Gravesande, Willem Jacob 's, Physices elementa mathematica, experimentis confirmata sive introductio ad philosophiam Newtonianam; Tom. 1

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            <s xml:id="echoid-s786" xml:space="preserve">
              <pb o="11" file="0041" n="41" rhead="MATHEMATICA. LIB. I. CAP. IV."/>
            cunda infinite exigua eſt reſpectu primæ, demonſtramus enim angulum quemcun-
              <lb/>
            quein ſecunda ſuperari ab angulo quocunque, id eſt, utcumque exiguo, in prima.</s>
            <s xml:id="echoid-s787" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s788" xml:space="preserve">Sit c tertia proportionalis ipſis a & </s>
            <s xml:id="echoid-s789" xml:space="preserve">b, utcunque ſumtis; </s>
            <s xml:id="echoid-s790" xml:space="preserve">ergo ac = bb.
              <lb/>
            </s>
            <s xml:id="echoid-s791" xml:space="preserve">Multiplicando per c æquationem ax = yy, habemus acx = yyc, id eſt bbx = yyc. </s>
            <s xml:id="echoid-s792" xml:space="preserve">
              <lb/>
            in ſecunda curva bbx valet z
              <emph style="super">3</emph>
            ; </s>
            <s xml:id="echoid-s793" xml:space="preserve">ergo z
              <emph style="super">3</emph>
            = yyc, ſi abſciſſa x fuerit eademiu
              <lb/>
            utraque curva.</s>
            <s xml:id="echoid-s794" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s795" xml:space="preserve">Ex æquatione hac deducimus z, c:</s>
            <s xml:id="echoid-s796" xml:space="preserve">:yy, z
              <emph style="super">2</emph>
            : </s>
            <s xml:id="echoid-s797" xml:space="preserve">unde patet yy ſuperari à zz, id
              <lb/>
            eſt, y minorem eſſe z, quamdiu hæc ac ſuperatur, unde ſequitur curvam ſe-
              <lb/>
            cundam dum ex A profluit, antequam z valeat c, inter tangentem & </s>
            <s xml:id="echoid-s798" xml:space="preserve">curvam
              <lb/>
            primam dari quod univerſaliter obtineri hac demonſtratione conſtat.</s>
            <s xml:id="echoid-s799" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s800" xml:space="preserve">Ponamus nunc tertiam dari curvam AI, cujus axis etiam eſt AD, & </s>
            <s xml:id="echoid-s801" xml:space="preserve">cujus æ-
              <lb/>
              <note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">42.</note>
            quatio, manentibus iiſdem abſciſſis x, ſit dx = u
              <emph style="super">4</emph>
            ; </s>
            <s xml:id="echoid-s802" xml:space="preserve">u eſt ordinata quæ-
              <lb/>
            cunque; </s>
            <s xml:id="echoid-s803" xml:space="preserve">& </s>
            <s xml:id="echoid-s804" xml:space="preserve">d linea determinata; </s>
            <s xml:id="echoid-s805" xml:space="preserve">hanc ſi augeamus, mutamus curvam & </s>
            <s xml:id="echoid-s806" xml:space="preserve">mi-
              <lb/>
            nuimus angulum quem curva cum tangente AF efficit; </s>
            <s xml:id="echoid-s807" xml:space="preserve">formaturque hiſce
              <lb/>
            curvis tertia claſſis angulorum, qui in infinitum minui poſſunt, & </s>
            <s xml:id="echoid-s808" xml:space="preserve">in qua
              <lb/>
            nullus datur angulus, qui non ſuperetur ab angulo quocunque in ſecunda.</s>
            <s xml:id="echoid-s809" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s810" xml:space="preserve">Datis b & </s>
            <s xml:id="echoid-s811" xml:space="preserve">d quibuſcunque, ſit bb ad dd; </s>
            <s xml:id="echoid-s812" xml:space="preserve">ut dad quartam quam dicamus e; </s>
            <s xml:id="echoid-s813" xml:space="preserve">erit
              <lb/>
            ergo bbe = d
              <emph style="super">3</emph>
            , & </s>
            <s xml:id="echoid-s814" xml:space="preserve">æquatio curvæ bbx = z
              <emph style="super">3</emph>
            mutabitur in hanc bbex = d
              <emph style="super">3</emph>
            x
              <lb/>
            = z
              <emph style="super">3</emph>
            e; </s>
            <s xml:id="echoid-s815" xml:space="preserve">ideoque z
              <emph style="super">3</emph>
            e = u
              <emph style="super">4</emph>
            , ſi agatur de iiſdem abſciſſis in utraque curva; </s>
            <s xml:id="echoid-s816" xml:space="preserve">id-
              <lb/>
            circo u, e:</s>
            <s xml:id="echoid-s817" xml:space="preserve">:z
              <emph style="super">3</emph>
            , u
              <emph style="super">3</emph>
            ; </s>
            <s xml:id="echoid-s818" xml:space="preserve">ergo u ſuperat z, quamdiu e ſuperat u, &</s>
            <s xml:id="echoid-s819" xml:space="preserve">, exeundo ex A,
              <lb/>
            curva, cujus abſciſſæ ſunt u, tranſit inter AF & </s>
            <s xml:id="echoid-s820" xml:space="preserve">aliam curvam Q. </s>
            <s xml:id="echoid-s821" xml:space="preserve">D. </s>
            <s xml:id="echoid-s822" xml:space="preserve">E.</s>
            <s xml:id="echoid-s823" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s824" xml:space="preserve">Curvæ, quarum æquatio eſt f
              <emph style="super">4</emph>
            x = t
              <emph style="super">3</emph>
            poſita f quantitate determinatâ in ſin-
              <lb/>
            gulis curvis, & </s>
            <s xml:id="echoid-s825" xml:space="preserve">t ordinata quæcunque, dabunt novam claſſem angulorum mino-
              <lb/>
              <note position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">43.</note>
            rum omnibus memoratis, & </s>
            <s xml:id="echoid-s826" xml:space="preserve">eodem modo claſſes in infinitum formari poſſunt,
              <lb/>
            ſemperque omnes anguli in claſſe quacunque ſuperantur ab omnibus angulis in claſ-
              <lb/>
            ſe præcedenti, & </s>
            <s xml:id="echoid-s827" xml:space="preserve">ſuperant omnes angulos in claſſe ſequenti.</s>
            <s xml:id="echoid-s828" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s829" xml:space="preserve">Inter duas claſſes quaſcunque datur ſeries infinita claſſium; </s>
            <s xml:id="echoid-s830" xml:space="preserve">quæ omnes eandem
              <lb/>
              <note position="right" xlink:label="note-0041-03" xlink:href="note-0041-03a" xml:space="preserve">44.</note>
            proprietatem babent, ut angulus quicunque unius ſit infinite parvus reſpectu angulo-
              <lb/>
            rum claſſis præcedentis, id eſt, ut ab omnibus ſuperetur, & </s>
            <s xml:id="echoid-s831" xml:space="preserve">infinite magnusre-
              <lb/>
            ſpectu claſſis ſequentis, cujus omnes angulos ſuperat.</s>
            <s xml:id="echoid-s832" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s833" xml:space="preserve">Curvæ ax = yy & </s>
            <s xml:id="echoid-s834" xml:space="preserve">bbx = z
              <emph style="super">3</emph>
            claſſes formant diverſas; </s>
            <s xml:id="echoid-s835" xml:space="preserve">quia ordinatarum di-
              <lb/>
            menſio z
              <emph style="super">3</emph>
            id ſecunda unitate ſuperat dimenſionem y
              <emph style="super">2</emph>
            primæ curvæ; </s>
            <s xml:id="echoid-s836" xml:space="preserve">demon-
              <lb/>
            ſtrabimus autem claſſes differre, quantumvis parum hæ dimenſiones differant,
              <lb/>
            unde conſtabit propoſitum: </s>
            <s xml:id="echoid-s837" xml:space="preserve">quia inter hoſce numeros 2 & </s>
            <s xml:id="echoid-s838" xml:space="preserve">3, & </s>
            <s xml:id="echoid-s839" xml:space="preserve">alios quoſ-
              <lb/>
            cunque, innumeridaripoſſunt, quiinter ſe differunt, quorum nulli, quantumvis
              <lb/>
            parum differentes, dari poſſunt, inter quos iterum non alii innumeri dari
              <lb/>
            poſſint.</s>
            <s xml:id="echoid-s840" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s841" xml:space="preserve">Sit ax = yy & </s>
            <s xml:id="echoid-s842" xml:space="preserve">g 1 {1/10} x = s 2 {1/10} id eſt, g {11/10} x = s {21/10}; </s>
            <s xml:id="echoid-s843" xml:space="preserve">ordinatas deſignat s, & </s>
            <s xml:id="echoid-s844" xml:space="preserve">g
              <lb/>
            conſtantem lineam, quamdiu curva non mutatur. </s>
            <s xml:id="echoid-s845" xml:space="preserve">Fiat ut a ad g, ita g {1/10} ad
              <lb/>
            quartam quantitatem, quæ dicatur b {1/10}; </s>
            <s xml:id="echoid-s846" xml:space="preserve">ergo g{11/10} = ab{1/10}; </s>
            <s xml:id="echoid-s847" xml:space="preserve">multiplican-
              <lb/>
            do per b {1/10} æquationem ax = yy datur ab {1/10} x = g {11/10} x = y
              <emph style="super">2</emph>
            b {1/10} = s {21/10};
              <lb/>
            </s>
            <s xml:id="echoid-s848" xml:space="preserve">under deducimus s {1/10}, b {1/10}:</s>
            <s xml:id="echoid-s849" xml:space="preserve">: yy. </s>
            <s xml:id="echoid-s850" xml:space="preserve">ss. </s>
            <s xml:id="echoid-s851" xml:space="preserve">Idcirco in viciniis puncti A, ubi s
              <lb/>
            neceſſario minor eſt determinatâ b, erit etiam y minor s unde liquet quod de
              <lb/>
            angulis dictum.</s>
            <s xml:id="echoid-s852" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s853" xml:space="preserve">Inter duas claſſes quaſcunque, quantitatum, quæ in infinitum differunt, da-
              <lb/>
              <note position="right" xlink:label="note-0041-04" xlink:href="note-0041-04a" xml:space="preserve">45.</note>
            ri in infinitum claſſes intermedias ex conſideratione mediarum proportiona-
              <lb/>
            lium etiam deducitur.</s>
            <s xml:id="echoid-s854" xml:space="preserve"/>
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