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-s3148" xml:space="preserve">
              <pb o="75" file="0127" n="138" rhead="MATHEMATICA. LIB. I. CAP XIX."/>
            bemus ergo etiam demonſtratam propoſitionem in n, 386. </s>
            <s xml:id="echoid-s3149" xml:space="preserve">memoratam.</s>
            <s xml:id="echoid-s3150" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3151" xml:space="preserve">Supereſt ut, quæ de evolutione Cycloidis in n. </s>
            <s xml:id="echoid-s3152" xml:space="preserve">283. </s>
            <s xml:id="echoid-s3153" xml:space="preserve">dicta ſunt, de mon-
              <lb/>
            ſtremus.</s>
            <s xml:id="echoid-s3154" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3155" xml:space="preserve">Detur iterum eadem Cycloïs ADB; </s>
            <s xml:id="echoid-s3156" xml:space="preserve">baſis AF; </s>
            <s xml:id="echoid-s3157" xml:space="preserve">axis FB; </s>
            <s xml:id="echoid-s3158" xml:space="preserve">FEB ſe-
              <lb/>
              <note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">318.</note>
            micirculus. </s>
            <s xml:id="echoid-s3159" xml:space="preserve">Producatur BF ad Cita, ut BF & </s>
            <s xml:id="echoid-s3160" xml:space="preserve">FC ſint æquales; </s>
            <s xml:id="echoid-s3161" xml:space="preserve">formato-
              <lb/>
            que parallelogrammo AfCF; </s>
            <s xml:id="echoid-s3162" xml:space="preserve">detur ſemicirculus A mf, qui ſemicirculo FEB,
              <lb/>
            æqualis erit; </s>
            <s xml:id="echoid-s3163" xml:space="preserve">ut & </s>
            <s xml:id="echoid-s3164" xml:space="preserve">ſemi cycloïs A qC, cujus axis eſt A f & </s>
            <s xml:id="echoid-s3165" xml:space="preserve">quæ æqualis
              <lb/>
            eſt ſemi-cycloïdi ADB. </s>
            <s xml:id="echoid-s3166" xml:space="preserve">Concipiamus etiam filum fixum in C & </s>
            <s xml:id="echoid-s3167" xml:space="preserve">cycloïdi
              <lb/>
            C q A applicatum, evolvi.</s>
            <s xml:id="echoid-s3168" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3169" xml:space="preserve">Ponamus filum ad hunc perveniſſe ſitum, ut cum cycloïde tantum con-
              <lb/>
            veniat à C ad q, & </s>
            <s xml:id="echoid-s3170" xml:space="preserve">ulterius protendi juxta tangentem ad curvam in q: </s>
            <s xml:id="echoid-s3171" xml:space="preserve">ſi
              <lb/>
            linea q Q æqualis ſit arcui q A, cui filum, nunc tenſum, fuit applicatum, e-
              <lb/>
            rit Q fili extremitas.</s>
            <s xml:id="echoid-s3172" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3173" xml:space="preserve">Ducatur q p ad baſin parallela ſemicirculum A m f ſecans in m, ex quo
              <lb/>
            puncto ducatur linea m A ad A, ſunt m A & </s>
            <s xml:id="echoid-s3174" xml:space="preserve">q N parallelæ & </s>
            <s xml:id="echoid-s3175" xml:space="preserve">æquales;</s>
            <s xml:id="echoid-s3176" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">285 316.</note>
            ſed q A, ideoque q Q dupla eſt m A aut qN; </s>
            <s xml:id="echoid-s3177" xml:space="preserve">ſunt ergo æquales N q, NQ;
              <lb/>
            </s>
            <s xml:id="echoid-s3178" xml:space="preserve">idcirco ſi per Q ad AF & </s>
            <s xml:id="echoid-s3179" xml:space="preserve">p q detur parallela QP, erunt æquales PF, A p; </s>
            <s xml:id="echoid-s3180" xml:space="preserve">
              <lb/>
            ergo etiam erunt æquales arcus FM, A m; </s>
            <s xml:id="echoid-s3181" xml:space="preserve">ut & </s>
            <s xml:id="echoid-s3182" xml:space="preserve">anguli MFA, m AF ;</s>
            <s xml:id="echoid-s3183" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0127-03" xlink:href="note-0127-03a" xml:space="preserve">32 27
                <lb/>
              El. III.</note>
            & </s>
            <s xml:id="echoid-s3184" xml:space="preserve">eſt FM, parallela A m , ut & </s>
            <s xml:id="echoid-s3185" xml:space="preserve">Q q; </s>
            <s xml:id="echoid-s3186" xml:space="preserve">unde ſequitur FMQN eſſe paral.</s>
            <s xml:id="echoid-s3187" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0127-04" xlink:href="note-0127-04a" xml:space="preserve">27. El. I.</note>
            lelogrammum, & </s>
            <s xml:id="echoid-s3188" xml:space="preserve">æquales eſie FN, QM; </s>
            <s xml:id="echoid-s3189" xml:space="preserve">ſunt etiam æquales qm, AN,
              <lb/>
            in parallelogrammo m AN q.</s>
            <s xml:id="echoid-s3190" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3191" xml:space="preserve">Linea mq, aut AN, æqualis eſt arcui A m , aut arcui FM; </s>
            <s xml:id="echoid-s3192" xml:space="preserve">AF,
              <note symbol="*" position="right" xlink:label="note-0127-05" xlink:href="note-0127-05a" xml:space="preserve">325.</note>
            qualis eſt ſemicirculo FMB ; </s>
            <s xml:id="echoid-s3193" xml:space="preserve">idcirco NF, aut QM, æqualis eſt
              <note symbol="*" position="right" xlink:label="note-0127-06" xlink:href="note-0127-06a" xml:space="preserve">282. 315.</note>
            MEB, & </s>
            <s xml:id="echoid-s3194" xml:space="preserve">punctum Q, ideſt fili extremitas datur in cycloïde ADB ,
              <note symbol="*" position="right" xlink:label="note-0127-07" xlink:href="note-0127-07a" xml:space="preserve">315.</note>
            integram extremitas hæc percurret dum totum filum evolvitur.</s>
            <s xml:id="echoid-s3195" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div493" type="section" level="1" n="157">
          <head xml:id="echoid-head226" xml:space="preserve">SHOLIUM 4.</head>
          <head xml:id="echoid-head227" style="it" xml:space="preserve">De linea celerrimi deſcenſus.</head>
          <p>
            <s xml:id="echoid-s3196" xml:space="preserve">Monuimus ſuperius , quod ex deinde demonſtratis etiam patuit ; </s>
            <s xml:id="echoid-s3197" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0127-08" xlink:href="note-0127-08a" xml:space="preserve">289.</note>
            per arcus circuli exiguos breviori tempore deſcendere, quàm per ho-
              <lb/>
              <note symbol="*" position="right" xlink:label="note-0127-09" xlink:href="note-0127-09a" xml:space="preserve">307.</note>
            rum arcuum chordas. </s>
            <s xml:id="echoid-s3198" xml:space="preserve">Unde patet corpus quod à puncto ad punctum de-
              <lb/>
            ſcendit, quando puncta ambo non in eadem verticali dantur, ut viam ſuam
              <lb/>
            breviſſimo tempore peragat, non debere per lineam rectam incedere. </s>
            <s xml:id="echoid-s3199" xml:space="preserve">Quam-
              <lb/>
            nam autem lineam ſequi debeat, lubet hìc demonſtrate; </s>
            <s xml:id="echoid-s3200" xml:space="preserve">quia ad hoc uſu ve-
              <lb/>
            niunt quæ in ſuperiori ſcholio de Cycloïde demonſtrata ſunt.</s>
            <s xml:id="echoid-s3201" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3202" xml:space="preserve">Sint puncta duo A & </s>
            <s xml:id="echoid-s3203" xml:space="preserve">B, lineâ CD ſeparata; </s>
            <s xml:id="echoid-s3204" xml:space="preserve">moveatur punctum & </s>
            <s xml:id="echoid-s3205" xml:space="preserve">ab A
              <lb/>
              <note position="right" xlink:label="note-0127-10" xlink:href="note-0127-10a" xml:space="preserve">319.</note>
            tendat ad B; </s>
            <s xml:id="echoid-s3206" xml:space="preserve">ſed ea lege, ut antequam ad lineam CD perveniat, feratur velo-
              <lb/>
              <note position="right" xlink:label="note-0127-11" xlink:href="note-0127-11a" xml:space="preserve">TAB. XII.
                <lb/>
              fig. 6.</note>
            citate quam dicimus v, ubi autem tranſivit lineam hanc incedat celeritate
              <lb/>
            majori quam vocamus c: </s>
            <s xml:id="echoid-s3207" xml:space="preserve">Ponamus ulterius punctum velocitatibus ſingulis
              <lb/>
            rectas vias percurrere; </s>
            <s xml:id="echoid-s3208" xml:space="preserve">ideòque moveri per rectam AB, aut lineas AE, EB
              <lb/>
            peragrare: </s>
            <s xml:id="echoid-s3209" xml:space="preserve">determinandum, quomodo motum dirigere debeat, ut tempore
              <lb/>
            omnium breviſſimo perveniat ex A in B,</s>
          </p>
          <p>
            <s xml:id="echoid-s3210" xml:space="preserve">Ponamus tempus quo corpus, velocitate v, lineam quamcunque percur-
              <lb/>
            rit ipſâ lineâ percurfâ repræſentari; </s>
            <s xml:id="echoid-s3211" xml:space="preserve">tempus quo linea percurritur, velocita-
              <lb/>
            te aliâ majori, eo brevius eſt, quo velocitas major eſt, & </s>
            <s xml:id="echoid-s3212" xml:space="preserve">minuitur in ratio-
              <lb/>
            ne in qua velocitas augetur; </s>
            <s xml:id="echoid-s3213" xml:space="preserve">tempus ergo, in quo linea quæcunque, veloci-
              <lb/>
            tate c percurritur, repræſentabitur lineâ minore ipſâ percurſâ, & </s>
            <s xml:id="echoid-s3214" xml:space="preserve">quæ ad percur-
              <lb/>
            ſam habet rationem quæ datur inter v & </s>
            <s xml:id="echoid-s3215" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3216" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3217" xml:space="preserve">Si punctum eat per AE & </s>
            <s xml:id="echoid-s3218" xml:space="preserve">EB tempus motus per AE; </s>
            <s xml:id="echoid-s3219" xml:space="preserve">quia velocitate </s>
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