Theodosius <Bithynius>; Clavius, Christoph, Theodosii Tripolitae Sphaericorum libri tres

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        <div xml:id="echoid-div282" type="section" level="1" n="130">
          <p>
            <s xml:id="echoid-s3488" xml:space="preserve">
              <pb o="90" file="102" n="102" rhead=""/>
            ad angulũ IGE, vt mox demonſtrabimus: </s>
            <s xml:id="echoid-s3489" xml:space="preserve">Eſt autem angulus OHE, angulo
              <lb/>
              <note position="left" xlink:label="note-102-01" xlink:href="note-102-01a" xml:space="preserve">10. vndec.</note>
            BGC, æqualis; </s>
            <s xml:id="echoid-s3490" xml:space="preserve">(ſunt enim rectæ OH, BG, communes ſectiones planorum pa-
              <lb/>
            rallelorũ OE, BC, factæ à plano AB, parallelæ; </s>
            <s xml:id="echoid-s3491" xml:space="preserve">necnõ & </s>
            <s xml:id="echoid-s3492" xml:space="preserve">rectæ EH, CG, com-
              <lb/>
              <note position="left" xlink:label="note-102-02" xlink:href="note-102-02a" xml:space="preserve">15. vndec.</note>
            munes ſectiones eorundem planorum factæ à plano AE.) </s>
            <s xml:id="echoid-s3493" xml:space="preserve">erit quoque maior
              <lb/>
            proportio rectę GI, ad rectam IK, hoc eſt, ad rectam ſibi æqualem IH, quàm
              <lb/>
            anguli BGC, ad angulum DGE: </s>
            <s xml:id="echoid-s3494" xml:space="preserve">Vt autem angulus BGC, ad angulum DGE,
              <lb/>
            ita eſt arcus BC, ad arcum DE. </s>
            <s xml:id="echoid-s3495" xml:space="preserve">Maior igitur proportio quoq; </s>
            <s xml:id="echoid-s3496" xml:space="preserve">erit rectæ GI,
              <lb/>
              <note position="left" xlink:label="note-102-03" xlink:href="note-102-03a" xml:space="preserve">33. ſexti.</note>
            ad rectam IH, quàm arcus BC, ad arcum DE: </s>
            <s xml:id="echoid-s3497" xml:space="preserve">Eſt autem, vt GI, ad IH, ita
              <lb/>
              <note position="left" xlink:label="note-102-04" xlink:href="note-102-04a" xml:space="preserve">4. ſexti.</note>
            GD, ad DN, hoc eſt, ita tota diameter DL, ad totam diametrum DM.
              <lb/>
            </s>
            <s xml:id="echoid-s3498" xml:space="preserve">
              <note position="left" xlink:label="note-102-05" xlink:href="note-102-05a" xml:space="preserve">15. quinti.</note>
            (ſunt enim DN, OH, communes ſectiones planorum parallelorum DF,
              <lb/>
            OE, factæ à plano AB, parallelæ.) </s>
            <s xml:id="echoid-s3499" xml:space="preserve">Igitur maior quoque proportio erit DL,
              <lb/>
              <note position="left" xlink:label="note-102-06" xlink:href="note-102-06a" xml:space="preserve">16. vndec.</note>
            diametri ſphæræ ad DM, diametrum paralleli DF, quàm arcus BC, ad arcum
              <lb/>
            DE. </s>
            <s xml:id="echoid-s3500" xml:space="preserve">Quapropter, ſi polus parallelorum ſit in circunferentia maximi circu-
              <lb/>
            li, &</s>
            <s xml:id="echoid-s3501" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3502" xml:space="preserve">Quod demon ſtrandum erat.</s>
            <s xml:id="echoid-s3503" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div284" type="section" level="1" n="131">
          <head xml:id="echoid-head145" xml:space="preserve">LEMMA.</head>
          <p style="it">
            <s xml:id="echoid-s3504" xml:space="preserve">_QVOD_ autem maior ſit proportio rectæ _GI,_ ad rectam _IK,_ quàm anguli _IKE,_
              <lb/>
            ad angulum _
              <emph style="sc">Ig</emph>
            E,_ hoc theoremate propoſito demonſtrabimus.</s>
            <s xml:id="echoid-s3505" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3506" xml:space="preserve">IN omni triangulo rectangulo, ſi ab vno acutorum angulorum
              <lb/>
            vtcunque ad latus oppoſitum linea recta ducatur; </s>
            <s xml:id="echoid-s3507" xml:space="preserve">erit maior propor-
              <lb/>
            tio huius lateris ad eius ſegmentum, quod prope angulum rectum
              <lb/>
            exiſtit, quàm anguli acuti, quem linea ducta cum prædicto latere, ef-
              <lb/>
            fecit, ad reliquum angulum acutum trianguli.</s>
            <s xml:id="echoid-s3508" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3509" xml:space="preserve">SIT triangulum rectangulum EGI, habens angulum I, rectum,
              <lb/>
            ducaturque ab angulo acuto IEG, ad latus oppoſitum GI, recta li-
              <lb/>
            nea EK, vtcunque. </s>
            <s xml:id="echoid-s3510" xml:space="preserve">Dico maiorem eſſe proportionem rectæ GI, ad IK,
              <lb/>
            quàm anguli acuti IKE, ad angulum acutum IGE. </s>
            <s xml:id="echoid-s3511" xml:space="preserve">Ducatur enim
              <lb/>
            per G, recta GA, ipſi EK, parallela, occurrens rectæ IE, protractæ
              <lb/>
              <note position="left" xlink:label="note-102-07" xlink:href="note-102-07a" xml:space="preserve">31.primi.</note>
              <figure xlink:label="fig-102-01" xlink:href="fig-102-01a" number="106">
                <image file="102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/102-01"/>
              </figure>
            in A. </s>
            <s xml:id="echoid-s3512" xml:space="preserve">Et quoniam angulus I, rectus eſt,
              <lb/>
            erit angulus IEG, acutus, & </s>
            <s xml:id="echoid-s3513" xml:space="preserve">propte-
              <lb/>
            rea AEG, obtuſus. </s>
            <s xml:id="echoid-s3514" xml:space="preserve">Latus igitur EG,
              <lb/>
            in triangulo GEI, maius eſt latere GI;
              <lb/>
            </s>
            <s xml:id="echoid-s3515" xml:space="preserve">
              <note position="left" xlink:label="note-102-08" xlink:href="note-102-08a" xml:space="preserve">19.primi.</note>
            in triangulo verò AEG, minus latere
              <lb/>
            AG. </s>
            <s xml:id="echoid-s3516" xml:space="preserve">Quare arcus circuli ex centro G,
              <lb/>
            ad interuallum GE, deſcriptus ſecabit
              <lb/>
            rectam GI, productam vltra I, nempe
              <lb/>
            in B, rectam vero GA, citra A, vt
              <lb/>
            in C. </s>
            <s xml:id="echoid-s3517" xml:space="preserve">Quoniam igitur triangulum GAE,
              <lb/>
            maius eſt ſectore GCE, maior erit proportio trianguli GAE, ad
              <lb/>
            triangulum GEI, quàm ſectoris GCE, ad triangulum GEI: </s>
            <s xml:id="echoid-s3518" xml:space="preserve">Eſt
              <lb/>
              <note position="left" xlink:label="note-102-09" xlink:href="note-102-09a" xml:space="preserve">3.quinti.</note>
            autcm maior adhuc proportio ſectoris GCE, ad triangulum GEI, quàm
              <lb/>
              <note position="left" xlink:label="note-102-10" xlink:href="note-102-10a" xml:space="preserve">3.quinti.</note>
            </s>
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