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

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            <s xml:id="echoid-s3940" xml:space="preserve">
              <pb o="461" file="0181" n="191" rhead="ET HYPERBOLÆ QUADRATURA."/>
            ope ſolius regulæ & </s>
            <s xml:id="echoid-s3941" xml:space="preserve">circini peracta, hanc in his non ſolum
              <lb/>
            eſſe impoſſibilem ſed etiam in omnibus problematis quæ ad
              <lb/>
            æquationem quadratic
              <unsure/>
            am reduci non poſſunt, ſicut facile
              <lb/>
            demonſtrari poſſet; </s>
            <s xml:id="echoid-s3942" xml:space="preserve">& </s>
            <s xml:id="echoid-s3943" xml:space="preserve">ſi per geometricum intelligatur redu-
              <lb/>
            ctio problematis ad æquationem analyticam, omnia hæc
              <lb/>
            problemata ſunt geometrice impoſſibilia, cum ex hic demon-
              <lb/>
            ſtratis, manifeſtum ſit talem reductionem fieri non poſſe:
              <lb/>
            </s>
            <s xml:id="echoid-s3944" xml:space="preserve">ſi verò per geometricum intelligatur methodus omnium poſ-
              <lb/>
            ſibilium ſimpliciſſima; </s>
            <s xml:id="echoid-s3945" xml:space="preserve">invenietur fortaſſe poſt maturam con-
              <lb/>
            ſiderationem omnia prædicta problemata eſſe geometriciſſi-
              <lb/>
            mè reſoluta, diligenter animadvertendum totam ſerierum
              <lb/>
            convergentium doctrinam poſſe etiam nullo negotio applicari
              <lb/>
            ſeriebus ſimplicibus. </s>
            <s xml:id="echoid-s3946" xml:space="preserve">Sit enim ſeries A, B, C, D, E, &</s>
            <s xml:id="echoid-s3947" xml:space="preserve">c,
              <lb/>
            talis naturæ ut tertius terminus C eodem modo
              <lb/>
              <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">
                <lb/>
              A
                <lb/>
              B
                <lb/>
              C
                <lb/>
              D
                <lb/>
              E
                <lb/>
              Z
                <lb/>
              </note>
            componatur ex primo & </s>
            <s xml:id="echoid-s3948" xml:space="preserve">ſecundo A, B, quo
              <lb/>
            quartus D componitur ex ſecundo & </s>
            <s xml:id="echoid-s3949" xml:space="preserve">tertio B, C,
              <lb/>
            & </s>
            <s xml:id="echoid-s3950" xml:space="preserve">quintus E ex tertio & </s>
            <s xml:id="echoid-s3951" xml:space="preserve">quarto C, D, & </s>
            <s xml:id="echoid-s3952" xml:space="preserve">ſic dein-
              <lb/>
            ceps in infinitum; </s>
            <s xml:id="echoid-s3953" xml:space="preserve">ſitque differentia anteceden-
              <lb/>
            tium A, B, major ſemper differentia immediatè
              <lb/>
            ſequentium B, C; </s>
            <s xml:id="echoid-s3954" xml:space="preserve">ſupponamus hanc ſeriem ita in infinitum
              <lb/>
            continuari donec duorum terminorum immediate ſe invicem
              <lb/>
            ſequentium nulla ſit differentia, ſitque unus ex illis terminis
              <lb/>
            z, quem ſeriei terminationem appellamus: </s>
            <s xml:id="echoid-s3955" xml:space="preserve">dico z eodem
              <lb/>
            modo componi ex A & </s>
            <s xml:id="echoid-s3956" xml:space="preserve">B quo ex B & </s>
            <s xml:id="echoid-s3957" xml:space="preserve">C vel C & </s>
            <s xml:id="echoid-s3958" xml:space="preserve">D; </s>
            <s xml:id="echoid-s3959" xml:space="preserve">de-
              <lb/>
            monſtratio vix differt ab hujus 10 & </s>
            <s xml:id="echoid-s3960" xml:space="preserve">ejus conſectario: </s>
            <s xml:id="echoid-s3961" xml:space="preserve">hac
              <lb/>
            ratione ſi ponatur triangulum, ſectori circulari vel elliptico
              <lb/>
            inſcriptum, vel ſectori hyperbolico circumſcriptum a, & </s>
            <s xml:id="echoid-s3962" xml:space="preserve">
              <lb/>
            trapezium, ſectori circulari vel elliptico regulariter in-
              <lb/>
            ſcriptum vel hyperbolico regulariter circumſcriptum b;
              <lb/>
            </s>
            <s xml:id="echoid-s3963" xml:space="preserve">erit hexagonum ſectori circulari vel elliptico regulari-
              <lb/>
            ter inſcriptum vel hyperbolico regulariter circumſcri-
              <lb/>
            ptum Vq {2 b3/a + b;</s>
            <s xml:id="echoid-s3964" xml:space="preserve">} & </s>
            <s xml:id="echoid-s3965" xml:space="preserve">proinde ſector circuli, ellipſeos vel hyperbo-
              <lb/>
            læ eodem modo componitur ex a & </s>
            <s xml:id="echoid-s3966" xml:space="preserve">b quo ex b & </s>
            <s xml:id="echoid-s3967" xml:space="preserve">Vq {2 b3/a + b;</s>
            <s xml:id="echoid-s3968" xml:space="preserve">}
              <lb/>
            atque hinc etiam demonſtrari poteſt, quod ratio inter ſecto-
              <lb/>
            rem & </s>
            <s xml:id="echoid-s3969" xml:space="preserve">ejus triangulum datum non ſit analytica, </s>
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