Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

Table of Notes

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          <pb o="419" file="0137" n="146" rhead="ET HYPERBOLÆ QUADRATURA."/>
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        <div xml:id="echoid-div154" type="section" level="1" n="70">
          <head xml:id="echoid-head105" xml:space="preserve">SCHOLIUM.</head>
          <p>
            <s xml:id="echoid-s2879" xml:space="preserve">Duæ præcedentes propoſitiones eodem modo demon-
              <lb/>
            ſtrari poſſunt de duobus quibuſcunque polygonis
              <lb/>
            complicatis loco polygonorum complicatorum ABIP,
              <lb/>
            A B D L P; </s>
            <s xml:id="echoid-s2880" xml:space="preserve">polygonum enim à tangentibus comprehenſum
              <lb/>
            tot continet æqualia trapezia, quot continet polygonum à
              <lb/>
            ſubtendentibus comprehenſum æqualia triangula: </s>
            <s xml:id="echoid-s2881" xml:space="preserve">atque hinc
              <lb/>
            evidens eſt has polygonorum analogias ita ſe habere in infi-
              <lb/>
            nitum, ducendo nimirum rectas AN, AK, AG, AC, per
              <lb/>
            puncta R, T, S, V, & </s>
            <s xml:id="echoid-s2882" xml:space="preserve">adhuc alia & </s>
            <s xml:id="echoid-s2883" xml:space="preserve">alia polygona intra & </s>
            <s xml:id="echoid-s2884" xml:space="preserve">
              <lb/>
            extra ſemper ſcribendo: </s>
            <s xml:id="echoid-s2885" xml:space="preserve">notandum nos appellare hanc poly-
              <lb/>
            gonorum inſcriptionem & </s>
            <s xml:id="echoid-s2886" xml:space="preserve">circumſcriptionem, inſcriptionem
              <lb/>
            & </s>
            <s xml:id="echoid-s2887" xml:space="preserve">circumſcriptionem ſubduplam, ex prædictis patet (ſi po-
              <lb/>
            natur triangulum A B P =
              <emph style="super">a</emph>
            , & </s>
            <s xml:id="echoid-s2888" xml:space="preserve">trapezium A B F P =
              <emph style="super">b</emph>
            ) tra-
              <lb/>
            pezium A B I P eſſe vqab & </s>
            <s xml:id="echoid-s2889" xml:space="preserve">polygonum A B D L P {2ab/a + vqab}:
              <lb/>
            </s>
            <s xml:id="echoid-s2890" xml:space="preserve">eodem modo poſito trapezio A B I P =
              <emph style="super">c</emph>
            , & </s>
            <s xml:id="echoid-s2891" xml:space="preserve">polygono
              <lb/>
            A B D L P =
              <emph style="super">d</emph>
            , erit polygonum A B E I O P = vqcd & </s>
            <s xml:id="echoid-s2892" xml:space="preserve">po-
              <lb/>
            lygonum A B C G K N P = {2cd/c + vqcd,}, ita ut evidens ſit hanc
              <lb/>
            polygonorum ſeriem eſſe convergentem; </s>
            <s xml:id="echoid-s2893" xml:space="preserve">atque in infinitum
              <lb/>
            illam continuando, manifeſtum eſt tandem exhiberi quanti-
              <lb/>
            tatem ſectori circulari, elliptico vel hyperbolico A B E I O P
              <lb/>
            æqualem; </s>
            <s xml:id="echoid-s2894" xml:space="preserve">differentia enim polygonorum complicatorum in
              <lb/>
            ſeriei continuatione ſemper diminuitur, ita ut omni exhibita
              <lb/>
            quantitate fieri poſſit minor, ut in ſequentis theorematis
              <lb/>
            Scholio demonſtrabimus: </s>
            <s xml:id="echoid-s2895" xml:space="preserve">ſi igitur prædicta polygonorum ſe-
              <lb/>
            ries terminari poſſet, hoc eſt, ſi inveniretur ultimum illud
              <lb/>
            polygonum inſcriptum (ſi ita loqui liceat) æquale ultimo
              <lb/>
            illi polygono circumſcripto, daretur infallibiliter circuli & </s>
            <s xml:id="echoid-s2896" xml:space="preserve">
              <lb/>
            hyperbolæ quadratura: </s>
            <s xml:id="echoid-s2897" xml:space="preserve">ſed quoniam difficile eſt, & </s>
            <s xml:id="echoid-s2898" xml:space="preserve">in geo-
              <lb/>
            metria omnino fortaſſe inauditum tales ſeries terminare; </s>
            <s xml:id="echoid-s2899" xml:space="preserve">præ-
              <lb/>
            mittendæ ſunt quædam propoſitiones è quibus inveniri poſ-
              <lb/>
            ſint hujuſmodi aliquot ſerierum terminationes, & </s>
            <s xml:id="echoid-s2900" xml:space="preserve">tandem </s>
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