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

Table of contents

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[41.] PROPOSITIO XVIII.
[42.] PROPOSITIO XIX.
[43.] PROPOSITIO XX.
[44.] PROPOSITIO XXI.
[45.] PROPOSITIO XXII.
[46.] LEMMA.
[47.] PROPOSITIO XXIII.
[48.] PROPOSITIO XXIV.
[49.] PROPOSITIO XXV.
[50.] PROPOSITIO XXVI.
[51.] HOROLOGII OSCILLATORII PARS TERTIA.
[52.] DEFINITIONES. I.
[53.] II.
[54.] III.
[56.] PROPOSITIOI.
[57.] PROPOSITIO II.
[58.] PROPOSITIO III.
[59.] PROPOSITIO IV.
[60.] PROPOSITIO V.
[61.] PROPOSITIO VI.
[62.] PROPOSITIO VII.
[63.] PROPOSITIO VIII.
[64.] PROPOSITIO IX.
[65.] Conoidis parabolici ſuperficiei curvæ circulum æqualem invenire.
[66.] Sphæroidis oblongi ſuperſiciei circulum æqualem invenire.
[67.] Sphæroidis lati ſive compreſſi ſuperficiei circulum æqualem invenire.
[68.] Conoidis hyperbolici ſuperficiei curvæ circulum æqualem invenire.
[69.] Curvæ parabolicæ æqualem rectam lineam invenire.
[70.] PROPOSITIO X.
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            <s xml:id="echoid-s2194" xml:space="preserve">
              <pb o="99" file="0147" n="159" rhead="HOROLOG. OSCILLATOR."/>
            penitiùs quam cyclois cognita ſit. </s>
            <s xml:id="echoid-s2195" xml:space="preserve">Methodum vero noſtram,
              <lb/>
              <note position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">
                <emph style="sc">De linea-</emph>
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                <emph style="sc">RUM CUR-</emph>
                <lb/>
                <emph style="sc">VARUM</emph>
                <lb/>
                <emph style="sc">EVOLUT@@-</emph>
                <lb/>
                <emph style="sc">NE</emph>
              .</note>
            qua in hac metienda uſi ſumus, in aliis quoque experiri li-
              <lb/>
            buit, de quibus porro nunc agemus.</s>
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        <div xml:id="echoid-div172" type="section" level="1" n="63">
          <head xml:id="echoid-head87" xml:space="preserve">PROPOSITIO VIII.</head>
          <p style="it">
            <s xml:id="echoid-s2197" xml:space="preserve">CUjus lineæ evolutione parabola deſcribatur os-
              <lb/>
            tendere.</s>
            <s xml:id="echoid-s2198" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2199" xml:space="preserve">Sit paraboloides A B, cujus axis A D; </s>
            <s xml:id="echoid-s2200" xml:space="preserve">vertex A; </s>
            <s xml:id="echoid-s2201" xml:space="preserve">pro-
              <lb/>
              <note position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">TAB. XIII,
                <lb/>
              Fig. 1.</note>
            prietas autem iſta, ut ordinatim ad axem applicatâ B D,
              <lb/>
            cubus abſciſſæ ad verticem D A æquetur ſolido, baſin ha-
              <lb/>
            benti quadratum D B, altitudinem vero æqualem lineæ cui-
              <lb/>
            dam datæ M; </s>
            <s xml:id="echoid-s2202" xml:space="preserve">quæ quidem curva pridem geometris nota
              <lb/>
            fuit; </s>
            <s xml:id="echoid-s2203" xml:space="preserve">& </s>
            <s xml:id="echoid-s2204" xml:space="preserve">ponatur axi D E juncta in directum A E, quæ ha-
              <lb/>
            beat {8/27} ipſius M. </s>
            <s xml:id="echoid-s2205" xml:space="preserve">Jam ſi filum continuum circa E A B ap-
              <lb/>
            plicetur, idque ab E evolvi incipiat, dico deſcriptam ex
              <lb/>
            evolutione eſſe parabolam E F, cujus axis E A G, vertex
              <lb/>
            E, latus rectum æquale duplæ E A.</s>
            <s xml:id="echoid-s2206" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2207" xml:space="preserve">Sumpto enim in curva A B puncto quolibet B, ducatur
              <lb/>
            quæ in ipſo tangat curvam recta B G, occurrens axi E A
              <lb/>
            in G. </s>
            <s xml:id="echoid-s2208" xml:space="preserve">& </s>
            <s xml:id="echoid-s2209" xml:space="preserve">ex G ducatur porro G F, quæ ad rectos angulos
              <lb/>
            occurrat parabolæ E F in F; </s>
            <s xml:id="echoid-s2210" xml:space="preserve">& </s>
            <s xml:id="echoid-s2211" xml:space="preserve">ſit ipſi G F perpendicula-
              <lb/>
            ris F H, quæ parabolam in F continget; </s>
            <s xml:id="echoid-s2212" xml:space="preserve">& </s>
            <s xml:id="echoid-s2213" xml:space="preserve">denique F K
              <lb/>
            ordinatim ad axem E G applicetur.</s>
            <s xml:id="echoid-s2214" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2215" xml:space="preserve">Eſt igitur K G æqualis dimidio lateri recto, hoc eſt, ipſi
              <lb/>
            E A; </s>
            <s xml:id="echoid-s2216" xml:space="preserve">ac proinde, additâ vel ablatâ utrimque A K, erit
              <lb/>
            E K æqualis A G. </s>
            <s xml:id="echoid-s2217" xml:space="preserve">Eſt autem A G triens ipſius A D, quo-
              <lb/>
            niam B G tangit paraboloidem in B: </s>
            <s xml:id="echoid-s2218" xml:space="preserve">illud enim ex natura
              <lb/>
            curvæ hujus facile demonſtrari poteſt. </s>
            <s xml:id="echoid-s2219" xml:space="preserve">Ergo & </s>
            <s xml:id="echoid-s2220" xml:space="preserve">E K æqualis
              <lb/>
            eſt trienti A D: </s>
            <s xml:id="echoid-s2221" xml:space="preserve">& </s>
            <s xml:id="echoid-s2222" xml:space="preserve">K H, quæ ex natura parabolæ dupla eſt
              <lb/>
            K E, æquabitur duabus tertiis A D. </s>
            <s xml:id="echoid-s2223" xml:space="preserve">Itaque cubus ex K H
              <lb/>
            æqualis eſt {8/27} cubi ex A D, hoc eſt, ſolido baſin habenti
              <lb/>
            quadratum D B, altitudinem vero æqualem {8/27} M, hoc eſt,
              <lb/>
            ipſi A E. </s>
            <s xml:id="echoid-s2224" xml:space="preserve">Quamobrem ut quadratum D B ad quadratum
              <lb/>
            K H, ita erit K H longitudine ad A E, hoc eſt ad K G.
              <lb/>
            </s>
            <s xml:id="echoid-s2225" xml:space="preserve">Erat autem K H æqualis {@/3} A D, hoc eſt ipſi G D. </s>
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