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

Table of figures

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          <p>
            <s xml:id="echoid-s2159" xml:space="preserve">
              <pb o="41" file="0059" n="59" rhead=""/>
            erit. </s>
            <s xml:id="echoid-s2160" xml:space="preserve">‖ Sin punctum Y extra datum angulum exiſtat, evidens eſt tan-
              <lb/>
            tùm uno modo problemati ſatisfactum iri; </s>
            <s xml:id="echoid-s2161" xml:space="preserve">quódque per alteram in-
              <lb/>
            erſectionem, & </s>
            <s xml:id="echoid-s2162" xml:space="preserve">Y, ducta recta ad angulum pertinet dato verticalem.
              <lb/>
            </s>
            <s xml:id="echoid-s2163" xml:space="preserve">hæc, inq̀uam, tantillùm attendenti manifeſtè conſtabunt; </s>
            <s xml:id="echoid-s2164" xml:space="preserve">nihil ut ſit
              <lb/>
            opus hic plura verba conſumere. </s>
            <s xml:id="echoid-s2165" xml:space="preserve">verùm ut in horum caſuum primo
              <lb/>
            conſtet (id quod pro ſequentibus ex uſu erit cognoſcere) quando
              <lb/>
            dictus circulus _byperbolem_ contingit; </s>
            <s xml:id="echoid-s2166" xml:space="preserve">ſeu quando tantùm una per
              <lb/>
            Y recta quantitatis ejuſdem interſeri poſſit, hoc adnectemus _Theo-_
              <lb/>
            _rema._</s>
            <s xml:id="echoid-s2167" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2168" xml:space="preserve">IX. </s>
            <s xml:id="echoid-s2169" xml:space="preserve">Si à puncto quovis Y intra rectum angulum XPF exiſtente
              <lb/>
              <note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">Fig. 54.</note>
            demittantur ad ejuſdem anguli latera perpendiculares YB, YD; </s>
            <s xml:id="echoid-s2170" xml:space="preserve">ac
              <lb/>
            inter YB, YD proportione mediæ ſint rectæ BN, GD; </s>
            <s xml:id="echoid-s2171" xml:space="preserve">per puncta
              <lb/>
            N, Y, G tranſibit recta cunctarum minima, quæ per Y ductæ angu-
              <lb/>
            lum XPF ſubtendere poſſunt.</s>
            <s xml:id="echoid-s2172" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2173" xml:space="preserve">Quòd NYG ſit una recta patet, quoniam eſt YB. </s>
            <s xml:id="echoid-s2174" xml:space="preserve">BN :</s>
            <s xml:id="echoid-s2175" xml:space="preserve">: GD.
              <lb/>
            </s>
            <s xml:id="echoid-s2176" xml:space="preserve">DY (ex conſtructione nimirum) porrò per Y tranſeat alia quæcunque
              <lb/>
            recta LYM; </s>
            <s xml:id="echoid-s2177" xml:space="preserve">& </s>
            <s xml:id="echoid-s2178" xml:space="preserve">NH ad GN, MH ad PF perpendiculares concur-
              <lb/>
            rant in H. </s>
            <s xml:id="echoid-s2179" xml:space="preserve">item HA ad NG parallela ducatur; </s>
            <s xml:id="echoid-s2180" xml:space="preserve">& </s>
            <s xml:id="echoid-s2181" xml:space="preserve">GS ad PF; </s>
            <s xml:id="echoid-s2182" xml:space="preserve">
              <lb/>
            denuóque connectatur GH. </s>
            <s xml:id="echoid-s2183" xml:space="preserve">Jam patet triangula GDY, YBN,
              <lb/>
            HMN, HMR ſimilia fore; </s>
            <s xml:id="echoid-s2184" xml:space="preserve">quódque proptereà eſt MN. </s>
            <s xml:id="echoid-s2185" xml:space="preserve">MR :</s>
            <s xml:id="echoid-s2186" xml:space="preserve">:
              <lb/>
            MN q. </s>
            <s xml:id="echoid-s2187" xml:space="preserve">MHq :</s>
            <s xml:id="echoid-s2188" xml:space="preserve">: DGq. </s>
            <s xml:id="echoid-s2189" xml:space="preserve">YDq. </s>
            <s xml:id="echoid-s2190" xml:space="preserve">item (ob BN, DG, YD {.</s>
            <s xml:id="echoid-s2191" xml:space="preserve">./.</s>
            <s xml:id="echoid-s2192" xml:space="preserve">.})
              <lb/>
            eſt BN. </s>
            <s xml:id="echoid-s2193" xml:space="preserve">YD :</s>
            <s xml:id="echoid-s2194" xml:space="preserve">: DGq. </s>
            <s xml:id="echoid-s2195" xml:space="preserve">YDq. </s>
            <s xml:id="echoid-s2196" xml:space="preserve">hoc eſt YN. </s>
            <s xml:id="echoid-s2197" xml:space="preserve">YG (vel MN. </s>
            <s xml:id="echoid-s2198" xml:space="preserve">GS)
              <lb/>
            :</s>
            <s xml:id="echoid-s2199" xml:space="preserve">: DGq. </s>
            <s xml:id="echoid-s2200" xml:space="preserve">YDq. </s>
            <s xml:id="echoid-s2201" xml:space="preserve">ergò eſt MN. </s>
            <s xml:id="echoid-s2202" xml:space="preserve">MR :</s>
            <s xml:id="echoid-s2203" xml:space="preserve">: MN. </s>
            <s xml:id="echoid-s2204" xml:space="preserve">GS. </s>
            <s xml:id="echoid-s2205" xml:space="preserve">adeóque MR
              <lb/>
            = GS. </s>
            <s xml:id="echoid-s2206" xml:space="preserve">itaque major eſt GS ipsâ MT; </s>
            <s xml:id="echoid-s2207" xml:space="preserve">abeóque rectæ GH, LM
              <lb/>
            protractæ concurrent; </s>
            <s xml:id="echoid-s2208" xml:space="preserve">puta ad Z. </s>
            <s xml:id="echoid-s2209" xml:space="preserve">ergò LM. </s>
            <s xml:id="echoid-s2210" xml:space="preserve">GH :</s>
            <s xml:id="echoid-s2211" xml:space="preserve">: LZ. </s>
            <s xml:id="echoid-s2212" xml:space="preserve">GZ. </s>
            <s xml:id="echoid-s2213" xml:space="preserve">
              <lb/>
            verùm propter angulum LGH recto P majorem, eſt LZ &</s>
            <s xml:id="echoid-s2214" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2215" xml:space="preserve">GZ. </s>
            <s xml:id="echoid-s2216" xml:space="preserve">
              <lb/>
            quare LM &</s>
            <s xml:id="echoid-s2217" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2218" xml:space="preserve">GH. </s>
            <s xml:id="echoid-s2219" xml:space="preserve">aſt ob angulum rectum GNH eſt GH &</s>
            <s xml:id="echoid-s2220" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2221" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s2222" xml:space="preserve">
              <lb/>
            quare magìs eſt LM &</s>
            <s xml:id="echoid-s2223" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2224" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s2225" xml:space="preserve">eodémque modo quævis per Y ducta
              <lb/>
            major oſtendetur ipsâ GN : </s>
            <s xml:id="echoid-s2226" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2227" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2228" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2229" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2230" xml:space="preserve">X. </s>
            <s xml:id="echoid-s2231" xml:space="preserve">Hinc etiam ſi GN ſit in ratione YB ad YN quarta proportio-
              <lb/>
            nalis; </s>
            <s xml:id="echoid-s2232" xml:space="preserve">erit GN minima. </s>
            <s xml:id="echoid-s2233" xml:space="preserve">nam indè conſequetur fore YB, BN, GD,
              <lb/>
            YD {.</s>
            <s xml:id="echoid-s2234" xml:space="preserve">./.</s>
            <s xml:id="echoid-s2235" xml:space="preserve">.}. </s>
            <s xml:id="echoid-s2236" xml:space="preserve">Etenim erit YNq. </s>
            <s xml:id="echoid-s2237" xml:space="preserve">YBq :</s>
            <s xml:id="echoid-s2238" xml:space="preserve">: GN. </s>
            <s xml:id="echoid-s2239" xml:space="preserve">YN. </s>
            <s xml:id="echoid-s2240" xml:space="preserve">& </s>
            <s xml:id="echoid-s2241" xml:space="preserve">dividendo
              <lb/>
            BNq. </s>
            <s xml:id="echoid-s2242" xml:space="preserve">YBq :</s>
            <s xml:id="echoid-s2243" xml:space="preserve">: GY. </s>
            <s xml:id="echoid-s2244" xml:space="preserve">YN :</s>
            <s xml:id="echoid-s2245" xml:space="preserve">: DY . </s>
            <s xml:id="echoid-s2246" xml:space="preserve">BN. </s>
            <s xml:id="echoid-s2247" xml:space="preserve">ac indè YBq x DY =
              <lb/>
            BNcub; </s>
            <s xml:id="echoid-s2248" xml:space="preserve">velDY = {BN cub/YBq}. </s>
            <s xml:id="echoid-s2249" xml:space="preserve">itáque DY eſt quarta proportionalis in ra-
              <lb/>
            tione YB ad BN.</s>
            <s xml:id="echoid-s2250" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2251" xml:space="preserve">XI. </s>
            <s xml:id="echoid-s2252" xml:space="preserve">Subnotari poteſt autem, quòd minimæ GN propiores </s>
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