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-div381" type="section" level="1" n="41">
          <p>
            <s xml:id="echoid-s12659" xml:space="preserve">
              <pb o="89" file="0267" n="282" rhead=""/>
            CG = KL x LZ; </s>
            <s xml:id="echoid-s12660" xml:space="preserve">& </s>
            <s xml:id="echoid-s12661" xml:space="preserve">AB x BF = IK x KZ, & </s>
            <s xml:id="echoid-s12662" xml:space="preserve">VA x AE =
              <lb/>
            DI x IZ. </s>
            <s xml:id="echoid-s12663" xml:space="preserve">Verùm ſumma CD x DH + BC x CG + AB x
              <lb/>
            BF + VA x AE à ſpatio VDH minimè differt; </s>
            <s xml:id="echoid-s12664" xml:space="preserve">& </s>
            <s xml:id="echoid-s12665" xml:space="preserve">ſumma LH x
              <lb/>
            DO + KL x LZ + IK x KZ + DI x IZ à ſpatio DHO mi-
              <lb/>
            nimè differt. </s>
            <s xml:id="echoid-s12666" xml:space="preserve">itaque ſpatio VDH, DHO æquantur.</s>
            <s xml:id="echoid-s12667" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12668" xml:space="preserve">Hoc _perutile Theorema_ doctiſſimo Viro D. </s>
            <s xml:id="echoid-s12669" xml:space="preserve">_Gregorio Aberdonenſi_
              <lb/>
            debetur; </s>
            <s xml:id="echoid-s12670" xml:space="preserve">cui ſequentia ſubnectimus.</s>
            <s xml:id="echoid-s12671" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12672" xml:space="preserve">XI. </s>
            <s xml:id="echoid-s12673" xml:space="preserve">Iiſdem poſitis; </s>
            <s xml:id="echoid-s12674" xml:space="preserve">ſolidum ex ſpatio DHO circa axem VDR
              <lb/>
            rotato factum duplum erit ſolidi facti ex ſpatio VDH itidem circa ax-
              <lb/>
              <note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">Fig. 125.</note>
            em VD rotato.</s>
            <s xml:id="echoid-s12675" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12676" xml:space="preserve">Nam eſt HL. </s>
            <s xml:id="echoid-s12677" xml:space="preserve">LG:</s>
            <s xml:id="echoid-s12678" xml:space="preserve">: (DH. </s>
            <s xml:id="echoid-s12679" xml:space="preserve">DT:</s>
            <s xml:id="echoid-s12680" xml:space="preserve">: DH. </s>
            <s xml:id="echoid-s12681" xml:space="preserve">HO:</s>
            <s xml:id="echoid-s12682" xml:space="preserve">:) DHq.
              <lb/>
            </s>
            <s xml:id="echoid-s12683" xml:space="preserve">DH x HO. </s>
            <s xml:id="echoid-s12684" xml:space="preserve">unde HL x DH x HO = LG x DHq = CD x
              <lb/>
            DHq. </s>
            <s xml:id="echoid-s12685" xml:space="preserve">Similíque diſcurſu ſunt LK x DL x LZ = BC x CGq. </s>
            <s xml:id="echoid-s12686" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s12687" xml:space="preserve">KI x DK x KZ = AB x BFq. </s>
            <s xml:id="echoid-s12688" xml:space="preserve">& </s>
            <s xml:id="echoid-s12689" xml:space="preserve">demum ID x DI x IZ =
              <lb/>
            VA x AEq. </s>
            <s xml:id="echoid-s12690" xml:space="preserve">Eſt autem (ut vulgò notatum habetur) ſumma CD
              <lb/>
            x DHq + BCB x CGq + AB x BFq + VA x AEq dupla
              <lb/>
            ſummæ DI x IE + DK x KF + DL x LG, &</s>
            <s xml:id="echoid-s12691" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12692" xml:space="preserve">Quare ſolidum
              <lb/>
            ex ſpatio HDO circa axem DR converſo factum duplum eſt ſolidi,
              <lb/>
            quod è ſpatio VDH circa VD converſo producitur.</s>
            <s xml:id="echoid-s12693" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12694" xml:space="preserve">XII. </s>
            <s xml:id="echoid-s12695" xml:space="preserve">Hinc, ſumma DI x IZ + DK x KZ + DL x LZ, &</s>
            <s xml:id="echoid-s12696" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s12697" xml:space="preserve">æquatur ſummæ quadratorum ex applicatis ad VD; </s>
            <s xml:id="echoid-s12698" xml:space="preserve">ſcilicet ipſis AEq
              <lb/>
            + BFq + CGq, &</s>
            <s xml:id="echoid-s12699" xml:space="preserve">c.</s>
            <s xml:id="echoid-s12700" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12701" xml:space="preserve">XIII. </s>
            <s xml:id="echoid-s12702" xml:space="preserve">Simili ratiocinio conſtabit ſummam DIq x IZ + DKq x
              <lb/>
            KZ + DLq x LZ, &</s>
            <s xml:id="echoid-s12703" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12704" xml:space="preserve">triplam eſſe ſummæ DIq x IE + DKq
              <lb/>
            x KF + DLq x LG, &</s>
            <s xml:id="echoid-s12705" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12706" xml:space="preserve">hòc eſt æqualem ſummæ cuborum ab
              <lb/>
            omnibus AE, BF, CG, &</s>
            <s xml:id="echoid-s12707" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12708" xml:space="preserve">ad VD applicatis. </s>
            <s xml:id="echoid-s12709" xml:space="preserve">Idem quoad _re-_
              <lb/>
            _liquas poteſtates_ obſervabilis eſt Concluſionum tenor.</s>
            <s xml:id="echoid-s12710" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12711" xml:space="preserve">XIV. </s>
            <s xml:id="echoid-s12712" xml:space="preserve">Iiſdem poſitis; </s>
            <s xml:id="echoid-s12713" xml:space="preserve">ſi DXH ſit linea talis, ut quævis ad DH
              <lb/>
            o
              <unsure/>
            rdinata, ceu IX, ſit media proportionalis inter ſibi congruas ordi-
              <lb/>
            natas IE, IZ; </s>
            <s xml:id="echoid-s12714" xml:space="preserve">erìt ſolidum ex ſpatio VDH circa axem DH rotato
              <lb/>
            duplum ſolidi ex ſpatio DXH circa eundem axem DH converſo pro-
              <lb/>
            creati.</s>
            <s xml:id="echoid-s12715" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12716" xml:space="preserve">Nam ob VA x AE = DI x IZ, erit VA x AE x EI = DI x IZ x IE = ID x
              <lb/>
            IXq. </s>
            <s xml:id="echoid-s12717" xml:space="preserve">Similíque de cauſa AB x BF x FK = IK x KXq; </s>
            <s xml:id="echoid-s12718" xml:space="preserve">& </s>
            <s xml:id="echoid-s12719" xml:space="preserve">BC
              <lb/>
              <note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">In 10. hujus.</note>
            x CG x GL = KL x LXq, &</s>
            <s xml:id="echoid-s12720" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12721" xml:space="preserve">Eſt autem ſumma VA x AE
              <lb/>
            x EI + AB x BF x FK + BC x CG x GL, &</s>
            <s xml:id="echoid-s12722" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12723" xml:space="preserve">Subdupla </s>
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