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

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            <s xml:id="echoid-s2565" xml:space="preserve">
              <pb o="112" file="0164" n="178" rhead="CHRISTIANI HUGENII"/>
            niam curva, ad quam ſunt puncta T, V, geometrica eſt.
              <lb/>
            </s>
            <s xml:id="echoid-s2566" xml:space="preserve">
              <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" 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>
            Ratio igitur Y K ad K T data erit, adeoque & </s>
            <s xml:id="echoid-s2567" xml:space="preserve">V X ad
              <lb/>
            X T. </s>
            <s xml:id="echoid-s2568" xml:space="preserve">ex qua etiam rationem L K ad N M dari oſtendimus.</s>
            <s xml:id="echoid-s2569" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2570" xml:space="preserve">Quænam vero ſit linea ad quam ſunt puncta T, V, in-
              <lb/>
            venitur ponendo certum punctum S in recta K L, & </s>
            <s xml:id="echoid-s2571" xml:space="preserve">vocan-
              <lb/>
            do S K, x; </s>
            <s xml:id="echoid-s2572" xml:space="preserve">K T, y. </s>
            <s xml:id="echoid-s2573" xml:space="preserve">Nam quia data eſt curva A B F,
              <lb/>
            eique B M ad angulos rectos ducta, invenietur inde quanti-
              <lb/>
            tas lineæ K M, per methodum tangentium à Carteſio traditam,
              <lb/>
            quæ ipſi K T, ſive y æquabitur, & </s>
            <s xml:id="echoid-s2574" xml:space="preserve">ex ea æquatione, natura
              <lb/>
            curvæ T V innoteſcet, ad quam deinde tangens ducenda
              <lb/>
            eſt. </s>
            <s xml:id="echoid-s2575" xml:space="preserve">Sed clariora omnia fient ſequenti exemplo.</s>
            <s xml:id="echoid-s2576" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s2577" xml:space="preserve">Sit A B F paraboloides illa, cui ſuperius rectam æqua-
              <lb/>
              <note position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">TAB. XVI.
                <lb/>
              Fig. 3.</note>
            lem invenimus; </s>
            <s xml:id="echoid-s2578" xml:space="preserve">in qua nempe cubi perpendicularium in
              <lb/>
            rectam S K, ſint inter ſe ſicut quadrata ex ipſa S K abſciſ-
              <lb/>
            ſarum. </s>
            <s xml:id="echoid-s2579" xml:space="preserve">Et oporteat invenire curvam C D E cujus evolu-
              <lb/>
            tione paraboloides S B F deſcribatur.</s>
            <s xml:id="echoid-s2580" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2581" xml:space="preserve">Hic primum ratio B O ad B P facile invenitur, quia
              <lb/>
            tangentem paraboloidis in puncto B duci ſcimus, ſumpta S H
              <lb/>
            æquali {1/2} S K. </s>
            <s xml:id="echoid-s2582" xml:space="preserve">Cui tangenti cum B M ad angulos rectos in-
              <lb/>
            ſiſtat, dantur jam lineæ M H, H K, ac proinde earum in-
              <lb/>
            ter ſe ratio, quæ eſt eadem quæ O B ad B P.</s>
            <s xml:id="echoid-s2583" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2584" xml:space="preserve">Ut autem ratio B P, ſive K L ad M N innoteſcat, po-
              <lb/>
            nantur ad K L perpendiculares rectæ K T, L V, æquales
              <lb/>
            ſingulis K M, L N, ſitque V X parallela L K. </s>
            <s xml:id="echoid-s2585" xml:space="preserve">Jam quia
              <lb/>
            ex duabus ſimul K L, L N, auferendo K M, relinquitur
              <lb/>
            M N ; </s>
            <s xml:id="echoid-s2586" xml:space="preserve">hoc eſt, auferendo ex duabus X V, V L, ſive</s>
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          <p>
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              <lb/>
              <note symbol="*" position="foot" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">In Exemplari ſuo ad marginem ſcripſit Auctor. ſupponitur hic rectam L N
                <lb/>
              majorem eſſe quam K M, quod melius fuerat antea probari, etſi verum eſt.
                <lb/>
              Demonſtratio autem haud difficilis eſt, ſit abſciſſa S K = x; perpendicularis K B
                <lb/>
              = u; Tatus rectum paraboloidis = a. Quia S H = {1/2} SK, eſt H K = {3/2} S K
                <lb/>
              ({3/2}x). Propter angulum rectum H B M, triangula rectangula H B K, K B M
                <lb/>
              ſimilia ſunt, & H K ({3/2}x), K B (u), K M, ſunt in continua proportione; ergo
                <lb/>
              K M = {2uu/3x}, cujus quadratum eſt {4u
                <emph style="super">4.</emph>
              /9xx} = {4au
                <emph style="super">4.</emph>
              /9axx}; ſed ut notavit auctor ex natu-
                <lb/>
              ra Paraboloidis A B F, u
                <emph style="super">3</emph>
              = axx; ergo quadratum lineæ K M = {4au
                <emph style="super">4</emph>
              /9axx} = {4au
                <emph style="super">4</emph>
              /9u
                <emph style="super">3</emph>
              } =
                <lb/>
              {4/9} a u unde ſequitur ipſam K M, augeri ſi creſcat B K (u). Cum autem L F exce-
                <lb/>
              dat B K, L N ſuperabit K M, quod demonſtrandum erat.</note>
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