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

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            <s xml:space="preserve">
              <pb o="80" file="0132" n="132" rhead="THEORIÆ"/>
            poſſe eſſe utcunque parvam, facile patet. </s>
            <s xml:space="preserve">Sit in fig. </s>
            <s xml:space="preserve">15. </s>
            <s xml:space="preserve">MQ
              <lb/>
              <note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">que magnam
                <lb/>
              vel parvam:
                <lb/>
              partis ſecundæ
                <lb/>
              demonſtratio.</note>
            ſegmentum axis utcunque parvum, vel magnum; </s>
            <s xml:space="preserve">ac detur area
              <lb/>
            utcunque magna, vel parva. </s>
            <s xml:space="preserve">Ea applicata ad MQ exhibebit
              <lb/>
            quandam altitudinem MN ita, ut, ducta NR parallela MQ,
              <lb/>
            ſit MNRQ æqualis areæ datæ, adeoque aſſumpta QS dupla
              <lb/>
            QR, area trianguli MSQ erit itidem æqualis areæ datæ.
              <lb/>
            </s>
            <s xml:space="preserve">Jam vero pro ſecundo caſu ſatis pater, poſſe curvam tranſi-
              <lb/>
            re infra rectam NR, uti tranſit XZ, cujus area idcirco eſſet
              <lb/>
            minor, quam area MNRQ; </s>
            <s xml:space="preserve">nam eſſet ejus pars. </s>
            <s xml:space="preserve">Quin im-
              <lb/>
            mo licet ordinata QV ſit utcunque magna; </s>
            <s xml:space="preserve">facile patet, poſ-
              <lb/>
            ſe arcum M a V ita accedere ad rectas MQ, QV; </s>
            <s xml:space="preserve">ut area
              <lb/>
            incluſa iis rectis, & </s>
            <s xml:space="preserve">ipſa curva, minuatur infra quoſcunque
              <lb/>
            determinatos limites. </s>
            <s xml:space="preserve">Poteſt enim jacere totus arcus intra duo
              <lb/>
            triangula Q a M, Q a V, quorum altitudines cum minui poſ-
              <lb/>
            ſint, quantum libuerit, ſtantibus baſibus MQ, QV, poteſt u-
              <lb/>
            tique area ultra quoſcunque limites imminui. </s>
            <s xml:space="preserve">Poſſet autem
              <lb/>
            ea area eſſe minor quacunque data; </s>
            <s xml:space="preserve">etiamſi QV eſſet aſym-
              <lb/>
            ptotus, qua de re paullo inferius.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">174. </s>
            <s xml:space="preserve">Pro primo autem caſu vel curva ſecet axem extra
              <lb/>
              <note position="left" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">Demonſtratio
                <lb/>
              primæ.</note>
            MQ, ut in T, vel in altero extremo, ut in M; </s>
            <s xml:space="preserve">fieri pote-
              <lb/>
            rit, ut ejus arcus TV, vel MV tranſeat per aliquod pun-
              <lb/>
            ctum V jacens ultra S, vel etiam per ipſum S ita, ut cur-
              <lb/>
            vatura illum ferat, quemadmodum figura exhibet, extra trian-
              <lb/>
            gulum MSQ, quo caſu patet, aream curvæ reſpondentem in-
              <lb/>
            tervallo MQ fore majorem, quam ſit area trianguli MSQ,
              <lb/>
            adeoque quam ſit area data; </s>
            <s xml:space="preserve">erit enim ejus trianguli area pars
              <lb/>
            areæ pertinentis ad curvam. </s>
            <s xml:space="preserve">Quod ſi curva etiam ſecaret ali-
              <lb/>
            cubi axem, ut in H inter M, & </s>
            <s xml:space="preserve">Q, tum vero fieri poſſet, ut
              <lb/>
            area reſpondens alteri e ſegmentis MH, QH eſſet major,
              <lb/>
            quam area data ſimul, & </s>
            <s xml:space="preserve">area alia aſſumpta, qua area aſſumpta
              <lb/>
            eſſet minor area reſpondens ſegmento alteri, adeoque exceſſus
              <lb/>
            prioris ſupra poſteriorem remaneret major, quam area data.</s>
            <s xml:space="preserve"/>
          </p>
          <p>
            <s xml:space="preserve">175. </s>
            <s xml:space="preserve">Area aſymptotica clauſa inter aſymptotum, & </s>
            <s xml:space="preserve">ordina-
              <lb/>
              <note position="left" xlink:label="note-0132-03" xlink:href="note-0132-03a" xml:space="preserve">Aream aſym-
                <lb/>
              ptoticam poſſe
                <lb/>
              eſſe infinitam,
                <lb/>
              vel finitam ma-
                <lb/>
              gnitudinis cu-
                <lb/>
              juſcunque.</note>
            tam quamvis, ut in fig. </s>
            <s xml:space="preserve">I BA ag, poteſt eſſe vel infi-
              <lb/>
            nita, vel finita magnitudinis cujuſvis ingentis, vel exiguæ.
              <lb/>
            </s>
            <s xml:space="preserve">Id quidem etiam geometrice demonſtrari poteſt, ſed multo
              <lb/>
            facilius demonſtratur calculo integrali admodum elementari; </s>
            <s xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0132-04" xlink:href="note-0132-04a" xml:space="preserve">Fig. 1.</note>
            & </s>
            <s xml:space="preserve">in Geometriæ ſublimioris elementis habentur theoremata,
              <lb/>
            ex quibus id admodum facile deducitur . </s>
            <s xml:space="preserve">Generaliter
              <note symbol="(l)" position="foot" xlink:label="note-0132-05" xlink:href="note-0132-05a" xml:space="preserve">Sit A a in Fig. I = x, ag = y; ac ſit x
                <emph style="super">m</emph>
              y
                <emph style="super">n</emph>
              = I; erit y = x
                <emph style="super">-{m/n}</emph>
              ,
                <lb/>
              v d x elementum areæ = x
                <emph style="super">-{m/n}</emph>
              d x, cujus integrale {n/n-m} x
                <emph style="super">{n-m}/n</emph>
              + A,</note>
            </s>
          </p>
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