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-s12723" xml:space="preserve">
              <pb o="90" file="0268" n="283" rhead=""/>
            mæ VDq + EIq + FKq + GLq; </s>
            <s xml:id="echoid-s12724" xml:space="preserve">ergò ſumma IXq +
              <lb/>
            KXq + LXq + HXq, ſubdupla eſt ſummæ VDq - EIq +
              <lb/>
            FKq + GLq. </s>
            <s xml:id="echoid-s12725" xml:space="preserve">Vnde liquet Propoſitum.</s>
            <s xml:id="echoid-s12726" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12727" xml:space="preserve">XV. </s>
            <s xml:id="echoid-s12728" xml:space="preserve">Quòd ſi curva DXH talis concipiatur, ut ſit ordinata quæpiam,
              <lb/>
            ceu IX, inter congruas ordinatas IE, IZ bimedia *; </s>
            <s xml:id="echoid-s12729" xml:space="preserve">erit ſumma cubo-
              <lb/>
            rum ex IX, KX, LX, &</s>
            <s xml:id="echoid-s12730" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12731" xml:space="preserve">ſubtripla cuborum ex DV, IE, KF, &</s>
            <s xml:id="echoid-s12732" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12733" xml:space="preserve">Sin IX
              <lb/>
            ſit trimed. </s>
            <s xml:id="echoid-s12734" xml:space="preserve">* erit IXqq + KXqq + LXqq, &</s>
            <s xml:id="echoid-s12735" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12736" xml:space="preserve">= {DVqq + IEqq + KFqq/4}
              <lb/>
            &</s>
            <s xml:id="echoid-s12737" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12738" xml:space="preserve">ac ità porrò quoad cæteras poteſtates. </s>
            <s xml:id="echoid-s12739" xml:space="preserve">* _Not._ </s>
            <s xml:id="echoid-s12740" xml:space="preserve">bimediam ap-
              <lb/>
            pello, quæ duarum mediarum proportionalium prima; </s>
            <s xml:id="echoid-s12741" xml:space="preserve">trimediam,
              <lb/>
            quæ trium prima eſt, &</s>
            <s xml:id="echoid-s12742" xml:space="preserve">c.</s>
            <s xml:id="echoid-s12743" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12744" xml:space="preserve">Hæc ſimili ratione colliguntur, ac comprobantur. </s>
            <s xml:id="echoid-s12745" xml:space="preserve">piget χοχχὺζι
              <unsure/>
            ν.</s>
            <s xml:id="echoid-s12746" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12747" xml:space="preserve">XVI. </s>
            <s xml:id="echoid-s12748" xml:space="preserve">Sit porrò linea VYQ talis, ut ordinata AY ipſi AT; </s>
            <s xml:id="echoid-s12749" xml:space="preserve">& </s>
            <s xml:id="echoid-s12750" xml:space="preserve">
              <lb/>
            ordinata BY ipſi BT, &</s>
            <s xml:id="echoid-s12751" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12752" xml:space="preserve">æquentur; </s>
            <s xml:id="echoid-s12753" xml:space="preserve">erit IZq + KZq + LZq,
              <lb/>
            &</s>
            <s xml:id="echoid-s12754" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12755" xml:space="preserve">(ſumma quadratorum ex ordinatis à curva DZO ad rectam DH)
              <lb/>
            æqualis ſummæ VA x AE x AY + AB x BF x BY + BC x CG
              <lb/>
            x CY, &</s>
            <s xml:id="echoid-s12756" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12757" xml:space="preserve">(hoc eſt figuræ VDH in figuram VDQ ductæ).</s>
            <s xml:id="echoid-s12758" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12759" xml:space="preserve">XVII. </s>
            <s xml:id="echoid-s12760" xml:space="preserve">Item, ſumma IZ. </s>
            <s xml:id="echoid-s12761" xml:space="preserve">cub. </s>
            <s xml:id="echoid-s12762" xml:space="preserve">+ KZ cub. </s>
            <s xml:id="echoid-s12763" xml:space="preserve">+ LZ cub. </s>
            <s xml:id="echoid-s12764" xml:space="preserve">&</s>
            <s xml:id="echoid-s12765" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12766" xml:space="preserve">=
              <lb/>
            VA x AE x AYq + AB x BE x BYq + BC x CG x CYq,
              <lb/>
            &</s>
            <s xml:id="echoid-s12767" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12768" xml:space="preserve">_hoc eſt figuræ_ VDH _in figuræ_ VDQ _quadrata ductæ_). </s>
            <s xml:id="echoid-s12769" xml:space="preserve">Simi-
              <lb/>
            lis & </s>
            <s xml:id="echoid-s12770" xml:space="preserve">aliarum _poteſtatum_ eſt ratio.</s>
            <s xml:id="echoid-s12771" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12772" xml:space="preserve">Ad ſuperiorum normam hæc facilè colliges.</s>
            <s xml:id="echoid-s12773" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12774" xml:space="preserve">XVIII. </s>
            <s xml:id="echoid-s12775" xml:space="preserve">Eadem vera ſunt, & </s>
            <s xml:id="echoid-s12776" xml:space="preserve">omnino ſimiliratione comprobantur,
              <lb/>
            Etiam ſi curvæ VH convexa rectæ VD obvertantur. </s>
            <s xml:id="echoid-s12777" xml:space="preserve">Nempe, ſi linea
              <lb/>
              <note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">Fig. 126.</note>
            DZO talis ſit, ut ductâ per quodvis in curva VH punctum E tangente
              <lb/>
            ET, & </s>
            <s xml:id="echoid-s12778" xml:space="preserve">EA ad HD parallelâ, ac EIZ ad VD parallelâ, ſit perpetim IZ =
              <lb/>
            AT; </s>
            <s xml:id="echoid-s12779" xml:space="preserve">erit ſpatium DHO ſpatio VDH æquale; </s>
            <s xml:id="echoid-s12780" xml:space="preserve">& </s>
            <s xml:id="echoid-s12781" xml:space="preserve">ſolidum factum ex ſpa-
              <lb/>
            DHO circa axem VR converſo duplum erit ſolidi ex ſpatio VDH
              <lb/>
            circa eundem axem VD rotato producti. </s>
            <s xml:id="echoid-s12782" xml:space="preserve">quin & </s>
            <s xml:id="echoid-s12783" xml:space="preserve">reliqua pari modo
              <lb/>
            convenient.</s>
            <s xml:id="echoid-s12784" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12785" xml:space="preserve">XIX Porrò, ſit curva quæpiam AMB, cujus axis AD, & </s>
            <s xml:id="echoid-s12786" xml:space="preserve">huic
              <lb/>
            perpendicularis BD; </s>
            <s xml:id="echoid-s12787" xml:space="preserve">tum alia ſit linea KZL talis, ut ſumpto in cur-
              <lb/>
            va AB utcunque puncto M; </s>
            <s xml:id="echoid-s12788" xml:space="preserve">& </s>
            <s xml:id="echoid-s12789" xml:space="preserve">per hoc ductis rectâ MT curvam
              <lb/>
              <note position="left" xlink:label="note-0268-02" xlink:href="note-0268-02a" xml:space="preserve">Fig. 127.</note>
            AB tangente, rectâ MFZ ad DB parallelâ (quæ lineam KL ſecet
              <lb/>
            in Z, rectam AD in F) datâque quâdam lineâ R; </s>
            <s xml:id="echoid-s12790" xml:space="preserve">ſit TF. </s>
            <s xml:id="echoid-s12791" xml:space="preserve">FM:</s>
            <s xml:id="echoid-s12792" xml:space="preserve"/>
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