lb/>
                <emph style="sc">NE.</emph>
              </note>
            ad H L; </s>
            <s xml:id="echoid-s2491" xml:space="preserve">& </s>
            <s xml:id="echoid-s2492" xml:space="preserve">dividendo, B M ad M G, eandem quæ N L
              <lb/>
            ad L H, ſive M K ad K H; </s>
            <s xml:id="echoid-s2493" xml:space="preserve">nam L H, K H pro eadem
              <lb/>
            habentur, propter propinquitatem punctorum B, F. </s>
            <s xml:id="echoid-s2494" xml:space="preserve">Data
              <lb/>
            autem eſt ratio M K ad K H, dato puncto B; </s>
            <s xml:id="echoid-s2495" xml:space="preserve">quoniam
              <lb/>
            tam M K, quam K H dantur magnitudine; </s>
            <s xml:id="echoid-s2496" xml:space="preserve">nam M K
              <lb/>
            æquatur dimidio lateri recto, K H vero duplæ K A. </s>
            <s xml:id="echoid-s2497" xml:space="preserve">Dataque
              <lb/>
            etiam eſt poſitione & </s>
            <s xml:id="echoid-s2498" xml:space="preserve">magnitudine recta B M. </s>
            <s xml:id="echoid-s2499" xml:space="preserve">Ergo & </s>
            <s xml:id="echoid-s2500" xml:space="preserve">M G
              <lb/>
            data erit, adeoque & </s>
            <s xml:id="echoid-s2501" xml:space="preserve">punctum G, ſive D, in curva R D E;
              <lb/>
            </s>
            <s xml:id="echoid-s2502" xml:space="preserve">quod nempe invenitur productâ B M uſque in G, ut ſit
              <lb/>
            B M ad M G ſicut {1/2} lateris recti ad duplam K A.</s>
            <s xml:id="echoid-s2503" xml:space="preserve"/>
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            <s xml:id="echoid-s2504" xml:space="preserve">Et ſic quidem, adſumptis in parabola A B F aliis quotli-
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            bet punctis præter B, totidem quoque puncta lineæ R D E,
              <lb/>
            ſimili ratione, invenientur; </s>
            <s xml:id="echoid-s2505" xml:space="preserve">atque hoc ipſo lineam R D E
              <lb/>
            geometricam eſſe conſtat, unáque proprietas ejus innoteſcit,
              <lb/>
            ex qua cæteræ deduci poſſunt. </s>
            <s xml:id="echoid-s2506" xml:space="preserve">Ut ſi inquirere deinde veli-
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            mus, quanam æquatione exprimatur relatio punctorum
              <lb/>
            omnium curvæ C D E ad rectam A Q: </s>
            <s xml:id="echoid-s2507" xml:space="preserve">ducta in hanc perpen-
              <lb/>
            diculari D Q, vocatoque latere recto parabolæ A B F, a;
              <lb/>
            </s>
            <s xml:id="echoid-s2508" xml:space="preserve">A K, b; </s>
            <s xml:id="echoid-s2509" xml:space="preserve">A Q, x; </s>
            <s xml:id="echoid-s2510" xml:space="preserve">Q D, y. </s>
            <s xml:id="echoid-s2511" xml:space="preserve">Quoniam ratio B M ad M D,
              <lb/>
            hoc eſt, K M ad M Q, eſt ea quæ {1/2} a ad 2 b, eſtque ipſa
              <lb/>
            K M = {1/2} a, erit & </s>
            <s xml:id="echoid-s2512" xml:space="preserve">M Q æqualis 2 b. </s>
            <s xml:id="echoid-s2513" xml:space="preserve">Eſt autem M A = {1/2}
              <lb/>
            a + b. </s>
            <s xml:id="echoid-s2514" xml:space="preserve">ergo A Q ſive x æqualis 3 b + {1/2} a. </s>
            <s xml:id="echoid-s2515" xml:space="preserve">Unde b = {1/3} x
              <lb/>
            -{1/6} a. </s>
            <s xml:id="echoid-s2516" xml:space="preserve">Porro quoniam, ſicut quadratum M K, hoc eſt, {1/4} a a
              <lb/>
            ad quadratum K B, hoc eſt, a b, ita qu. </s>
            <s xml:id="echoid-s2517" xml:space="preserve">M Q, hoc eſt,
              <lb/>
            4 b b ad qu. </s>
            <s xml:id="echoid-s2518" xml:space="preserve">Q D; </s>
            <s xml:id="echoid-s2519" xml:space="preserve">erit qu. </s>
            <s xml:id="echoid-s2520" xml:space="preserve">Q D, ſive y y = {16b
              <emph style="super">3</emph>
            /4}. </s>
            <s xml:id="echoid-s2521" xml:space="preserve">Ubi, ſi in
              <lb/>
            locum b ſubſtituatur {1/3} x - {1/6}a, quod illi æquale inventum eſt,
              <lb/>
            fiet y y = 16. </s>
            <s xml:id="echoid-s2522" xml:space="preserve">cub. </s>
            <s xml:id="echoid-s2523" xml:space="preserve">{1/3} x - {1/6} a diviſis per a. </s>
            <s xml:id="echoid-s2524" xml:space="preserve">Ac proinde {27/16} a y y
              <lb/>
            = cubo ab x - {1/2} a. </s>
            <s xml:id="echoid-s2525" xml:space="preserve">Accipiatur A R in axe parabolæ = {1/2} a; </s>
            <s xml:id="echoid-s2526" xml:space="preserve">
              <lb/>
            eritque R Q = x - {1/2} a. </s>
            <s xml:id="echoid-s2527" xml:space="preserve">Curvam igitur C D ejus naturæ eſſe
              <lb/>
            liquet, ut ſemper cubus lineæ R Q æquetur parallelepipedo,
              <lb/>
            cujus baſis qu. </s>
            <s xml:id="echoid-s2528" xml:space="preserve">Q D, altitudo {27/16} a; </s>
            <s xml:id="echoid-s2529" xml:space="preserve">ac proinde ipſam para-
              <lb/>
            boloidem eſſe, cujus evolutione deſcribi parabolam A B ſu-
              <lb/>
            pra oſtendimus; </s>
            <s xml:id="echoid-s2530" xml:space="preserve">cujus nimirum paraboloidis latus rectum æ-
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            quetur {27/16} lateris recti parabolæ A B. </s>
            <s xml:id="echoid-s2531" xml:space="preserve">tunc enim hujus latus
              <lb/>
            rectum æquale fit {15/27} lateris recti paraboloidis, quemadmo-
              <lb/>
            dum ibi fuit deſinitum.</s>
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