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

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        <div xml:id="echoid-div58" type="section" level="1" n="14">
          <p>
            <s xml:id="echoid-s2342" xml:space="preserve">
              <pb o="43" file="0061" n="61" rhead=""/>
            jam aliquoties inſinuatâ; </s>
            <s xml:id="echoid-s2343" xml:space="preserve">ſcilicet ut ſit AB. </s>
            <s xml:id="echoid-s2344" xml:space="preserve">YB :</s>
            <s xml:id="echoid-s2345" xml:space="preserve">: √ Iq -Rq. </s>
            <s xml:id="echoid-s2346" xml:space="preserve">I;
              <lb/>
            </s>
            <s xml:id="echoid-s2347" xml:space="preserve">& </s>
            <s xml:id="echoid-s2348" xml:space="preserve">deſignetur quilibet refractus KN; </s>
            <s xml:id="echoid-s2349" xml:space="preserve">tum continuetur ratio YB ad
              <lb/>
            BN; </s>
            <s xml:id="echoid-s2350" xml:space="preserve">ut ſit ad has proportione quarta BP; </s>
            <s xml:id="echoid-s2351" xml:space="preserve">& </s>
            <s xml:id="echoid-s2352" xml:space="preserve">per punctum P du-
              <lb/>
              <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">Fig. 57, 58.</note>
            catur recta PZ ad AB parallela; </s>
            <s xml:id="echoid-s2353" xml:space="preserve">refracto KN occurrens in Z; </s>
            <s xml:id="echoid-s2354" xml:space="preserve">dico
              <lb/>
            nullum alium refractum per Z traſire. </s>
            <s xml:id="echoid-s2355" xml:space="preserve">Nam ſi ſieri poteſt tranſeat
              <lb/>
            alius ZR; </s>
            <s xml:id="echoid-s2356" xml:space="preserve">& </s>
            <s xml:id="echoid-s2357" xml:space="preserve">per Y traducantur rectæ NYG, RYS ; </s>
            <s xml:id="echoid-s2358" xml:space="preserve">è præmon-
              <lb/>
            ſtratis apparet quòd ſit RS = NG. </s>
            <s xml:id="echoid-s2359" xml:space="preserve">item è prædictis manifeſtum
              <note symbol="*" position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">12. Lect. 4.</note>
            quò RS &</s>
            <s xml:id="echoid-s2360" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2361" xml:space="preserve">N G. </s>
            <s xml:id="echoid-s2362" xml:space="preserve">quæ repugnant.</s>
            <s xml:id="echoid-s2363" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">9 hujus Lect.</note>
          <p>
            <s xml:id="echoid-s2364" xml:space="preserve">XVI. </s>
            <s xml:id="echoid-s2365" xml:space="preserve">Non diſpari ratione, quoad caſum ſecundum, deſignetur quilibet
              <lb/>
            refractus KN; </s>
            <s xml:id="echoid-s2366" xml:space="preserve">& </s>
            <s xml:id="echoid-s2367" xml:space="preserve">fiat KB . </s>
            <s xml:id="echoid-s2368" xml:space="preserve">GB :</s>
            <s xml:id="echoid-s2369" xml:space="preserve">: √ Rq - Iq. </s>
            <s xml:id="echoid-s2370" xml:space="preserve">R; </s>
            <s xml:id="echoid-s2371" xml:space="preserve">tum adnexâ
              <lb/>
            GN, ad ipſas NG, GB ſumatur tertia proportionalis V; </s>
            <s xml:id="echoid-s2372" xml:space="preserve">& </s>
            <s xml:id="echoid-s2373" xml:space="preserve">ſiat
              <lb/>
            NG. </s>
            <s xml:id="echoid-s2374" xml:space="preserve">V :</s>
            <s xml:id="echoid-s2375" xml:space="preserve">: BN. </s>
            <s xml:id="echoid-s2376" xml:space="preserve">NP; </s>
            <s xml:id="echoid-s2377" xml:space="preserve">& </s>
            <s xml:id="echoid-s2378" xml:space="preserve">per punctum P ducatur PY ad BA pa-
              <lb/>
            rallela refractum NK decuſſans in Z; </s>
            <s xml:id="echoid-s2379" xml:space="preserve">dico nullum alium refractum
              <lb/>
            per ipſum Z meare. </s>
            <s xml:id="echoid-s2380" xml:space="preserve">Nam, ſi neges, tranſeat alius ZR; </s>
            <s xml:id="echoid-s2381" xml:space="preserve">& </s>
            <s xml:id="echoid-s2382" xml:space="preserve">per
              <lb/>
            Y trajiciatur RY S; </s>
            <s xml:id="echoid-s2383" xml:space="preserve">& </s>
            <s xml:id="echoid-s2384" xml:space="preserve">quoniam ZP . </s>
            <s xml:id="echoid-s2385" xml:space="preserve">YP :</s>
            <s xml:id="echoid-s2386" xml:space="preserve">: KB. </s>
            <s xml:id="echoid-s2387" xml:space="preserve">GB :</s>
            <s xml:id="echoid-s2388" xml:space="preserve">: √ Rq
              <lb/>
            - Iq. </s>
            <s xml:id="echoid-s2389" xml:space="preserve">R. </s>
            <s xml:id="echoid-s2390" xml:space="preserve">ex * antedictis apparet fore RS = NG. </s>
            <s xml:id="echoid-s2391" xml:space="preserve">quinetiam ob
              <lb/>
              <note position="right" xlink:label="note-0061-04" xlink:href="note-0061-04a" xml:space="preserve">* 14. Lect.4.</note>
            NGq . </s>
            <s xml:id="echoid-s2392" xml:space="preserve">GBq :</s>
            <s xml:id="echoid-s2393" xml:space="preserve">: NG. </s>
            <s xml:id="echoid-s2394" xml:space="preserve">V :</s>
            <s xml:id="echoid-s2395" xml:space="preserve">: BN . </s>
            <s xml:id="echoid-s2396" xml:space="preserve">NP . </s>
            <s xml:id="echoid-s2397" xml:space="preserve">erit dividendo NBq.
              <lb/>
            </s>
            <s xml:id="echoid-s2398" xml:space="preserve">GBq :</s>
            <s xml:id="echoid-s2399" xml:space="preserve">: BP . </s>
            <s xml:id="echoid-s2400" xml:space="preserve">NP . </s>
            <s xml:id="echoid-s2401" xml:space="preserve">hoc eſt NPq. </s>
            <s xml:id="echoid-s2402" xml:space="preserve">PYq :</s>
            <s xml:id="echoid-s2403" xml:space="preserve">: BP . </s>
            <s xml:id="echoid-s2404" xml:space="preserve">NP; </s>
            <s xml:id="echoid-s2405" xml:space="preserve">inde facile
              <lb/>
            deducitur eſſe BP quartam proportionalem in ratione YP ad PN ; </s>
            <s xml:id="echoid-s2406" xml:space="preserve">
              <lb/>
            conſequentérque fore RS minimâ NG majorem . </s>
            <s xml:id="echoid-s2407" xml:space="preserve">quod adver-
              <lb/>
            ſatur oſtenſis . </s>
            <s xml:id="echoid-s2408" xml:space="preserve">itaque potiùs per Z nullus alius tranſit reſractus: </s>
            <s xml:id="echoid-s2409" xml:space="preserve">
              <lb/>
            Q. </s>
            <s xml:id="echoid-s2410" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2411" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2412" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2413" xml:space="preserve">XV I. </s>
            <s xml:id="echoid-s2414" xml:space="preserve">Prætereà, ſi refractum NKZ interſecet alius quilibet M I,
              <lb/>
            ad rectiorem pertinens incidentem (hoc eſt ut incidentiæ punctum M
              <lb/>
            inter B, & </s>
            <s xml:id="echoid-s2415" xml:space="preserve">N jaceat) interſectio X ſolitario puncto Z citerior erit
              <lb/>
            (ſeu perpendiculari KB propinquior) . </s>
            <s xml:id="echoid-s2416" xml:space="preserve">Nam ab X demittatur per-
              <lb/>
              <note position="right" xlink:label="note-0061-05" xlink:href="note-0061-05a" xml:space="preserve">Fig. 59, 60.</note>
            pendicularis XQ. </s>
            <s xml:id="echoid-s2417" xml:space="preserve">ipſam NG ſecans in γ ; </s>
            <s xml:id="echoid-s2418" xml:space="preserve">& </s>
            <s xml:id="echoid-s2419" xml:space="preserve">(in primo caſu) per
              <lb/>
            M, Y traducatur recta MY H. </s>
            <s xml:id="echoid-s2420" xml:space="preserve">ergò MH = Nγ. </s>
            <s xml:id="echoid-s2421" xml:space="preserve">quare minima earum
              <lb/>
            quæ per Y angulo XQF interſeri poſſuntinter puncta M, N cadet
              <lb/>
            (utì nuper admonitum, & </s>
            <s xml:id="echoid-s2422" xml:space="preserve">adſtructum). </s>
            <s xml:id="echoid-s2423" xml:space="preserve">puta ad φ. </s>
            <s xml:id="echoid-s2424" xml:space="preserve">ergò quum ſit
              <lb/>
            BP quarta proportionalis in ratione YB ad BN ; </s>
            <s xml:id="echoid-s2425" xml:space="preserve">& </s>
            <s xml:id="echoid-s2426" xml:space="preserve">BQ quarta
              <lb/>
            proportionalis in ratione YB ad B φ, erit PB &</s>
            <s xml:id="echoid-s2427" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2428" xml:space="preserve">QB; </s>
            <s xml:id="echoid-s2429" xml:space="preserve">adeóque
              <lb/>
            recta XQ rectis ZP, KB interjacet : </s>
            <s xml:id="echoid-s2430" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2431" xml:space="preserve">E . </s>
            <s xml:id="echoid-s2432" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2433" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2434" xml:space="preserve">In ſecundo caſu, per γ trajiciatur recta M γ H. </s>
            <s xml:id="echoid-s2435" xml:space="preserve">ergò cùm ſit
              <lb/>
            Q X. </s>
            <s xml:id="echoid-s2436" xml:space="preserve">Q γ :</s>
            <s xml:id="echoid-s2437" xml:space="preserve">: PZ. </s>
            <s xml:id="echoid-s2438" xml:space="preserve">PY :</s>
            <s xml:id="echoid-s2439" xml:space="preserve">: √ Rq - Iq. </s>
            <s xml:id="echoid-s2440" xml:space="preserve">R. </s>
            <s xml:id="echoid-s2441" xml:space="preserve">erit HM = GN. </s>
            <s xml:id="echoid-s2442" xml:space="preserve">ergò
              <lb/>
            minima per γ ducibilium angulo AB F intercipienda punctis M, N
              <lb/>
            intercider; </s>
            <s xml:id="echoid-s2443" xml:space="preserve">puta ad φ. </s>
            <s xml:id="echoid-s2444" xml:space="preserve">quare QB quarta proportionalis erit in ratione
              <lb/>
            γ Qad Q φ; </s>
            <s xml:id="echoid-s2445" xml:space="preserve">& </s>
            <s xml:id="echoid-s2446" xml:space="preserve">eſt γ Q. </s>
            <s xml:id="echoid-s2447" xml:space="preserve">Qφ &</s>
            <s xml:id="echoid-s2448" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s2449" xml:space="preserve">(γ Q. </s>
            <s xml:id="echoid-s2450" xml:space="preserve">QN.) </s>
            <s xml:id="echoid-s2451" xml:space="preserve">:</s>
            <s xml:id="echoid-s2452" xml:space="preserve">: YP. </s>
            <s xml:id="echoid-s2453" xml:space="preserve">PN . </s>
            <s xml:id="echoid-s2454" xml:space="preserve">&</s>
            <s xml:id="echoid-s2455" xml:space="preserve"/>
          </p>
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