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

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          <p>
            <s xml:id="echoid-s3219" xml:space="preserve">
              <pb o="76" file="0128" n="139" rhead="PHYSICES ELEMENTA"/>
            percurritur linea hæc, hac ipſâ lineâ repræſentatur; </s>
            <s xml:id="echoid-s3220" xml:space="preserve">tempus quo EB peragratur,
              <lb/>
            repræſentatur lineâ EF, quæ ſe habet ad EB, ut v ad c. </s>
            <s xml:id="echoid-s3221" xml:space="preserve">Punctum ve-
              <lb/>
            ro F determinatur ſi ex B ad CD ducatur BD perpendicularis, fiatque c, v:</s>
            <s xml:id="echoid-s3222" xml:space="preserve">:
              <lb/>
            BD, LD & </s>
            <s xml:id="echoid-s3223" xml:space="preserve">per L ad DC ducatur parallela, ſecabit hæc BE in puncto F:
              <lb/>
            </s>
            <s xml:id="echoid-s3224" xml:space="preserve">nam propter parallelas ED, FL, habemus BD, LD :</s>
            <s xml:id="echoid-s3225" xml:space="preserve">: BE, FE.</s>
            <s xml:id="echoid-s3226" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3227" xml:space="preserve">Ex hac demoſiſtratione etiam ſequitur, ſi punctum per lineas alias AM,
              <lb/>
            MB, progrediatur, quarum ultima ſecat LF in N, tempus motus repræ-
              <lb/>
            ſentari lineis AM, MN, ita ut determinandum ſit per quod punctum lineæ
              <lb/>
            CD punctum mobile tranſeat, quando ſumma talium linearum tempora repræ-
              <lb/>
            ſentantium eſt omnium minima; </s>
            <s xml:id="echoid-s3228" xml:space="preserve">quod ut fiat ad ſequentia attendendum.</s>
            <s xml:id="echoid-s3229" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3230" xml:space="preserve">Summas ab utraque parte recedendo à puncto quæſito augeri continuo;
              <lb/>
            </s>
            <s xml:id="echoid-s3231" xml:space="preserve">ideoque in eo puncto ſolo ſummas vicinas eſſe æquales, idcirco ſi punctum
              <lb/>
            hoc ſit E erunt æquales AE + EF & </s>
            <s xml:id="echoid-s3232" xml:space="preserve">Ae + ef ex qua æqualitate ſitus pun-
              <lb/>
              <note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">TAB XII.
                <lb/>
              fig. 7.</note>
            cti E deducendus eſt.</s>
            <s xml:id="echoid-s3233" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3234" xml:space="preserve">Centro A, radio A e deſcribatur circuli arcus eb; </s>
            <s xml:id="echoid-s3235" xml:space="preserve">Centro Bradiis Bf, & </s>
            <s xml:id="echoid-s3236" xml:space="preserve">
              <lb/>
            BE deſcribantur arcus E i, fg, eruntque æquales Ab + Eg & </s>
            <s xml:id="echoid-s3237" xml:space="preserve">A e + i f ſubtra-
              <lb/>
            ctis hiſce quantitatibus æqualibus ex AE + EF = Ae + ef, reſtanth E + gF = ei.
              <lb/>
            </s>
            <s xml:id="echoid-s3238" xml:space="preserve">Unde deducimus. </s>
            <s xml:id="echoid-s3239" xml:space="preserve">hE = ei-gF. </s>
            <s xml:id="echoid-s3240" xml:space="preserve">Propter triangula ſimilia eiE, fgF, & </s>
            <s xml:id="echoid-s3241" xml:space="preserve">Bfg,
              <lb/>
            Bie ut & </s>
            <s xml:id="echoid-s3242" xml:space="preserve">BFL, BED,</s>
          </p>
          <p>
            <s xml:id="echoid-s3243" xml:space="preserve">ei, g F :</s>
            <s xml:id="echoid-s3244" xml:space="preserve">: Ei, fg, :</s>
            <s xml:id="echoid-s3245" xml:space="preserve">: BE, Bf aut BF (difterentia enim eſt infi nite exigua)
              <lb/>
            :</s>
            <s xml:id="echoid-s3246" xml:space="preserve">: BD, BL. </s>
            <s xml:id="echoid-s3247" xml:space="preserve">Dividendo</s>
          </p>
          <p>
            <s xml:id="echoid-s3248" xml:space="preserve">ei, ei g F = bE:</s>
            <s xml:id="echoid-s3249" xml:space="preserve">: BD, BD-BL = LD; </s>
            <s xml:id="echoid-s3250" xml:space="preserve">id eſt ut velocitas infra lineam ad
              <lb/>
            velocitatem ſupra lineam.</s>
            <s xml:id="echoid-s3251" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3252" xml:space="preserve">Triangula eiE, ENO, ſunt ſimilia ut & </s>
            <s xml:id="echoid-s3253" xml:space="preserve">eb E & </s>
            <s xml:id="echoid-s3254" xml:space="preserve">e MP; </s>
            <s xml:id="echoid-s3255" xml:space="preserve">ergo</s>
          </p>
          <p>
            <s xml:id="echoid-s3256" xml:space="preserve">ei, Ee:</s>
            <s xml:id="echoid-s3257" xml:space="preserve">: EO, EN</s>
          </p>
          <p>
            <s xml:id="echoid-s3258" xml:space="preserve">bE, Ee:</s>
            <s xml:id="echoid-s3259" xml:space="preserve">: EP, Me = EN nam ſunt radii ejuſdem circuli; </s>
            <s xml:id="echoid-s3260" xml:space="preserve">E e c-
              <lb/>
            nim eſt infinite exigua.</s>
            <s xml:id="echoid-s3261" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3262" xml:space="preserve">Ex æquo ei, bE:</s>
            <s xml:id="echoid-s3263" xml:space="preserve">: EO, EP. </s>
            <s xml:id="echoid-s3264" xml:space="preserve">Sunt autem hæ lineæ coſinus angulorum
              <lb/>
              <note position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">320.</note>
            quos directiones motuum efficiunt cum linea CD quæ ſpatia ſeparat in quibus velo-
              <lb/>
            eitates differunt: </s>
            <s xml:id="echoid-s3265" xml:space="preserve">qui ergo coſinus directionum ſunt inter ut velocitates in ipſis
              <lb/>
            iltis directionibus, quando tempus eſt omnium breviſſimum.</s>
            <s xml:id="echoid-s3266" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3267" xml:space="preserve">Moveatur iterum corpus ex A & </s>
            <s xml:id="echoid-s3268" xml:space="preserve">tendat ad B, ea conditioneut dum tranſ-
              <lb/>
              <note position="left" xlink:label="note-0128-03" xlink:href="note-0128-03a" xml:space="preserve">321.</note>
            it lineas CD, IL, MN, OP, ſingulis vicibus velocitatem mutet, quæ-
              <lb/>
              <note position="left" xlink:label="note-0128-04" xlink:href="note-0128-04a" xml:space="preserve">TAB. XII.
                <lb/>
              fig. 8.</note>
            ritur qua lege movetur, poſitis hiſce lineis parallelis, ut tempore breviſſimo
              <lb/>
            ex A ad B perveniat.</s>
            <s xml:id="echoid-s3269" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3270" xml:space="preserve">Requiritur ut corpus ex A ad F perveniat tempore breviſſimo poſſibili,
              <lb/>
            ut & </s>
            <s xml:id="echoid-s3271" xml:space="preserve">ex E ad G, ex F ad H, & </s>
            <s xml:id="echoid-s3272" xml:space="preserve">ex G ad B, aliter enim in toto motu tem-
              <lb/>
            pus brevius dari poteſt. </s>
            <s xml:id="echoid-s3273" xml:space="preserve">Ideò coſinus angulorum quos motus directiones AE,
              <lb/>
              <note position="left" xlink:label="note-0128-05" xlink:href="note-0128-05a" xml:space="preserve">322.</note>
            EF, FG, GH, HB, efficiunt cum lineis, parallelis inter ſe, ſeparantibus ſpa-
              <lb/>
            tia in quibus diverſa eſt velocitas, ſunt reſpectivè inter ſe ut velocitates quibus ſin-
              <lb/>
            gu'a percurruntur.</s>
            <s xml:id="echoid-s3274" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3275" xml:space="preserve">Conſideremus nunc corpus quod gravitate deſcendit. </s>
            <s xml:id="echoid-s3276" xml:space="preserve">Celeritas continuo
              <lb/>
            deſcendendo augetur, & </s>
            <s xml:id="echoid-s3277" xml:space="preserve">ad eandem profunditatem ubique eſt eadem ,
              <note symbol="*" position="left" xlink:label="note-0128-06" xlink:href="note-0128-06a" xml:space="preserve">271.</note>
            meris ergo, & </s>
            <s xml:id="echoid-s3278" xml:space="preserve">inter ſe infinitè parum diſtantibus, planis horizontalibus divi-
              <lb/>
            duntur ſpatia in quibus celeritas variat: </s>
            <s xml:id="echoid-s3279" xml:space="preserve">Linea ergo celerrimi deſcenſus inter
              <lb/>
              <note position="left" xlink:label="note-0128-07" xlink:href="note-0128-07a" xml:space="preserve">323.</note>
            duo puncta eſt cujus tangens ubique cum borizonte eſſicit angulum, cujus coſinus
              <lb/>
            velocitati cadendo acquiſitæ proportionalis eſt , id eſt radici quadratæ
              <note symbol="*" position="left" xlink:label="note-0128-08" xlink:href="note-0128-08a" xml:space="preserve">322.</note>
            dinis per quam corpus cecidit . </s>
            <s xml:id="echoid-s3280" xml:space="preserve">Hanc autem eſſe Cycloïdis proprietatem
              <note symbol="*" position="left" xlink:label="note-0128-09" xlink:href="note-0128-09a" xml:space="preserve">255-271.</note>
            monſtramus.</s>
            <s xml:id="echoid-s3281" xml:space="preserve"/>
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