Bošković, Ruđer Josip, Theoria philosophiae naturalis redacta ad unicam legem virium in natura existentium

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          <head xml:space="preserve">§. III.</head>
          <head style="it" xml:space="preserve">Solutio analytica Problematis determinantis
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          naturam Legis Virium. </head>
          <p>
            <s xml:space="preserve">25. </s>
            <s xml:space="preserve">UT haſce conditiones impleamus, formulam invenie-
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              <note position="right" xlink:label="note-0329-01" xlink:href="note-0329-01a" xml:space="preserve">Denominatio,
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              ac præparatio.</note>
            mus algebraicam, quæ ipſam continebit legem no-
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            ſtram, ſed hic elementa communia vulgaris Carte-
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            ſianæ algebræ ſupponemus ut nota, ſine quibus res omnino con-
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            fici nequaquam poteſt. </s>
            <s xml:space="preserve">Dicatur autem ordinata y, abſciſſa x,
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            ac ponatur x x = z. </s>
            <s xml:space="preserve">Capiantur omnium AE, AG, AI &</s>
            <s xml:space="preserve">c
              <lb/>
              <note position="right" xlink:label="note-0329-02" xlink:href="note-0329-02a" xml:space="preserve">Fig. 1.</note>
            valores cum ſigno negativo, & </s>
            <s xml:space="preserve">ſumma quadratorum omnium
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            ejuſmodi valorum dicatur a, ſumma productorum e binis
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            quibuſque quadratis b, ſumma productorum e ternis c, & </s>
            <s xml:space="preserve">
              <lb/>
            ita porro; </s>
            <s xml:space="preserve">productum autem ex omnibus dicatur f. </s>
            <s xml:space="preserve">Nume-
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            rus eorundem valorum dicatur m. </s>
            <s xml:space="preserve">His poſitis ponatur z
              <emph style="super">m</emph>
            +
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            az
              <emph style="super">m - 1</emph>
            + bz
              <emph style="super">m - 2</emph>
            + cz
              <emph style="super">m - 3</emph>
            &</s>
            <s xml:space="preserve">c.</s>
            <s xml:space="preserve">..</s>
            <s xml:space="preserve">.. </s>
            <s xml:space="preserve">+ f = P. </s>
            <s xml:space="preserve">Si ponatur P
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            = o, patet æquationis ejus omnes radices fore reales, & </s>
            <s xml:space="preserve">poſi-
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            tivas, nimirum ſola illa quadrata quantitatum AE, AG, AI
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            &</s>
            <s xml:space="preserve">c, qui erunt valores ipſius z; </s>
            <s xml:space="preserve">adeoque cum ob x x = z, ſit
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            x = ± √ z, patet, valores x fore tam AE, AG, AI poſi-
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            tivas, quam AE', AG' &</s>
            <s xml:space="preserve">c negativas.</s>
            <s xml:space="preserve"/>
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          <p>
            <s xml:space="preserve">26. </s>
            <s xml:space="preserve">Deinde ſumatur quæcunque quantitas data per z, & </s>
            <s xml:space="preserve">con-
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              <note position="right" xlink:label="note-0329-03" xlink:href="note-0329-03a" xml:space="preserve">Aſſumptio cu-
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              juſdam valoris
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              ad rem idonei.</note>
            ſtantes quomodocunque, dummodo non habeat ullum diviſo-
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            rem communem cum P, ne evaneſcente z, eadem evaneſcat,
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            ac facta x infiniteſima ordinis primi, evadat infiniteſima ordi-
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            nis ejuſdem, vel inferioris, ut erit quæcunque formula z
              <emph style="super">r</emph>
            +
              <lb/>
            gz
              <emph style="super">r - 1</emph>
            + bz
              <emph style="super">r - 2</emph>
            &</s>
            <s xml:space="preserve">c + l, quæ poſita = o habeat radices quot-
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            cunque imaginarias, & </s>
            <s xml:space="preserve">quotcunque, & </s>
            <s xml:space="preserve">quaſcunque reales,
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            (dummodo earum nulla ſit ex iis AE, AG, AI &</s>
            <s xml:space="preserve">c, ſive
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            poſitiva, ſive negativa) ſi deinde tota multiplicetur per z.
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            </s>
            <s xml:space="preserve">Ea dicatur Q.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">27. </s>
            <s xml:space="preserve">Si jam fiat P - Qf = o; </s>
            <s xml:space="preserve">dico, hanc æquationem ſatis-
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              <note position="right" xlink:label="note-0329-04" xlink:href="note-0329-04a" xml:space="preserve">Formula con-
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              tinens æqua-
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              tionem quæſi-
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              tam.</note>
            facere reliquis omnibus hujus curvæ conditionibus, & </s>
            <s xml:space="preserve">rite de-
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            terminato valore Q, poſſe infinitis modis ſatisfieri etiam po-
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            ſtremæ conditioni expoſitæ ſexto loco.</s>
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          <note symbol="(d)" position="foot" xml:space="preserve">Hæc ſolutio excerpta eſt ex diſſertati one De Lege Virium in Na-
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          tura exiſtentium. Accedit iis, quæ inde ſunt eruta, ſcholium 3 primo ad-
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          jectum in ha editione Veneta prima. Ipſum problema hic ſolvendum ha-
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          betur in ipſo hoc Opere parte 1 num. 117, ac ejus conditiones num. 118.</note>
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