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

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              <pb o="114" file="0166" n="166" rhead="THEORIÆ"/>
            diſtantiarum ſupra omnes ulteriores æquari progreſſui plani to-
              <lb/>
            ties ſumpto, quot puncta habentur, & </s>
            <s xml:space="preserve">in regreſſu deſtruitur e
              <lb/>
            contrario, quidquid in ejuſmodi progreſſu eſt factum, atque id-
              <lb/>
            circo ad æqualitatem reditur. </s>
            <s xml:space="preserve">Verum ut demonſtratio quam-
              <lb/>
            accuratiſſima evadat, exprimat in fig. </s>
            <s xml:space="preserve">36 recta AB planum
              <lb/>
              <note position="left" xlink:label="note-0166-01" xlink:href="note-0166-01a" xml:space="preserve">Fig. 36.</note>
            diſtantiarum æqualium, & </s>
            <s xml:space="preserve">CD planum ipſi parallelum, ac o-
              <lb/>
            mnia puncta diſtribui poterunt in claſſes tres, in quorum prima
              <lb/>
            ſint omnia puncta jacentia citra utrumque planum, ut punctum
              <lb/>
            E; </s>
            <s xml:space="preserve">in ſecunda omnia puncta jacentia inter utrumque, ut F,
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            in tertia omnia puncta adhuc jacentia ultra utrumque, ut G.
              <lb/>
            </s>
            <s xml:space="preserve">Rectæ autem per ipſa ductæ in directione data quacunque, oc-
              <lb/>
            currant rectæ AB in M, H, K, & </s>
            <s xml:space="preserve">rectæ CD in N, I,
              <lb/>
            L; </s>
            <s xml:space="preserve">ac ſit quædam recta directionis ejuſdem ipſis AB, CD oc-
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            currens in O, P. </s>
            <s xml:space="preserve">Patet, ipſam OP fore æqualem ipſis M ,
              <lb/>
            HI, KL. </s>
            <s xml:space="preserve">Dicatur jam ſumma omnium punctorum E primæ
              <lb/>
            claſſis E, & </s>
            <s xml:space="preserve">diſtantiarum omnium EM ſumma e; </s>
            <s xml:space="preserve">punctorum
              <lb/>
            F ſecundæ claſſis F, & </s>
            <s xml:space="preserve">diſtantiarum f; </s>
            <s xml:space="preserve">punctorum G tertiæ
              <lb/>
            claſſis ſumma G, & </s>
            <s xml:space="preserve">diſtantiarum earundem g; </s>
            <s xml:space="preserve">diſtantia vero
              <lb/>
            OP dicatur O. </s>
            <s xml:space="preserve">Patet, ſummam omnium MN fore ExO; </s>
            <s xml:space="preserve">
              <lb/>
            ſummam omnium HI fore, FxO; </s>
            <s xml:space="preserve">ſummam omnium KL
              <lb/>
            fore GxO; </s>
            <s xml:space="preserve">erit autem quævis EN = EM + MN; </s>
            <s xml:space="preserve">quæ-
              <lb/>
            vis FI = HI-FH; </s>
            <s xml:space="preserve">quævis GL = KG-KL. </s>
            <s xml:space="preserve">Qua-
              <lb/>
            re ſumma omnium EN erit e + ExO; </s>
            <s xml:space="preserve">ſumma omnium FI
              <lb/>
            = FxO-f, & </s>
            <s xml:space="preserve">ſumma omnium GL =g-GxO; </s>
            <s xml:space="preserve">
              <lb/>
            adeoque ſumma omnium diſtantiarum punctorum jacentium ci-
              <lb/>
            tra planum CD, primæ nimirum, ac ſecundæ claſſis, erit e
              <lb/>
            + ExO + FxO-f, & </s>
            <s xml:space="preserve">ſumma omnium jacentium ul-
              <lb/>
            tra, nimirum claſſis tertiæ, erit g-GxO. </s>
            <s xml:space="preserve">Quare exceſſus
              <lb/>
            prioris ſummæ ſupra ſecundam erit e + ExO + FxO-f
              <lb/>
            -g + GxO; </s>
            <s xml:space="preserve">adeoque ſi priu
              <emph style="super">s</emph>
            fuerit e=f + g; </s>
            <s xml:space="preserve">deleto
              <lb/>
            e-f-g, totus exceſſus erit ExO + FxO + GxO, ſive (E
              <lb/>
            + F + G)xO, ſumma omnium punctorum ducta in diſtan-
              <lb/>
            tiam planorum; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">vice verſa ſi is exceſſus reſpectu ſecundi pla-
              <lb/>
            ni BC fuerit æqualis huic ſummæ ductæ in diſtantiam O, o-
              <lb/>
            portebit, e-f-g æquetur nihilo, adeoque ſit e= f + g, ni-
              <lb/>
            mirum reſpectu primi plani AB ſummas diſtantiarum hinc,
              <lb/>
            & </s>
            <s xml:space="preserve">inde æquales eſſe.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">244. </s>
            <s xml:space="preserve">Si aliqua puncta ſint in altero ex iis planis, ea ſupe-
              <lb/>
              <note position="left" xlink:label="note-0166-02" xlink:href="note-0166-02a" xml:space="preserve">Complemen-
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              tum demon-
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              ſtrationis, ut
                <lb/>
              extendatur ad
                <lb/>
              omnes caſus.</note>
            rioribus formulis contineri poſſunt, concepta zero ſingulorum
              <lb/>
            diſtantia a plano, in quo jacent; </s>
            <s xml:space="preserve">ſed & </s>
            <s xml:space="preserve">ii caſus involvi facile
              <lb/>
            poſſent, concipiendo alias binas punctorum claſſes; </s>
            <s xml:space="preserve">quorum
              <lb/>
            priora ſint in priore plano A B, poſteriora in poſteriore CB,
              <lb/>
            quæ quidem nihil rem turbant: </s>
            <s xml:space="preserve">nam prioris claſſis diſtantiæ a
              <lb/>
            priore plano erunt omnes ſimul zero, & </s>
            <s xml:space="preserve">a poſteriore æquabun-
              <lb/>
            tur diſtantiæ O ductæ in eorum numerum, quæ ſumma acce-
              <lb/>
            dit priori ſummæ punctorum jacentium citra; </s>
            <s xml:space="preserve">poſterioris au-
              <lb/>
            tem claſſis diſtantiæ a priore erant prius ſimul æquales ſummæ
              <lb/>
            ipſorum ductæ itidem in O, & </s>
            <s xml:space="preserve">deinde fiunt nihil; </s>
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