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

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        <div xml:id="echoid-div483" type="section" level="1" n="156">
          <head xml:id="echoid-head224" xml:space="preserve">SCHOLIUM. 3.</head>
          <head xml:id="echoid-head225" style="it" xml:space="preserve">In quo quædam in boc capite memoratæ Cycloidis
            <lb/>
          proprietates demonſtrantur.</head>
          <p>
            <s xml:id="echoid-s3083" xml:space="preserve">Poſitâ cycloidis memoratâ formatione; </s>
            <s xml:id="echoid-s3084" xml:space="preserve">ſit circulus generator BEF. </s>
            <s xml:id="echoid-s3085" xml:space="preserve">
              <note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">34.</note>
            mus hunc perveniſſe ad punctum Gbaſeos, punctum F erit in f, poſito arcu Gf
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve"> 282.</note>
            lineæ GF æquali; </s>
            <s xml:id="echoid-s3086" xml:space="preserve">Punctum deſcribens erit in b, & </s>
            <s xml:id="echoid-s3087" xml:space="preserve">erit hoc punctum Cycloïdis-</s>
          </p>
          <note position="left" xml:space="preserve">TAB. XII.
            <lb/>
          fig. 4.</note>
          <p>
            <s xml:id="echoid-s3088" xml:space="preserve">Ducatur G c H diameter per punctum contactus, erit hæc ad baſin per-
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0126-04" xlink:href="note-0126-04a" xml:space="preserve">18. El. III.</note>
            pendicularis , & </s>
            <s xml:id="echoid-s3089" xml:space="preserve">parallela diametro BF. </s>
            <s xml:id="echoid-s3090" xml:space="preserve">Ductâ nunc b L, per punctum Cycloïdis b, baſi parallelâ, ſecante circulum FEB in E, & </s>
            <s xml:id="echoid-s3091" xml:space="preserve">GH in I; </s>
            <s xml:id="echoid-s3092" xml:space="preserve">ma-
              <lb/>
            nifeſtum eſt, propter æquales GI & </s>
            <s xml:id="echoid-s3093" xml:space="preserve">FL , in circulis æqualibus
              <note symbol="*" position="left" xlink:label="note-0126-05" xlink:href="note-0126-05a" xml:space="preserve">34. El. I.</note>
            eſſe b I, EL; </s>
            <s xml:id="echoid-s3094" xml:space="preserve">additâ utrimque IE æquales erunt b E, IL, cui æqualis
              <lb/>
            GF .</s>
            <s xml:id="echoid-s3095" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">34. El. I.</note>
          <p>
            <s xml:id="echoid-s3096" xml:space="preserve">Facile etiam liquet arcus G f, b H, EB, æquales eſſe inter ſe & </s>
            <s xml:id="echoid-s3097" xml:space="preserve">lineæ
              <lb/>
            GF; </s>
            <s xml:id="echoid-s3098" xml:space="preserve">ideoque lineæ b E.</s>
            <s xml:id="echoid-s3099" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3100" xml:space="preserve">Ex quibus hanc curvæ deducimus proprietatem, Si ex puncto quocunque Cy-
              <lb/>
              <note position="left" xlink:label="note-0126-07" xlink:href="note-0126-07a" xml:space="preserve">315.</note>
            cloidis ad baſin ducatur parallela, quæ ſemicirculum ſecat ſuper axe deſcriptum
              <lb/>
            ad partem curvæ, qualis linea hìc eſt b EL, erit hujus portio, inter Cycloi-
              <lb/>
            dem & </s>
            <s xml:id="echoid-s3101" xml:space="preserve">ſemicirculum intercepta, æqualis arcui ſemicir culi inter lineam memora-
              <lb/>
            tam & </s>
            <s xml:id="echoid-s3102" xml:space="preserve">verticem intercepto. </s>
            <s xml:id="echoid-s3103" xml:space="preserve">id eſt b E arcui EB æqualis eſt.</s>
            <s xml:id="echoid-s3104" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3105" xml:space="preserve">Sit Cycloïs ADB; </s>
            <s xml:id="echoid-s3106" xml:space="preserve">vertex B; </s>
            <s xml:id="echoid-s3107" xml:space="preserve">baſis AF; </s>
            <s xml:id="echoid-s3108" xml:space="preserve">axis BF, qui diameter eſt ſemi-
              <lb/>
              <note position="left" xlink:label="note-0126-08" xlink:href="note-0126-08a" xml:space="preserve">316.</note>
            circuli FMB.</s>
            <s xml:id="echoid-s3109" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">TAB. XII.
            <lb/>
          fig. 5.</note>
          <p>
            <s xml:id="echoid-s3110" xml:space="preserve">Sumtâ D d portione quacunque infinitè exigua Cycloïdis, poterit hæc
              <lb/>
            pro lineâ rectâ haberi, & </s>
            <s xml:id="echoid-s3111" xml:space="preserve">continuatâ formabit tangentem in puncto D aut d.
              <lb/>
            </s>
            <s xml:id="echoid-s3112" xml:space="preserve">Ducantur DL, dl, ad baſin parallelæ ſemicirculum ſecantes in E, e; </s>
            <s xml:id="echoid-s3113" xml:space="preserve">& </s>
            <s xml:id="echoid-s3114" xml:space="preserve">
              <lb/>
            ductâ B e continuetur hæc donec ſecet in b lineam DL; </s>
            <s xml:id="echoid-s3115" xml:space="preserve">ſit etiam BO ad ba-
              <lb/>
            ſin parallela, circulum tangens in B, & </s>
            <s xml:id="echoid-s3116" xml:space="preserve">quæ in O ſecatur lineâ eO, con-
              <lb/>
            tinuatione lineæ E e.</s>
            <s xml:id="echoid-s3117" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3118" xml:space="preserve">Triangula b Ee & </s>
            <s xml:id="echoid-s3119" xml:space="preserve">e OB, propter Bo & </s>
            <s xml:id="echoid-s3120" xml:space="preserve">hE parallelas ſunt ſimilia. </s>
            <s xml:id="echoid-s3121" xml:space="preserve">La-
              <lb/>
            tera autem EO & </s>
            <s xml:id="echoid-s3122" xml:space="preserve">OB ſunt æqualia ; </s>
            <s xml:id="echoid-s3123" xml:space="preserve">ergo & </s>
            <s xml:id="echoid-s3124" xml:space="preserve">æqualia e E, h E; </s>
            <s xml:id="echoid-s3125" xml:space="preserve">eſt eE
              <note symbol="*" position="left" xlink:label="note-0126-10" xlink:href="note-0126-10a" xml:space="preserve">36. El III.</note>
            cuum B e BE, aut linearum de, DE, differentia ; </s>
            <s xml:id="echoid-s3126" xml:space="preserve">quæ eadem
              <note symbol="*" position="left" xlink:label="note-0126-11" xlink:href="note-0126-11a" xml:space="preserve">315</note>
            tia eſt ideò etiam h E, quare ſunt æquales parallelæ D h, de; </s>
            <s xml:id="echoid-s3127" xml:space="preserve">ſuntetiam id-
              <lb/>
            circo æquales & </s>
            <s xml:id="echoid-s3128" xml:space="preserve">parallelæ D d, b e . </s>
            <s xml:id="echoid-s3129" xml:space="preserve">id eſt tangens in d parallela chordæ e
              <note symbol="*" position="left" xlink:label="note-0126-12" xlink:href="note-0126-12a" xml:space="preserve">33. El I.</note>
            quam Cycloïdis proprietatem ſuperius indicavimus in n. </s>
            <s xml:id="echoid-s3130" xml:space="preserve">285.</s>
            <s xml:id="echoid-s3131" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3132" xml:space="preserve">Iiſdem poſitis ducatur FE i; </s>
            <s xml:id="echoid-s3133" xml:space="preserve">erit hæc ad BE aut B b (propter augulum
              <lb/>
              <note position="left" xlink:label="note-0126-13" xlink:href="note-0126-13a" xml:space="preserve">317.</note>
            infinite exiguum e BE) perpendicularis , dividetque baſin trianguli
              <note symbol="*" position="left" xlink:label="note-0126-14" xlink:href="note-0126-14a" xml:space="preserve">31. El. III,</note>
            les b E e in duas partes æquales ita, ut ei ſit dimidium ipſius eb aut d D.
              <lb/>
            </s>
            <s xml:id="echoid-s3134" xml:space="preserve">Eſt verò ei differentia inter chordas BE, Be; </s>
            <s xml:id="echoid-s3135" xml:space="preserve">nam ſi centro B, radio BE,
              <lb/>
            circulus deſcribatur coincidet hic cum Ei, quæ infinite exigua eſt; </s>
            <s xml:id="echoid-s3136" xml:space="preserve">& </s>
            <s xml:id="echoid-s3137" xml:space="preserve">D d
              <lb/>
            eſt differentia arcuum Cycloidis DB, dB.</s>
            <s xml:id="echoid-s3138" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3139" xml:space="preserve">Concipiamus nunc lineam ad baſim Cycloidis AF parallelam moveri à
              <lb/>
            B ad F, aliamque lineam interea circa B ita rotari, ut continuo tranſeat
              <lb/>
            per interſectionem primæ cum ſemicirculo. </s>
            <s xml:id="echoid-s3140" xml:space="preserve">Ubi prima Ex. </s>
            <s xml:id="echoid-s3141" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s3142" xml:space="preserve">pervenit
              <lb/>
            ad dl erit ſecunda in B e, translatâ primâ ad DL rotatur ſecunda ut ſit in
              <lb/>
            BE. </s>
            <s xml:id="echoid-s3143" xml:space="preserve">In hoc motu, commune initium habent, & </s>
            <s xml:id="echoid-s3144" xml:space="preserve">continuo augentur, arcus
              <lb/>
            Cycloïdis DB & </s>
            <s xml:id="echoid-s3145" xml:space="preserve">chorda EB; </s>
            <s xml:id="echoid-s3146" xml:space="preserve">ſed illius augmentum ſemper duplum eſt au-
              <lb/>
            gmentihujus, quare & </s>
            <s xml:id="echoid-s3147" xml:space="preserve">integer arcus qui eſt ſumma augmentorum, erit du-
              <lb/>
            plusintegræ chordæ, quæ etiam ſummam valet augmentorum ſuorum. </s>
            <s xml:id="echoid-s3148" xml:space="preserve"/>
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