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

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        <div xml:id="echoid-div198" type="section" level="1" n="71">
          <pb o="116" file="0172" n="188" rhead="CHRISTIANI HUGENII"/>
          <p>
            <s xml:id="echoid-s2679" xml:space="preserve">Præter haſce autem paraboloides lineas, alias item inve-
              <lb/>
              <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" 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>
            nimus, à quibus, non abſimili conſtructione, deducuntur
              <lb/>
            curvæ rectis comparabiles. </s>
            <s xml:id="echoid-s2680" xml:space="preserve">Aſſimilantur autem hyperbolis,
              <lb/>
            eo quod aſymptotos ſuas habent, ſed tantum angulum re-
              <lb/>
            ctum conſtituentes. </s>
            <s xml:id="echoid-s2681" xml:space="preserve">Et harum primam quidem ſtatuimus hy-
              <lb/>
            perbolam ipſam, quæ eſt è coni ſectione.</s>
            <s xml:id="echoid-s2682" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2683" xml:space="preserve">Reliquarum vero naturam ut explicemus; </s>
            <s xml:id="echoid-s2684" xml:space="preserve">ſunto P S, S K,
              <lb/>
              <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">TAB. XVII.
                <lb/>
              Fig. 3.</note>
            aſymptoti curvæ A B, rectum angulum comprehendentes,
              <lb/>
            & </s>
            <s xml:id="echoid-s2685" xml:space="preserve">à curvæ puncto quolibet B ducatur B K parallela P S,
              <lb/>
            ſitque S K = x; </s>
            <s xml:id="echoid-s2686" xml:space="preserve">K B = y. </s>
            <s xml:id="echoid-s2687" xml:space="preserve">Si igitur hyperbola ſit A B,
              <lb/>
            ſcimus rectangulum linearum S K, K B, hoc eſt, rectan-
              <lb/>
            gulum x y ſemper eidem quadrato æquale eſſe, quod voce-
              <lb/>
            tur a a.</s>
            <s xml:id="echoid-s2688" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2689" xml:space="preserve">Proxima vero hyperboloidum erit, in quaſolidum ex qua-
              <lb/>
            drato lineæ S K, in altitudinem K B ductum, hoc eſt, ſo-
              <lb/>
            lidum x x y, cubo certo æquabitur, qui vocetur a
              <emph style="super">3</emph>
            . </s>
            <s xml:id="echoid-s2690" xml:space="preserve">Atque
              <lb/>
            ita innumeræ aliæ hujus generis hyperboloides exiſtunt, qua-
              <lb/>
            rum proprietatem ſequens tabella fingulis æquationibus ex-
              <lb/>
            hibet, ſimulque rationem conſtruendi curvam D C, cujus
              <lb/>
            evolutione quæque generetur.</s>
            <s xml:id="echoid-s2691" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          # x y = a
            <emph style="super">2</emph>
          # # {1/2} B M + {1/2} B Z
            <lb/>
          # x
            <emph style="super">2</emph>
          y = a
            <emph style="super">3</emph>
          # # {2/3} B M + {1/3} B Z
            <lb/>
          Si # x y
            <emph style="super">2</emph>
          = a
            <emph style="super">3</emph>
          # Erit # {1/3} B M + {2/3} B Z # = B D
            <lb/>
          # x
            <emph style="super">3</emph>
          y = a
            <emph style="super">4</emph>
          # # {3/4} B M + {1/4} B Z
            <lb/>
          # x y
            <emph style="super">3</emph>
          = a
            <emph style="super">4</emph>
          # # {1/4} B M + {3/4} B Z
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s2692" xml:space="preserve">Recta D B M Z curvam A B, ut antea quoque, ſecat
              <lb/>
            ad angulos rectos, occurritque aſymptotis S K, S P, in M
              <lb/>
            & </s>
            <s xml:id="echoid-s2693" xml:space="preserve">Z. </s>
            <s xml:id="echoid-s2694" xml:space="preserve">Si igitur exempli gratia hyperbola fuerit A B, cujus
              <lb/>
            æquatio eſt x y = a
              <emph style="super">2</emph>
            , ſumetur B D = {1/2} B M + {1/2} B Z,
              <lb/>
            quemadmodum tabella præcipit. </s>
            <s xml:id="echoid-s2695" xml:space="preserve">Eritque punctum Din cur-
              <lb/>
            va D C quæſita, cujus alia quotlibet puncta ſic inveniri po-
              <lb/>
            terunt, & </s>
            <s xml:id="echoid-s2696" xml:space="preserve">portio ejus quælibet rectæ lineæ adæquari. </s>
            <s xml:id="echoid-s2697" xml:space="preserve">Et
              <lb/>
            hæc quidem eadem illa eſt curva, cujus relationem ad axem
              <lb/>
            hyperbolæ ſuperius æquatione expreſſimus. </s>
            <s xml:id="echoid-s2698" xml:space="preserve">Conſtructio au-
              <lb/>
            tem tabellæ hujus plane eadem eſt quæ ſuperioris.</s>
            <s xml:id="echoid-s2699" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2700" xml:space="preserve">Cæterum, quoniam tum ad harum curvarum, tum ad </s>
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