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-s14250" xml:space="preserve">III. </s>
            <s xml:id="echoid-s14251" xml:space="preserve">Curva AX X talis ſit, ut PX ſecanti CS (vel CT) æquetur;
              <lb/>
            </s>
            <s xml:id="echoid-s14252" xml:space="preserve">_ſpatium_ AC PX hoc eſt _Summa ſecamium ad arcum_ AM pertinen-
              <lb/>
            tium, & </s>
            <s xml:id="echoid-s14253" xml:space="preserve">ad CB applicatarum) æquatur _duplo ſectori_ ACM.</s>
            <s xml:id="echoid-s14254" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14255" xml:space="preserve">Nam _ſpatium_ AF MX _segmenti_ AFM _duplum_ eſt; </s>
            <s xml:id="echoid-s14256" xml:space="preserve">& </s>
            <s xml:id="echoid-s14257" xml:space="preserve">
              <note position="right" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">Fig. 166.</note>
              <note symbol="(a)" position="right" xlink:label="note-0289-02" xlink:href="note-0289-02a" xml:space="preserve">10. Lect.
                <lb/>
              XI.</note>
            _angulum_ FC PM _Trianguli_ FCM. </s>
            <s xml:id="echoid-s14258" xml:space="preserve">ergo _totum ſpatium_ ACPX
              <lb/>
            totius _ſectoris_ ACM duplum eſt.</s>
            <s xml:id="echoid-s14259" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14260" xml:space="preserve">Etiam hoc è 16. </s>
            <s xml:id="echoid-s14261" xml:space="preserve">hujus duodecimæ Lectionis apertè conſtat.</s>
            <s xml:id="echoid-s14262" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14263" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s14264" xml:space="preserve">Curva CVV talis ſit, ut PV _Tangenti_ AS æquetur; </s>
            <s xml:id="echoid-s14265" xml:space="preserve">erit
              <lb/>
            _ſpatium_ CVP (hoc eſt _ſumma tangentium ad arcum_ AM _pertinen-_
              <lb/>
              <note position="right" xlink:label="note-0289-03" xlink:href="note-0289-03a" xml:space="preserve">Fig. 166.</note>
            _tium_, & </s>
            <s xml:id="echoid-s14266" xml:space="preserve">ad rectam CB applicatarum) æquale _ſemiſſi quadrati ex_
              <lb/>
            _ſubtenſa_ AM.</s>
            <s xml:id="echoid-s14267" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14268" xml:space="preserve">Manifeſtè conſectatur ex ſeptima undecimæ Lectionis.</s>
            <s xml:id="echoid-s14269" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14270" xml:space="preserve">V. </s>
            <s xml:id="echoid-s14271" xml:space="preserve">Acceptâ CQ = CP; </s>
            <s xml:id="echoid-s14272" xml:space="preserve">& </s>
            <s xml:id="echoid-s14273" xml:space="preserve">ductâ QO ad CE parallelâ (quæ
              <lb/>
            _byperbolæ_ LE occurrat in O) erit _ſpatium byperbolicum_ PL OQ du-
              <lb/>
            ctum in _radium_ CB (ſeu _cylindricum ad_ bafin PLOQ, altitudine
              <lb/>
            BC (duplum _ſummæ quadratorum_ ex rectis CS, ſeu PX ad _arcum_
              <lb/>
              <note position="right" xlink:label="note-0289-04" xlink:href="note-0289-04a" xml:space="preserve">Fig. 166.</note>
            AM pertinentibus, & </s>
            <s xml:id="echoid-s14274" xml:space="preserve">ad rectam CB applicatis.</s>
            <s xml:id="echoid-s14275" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14276" xml:space="preserve">Nam quia PL. </s>
            <s xml:id="echoid-s14277" xml:space="preserve">QO:</s>
            <s xml:id="echoid-s14278" xml:space="preserve">: (BQ. </s>
            <s xml:id="echoid-s14279" xml:space="preserve">BP. </s>
            <s xml:id="echoid-s14280" xml:space="preserve">hoc eſt:</s>
            <s xml:id="echoid-s14281" xml:space="preserve">:) BC + CP.
              <lb/>
            </s>
            <s xml:id="echoid-s14282" xml:space="preserve">BC - CP; </s>
            <s xml:id="echoid-s14283" xml:space="preserve">erit componendo PL + QO. </s>
            <s xml:id="echoid-s14284" xml:space="preserve">QO:</s>
            <s xml:id="echoid-s14285" xml:space="preserve">: 2 BC. </s>
            <s xml:id="echoid-s14286" xml:space="preserve">BC
              <lb/>
            - CP. </s>
            <s xml:id="echoid-s14287" xml:space="preserve">item eſt QO. </s>
            <s xml:id="echoid-s14288" xml:space="preserve">BC:</s>
            <s xml:id="echoid-s14289" xml:space="preserve">: BC. </s>
            <s xml:id="echoid-s14290" xml:space="preserve">BC + CP; </s>
            <s xml:id="echoid-s14291" xml:space="preserve">ergò (pares ra-
              <lb/>
            tiones adjungendo) eſt PL + QO. </s>
            <s xml:id="echoid-s14292" xml:space="preserve">QO + QO. </s>
            <s xml:id="echoid-s14293" xml:space="preserve">BC = 2 BC. </s>
            <s xml:id="echoid-s14294" xml:space="preserve">
              <lb/>
            BC - CP + BC. </s>
            <s xml:id="echoid-s14295" xml:space="preserve">BC + CP; </s>
            <s xml:id="echoid-s14296" xml:space="preserve">hoc eſt PL + QO. </s>
            <s xml:id="echoid-s14297" xml:space="preserve">BC:</s>
            <s xml:id="echoid-s14298" xml:space="preserve">:
              <lb/>
            2 BCq. </s>
            <s xml:id="echoid-s14299" xml:space="preserve">BCq - CPQ (hoc eſt:</s>
            <s xml:id="echoid-s14300" xml:space="preserve">:) 2 BCq. </s>
            <s xml:id="echoid-s14301" xml:space="preserve">PMq. </s>
            <s xml:id="echoid-s14302" xml:space="preserve">verùm
              <lb/>
            eſt PXq. </s>
            <s xml:id="echoid-s14303" xml:space="preserve">BCq:</s>
            <s xml:id="echoid-s14304" xml:space="preserve">: BCq. </s>
            <s xml:id="echoid-s14305" xml:space="preserve">PMq. </s>
            <s xml:id="echoid-s14306" xml:space="preserve">vel(antecedentes duplando)2 PXq. </s>
            <s xml:id="echoid-s14307" xml:space="preserve">
              <lb/>
            BCq:</s>
            <s xml:id="echoid-s14308" xml:space="preserve">: 2BCq.</s>
            <s xml:id="echoid-s14309" xml:space="preserve">PMq. </s>
            <s xml:id="echoid-s14310" xml:space="preserve">ergò PL + QO. </s>
            <s xml:id="echoid-s14311" xml:space="preserve">BC:</s>
            <s xml:id="echoid-s14312" xml:space="preserve">: 2 PXq. </s>
            <s xml:id="echoid-s14313" xml:space="preserve">BCq. </s>
            <s xml:id="echoid-s14314" xml:space="preserve">vel PL x BC +
              <lb/>
            QOxBC.</s>
            <s xml:id="echoid-s14315" xml:space="preserve">BCq:</s>
            <s xml:id="echoid-s14316" xml:space="preserve">:2PXq. </s>
            <s xml:id="echoid-s14317" xml:space="preserve">BCq. </s>
            <s xml:id="echoid-s14318" xml:space="preserve">quare PL x BC + QO x BC = 2PXq. </s>
            <s xml:id="echoid-s14319" xml:space="preserve">
              <lb/>
            itaque BC in omnes PL + QO ducta adæquat omnia totidem PXq. </s>
            <s xml:id="echoid-s14320" xml:space="preserve">
              <lb/>
            unde conſtat Propoſitum.</s>
            <s xml:id="echoid-s14321" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14322" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s14323" xml:space="preserve">Hinc ſpatium αγψμ (hoc eſt _ſumma ſecantium in arcu_ AM
              <lb/>
              <note position="right" xlink:label="note-0289-05" xlink:href="note-0289-05a" xml:space="preserve">Fig. 167.</note>
            ad αβ applicatarum) æquatur _ſubduple ſpatio byperbolico_ PLOQ.</s>
            <s xml:id="echoid-s14324" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14325" xml:space="preserve">Nam ſumatur arcus MN indefinitê parvus, & </s>
            <s xml:id="echoid-s14326" xml:space="preserve">huic æqualis recta μ ν,
              <lb/>
            ducatúrque recta NR ad AC parallela. </s>
            <s xml:id="echoid-s14327" xml:space="preserve">Eſtque MN. </s>
            <s xml:id="echoid-s14328" xml:space="preserve">MR:</s>
            <s xml:id="echoid-s14329" xml:space="preserve">: (MC.
              <lb/>
            </s>
            <s xml:id="echoid-s14330" xml:space="preserve">CF:</s>
            <s xml:id="echoid-s14331" xml:space="preserve">: CS. </s>
            <s xml:id="echoid-s14332" xml:space="preserve">CA:</s>
            <s xml:id="echoid-s14333" xml:space="preserve">: PX. </s>
            <s xml:id="echoid-s14334" xml:space="preserve">CA:</s>
            <s xml:id="echoid-s14335" xml:space="preserve">:) PXq. </s>
            <s xml:id="echoid-s14336" xml:space="preserve">PX x CA. </s>
            <s xml:id="echoid-s14337" xml:space="preserve">adeóque
              <lb/>
            MN x PX x CA = MR x PXq. </s>
            <s xml:id="echoid-s14338" xml:space="preserve">ſeu μν x μψ x CA = MR x
              <lb/>
            PXq. </s>
            <s xml:id="echoid-s14339" xml:space="preserve">atqui (ex præcedente) omnium MR x PXq ſumma ſpatii
              <lb/>
            PL OQ in CA ducti ſubdupla eſt. </s>
            <s xml:id="echoid-s14340" xml:space="preserve">Ergò omnia totidem μν x μ ψ
              <lb/>
            in CA ducta eidem ſubduplo æquantur. </s>
            <s xml:id="echoid-s14341" xml:space="preserve">quare ſpatium αγψμ </s>
          </p>
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