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-s4306" xml:space="preserve">V. </s>
            <s xml:id="echoid-s4307" xml:space="preserve">Horum caſuum primus ad unum duntaxat ab una axis parte radi-
              <lb/>
            um pertinet, qui reliquos aliis caſibus convenientes medius diſterminat.
              <lb/>
            </s>
            <s xml:id="echoid-s4308" xml:space="preserve">
              <note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">Fig. 101.</note>
            de poſteribus itaque duobus ſeparatim paullò diſpiciamus, Sit jam
              <lb/>
            itaque primò AC = AG = A γ; </s>
            <s xml:id="echoid-s4309" xml:space="preserve">unde quilibet incidens cavo GB γ
              <lb/>
            radius (ut AN) major erit quam AC; </s>
            <s xml:id="echoid-s4310" xml:space="preserve">hujus itaque reflexus axem
              <lb/>
            ſecet puncto K; </s>
            <s xml:id="echoid-s4311" xml:space="preserve">dico, ſi ſemidiameter CB dividatur in Z; </s>
            <s xml:id="echoid-s4312" xml:space="preserve">ut ſit CZ.
              <lb/>
            </s>
            <s xml:id="echoid-s4313" xml:space="preserve">ZB:</s>
            <s xml:id="echoid-s4314" xml:space="preserve">: AC. </s>
            <s xml:id="echoid-s4315" xml:space="preserve">AB; </s>
            <s xml:id="echoid-s4316" xml:space="preserve">fore CK &</s>
            <s xml:id="echoid-s4317" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s4318" xml:space="preserve">CZ. </s>
            <s xml:id="echoid-s4319" xml:space="preserve">etenim ob angulum ANK
              <lb/>
              <note position="left" xlink:label="note-0088-02" xlink:href="note-0088-02a" xml:space="preserve">Fig. 100.</note>
            biſectum, erit AC. </s>
            <s xml:id="echoid-s4320" xml:space="preserve">CK:</s>
            <s xml:id="echoid-s4321" xml:space="preserve">: AN. </s>
            <s xml:id="echoid-s4322" xml:space="preserve">NK. </s>
            <s xml:id="echoid-s4323" xml:space="preserve">vel permutando AC. </s>
            <s xml:id="echoid-s4324" xml:space="preserve">AN
              <lb/>
            :</s>
            <s xml:id="echoid-s4325" xml:space="preserve">: CK. </s>
            <s xml:id="echoid-s4326" xml:space="preserve">NK. </s>
            <s xml:id="echoid-s4327" xml:space="preserve">eſt autem AC. </s>
            <s xml:id="echoid-s4328" xml:space="preserve">AB &</s>
            <s xml:id="echoid-s4329" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4330" xml:space="preserve">AC. </s>
            <s xml:id="echoid-s4331" xml:space="preserve">AN ergo AC. </s>
            <s xml:id="echoid-s4332" xml:space="preserve">AB
              <lb/>
            &</s>
            <s xml:id="echoid-s4333" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4334" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4335" xml:space="preserve">NK &</s>
            <s xml:id="echoid-s4336" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4337" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4338" xml:space="preserve">BK. </s>
            <s xml:id="echoid-s4339" xml:space="preserve">ergo cùm ſit, ex hypotheſi, CZ. </s>
            <s xml:id="echoid-s4340" xml:space="preserve">ZB
              <lb/>
            :</s>
            <s xml:id="echoid-s4341" xml:space="preserve">: AC. </s>
            <s xml:id="echoid-s4342" xml:space="preserve">AB; </s>
            <s xml:id="echoid-s4343" xml:space="preserve">erit CZ. </s>
            <s xml:id="echoid-s4344" xml:space="preserve">ZB &</s>
            <s xml:id="echoid-s4345" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4346" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4347" xml:space="preserve">BK. </s>
            <s xml:id="echoid-s4348" xml:space="preserve">componendóque CB.
              <lb/>
            </s>
            <s xml:id="echoid-s4349" xml:space="preserve">ZB. </s>
            <s xml:id="echoid-s4350" xml:space="preserve">&</s>
            <s xml:id="echoid-s4351" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4352" xml:space="preserve">CB. </s>
            <s xml:id="echoid-s4353" xml:space="preserve">KB. </s>
            <s xml:id="echoid-s4354" xml:space="preserve">unde ZB &</s>
            <s xml:id="echoid-s4355" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s4356" xml:space="preserve">KB; </s>
            <s xml:id="echoid-s4357" xml:space="preserve">ſeu CZ &</s>
            <s xml:id="echoid-s4358" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4359" xml:space="preserve">CK; </s>
            <s xml:id="echoid-s4360" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s4361" xml:space="preserve">
              <lb/>
            E. </s>
            <s xml:id="echoid-s4362" xml:space="preserve">D.</s>
            <s xml:id="echoid-s4363" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4364" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s4365" xml:space="preserve">Hinc punctum Z eſt limes infra quem, Verſus centrum, nullus
              <lb/>
            reflexus axem interſecat.</s>
            <s xml:id="echoid-s4366" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4367" xml:space="preserve">_Coroll._ </s>
            <s xml:id="echoid-s4368" xml:space="preserve">Hinc ſi puncta Z, ζ ſint limites punctorum A, _a_ (quorum
              <lb/>
            A remotius) erit CZ &</s>
            <s xml:id="echoid-s4369" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s4370" xml:space="preserve">C ζ.</s>
            <s xml:id="echoid-s4371" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4372" xml:space="preserve">Nam BC. </s>
            <s xml:id="echoid-s4373" xml:space="preserve">AC &</s>
            <s xml:id="echoid-s4374" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4375" xml:space="preserve">BC. </s>
            <s xml:id="echoid-s4376" xml:space="preserve">_a_ C. </s>
            <s xml:id="echoid-s4377" xml:space="preserve">componendóque AB. </s>
            <s xml:id="echoid-s4378" xml:space="preserve">AC &</s>
            <s xml:id="echoid-s4379" xml:space="preserve">lt;
              <lb/>
            </s>
            <s xml:id="echoid-s4380" xml:space="preserve">_a_ B. </s>
            <s xml:id="echoid-s4381" xml:space="preserve">_a_ C. </s>
            <s xml:id="echoid-s4382" xml:space="preserve">hoc eſt ZB. </s>
            <s xml:id="echoid-s4383" xml:space="preserve">ZC &</s>
            <s xml:id="echoid-s4384" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4385" xml:space="preserve">ζ B. </s>
            <s xml:id="echoid-s4386" xml:space="preserve">ζ C. </s>
            <s xml:id="echoid-s4387" xml:space="preserve">vel compoſitè CB. </s>
            <s xml:id="echoid-s4388" xml:space="preserve">ZC
              <lb/>
            &</s>
            <s xml:id="echoid-s4389" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s4390" xml:space="preserve">CB. </s>
            <s xml:id="echoid-s4391" xml:space="preserve">ζ C. </s>
            <s xml:id="echoid-s4392" xml:space="preserve">ergò ZC &</s>
            <s xml:id="echoid-s4393" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s4394" xml:space="preserve">ζ C.</s>
            <s xml:id="echoid-s4395" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4396" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s4397" xml:space="preserve">Quinetiam erit in hoc caſu; </s>
            <s xml:id="echoid-s4398" xml:space="preserve">ANq - ACq. </s>
            <s xml:id="echoid-s4399" xml:space="preserve">CNq:</s>
            <s xml:id="echoid-s4400" xml:space="preserve">:
              <lb/>
            AC. </s>
            <s xml:id="echoid-s4401" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4402" xml:space="preserve">Nam ducatur KH ad CN parallela, protractæ AN
              <lb/>
            occurrens in H; </s>
            <s xml:id="echoid-s4403" xml:space="preserve">& </s>
            <s xml:id="echoid-s4404" xml:space="preserve">connectatur CP; </s>
            <s xml:id="echoid-s4405" xml:space="preserve">& </s>
            <s xml:id="echoid-s4406" xml:space="preserve">eodem planè modo quo ſu-
              <lb/>
            periùs (in iis quæ circa convexas partes attigimus) oſtendetur fore
              <lb/>
            AN x NP. </s>
            <s xml:id="echoid-s4407" xml:space="preserve">CNq:</s>
            <s xml:id="echoid-s4408" xml:space="preserve">: AK. </s>
            <s xml:id="echoid-s4409" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4410" xml:space="preserve">unde diviſim erit AN x NP -
              <lb/>
            CNq. </s>
            <s xml:id="echoid-s4411" xml:space="preserve">CNq:</s>
            <s xml:id="echoid-s4412" xml:space="preserve">: AC. </s>
            <s xml:id="echoid-s4413" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s4414" xml:space="preserve">eſt autem AN x NP = ANq -
              <lb/>
            AN x AP = ANq -: </s>
            <s xml:id="echoid-s4415" xml:space="preserve">ACq - CNq = ANq - ACq +
              <lb/>
            CNq; </s>
            <s xml:id="echoid-s4416" xml:space="preserve">adeóque AN x NP - CNq = ANq - ACq. </s>
            <s xml:id="echoid-s4417" xml:space="preserve">ergò
              <lb/>
            demum erit ANq - ACq. </s>
            <s xml:id="echoid-s4418" xml:space="preserve">CNq:</s>
            <s xml:id="echoid-s4419" xml:space="preserve">: AC. </s>
            <s xml:id="echoid-s4420" xml:space="preserve">CK: </s>
            <s xml:id="echoid-s4421" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s4422" xml:space="preserve">E. </s>
            <s xml:id="echoid-s4423" xml:space="preserve">D.</s>
            <s xml:id="echoid-s4424" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4425" xml:space="preserve">Notetur; </s>
            <s xml:id="echoid-s4426" xml:space="preserve">ſi fuerit AC minor ſemiſſe ſemidiametri circuli re-
              <lb/>
            flectentis, quòd punctum A duos focos habebit ad eaſdem centri par-
              <lb/>
            tes, quorum alter ad partes D, alter ad B pertinebit; </s>
            <s xml:id="echoid-s4427" xml:space="preserve">ſin AC major
              <lb/>
            fuerit iſtâ Semiſſe, focis qui ad diverſos vertices B, & </s>
            <s xml:id="echoid-s4428" xml:space="preserve">D pertinent,
              <lb/>
            centrum C interjacebit.</s>
            <s xml:id="echoid-s4429" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4430" xml:space="preserve">VIII. </s>
            <s xml:id="echoid-s4431" xml:space="preserve">Etiam hoc interſeram _Theorema_, præmiſſis conforme: </s>
            <s xml:id="echoid-s4432" xml:space="preserve">Si
              <lb/>
            fiat 2 CK. </s>
            <s xml:id="echoid-s4433" xml:space="preserve">CN:</s>
            <s xml:id="echoid-s4434" xml:space="preserve">: CN. </s>
            <s xml:id="echoid-s4435" xml:space="preserve">F; </s>
            <s xml:id="echoid-s4436" xml:space="preserve">itémque 2 CA. </s>
            <s xml:id="echoid-s4437" xml:space="preserve">CN:</s>
            <s xml:id="echoid-s4438" xml:space="preserve">: CN. </s>
            <s xml:id="echoid-s4439" xml:space="preserve">E;</s>
            <s xml:id="echoid-s4440" xml:space="preserve"/>
          </p>
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