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

Table of handwritten notes

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            <s xml:id="echoid-s2587" xml:space="preserve">
              <pb o="113" file="0167" n="182" rhead="HOROLOG. OSCILLATOR."/>
            X V, X K, ipſam K T; </s>
            <s xml:id="echoid-s2588" xml:space="preserve">hinc autem relinqui apparet V X
              <lb/>
              <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">
                <emph style="sc">De linea-</emph>
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                <emph style="sc">RUM CUR-</emph>
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                <emph style="sc">VARUM</emph>
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                <emph style="sc">EVOLUTIO</emph>
              -
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                <emph style="sc">NE</emph>
              .</note>
            & </s>
            <s xml:id="echoid-s2589" xml:space="preserve">X T: </s>
            <s xml:id="echoid-s2590" xml:space="preserve">erunt igitur hæ duæ V X, X T ipſi M N æqua-
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            les, ac proinde ratio K L ad M N eadem quæ V X ad
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            duas ſimul V X, X T. </s>
            <s xml:id="echoid-s2591" xml:space="preserve">Ut autem hæc ratio innoteſcat cum
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            intervallum K L eſt minimum; </s>
            <s xml:id="echoid-s2592" xml:space="preserve">oportet ſecundum prædicta
              <lb/>
            inquirere quis ſit locus, ſive linea ad quam ſunt puncta
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            T, V. </s>
            <s xml:id="echoid-s2593" xml:space="preserve">Quod ut fiat ſit latus rectum paraboloidis A B F = a;
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            </s>
            <s xml:id="echoid-s2594" xml:space="preserve">S K = x; </s>
            <s xml:id="echoid-s2595" xml:space="preserve">K T = y.</s>
            <s xml:id="echoid-s2596" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2597" xml:space="preserve">Quia igitur proportionales ſunt K H, K B, K M, eſt-
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            que H K = {1/2} x: </s>
            <s xml:id="echoid-s2598" xml:space="preserve">K B ex natura paraboloidis æqualis R.
              <lb/>
            </s>
            <s xml:id="echoid-s2599" xml:space="preserve">cub. </s>
            <s xml:id="echoid-s2600" xml:space="preserve">a x x: </s>
            <s xml:id="echoid-s2601" xml:space="preserve">fiet K M, hoc eſt K T = {2/3} R. </s>
            <s xml:id="echoid-s2602" xml:space="preserve">cub. </s>
            <s xml:id="echoid-s2603" xml:space="preserve">a a x = y,
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            ac proinde {8/27} a a x = y
              <emph style="super">3</emph>
            . </s>
            <s xml:id="echoid-s2604" xml:space="preserve">Unde patet locum punctorum T,
              <lb/>
            V, eſſe paraboloidem illam, quam cubicam vocant geome-
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            træ. </s>
            <s xml:id="echoid-s2605" xml:space="preserve">Cui proinde ad T tangens ducetur, ſumptâ S Y duplâ
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            ipſius S K, junctâque Y T. </s>
            <s xml:id="echoid-s2606" xml:space="preserve">Et jam quidem ratio V X ad
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            duas ſimul V X, X T, quam diximus eandem eſſe ac K L
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            ad M N, erit ea quæ Y K ad utramque ſimul Y K, K T. </s>
            <s xml:id="echoid-s2607" xml:space="preserve">
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            Hæc autem ratio data eſt, ergo & </s>
            <s xml:id="echoid-s2608" xml:space="preserve">ratio K L ad M N. </s>
            <s xml:id="echoid-s2609" xml:space="preserve">Sed
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            & </s>
            <s xml:id="echoid-s2610" xml:space="preserve">rationem O B ad P B datam eſſe oſtenſum eſt. </s>
            <s xml:id="echoid-s2611" xml:space="preserve">Ergo,
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            cum ex duabus hiſce componatur ratio B D ad D M, ut ſu-
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            pra patuit, dabitur & </s>
            <s xml:id="echoid-s2612" xml:space="preserve">hæc; </s>
            <s xml:id="echoid-s2613" xml:space="preserve">& </s>
            <s xml:id="echoid-s2614" xml:space="preserve">dividendo, ratio B M ad
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            M D; </s>
            <s xml:id="echoid-s2615" xml:space="preserve">adeoque & </s>
            <s xml:id="echoid-s2616" xml:space="preserve">punctum D in curva D E.</s>
            <s xml:id="echoid-s2617" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2618" xml:space="preserve">Ad conſtructionem autem breviſſimam hoc pacto hic per-
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            veniemus. </s>
            <s xml:id="echoid-s2619" xml:space="preserve">K T ſive K M dicta fuit y. </s>
            <s xml:id="echoid-s2620" xml:space="preserve">Itaque M H erit y
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            + {3/2} x. </s>
            <s xml:id="echoid-s2621" xml:space="preserve">Et M H ad H K, ſive O B ad B P, ut y + {3/2} x
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            ad {3/2} x. </s>
            <s xml:id="echoid-s2622" xml:space="preserve">ſive, ſumptis omnium duplis, ut 2 y + 3 x ad 3 x.
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            </s>
            <s xml:id="echoid-s2623" xml:space="preserve">Deinde quia Y K = 3 x, erit Y K ad Y K + K T, ſi-
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            ve per prædicta, K L ad M N, ut 3 x ad 3 x + y. </s>
            <s xml:id="echoid-s2624" xml:space="preserve">Atqui
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            ex rationibus O B ad B P, & </s>
            <s xml:id="echoid-s2625" xml:space="preserve">K L ad M N, componi di-
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            ximus rationem B D ad D M. </s>
            <s xml:id="echoid-s2626" xml:space="preserve">Ergo ratio B D ad D M erit
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            compoſita ex rationibus 2 y + 3 x ad 3 x, & </s>
            <s xml:id="echoid-s2627" xml:space="preserve">3 x ad 3 x
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            + y; </s>
            <s xml:id="echoid-s2628" xml:space="preserve">ideoque erit ea quæ 2 y + 3 x ad 3 x + y. </s>
            <s xml:id="echoid-s2629" xml:space="preserve">& </s>
            <s xml:id="echoid-s2630" xml:space="preserve">divi-
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            dendo, ratio B M ad M D, eadem quæ y ad 3 x + y.</s>
            <s xml:id="echoid-s2631" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2632" xml:space="preserve">Sit S Z perpendicularis ad S K, eique occurrat M B pro-
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            ducta in Z. </s>
            <s xml:id="echoid-s2633" xml:space="preserve">Quia ergo ratio B M ad M D inventa eſt ea quæ
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            y ad y + 3 x, hoc eſt quæ M K ad M K + 3 K S. </s>
            <s xml:id="echoid-s2634" xml:space="preserve">Sicut </s>
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