Barrow, Isaac, Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

Table of contents

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[31.] Lect. IV.
Page: 222 (29)
[32.] Lect. VII.
Page: 249 (56)
[33.] Lect. VIII.
Page: 256 (63)
[34.] Lect. IX.
Page: 263 (70)
[35.] Lect. X.
Page: 268 (75)
[36.] Exemp. I.
Page: 274 (81)
[37.] _Exemp_. II.
Page: 275 (82)
[38.] _Exemp_. III
Page: 275 (82)
[39.] Exemp. IV.
Page: 276 (83)
[40.] Eæemp. V.
Page: 277 (84)
[41.] Lect. XI.
Page: 278 (85)
[42.] APPENDICUL A.
Page: 287 (94)
[43.] Lect. XII.
Page: 298 (105)
[44.] APPENDICULA 1.
Page: 303 (110)
[45.] Præparatio Communis.
Page: 303 (110)
[46.] APPENDICULA 2.
Page: 308 (115)
[47.] Conicorum Superſicies dimetiendi Metbodus.
Page: 310 (117)
[48.] Exemplum.
Page: 311 (118)
[49.] Prop. 1.
Page: 312 (119)
[50.] Prop. 2.
Page: 312 (119)
[51.] Prop. 3.
Page: 313 (120)
[52.] Prop. 4.
Page: 313 (120)
[53.] APPENDICULA 3.
Page: 314 (121)
[54.] Problema I.
Page: 314 (121)
[55.] Exemp. I.
Page: 314 (121)
[56.] Exemp. II.
Page: 315 (122)
[57.] Probl. II.
Page: 315 (122)
[58.] Exemp. I.
Page: 315 (122)
[59.] _Exemp_. II.
Page: 316 (123)
[60.] _Probl_. III.
Page: 316 (123)
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          <p>
            <s xml:id="echoid-s12210" xml:space="preserve">
              <pb o="81" file="0259" n="274" rhead=""/>
            rectam AP ſecare ad T; </s>
            <s xml:id="echoid-s12211" xml:space="preserve">ut ipſius jam rectæ PT quantitatem exqui-
              <lb/>
              <note position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">Fig. 115.</note>
            ram; </s>
            <s xml:id="echoid-s12212" xml:space="preserve">curvæ arcum MN indefinitè parvum ſtatuo; </s>
            <s xml:id="echoid-s12213" xml:space="preserve">tum duco rectas
              <lb/>
            NQ ad MP, & </s>
            <s xml:id="echoid-s12214" xml:space="preserve">NR ad AP parallelas; </s>
            <s xml:id="echoid-s12215" xml:space="preserve">nomino MP = _m_; </s>
            <s xml:id="echoid-s12216" xml:space="preserve">PT
              <lb/>
            = _t_; </s>
            <s xml:id="echoid-s12217" xml:space="preserve">MR = _a_; </s>
            <s xml:id="echoid-s12218" xml:space="preserve">NR = _e_; </s>
            <s xml:id="echoid-s12219" xml:space="preserve">reliquáſque rectas, ex ſpeciali curvæ
              <lb/>
            natura determinatas, utiles propoſito, nominibus deſigno; </s>
            <s xml:id="echoid-s12220" xml:space="preserve">ipſas au-
              <lb/>
            tem MR, NR (& </s>
            <s xml:id="echoid-s12221" xml:space="preserve">mediantibus illis ipſas MP, PT) per _æquationem_
              <lb/>
            è Calculo deprehenſam inter ſe comparo; </s>
            <s xml:id="echoid-s12222" xml:space="preserve">regulas interim has obſer-
              <lb/>
            vans. </s>
            <s xml:id="echoid-s12223" xml:space="preserve">1. </s>
            <s xml:id="echoid-s12224" xml:space="preserve">Inter computandum omnes abjicio terminos, in quibus
              <lb/>
            ipſarum _a_, vel _e_ poteſtas habetur, vel in quibus ipſæ ducuntur in ſe
              <lb/>
            (etenim iſti termini nihil valebunt).</s>
            <s xml:id="echoid-s12225" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12226" xml:space="preserve">2. </s>
            <s xml:id="echoid-s12227" xml:space="preserve">Poſt _æquationem constitutam_, omnes abjicio terminos, literis
              <lb/>
            conftantes quantitates notas, ſeu determinatas deſignantibus; </s>
            <s xml:id="echoid-s12228" xml:space="preserve">aut in
              <lb/>
            quibus non habentur _a_, vel _e_. </s>
            <s xml:id="echoid-s12229" xml:space="preserve">(etenim illi termini ſemper, ad unam
              <lb/>
            æquationis partem adducti, nihilum adæquabunt).</s>
            <s xml:id="echoid-s12230" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12231" xml:space="preserve">3. </s>
            <s xml:id="echoid-s12232" xml:space="preserve">Pro _a_ ipſam _m_; </s>
            <s xml:id="echoid-s12233" xml:space="preserve">(vel MP) pro _e_ ipſam _t_ (vel PT) ſubſtituo.
              <lb/>
            </s>
            <s xml:id="echoid-s12234" xml:space="preserve">Hinc demùm ipſius PT quantitas dignoſcetur.</s>
            <s xml:id="echoid-s12235" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12236" xml:space="preserve">Quòd ſi calculum ingrediatur curvæ cujuſpiam indefinita particula;
              <lb/>
            </s>
            <s xml:id="echoid-s12237" xml:space="preserve">ſubſtituatur ejus loco tangentis particula ritè ſumpta; </s>
            <s xml:id="echoid-s12238" xml:space="preserve">vel ei quævis
              <lb/>
            (ob indefinitam curvæ parvitatem) æquipollens recta.</s>
            <s xml:id="echoid-s12239" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12240" xml:space="preserve">Hæc autem è ſubnexis Exemplis clariùs eluceſcent.</s>
            <s xml:id="echoid-s12241" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div371" type="section" level="1" n="36">
          <head xml:id="echoid-head39" xml:space="preserve">Exemp. I.</head>
          <p>
            <s xml:id="echoid-s12242" xml:space="preserve">Angulus ABH rectus ſit; </s>
            <s xml:id="echoid-s12243" xml:space="preserve">& </s>
            <s xml:id="echoid-s12244" xml:space="preserve">ſit curva AMO talis, ut per A du-
              <lb/>
            ctâ utcunque rectâ AK, quæ rectam BH ſecet in K, curvam AMO
              <lb/>
              <note position="right" xlink:label="note-0259-02" xlink:href="note-0259-02a" xml:space="preserve">Fig. 116.</note>
            in M, ſit ſemper ſubtenſa AM æqualis abſciſſæ BK; </s>
            <s xml:id="echoid-s12245" xml:space="preserve">hujus curvæ ad
              <lb/>
            M tangens eſt deſignanda.</s>
            <s xml:id="echoid-s12246" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12247" xml:space="preserve">Fiant quæ ſuprà præſcripta ſunt, & </s>
            <s xml:id="echoid-s12248" xml:space="preserve">(ductâ ANL) nominetur
              <lb/>
            AB = _r_; </s>
            <s xml:id="echoid-s12249" xml:space="preserve">& </s>
            <s xml:id="echoid-s12250" xml:space="preserve">AP = _q_; </s>
            <s xml:id="echoid-s12251" xml:space="preserve">unde AQ = _q_ - _e_; </s>
            <s xml:id="echoid-s12252" xml:space="preserve">item QN = _m_ -
              <lb/>
            _a_. </s>
            <s xml:id="echoid-s12253" xml:space="preserve">ergò eſt _qq_ + _ee_ - 2 _qe_ + _mm_ + _aa_ - 2 _ma_ = (AQq
              <lb/>
            + QNq = ANq = ) BLq; </s>
            <s xml:id="echoid-s12254" xml:space="preserve">hoc eſt (rejectis, uti monitum eſt,
              <lb/>
            rejiciendis) _qq_ - 2 _qe_ + _mm_ - 2 _ma_ = BLq. </s>
            <s xml:id="echoid-s12255" xml:space="preserve">Porrò eſt
              <lb/>
            AQ. </s>
            <s xml:id="echoid-s12256" xml:space="preserve">QN:</s>
            <s xml:id="echoid-s12257" xml:space="preserve">: AB. </s>
            <s xml:id="echoid-s12258" xml:space="preserve">BL; </s>
            <s xml:id="echoid-s12259" xml:space="preserve">hoc eſt _q_ - _e. </s>
            <s xml:id="echoid-s12260" xml:space="preserve">m_ - _a_:</s>
            <s xml:id="echoid-s12261" xml:space="preserve">: _r._ </s>
            <s xml:id="echoid-s12262" xml:space="preserve">BL =
              <lb/>
            {_rm_ - _ra_.</s>
            <s xml:id="echoid-s12263" xml:space="preserve">/_q_ - _e_} quare {_rrmm_ + _rraa_ - 2 _rrma_/_qq_ + _ee_ - 2 _qe_.</s>
            <s xml:id="echoid-s12264" xml:space="preserve">} = BLq; </s>
            <s xml:id="echoid-s12265" xml:space="preserve">ſeu
              <unsure/>
              <lb/>
            (rejectis ſuperfluis) {_rrmm_ - 2 _rrma_/_qq_ - 2 _qe@_} = BLq = _qq_ - 2 _qe_ +
              <lb/>
            _mm_ - 2 _ma_. </s>
            <s xml:id="echoid-s12266" xml:space="preserve">vel _rrmm_ - 2 _rrma_ = _q_
              <emph style="sub">4</emph>
            - 2 _q_
              <emph style="sub">3</emph>
            _e_ + _qqmm_ - 2 _qqma_ - 2 _q_
              <emph style="sub">3</emph>
            _e_ +
              <lb/>
            4 _qqee_ - 2 _qmme_ + 4 _qmae_; </s>
            <s xml:id="echoid-s12267" xml:space="preserve">hoc eſt (abjectis iis, quæ </s>
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