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

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          <pb o="278" file="0330" n="330" rhead="SUPPLEMENTA. §. III."/>
          <p>
            <s xml:space="preserve">28. </s>
            <s xml:space="preserve">Nam inprimis, quoniam valores P, & </s>
            <s xml:space="preserve">Q poſiti = o
              <lb/>
              <note position="left" xlink:label="note-0330-01" xlink:href="note-0330-01a" xml:space="preserve">Æquationem
                <lb/>
              fore ſimplicem
                <lb/>
              non reſolubi-
                <lb/>
              lem in plures.</note>
            nullam habent radicem communem, nullum habebunt diviſo-
              <lb/>
            rem communem. </s>
            <s xml:space="preserve">Hinc hæc æquatio non poteſt per diviſio-
              <lb/>
            nem reduci ad binas, adeoque non eſt compoſita ex binis æqua-
              <lb/>
            tionibus, ſed ſimplex, & </s>
            <s xml:space="preserve">proinde ſimplicem quandam curvam
              <lb/>
            continuam exhibet, quæ ex aliis non componitur. </s>
            <s xml:space="preserve">Quod erat
              <lb/>
            primum.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">29. </s>
            <s xml:space="preserve">Deinde curva hujuſmodi ſecabit axem C' AC in iis o-
              <lb/>
              <note position="left" xlink:label="note-0330-02" xlink:href="note-0330-02a" xml:space="preserve">Exhibituram
                <lb/>
              datum nume-
                <lb/>
              rum interſecti-
                <lb/>
              onum curvæ,
                <lb/>
              in datis pun-
                <lb/>
              ctis.</note>
            mnibus, & </s>
            <s xml:space="preserve">ſolis punctis, E, G, I &</s>
            <s xml:space="preserve">c E', G', &</s>
            <s xml:space="preserve">c. </s>
            <s xml:space="preserve">Nam ea
              <lb/>
            ſecabit axem C' AC ſolum in iis punctis, in quibus y = o, & </s>
            <s xml:space="preserve">
              <lb/>
            ſecabit in omnibus. </s>
            <s xml:space="preserve">Porro ubi fuerit y = o, erit & </s>
            <s xml:space="preserve">Qy = o,
              <lb/>
            adeoque ob P - Q y = o; </s>
            <s xml:space="preserve">erit P = o. </s>
            <s xml:space="preserve">Id autem continget ſo-
              <lb/>
            lum in iis punctis, in quibus z fuerit una e radicibus æqua-
              <lb/>
            tionis P = o, nimirum, ut ſupra vidimus, in punctis E, G,
              <lb/>
            I, vel E', G', &</s>
            <s xml:space="preserve">c. </s>
            <s xml:space="preserve">Quare ſolum in his punctis evaneſcet y, & </s>
            <s xml:space="preserve">
              <lb/>
            curva axem ſecabit. </s>
            <s xml:space="preserve">Secaturam autem in his omnibus patet
              <lb/>
            ex eo, quod in his omnibus punctis erit P = o. </s>
            <s xml:space="preserve">Quare erit
              <lb/>
            etiam Qy = o. </s>
            <s xml:space="preserve">Non erit autem Q = o; </s>
            <s xml:space="preserve">cum nulla ſit radix
              <lb/>
            communis æquationum P = o, & </s>
            <s xml:space="preserve">Q = o. </s>
            <s xml:space="preserve">Quare erit y = o, & </s>
            <s xml:space="preserve">
              <lb/>
            curva axem ſecabit. </s>
            <s xml:space="preserve">Quod erat ſecundum.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">30. </s>
            <s xml:space="preserve">Præterea cum ſit P - Qx = o, erit y = {P/Q}; </s>
            <s xml:space="preserve">determinata
              <lb/>
              <note position="left" xlink:label="note-0330-03" xlink:href="note-0330-03a" xml:space="preserve">Singulas ordi-
                <lb/>
              natas reſponſu-
                <lb/>
              ras ſingulis ab-
                <lb/>
              ſciſſis.</note>
            autem utcunque abſciſſa x, habebitur determinata quædam z,
              <lb/>
            adeoque & </s>
            <s xml:space="preserve">P, Q erunt unicæ, & </s>
            <s xml:space="preserve">determinatæ. </s>
            <s xml:space="preserve">Erit igitur
              <lb/>
            etiam y unica, & </s>
            <s xml:space="preserve">determinata; </s>
            <s xml:space="preserve">ac proinde reſpondebunt ſin-
              <lb/>
            gulis abſciſſis z ſingulæ tantum ordinatæ y. </s>
            <s xml:space="preserve">Quod erat ter-
              <lb/>
            tium.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">31. </s>
            <s xml:space="preserve">Rurſus ſive x aſſumatur poſitiva, ſive negativa, dum-
              <lb/>
              <note position="left" xlink:label="note-0330-04" xlink:href="note-0330-04a" xml:space="preserve">Abſciſſis hinc
                <lb/>
              inde æqualibus
                <lb/>
              reſponſuras æ-
                <unsure/>
                <lb/>
              quales ordina-
                <lb/>
              tas.</note>
            modo ejuſdem longitudinis ſit, ſemper valor z = x x erit idem;
              <lb/>
            </s>
            <s xml:space="preserve">ac proinde valores tam P, quam Q erunt ſemper iidem. </s>
            <s xml:space="preserve">Qua-
              <lb/>
            re ſemper eadem y. </s>
            <s xml:space="preserve">Sumptis igitur abſciſſis z æqualibus hinc,
              <lb/>
            & </s>
            <s xml:space="preserve">inde ab A, altera poſitiva, altera negativa, reſpondebunt or-
              <lb/>
            dinatæ æquales. </s>
            <s xml:space="preserve">Quod erat quartum.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">32. </s>
            <s xml:space="preserve">Si autem x minuatur in infinitum, ſive ea poſitiva ſit,
              <lb/>
              <note position="left" xlink:label="note-0330-05" xlink:href="note-0330-05a" xml:space="preserve">Primum ar-
                <lb/>
              cum fore crus
                <lb/>
              aſymptoticum
                <lb/>
              cum area infi-
                <lb/>
              nita.</note>
            ſive negativa; </s>
            <s xml:space="preserve">ſemper z minuetur in infinitum, & </s>
            <s xml:space="preserve">evadet infi-
              <lb/>
            niteſima ordinis ſecundi. </s>
            <s xml:space="preserve">Quare in valore P decreſcent in infini-
              <lb/>
            tum omnes termini præter y, quia omnes præter eum multipli-
              <lb/>
            cantur per z, adeoque valor P erit adhuc finitus. </s>
            <s xml:space="preserve">Valor autem
              <lb/>
            Q, qui habet formulam ductam in z totam, minuetur in infi-
              <lb/>
            nitum, eritque infiniteſimus ordinis ſecundi. </s>
            <s xml:space="preserve">Igitur {P/Q} = y au-
              <lb/>
            gebitur in infinitum ita, ut evadat infinita ordinis ſecundi.
              <lb/>
            </s>
            <s xml:space="preserve">Quare curva habebit pro aſymptoto rectam AB, & </s>
            <s xml:space="preserve">area BAED
              <lb/>
            excreſcet in infinitum, & </s>
            <s xml:space="preserve">ſi ordinatæ y poſitivæ aſſumantur ad
              <lb/>
            partes AB, & </s>
            <s xml:space="preserve">exprimant vires repulſivas, arcus aſymptoticus
              <lb/>
            ED jacebit ad partes ipſas AB. </s>
            <s xml:space="preserve">Quod erat quintum.</s>
            <s xml:space="preserve"/>
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