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-s3993" xml:space="preserve">
              <pb o="66" file="0084" n="84" rhead=""/>
            centrum C) datáque ſint duo puncta A, X; </s>
            <s xml:id="echoid-s3994" xml:space="preserve">reperiendum eſt in
              <lb/>
            circumferentia punctum aliquod; </s>
            <s xml:id="echoid-s3995" xml:space="preserve">à quo ductæ ad A, X rectæ, altera ſit
              <lb/>
            alterius reflexa. </s>
            <s xml:id="echoid-s3996" xml:space="preserve">‖ Hocità perficimus:</s>
            <s xml:id="echoid-s3997" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3998" xml:space="preserve">Conjungantur rectæ AC, XC; </s>
            <s xml:id="echoid-s3999" xml:space="preserve">& </s>
            <s xml:id="echoid-s4000" xml:space="preserve">fiat (ſeorſim) ang. </s>
            <s xml:id="echoid-s4001" xml:space="preserve">δ = {1/2} ang.
              <lb/>
            </s>
            <s xml:id="echoid-s4002" xml:space="preserve">
              <note position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">Fig. 92, 93.</note>
            AC X. </s>
            <s xml:id="echoid-s4003" xml:space="preserve">& </s>
            <s xml:id="echoid-s4004" xml:space="preserve">in ξ δ crure anguli δ ſumpto liberè puncto π ducatur π V
              <lb/>
            ad ξ δ perpendicularis alterum crus ſecans in V; </s>
            <s xml:id="echoid-s4005" xml:space="preserve">& </s>
            <s xml:id="echoid-s4006" xml:space="preserve">in V π protracta
              <lb/>
            capiatur π γ = π V; </s>
            <s xml:id="echoid-s4007" xml:space="preserve">tum dividatur γ V in φ, ut ſit γ φ. </s>
            <s xml:id="echoid-s4008" xml:space="preserve">φ V:</s>
            <s xml:id="echoid-s4009" xml:space="preserve">: XC. </s>
            <s xml:id="echoid-s4010" xml:space="preserve">CA;
              <lb/>
            </s>
            <s xml:id="echoid-s4011" xml:space="preserve">perque punctum φ trajiciatur κ ξ ſic ut ſit κξ. </s>
            <s xml:id="echoid-s4012" xml:space="preserve">κ v:</s>
            <s xml:id="echoid-s4013" xml:space="preserve">: CX. </s>
            <s xml:id="echoid-s4014" xml:space="preserve">CN. </s>
            <s xml:id="echoid-s4015" xml:space="preserve">denique
              <lb/>
              <note position="left" xlink:label="note-0084-02" xlink:href="note-0084-02a" xml:space="preserve">L@m. pr@ced.</note>
            fiat angulus XCNæ qualis angulo ξ κ v; </s>
            <s xml:id="echoid-s4016" xml:space="preserve">erit punctum N quale deſi-
              <lb/>
            deramus. </s>
            <s xml:id="echoid-s4017" xml:space="preserve">Nam ducantur XN, ξ v; </s>
            <s xml:id="echoid-s4018" xml:space="preserve">& </s>
            <s xml:id="echoid-s4019" xml:space="preserve">fiat ang. </s>
            <s xml:id="echoid-s4020" xml:space="preserve">CNG = ang.
              <lb/>
            </s>
            <s xml:id="echoid-s4021" xml:space="preserve">κ v γ. </s>
            <s xml:id="echoid-s4022" xml:space="preserve">adſumatúrque PG = PN; </s>
            <s xml:id="echoid-s4023" xml:space="preserve">& </s>
            <s xml:id="echoid-s4024" xml:space="preserve">connectatur XG. </s>
            <s xml:id="echoid-s4025" xml:space="preserve">liquet jam
              <lb/>
            trangula XCN, ξ κ v ſimilia fore; </s>
            <s xml:id="echoid-s4026" xml:space="preserve">nec non ipſa CNF, κ v φ; </s>
            <s xml:id="echoid-s4027" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s4028" xml:space="preserve">ipſa XP F, ξ π φ; </s>
            <s xml:id="echoid-s4029" xml:space="preserve">ipsáque demùm XFN, ξ φ v aſſimilari. </s>
            <s xml:id="echoid-s4030" xml:space="preserve">quare
              <lb/>
            PF. </s>
            <s xml:id="echoid-s4031" xml:space="preserve">XF:</s>
            <s xml:id="echoid-s4032" xml:space="preserve">: π φ. </s>
            <s xml:id="echoid-s4033" xml:space="preserve">ξ φ. </s>
            <s xml:id="echoid-s4034" xml:space="preserve">& </s>
            <s xml:id="echoid-s4035" xml:space="preserve">XF. </s>
            <s xml:id="echoid-s4036" xml:space="preserve">FN:</s>
            <s xml:id="echoid-s4037" xml:space="preserve">: ξ φ. </s>
            <s xml:id="echoid-s4038" xml:space="preserve">φ V. </s>
            <s xml:id="echoid-s4039" xml:space="preserve">& </s>
            <s xml:id="echoid-s4040" xml:space="preserve">ex æquo PF. </s>
            <s xml:id="echoid-s4041" xml:space="preserve">
              <lb/>
            FN:</s>
            <s xml:id="echoid-s4042" xml:space="preserve">: π φ. </s>
            <s xml:id="echoid-s4043" xml:space="preserve">φ V. </s>
            <s xml:id="echoid-s4044" xml:space="preserve">& </s>
            <s xml:id="echoid-s4045" xml:space="preserve">antecedentes duplando 2 PF. </s>
            <s xml:id="echoid-s4046" xml:space="preserve">FN :</s>
            <s xml:id="echoid-s4047" xml:space="preserve">: 2 π φ. </s>
            <s xml:id="echoid-s4048" xml:space="preserve">φ V. </s>
            <s xml:id="echoid-s4049" xml:space="preserve">
              <lb/>
            componendóque 2 PF + FN. </s>
            <s xml:id="echoid-s4050" xml:space="preserve">FN:</s>
            <s xml:id="echoid-s4051" xml:space="preserve">: 2 π φ + φ V. </s>
            <s xml:id="echoid-s4052" xml:space="preserve">φ V. </s>
            <s xml:id="echoid-s4053" xml:space="preserve">hoc eſt
              <lb/>
            GF. </s>
            <s xml:id="echoid-s4054" xml:space="preserve">FN:</s>
            <s xml:id="echoid-s4055" xml:space="preserve">: γ φ. </s>
            <s xml:id="echoid-s4056" xml:space="preserve">φ V (hoc eſt):</s>
            <s xml:id="echoid-s4057" xml:space="preserve">: XC. </s>
            <s xml:id="echoid-s4058" xml:space="preserve">CA. </s>
            <s xml:id="echoid-s4059" xml:space="preserve">ducatur jam NL ad
              <lb/>
            XG parailela; </s>
            <s xml:id="echoid-s4060" xml:space="preserve">quare eſt ang. </s>
            <s xml:id="echoid-s4061" xml:space="preserve">ING = ang. </s>
            <s xml:id="echoid-s4062" xml:space="preserve">G = ang. </s>
            <s xml:id="echoid-s4063" xml:space="preserve">XNG; </s>
            <s xml:id="echoid-s4064" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s4065" xml:space="preserve">XG (XN). </s>
            <s xml:id="echoid-s4066" xml:space="preserve">NL:</s>
            <s xml:id="echoid-s4067" xml:space="preserve">: GF. </s>
            <s xml:id="echoid-s4068" xml:space="preserve">FN:</s>
            <s xml:id="echoid-s4069" xml:space="preserve">: XC. </s>
            <s xml:id="echoid-s4070" xml:space="preserve">CA. </s>
            <s xml:id="echoid-s4071" xml:space="preserve">porro fiàt ang. </s>
            <s xml:id="echoid-s4072" xml:space="preserve">
              <lb/>
            LNH = ang. </s>
            <s xml:id="echoid-s4073" xml:space="preserve">XCA; </s>
            <s xml:id="echoid-s4074" xml:space="preserve">& </s>
            <s xml:id="echoid-s4075" xml:space="preserve">HN protracta ipſi CA occurrat in M; </s>
            <s xml:id="echoid-s4076" xml:space="preserve">
              <lb/>
            cſtque proptereà triangulum HNLſimile triangulo HCM; </s>
            <s xml:id="echoid-s4077" xml:space="preserve">idcir-
              <lb/>
            cóque HC. </s>
            <s xml:id="echoid-s4078" xml:space="preserve">CM:</s>
            <s xml:id="echoid-s4079" xml:space="preserve">: HN. </s>
            <s xml:id="echoid-s4080" xml:space="preserve">NL. </s>
            <s xml:id="echoid-s4081" xml:space="preserve">ducatur denuò tangens NQ; </s>
            <s xml:id="echoid-s4082" xml:space="preserve">eſt-
              <lb/>
            que tum ang. </s>
            <s xml:id="echoid-s4083" xml:space="preserve">PNQ = rect - CNP = rect - κ V π = ang. </s>
            <s xml:id="echoid-s4084" xml:space="preserve">
              <lb/>
            δ = {1/2} XC A; </s>
            <s xml:id="echoid-s4085" xml:space="preserve">vel 2 ang. </s>
            <s xml:id="echoid-s4086" xml:space="preserve">PNQ = ang XCA = ang LNH. </s>
            <s xml:id="echoid-s4087" xml:space="preserve">
              <lb/>
            verùm erat priùs 2 ang. </s>
            <s xml:id="echoid-s4088" xml:space="preserve">XNF = ang. </s>
            <s xml:id="echoid-s4089" xml:space="preserve">XNL. </s>
            <s xml:id="echoid-s4090" xml:space="preserve">ergo 2 ang XNF
              <lb/>
            - 2 ang PNQ = ang XNL- ang. </s>
            <s xml:id="echoid-s4091" xml:space="preserve">LNH. </s>
            <s xml:id="echoid-s4092" xml:space="preserve">hoc eſt 2 ang XNQ
              <lb/>
            = ang. </s>
            <s xml:id="echoid-s4093" xml:space="preserve">XNH. </s>
            <s xml:id="echoid-s4094" xml:space="preserve">ergo tangens NQ biſecat angulum XNH; </s>
            <s xml:id="echoid-s4095" xml:space="preserve">indéque
              <lb/>
            conſectatur fore rectam HM ipſius XN reflexam; </s>
            <s xml:id="echoid-s4096" xml:space="preserve">ac ideò eſſe XC. </s>
            <s xml:id="echoid-s4097" xml:space="preserve">
              <lb/>
            HC:</s>
            <s xml:id="echoid-s4098" xml:space="preserve">: XN. </s>
            <s xml:id="echoid-s4099" xml:space="preserve">HN. </s>
            <s xml:id="echoid-s4100" xml:space="preserve">atqui fuit priùs HC. </s>
            <s xml:id="echoid-s4101" xml:space="preserve">CM:</s>
            <s xml:id="echoid-s4102" xml:space="preserve">: HN. </s>
            <s xml:id="echoid-s4103" xml:space="preserve">NL quare
              <lb/>
            jam erit ex æquo XC. </s>
            <s xml:id="echoid-s4104" xml:space="preserve">CM:</s>
            <s xml:id="echoid-s4105" xml:space="preserve">: XN. </s>
            <s xml:id="echoid-s4106" xml:space="preserve">NL (h@c eſt etiam è præmon-
              <lb/>
            ſtratis):</s>
            <s xml:id="echoid-s4107" xml:space="preserve">: XC. </s>
            <s xml:id="echoid-s4108" xml:space="preserve">CA. </s>
            <s xml:id="echoid-s4109" xml:space="preserve">unde CM = CA. </s>
            <s xml:id="echoid-s4110" xml:space="preserve">quapropter HM, ipſius
              <lb/>
            XN reflexa tranſit per A: </s>
            <s xml:id="echoid-s4111" xml:space="preserve">Quod propoſitum erat efficere.</s>
            <s xml:id="echoid-s4112" xml:space="preserve">‖</s>
          </p>
          <p>
            <s xml:id="echoid-s4113" xml:space="preserve">VIII. </s>
            <s xml:id="echoid-s4114" xml:space="preserve">Hujuſce _Problematis_ ità generaliùs propoſiti varii quidem
              <lb/>
            caſus ſunt (etenim vel data puncta jacent ambo extra circuh@ re-
              <lb/>
            flectentem; </s>
            <s xml:id="echoid-s4115" xml:space="preserve">vel utrumque poſitum eſt intra circulum; </s>
            <s xml:id="echoid-s4116" xml:space="preserve">vel unum intra
              <lb/>
            jacet, alterum extra; </s>
            <s xml:id="echoid-s4117" xml:space="preserve">quinetiam in horum caſuum unoquoque pluries
              <lb/>
            conficitur negotium) aſt ubique non abſimilis erit conſtructio; </s>
            <s xml:id="echoid-s4118" xml:space="preserve">ſanè
              <lb/>
            nimius eſſem; </s>
            <s xml:id="echoid-s4119" xml:space="preserve">meámque pariter ac veſtram patientiam macerarem
              <lb/>
            omnes intricati _Problematis_ nodos evolvendo; </s>
            <s xml:id="echoid-s4120" xml:space="preserve">ſuffecerit ejuſce ſpeci-
              <lb/>
            men aliquod protuliſſe.</s>
            <s xml:id="echoid-s4121" xml:space="preserve"/>
          </p>
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