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-s2092" xml:space="preserve">
              <pb o="40" file="0058" n="58" rhead=""/>
            tùm & </s>
            <s xml:id="echoid-s2093" xml:space="preserve">levi pede prætereo; </s>
            <s xml:id="echoid-s2094" xml:space="preserve">quoniam aliunde mox apparitura) ſit,
              <lb/>
            inquam, ejuſmodi quælibet interſectio N; </s>
            <s xml:id="echoid-s2095" xml:space="preserve">dico fore XN, ipſius AN
              <lb/>
              <note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">Fig. 51.</note>
            refractum. </s>
            <s xml:id="echoid-s2096" xml:space="preserve">Etenim eſt I. </s>
            <s xml:id="echoid-s2097" xml:space="preserve">R :</s>
            <s xml:id="echoid-s2098" xml:space="preserve">: AH. </s>
            <s xml:id="echoid-s2099" xml:space="preserve">AT. </s>
            <s xml:id="echoid-s2100" xml:space="preserve">hoc eſt ( quoniam AH,
              <lb/>
            KX ſunt ex conſtructione pares) I. </s>
            <s xml:id="echoid-s2101" xml:space="preserve">R :</s>
            <s xml:id="echoid-s2102" xml:space="preserve">: KX. </s>
            <s xml:id="echoid-s2103" xml:space="preserve">AT :</s>
            <s xml:id="echoid-s2104" xml:space="preserve">: NK. </s>
            <s xml:id="echoid-s2105" xml:space="preserve">NA.
              <lb/>
            </s>
            <s xml:id="echoid-s2106" xml:space="preserve">unde maniſeſtum, è præmonſtratis, eſt propoſitum.</s>
            <s xml:id="echoid-s2107" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2108" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s2109" xml:space="preserve">Veruntamen hujuſmodi conſtructiones _Geometrarum_ uſus
              <lb/>
            aut non libenter admittit, aut alias ſaltem exigit per lineas vulgo no-
              <lb/>
            tas, atque receptas; </s>
            <s xml:id="echoid-s2110" xml:space="preserve">itaque conſuetudini morem gerentes rem aliter
              <lb/>
            conficiemus; </s>
            <s xml:id="echoid-s2111" xml:space="preserve">huc utique faciens ſequens _Problema Lemmaticum_ præ-
              <lb/>
            mittentes: </s>
            <s xml:id="echoid-s2112" xml:space="preserve">Dato angulo recto XP F; </s>
            <s xml:id="echoid-s2113" xml:space="preserve">punctóque quovis Y; </s>
            <s xml:id="echoid-s2114" xml:space="preserve">per hoc
              <lb/>
            rectam duce e dati anguli cruribus occurentem, ſic ut ab iis intercep-
              <lb/>
            ta ſit a qualis datæ rectæ T. </s>
            <s xml:id="echoid-s2115" xml:space="preserve">‖ Expeditiſſimè quidem perſicitur hoc ope
              <lb/>
            _Concboidis_ alicujus polo Y deſcriptæ; </s>
            <s xml:id="echoid-s2116" xml:space="preserve">ſed enim quoniam & </s>
            <s xml:id="echoid-s2117" xml:space="preserve">iſte modus
              <lb/>
              <note position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">Fig. 52.</note>
            hand ità Geometricus cenſetur; </s>
            <s xml:id="echoid-s2118" xml:space="preserve">adhuc iiſdem Geometris obſequentes
              <lb/>
            ità propoſitum exequemur. </s>
            <s xml:id="echoid-s2119" xml:space="preserve">Ducatur YB ad PF perpendicularis; </s>
            <s xml:id="echoid-s2120" xml:space="preserve">& </s>
            <s xml:id="echoid-s2121" xml:space="preserve">
              <lb/>
            _Aſymptotis_ PX, PB ducatur _Hyperbola_ per Y tranſiens (ſi quidem
              <lb/>
            punctum Y exiſtat extra angulum datum, aut iſtius oppoſita (ſc. </s>
            <s xml:id="echoid-s2122" xml:space="preserve">pun-
              <lb/>
            ctum Y ſit intra dictum angulum) tum centro Y intervallo datam T
              <lb/>
            æquante deſcriptus circulus _Hyperbolam_ inteerſecet in K; </s>
            <s xml:id="echoid-s2123" xml:space="preserve">& </s>
            <s xml:id="echoid-s2124" xml:space="preserve">à K de-
              <lb/>
            mittatur KL ad BP perpendicularis; </s>
            <s xml:id="echoid-s2125" xml:space="preserve">accipiatur autem BN = PL;
              <lb/>
            </s>
            <s xml:id="echoid-s2126" xml:space="preserve">& </s>
            <s xml:id="echoid-s2127" xml:space="preserve">per NY trajiciatur recta NG. </s>
            <s xml:id="echoid-s2128" xml:space="preserve">dico factum; </s>
            <s xml:id="echoid-s2129" xml:space="preserve">vel eſſe NG parem
              <lb/>
            datæ T. </s>
            <s xml:id="echoid-s2130" xml:space="preserve">‖ Nam (ductâ YH ad PB parallelâ) ex _Hyperbolæ_ proprie-
              <lb/>
            tate eſt PL x LK = PB x BY. </s>
            <s xml:id="echoid-s2131" xml:space="preserve">adeóque cùm ſit ex conſtructione
              <lb/>
            BN = PL; </s>
            <s xml:id="echoid-s2132" xml:space="preserve">erit BN x LK :</s>
            <s xml:id="echoid-s2133" xml:space="preserve">: PB x BY. </s>
            <s xml:id="echoid-s2134" xml:space="preserve">adeóque BN. </s>
            <s xml:id="echoid-s2135" xml:space="preserve">BY :</s>
            <s xml:id="echoid-s2136" xml:space="preserve">:
              <lb/>
            PB. </s>
            <s xml:id="echoid-s2137" xml:space="preserve">LK. </s>
            <s xml:id="echoid-s2138" xml:space="preserve">eſt autem BN. </s>
            <s xml:id="echoid-s2139" xml:space="preserve">BY :</s>
            <s xml:id="echoid-s2140" xml:space="preserve">: DY. </s>
            <s xml:id="echoid-s2141" xml:space="preserve">DG. </s>
            <s xml:id="echoid-s2142" xml:space="preserve">ergo eſt PB. </s>
            <s xml:id="echoid-s2143" xml:space="preserve">LK :</s>
            <s xml:id="echoid-s2144" xml:space="preserve">:
              <lb/>
            DY. </s>
            <s xml:id="echoid-s2145" xml:space="preserve">DG. </s>
            <s xml:id="echoid-s2146" xml:space="preserve">quare cùm ſit PB = DY. </s>
            <s xml:id="echoid-s2147" xml:space="preserve">erit LK = DG. </s>
            <s xml:id="echoid-s2148" xml:space="preserve">adeóque
              <lb/>
            (pares LH, DP addendo, vel ſubtrahendo) eſt KH = GP. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">quin-
              <lb/>
            etiam eſt YH = LB = PN (communem nempe PB, vel LN
              <lb/>
            addendo) Ergò patet fore YK(vel T) æqualem ipſi GN: </s>
            <s xml:id="echoid-s2150" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2151" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2152" xml:space="preserve">F.</s>
            <s xml:id="echoid-s2153" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2154" xml:space="preserve">VIII. </s>
            <s xml:id="echoid-s2155" xml:space="preserve">Notandum eſt autem in caſu, quando punctum Y intra datum
              <lb/>
            angulum XPF exiſtit, quòd circulus ille centro Y deſcriptus ſubinde
              <lb/>
            deſignatam hyperbolem binis punctis ſecabit (quod enim pluribus haud
              <lb/>
            quoquam ſecabit univerſim haud ità pridem circa tales ad eadem con-
              <lb/>
            vexas curvas oſtendimus) quo caſu patet duas obvenire propoſiti ſolu-
              <lb/>
            tiones, aliquando rurſus ille dictus circulus _Hyperbolen_ continget; </s>
            <s xml:id="echoid-s2156" xml:space="preserve">& </s>
            <s xml:id="echoid-s2157" xml:space="preserve">
              <lb/>
            tum una tantùm per Y duci poterit recta, datam T adæquans; </s>
            <s xml:id="echoid-s2158" xml:space="preserve">illa
              <lb/>
            ſcilicet omnium quæ per Y dato angulo interſeri poſſunt minima.
              <lb/>
            </s>
            <s xml:id="echoid-s2159" xml:space="preserve">Quod ſi circulus Hyperbolæ non occurrat, _Problema_ prorſus </s>
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