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-div283" type="section" level="1" n="32">
          <p>
            <s xml:id="echoid-s10633" xml:space="preserve">
              <pb o="57" file="0235" n="250" rhead=""/>
            diſcurſu eſt DM x TD - DM x RD = MO x RD. </s>
            <s xml:id="echoid-s10634" xml:space="preserve">quapropter
              <lb/>
              <note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Fig. 63.</note>
            erit MN x SD. </s>
            <s xml:id="echoid-s10635" xml:space="preserve">MO x RD :</s>
            <s xml:id="echoid-s10636" xml:space="preserve">: TD - SD. </s>
            <s xml:id="echoid-s10637" xml:space="preserve">TD - RD. </s>
            <s xml:id="echoid-s10638" xml:space="preserve">hoc eſt
              <lb/>
            LG x SD. </s>
            <s xml:id="echoid-s10639" xml:space="preserve">KG x RD :</s>
            <s xml:id="echoid-s10640" xml:space="preserve">: TD - SD. </s>
            <s xml:id="echoid-s10641" xml:space="preserve">TD - RD; </s>
            <s xml:id="echoid-s10642" xml:space="preserve">vel (ad æqua-
              <lb/>
            tionem redigendo) LG x SD x TD - LG x SD x RD = KG x
              <lb/>
            RD x TD - KG x RD x SD; </s>
            <s xml:id="echoid-s10643" xml:space="preserve">tranſponendóque LG x SD x
              <lb/>
            TD + KG x RD x SD - LG x SD x RD = KG x RD x TD.
              <lb/>
            </s>
            <s xml:id="echoid-s10644" xml:space="preserve">hoc eſt LG x SD x TD + KL x SD x RD = KG x RD x TD. </s>
            <s xml:id="echoid-s10645" xml:space="preserve">
              <lb/>
            vel (ad analogiſmum reducendo) LG x TD + KL x RD. </s>
            <s xml:id="echoid-s10646" xml:space="preserve">KG x
              <lb/>
            TD :</s>
            <s xml:id="echoid-s10647" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s10648" xml:space="preserve">SD. </s>
            <s xml:id="echoid-s10649" xml:space="preserve">Quod erat Propoſitum.</s>
            <s xml:id="echoid-s10650" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10651" xml:space="preserve">V. </s>
            <s xml:id="echoid-s10652" xml:space="preserve">Quòd ſi puncta T, R non ad eaſdem puncti D partes ſita ſint,
              <lb/>
              <note position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">Fig. 64.</note>
            erit LG x RD - KL x TD. </s>
            <s xml:id="echoid-s10653" xml:space="preserve">KG x TD :</s>
            <s xml:id="echoid-s10654" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s10655" xml:space="preserve">SD.</s>
            <s xml:id="echoid-s10656" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10657" xml:space="preserve">Simili conſtabit id diſcurſu; </s>
            <s xml:id="echoid-s10658" xml:space="preserve">quem piget repetere.</s>
            <s xml:id="echoid-s10659" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10660" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s10661" xml:space="preserve">Sint quatuor continuè proportionalium ſeries æquinumeræ (qua-
              <lb/>
            les adſcriptas cernis) quarum cùm antecedentes primi, tum ultimi conſe-
              <lb/>
            quentes inter ſe proportionales ſint(A. </s>
            <s xml:id="echoid-s10662" xml:space="preserve">α:</s>
            <s xml:id="echoid-s10663" xml:space="preserve">: M. </s>
            <s xml:id="echoid-s10664" xml:space="preserve">μ; </s>
            <s xml:id="echoid-s10665" xml:space="preserve">& </s>
            <s xml:id="echoid-s10666" xml:space="preserve">F. </s>
            <s xml:id="echoid-s10667" xml:space="preserve">φ:</s>
            <s xml:id="echoid-s10668" xml:space="preserve">: S. </s>
            <s xml:id="echoid-s10669" xml:space="preserve">σ)
              <lb/>
            crunt ejuſdem ordinis quilibet accepti quatuor etiam inter ſe proportio-
              <lb/>
            nales (puta nempe, D. </s>
            <s xml:id="echoid-s10670" xml:space="preserve">δ:</s>
            <s xml:id="echoid-s10671" xml:space="preserve">: P. </s>
            <s xml:id="echoid-s10672" xml:space="preserve">π).</s>
            <s xml:id="echoid-s10673" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          A. # B. # C. # D. # E. # F.
            <lb/>
          α. # β. # γ. # δ. # @. # φ
            <lb/>
          M. # N. # O. # P. # R. # S.
            <lb/>
          μ. # ν. # ο. # π. # ς. # σ.
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s10674" xml:space="preserve">Sunt enim Aμ, Bγ, Cο, Dπ, Eς, Fσ, \\ & </s>
            <s xml:id="echoid-s10675" xml:space="preserve">αM, βN, γO, δP, @R, φ S,} Continuè propor-
              <lb/>
            tionales.</s>
            <s xml:id="echoid-s10676" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10677" xml:space="preserve">Cùm igitur ſit Aμ, = αM; </s>
            <s xml:id="echoid-s10678" xml:space="preserve">& </s>
            <s xml:id="echoid-s10679" xml:space="preserve">Fσ = φ S, liquidum eſt ſore D π
              <lb/>
            = σ P; </s>
            <s xml:id="echoid-s10680" xml:space="preserve">ac idcircò D. </s>
            <s xml:id="echoid-s10681" xml:space="preserve">δ:</s>
            <s xml:id="echoid-s10682" xml:space="preserve">: P. </s>
            <s xml:id="echoid-s10683" xml:space="preserve">π. </s>
            <s xml:id="echoid-s10684" xml:space="preserve">Ad utramque proportionalitatem
              <lb/>
            (tam Arithmeticam quàm Geometricam) æquè ſpectat hæc Con-
              <lb/>
            cluſio.</s>
            <s xml:id="echoid-s10685" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10686" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s10687" xml:space="preserve">Rectæ A B, CD parallelæ ſint; </s>
            <s xml:id="echoid-s10688" xml:space="preserve">hásque ſecet poſitione data
              <lb/>
              <note position="right" xlink:label="note-0235-04" xlink:href="note-0235-04a" xml:space="preserve">Fig. 65.</note>
            BD; </s>
            <s xml:id="echoid-s10689" xml:space="preserve">lineæ verò EBE; </s>
            <s xml:id="echoid-s10690" xml:space="preserve">FBF ita relatæ ſint, ut ductâ utcunque
              <lb/>
            recta PG ad DB parallelâ; </s>
            <s xml:id="echoid-s10691" xml:space="preserve">ſit ſemper PF eodem ordine media pro-
              <lb/>
            portionalis inter PG, PE; </s>
            <s xml:id="echoid-s10692" xml:space="preserve">tum per quodvis deſignatum lineæ EBE
              <lb/>
            punctum E tranſeat HE ipſis AB, CD parallela, sítque alia curva
              <lb/>
            KEK talis, ut ductâ utcunque QL itidem ad DB parallelâ, ſit </s>
          </p>
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