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

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          <p>
            <s xml:id="echoid-s12887" xml:space="preserve">
              <pb o="92" file="0270" n="285" rhead=""/>
            a
              <unsure/>
            quatur ipſi R x RM, vel R x QP. </s>
            <s xml:id="echoid-s12888" xml:space="preserve">itaque totum ſpatium ADE
              <lb/>
            quod ab ejuſmodi ſectoribus minimè differt adæquatur toti R x DK.
              <lb/>
            </s>
            <s xml:id="echoid-s12889" xml:space="preserve">quod erat Propoſitum.</s>
            <s xml:id="echoid-s12890" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12891" xml:space="preserve">XXIII. </s>
            <s xml:id="echoid-s12892" xml:space="preserve">Iiſdem, quoad cætera, poſitis atque paratis, ducantur KH
              <lb/>
              <note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Fig. 128.</note>
            ad KT, & </s>
            <s xml:id="echoid-s12893" xml:space="preserve">MI ad MS perpendiculares; </s>
            <s xml:id="echoid-s12894" xml:space="preserve">& </s>
            <s xml:id="echoid-s12895" xml:space="preserve">concipiatur jam curva
              <lb/>
            AE naturâ talis, ut ſit DE = √ DK x DH; </s>
            <s xml:id="echoid-s12896" xml:space="preserve">& </s>
            <s xml:id="echoid-s12897" xml:space="preserve">DF = √ DM x
              <lb/>
            DI; </s>
            <s xml:id="echoid-s12898" xml:space="preserve">ac ità perpetuò; </s>
            <s xml:id="echoid-s12899" xml:space="preserve">erit ſpatium ADE quadrati ex DK ſubqua-
              <lb/>
            druplum.</s>
            <s xml:id="echoid-s12900" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12901" xml:space="preserve">Nam eſt MP. </s>
            <s xml:id="echoid-s12902" xml:space="preserve">PK:</s>
            <s xml:id="echoid-s12903" xml:space="preserve">: DK. </s>
            <s xml:id="echoid-s12904" xml:space="preserve">DH:</s>
            <s xml:id="echoid-s12905" xml:space="preserve">: DKq. </s>
            <s xml:id="echoid-s12906" xml:space="preserve">DK x DH:</s>
            <s xml:id="echoid-s12907" xml:space="preserve">: DKq.
              <lb/>
            </s>
            <s xml:id="echoid-s12908" xml:space="preserve">DEq. </s>
            <s xml:id="echoid-s12909" xml:space="preserve">item DP. </s>
            <s xml:id="echoid-s12910" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s12911" xml:space="preserve">: DE. </s>
            <s xml:id="echoid-s12912" xml:space="preserve">EX; </s>
            <s xml:id="echoid-s12913" xml:space="preserve">hoc eſt DK. </s>
            <s xml:id="echoid-s12914" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s12915" xml:space="preserve">: DE. </s>
            <s xml:id="echoid-s12916" xml:space="preserve">
              <lb/>
            EX. </s>
            <s xml:id="echoid-s12917" xml:space="preserve">ergò MP x DK. </s>
            <s xml:id="echoid-s12918" xml:space="preserve">PK x PM:</s>
            <s xml:id="echoid-s12919" xml:space="preserve">: DKq x DE. </s>
            <s xml:id="echoid-s12920" xml:space="preserve">DEq x EX. </s>
            <s xml:id="echoid-s12921" xml:space="preserve">
              <lb/>
            hoc eſt DK PK:</s>
            <s xml:id="echoid-s12922" xml:space="preserve">: DKq. </s>
            <s xml:id="echoid-s12923" xml:space="preserve">DE x EX. </s>
            <s xml:id="echoid-s12924" xml:space="preserve">vel DKq. </s>
            <s xml:id="echoid-s12925" xml:space="preserve">DK x PK:</s>
            <s xml:id="echoid-s12926" xml:space="preserve">: DKq. </s>
            <s xml:id="echoid-s12927" xml:space="preserve">
              <lb/>
            DE x EX. </s>
            <s xml:id="echoid-s12928" xml:space="preserve">unde DK x PK = DE x EX. </s>
            <s xml:id="echoid-s12929" xml:space="preserve">Simili ratione DM x MR
              <lb/>
            (vel DP x PQ) = DF x FY. </s>
            <s xml:id="echoid-s12930" xml:space="preserve">Verúm omnia DK x PK, DP x
              <lb/>
            PQ, &</s>
            <s xml:id="echoid-s12931" xml:space="preserve">c æquantur ſemiſſi quadrati ex DK; </s>
            <s xml:id="echoid-s12932" xml:space="preserve">& </s>
            <s xml:id="echoid-s12933" xml:space="preserve">omnia DE x EX,
              <lb/>
            DF x FY, &</s>
            <s xml:id="echoid-s12934" xml:space="preserve">c æquantur _duplo ſpatio_ EDA; </s>
            <s xml:id="echoid-s12935" xml:space="preserve">unde manifeſte con-
              <lb/>
            ſequitur Propoſitum.</s>
            <s xml:id="echoid-s12936" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12937" xml:space="preserve">XXIV. </s>
            <s xml:id="echoid-s12938" xml:space="preserve">Sit curva quæpiam DOK, in qua punctum D; </s>
            <s xml:id="echoid-s12939" xml:space="preserve">cuique
              <lb/>
              <note position="left" xlink:label="note-0270-02" xlink:href="note-0270-02a" xml:space="preserve">Fig. 129.</note>
            ſubtendatur recta DK; </s>
            <s xml:id="echoid-s12940" xml:space="preserve">ſit item curva DZI talis, ut ſumpto in curva
              <lb/>
            DOK puncto quopiam M, connexâque DM; </s>
            <s xml:id="echoid-s12941" xml:space="preserve">& </s>
            <s xml:id="echoid-s12942" xml:space="preserve">ductâ DS ad DM
              <lb/>
            perpendiculari, & </s>
            <s xml:id="echoid-s12943" xml:space="preserve">MS curvam DOK tangente; </s>
            <s xml:id="echoid-s12944" xml:space="preserve">ſumptâ demum
              <lb/>
            DP = DM, & </s>
            <s xml:id="echoid-s12945" xml:space="preserve">ductâ PZ ad DK perpendiculari, ſit PZ = DS;
              <lb/>
            </s>
            <s xml:id="echoid-s12946" xml:space="preserve">erit _ſpatium_ DKI æquale _duplo ſpatio_ DKOD.</s>
            <s xml:id="echoid-s12947" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12948" xml:space="preserve">Nam recta KP concipiatur indefinitè parva; </s>
            <s xml:id="echoid-s12949" xml:space="preserve">& </s>
            <s xml:id="echoid-s12950" xml:space="preserve">DT ipſi DK per-
              <lb/>
            pendicularis ſit, & </s>
            <s xml:id="echoid-s12951" xml:space="preserve">KT curvam DOK tangat. </s>
            <s xml:id="echoid-s12952" xml:space="preserve">Eſt itaque (ducto
              <lb/>
            arcu MP) rurſus KP. </s>
            <s xml:id="echoid-s12953" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s12954" xml:space="preserve">: KD. </s>
            <s xml:id="echoid-s12955" xml:space="preserve">DT:</s>
            <s xml:id="echoid-s12956" xml:space="preserve">: KD. </s>
            <s xml:id="echoid-s12957" xml:space="preserve">KI. </s>
            <s xml:id="echoid-s12958" xml:space="preserve">unde KP x
              <lb/>
            KI = PM x KD. </s>
            <s xml:id="echoid-s12959" xml:space="preserve">Capiatur alia particula PQ, & </s>
            <s xml:id="echoid-s12960" xml:space="preserve">centro D per
              <lb/>
            Q ducatur arcus QN, quem ſecet ſubtenſa DM in R; </s>
            <s xml:id="echoid-s12961" xml:space="preserve">eſt ergòrur-
              <lb/>
            ſus MR. </s>
            <s xml:id="echoid-s12962" xml:space="preserve">RN:</s>
            <s xml:id="echoid-s12963" xml:space="preserve">: MD. </s>
            <s xml:id="echoid-s12964" xml:space="preserve">DS; </s>
            <s xml:id="echoid-s12965" xml:space="preserve">hoc eſt PQ. </s>
            <s xml:id="echoid-s12966" xml:space="preserve">RN:</s>
            <s xml:id="echoid-s12967" xml:space="preserve">: MD. </s>
            <s xml:id="echoid-s12968" xml:space="preserve">PZ qua-
              <lb/>
            re PQ x PZ = RN x MD; </s>
            <s xml:id="echoid-s12969" xml:space="preserve">ac ità continuò deinceps. </s>
            <s xml:id="echoid-s12970" xml:space="preserve">patet igitur
              <lb/>
            omnia ſimul rectangula KP x KI, PQ x PZ, &</s>
            <s xml:id="echoid-s12971" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12972" xml:space="preserve">æquari aggrega-
              <lb/>
            to omnium PM x KD, RN x MD, &</s>
            <s xml:id="echoid-s12973" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12974" xml:space="preserve">hoc eſt ſpatium DKI
              <lb/>
            duplo ſpatio DKOD æquari.</s>
            <s xml:id="echoid-s12975" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12976" xml:space="preserve">XXV. </s>
            <s xml:id="echoid-s12977" xml:space="preserve">Iiſdem quoad cætera poſitis atque paratis, ordinatæ PZ jam
              <lb/>
            æquales concipiantur ipſis MS reſpectivis; </s>
            <s xml:id="echoid-s12978" xml:space="preserve">& </s>
            <s xml:id="echoid-s12979" xml:space="preserve">ad rectam aſſumptam
              <lb/>
              <note position="left" xlink:label="note-0270-03" xlink:href="note-0270-03a" xml:space="preserve">Fig. 130.</note>
            X_k_, diſtantiáſque X_k_, X_m_, X_n_, &</s>
            <s xml:id="echoid-s12980" xml:space="preserve">c, æquales ipſis curvæ partibus
              <lb/>
            DOK, DOM, DON, &</s>
            <s xml:id="echoid-s12981" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12982" xml:space="preserve">applicentur rectæ _kd_, _md_, _nd_, &</s>
            <s xml:id="echoid-s12983" xml:space="preserve">c.</s>
            <s xml:id="echoid-s12984" xml:space="preserve"/>
          </p>
        </div>
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