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

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            <s xml:id="echoid-s6788" xml:space="preserve">
              <pb o="181" file="193" n="193" rhead=""/>
            ducto auferatur numerus procreatus ex multiplicatione duarum chordarum datarum in-
              <lb/>
            ter ſe; </s>
            <s xml:id="echoid-s6789" xml:space="preserve">reliquus autem numerus per diametrum diuidatur, relinquetur chorda, ex qua ſi per
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s6790" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6791" xml:space="preserve">inueſtigetur chorda arcus, qui cum relictæ chordæ arcu ſemicirculum conficit,
              <lb/>
            erit hæc inuenta ſubtendens arcum compoſitum ex duobus arcubus duarum chordarum
              <lb/>
            datarum. </s>
            <s xml:id="echoid-s6792" xml:space="preserve">Operatio hæc perſpicua eſt ex figura, & </s>
            <s xml:id="echoid-s6793" xml:space="preserve">demonſtrarione priori huius propoſ.</s>
            <s xml:id="echoid-s6794" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6795" xml:space="preserve">EADEM hæc operatio colligi poteſt ex poſteriori demonſtratione, vt manifeſtum eſt.</s>
            <s xml:id="echoid-s6796" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div517" type="section" level="1" n="251">
          <head xml:id="echoid-head278" xml:space="preserve">PROBL. 7. PROPOS. 15.</head>
          <note position="right" xml:space="preserve">Quo pa-
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          cto ex da-
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          ta chorda
            <lb/>
          reperiatur
            <lb/>
          chorda ſe-
            <lb/>
          miſſis arcus
            <lb/>
          datæ chor-
            <lb/>
          dæ.</note>
          <p>
            <s xml:id="echoid-s6797" xml:space="preserve">EX data chorda cuiuſuis arcus chordã ſemiſ-
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            ſis illius arcus inuenire.</s>
            <s xml:id="echoid-s6798" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6799" xml:space="preserve">IN circulo ABC, cuius centrum E, data ſit chorda BC, arcus BDC, cu-
              <lb/>
            ius ſemiſsis ſit arcus BD, eiusq́; </s>
            <s xml:id="echoid-s6800" xml:space="preserve">chorda BD, quam inuenire oporteat. </s>
            <s xml:id="echoid-s6801" xml:space="preserve">Ducta
              <lb/>
            diametro DG, ſecabitea, per lemma in definitionibus poſitum, rectam BC,
              <lb/>
            bifariam, ac proinde ad angulos rectos. </s>
            <s xml:id="echoid-s6802" xml:space="preserve">Iunctis autem rectis BA, BG; </s>
            <s xml:id="echoid-s6803" xml:space="preserve">erunt
              <lb/>
              <note position="right" xlink:label="note-193-02" xlink:href="note-193-02a" xml:space="preserve">3. tertij.</note>
            duo triangula ABC, EFC, æquiangula, cum angulus EFC, oſtenſus ſitre-
              <lb/>
            ctus, & </s>
            <s xml:id="echoid-s6804" xml:space="preserve">angulus ABC, ſit quoq; </s>
            <s xml:id="echoid-s6805" xml:space="preserve">rectus in ſemicirculo, at angulus C, commu
              <lb/>
              <note position="right" xlink:label="note-193-03" xlink:href="note-193-03a" xml:space="preserve">31. tertij.</note>
            nis. </s>
            <s xml:id="echoid-s6806" xml:space="preserve">Igitur erit, vt CF, ad FE, ita CB, ad BA: </s>
            <s xml:id="echoid-s6807" xml:space="preserve">& </s>
            <s xml:id="echoid-s6808" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-193-04" xlink:href="note-193-04a" xml:space="preserve">4. ſexti.</note>
              <figure xlink:label="fig-193-01" xlink:href="fig-193-01a" number="144">
                <image file="193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/193-01"/>
              </figure>
            permutando, vt CF, ad CB, ita FE, ad BA. </s>
            <s xml:id="echoid-s6809" xml:space="preserve">Cum
              <lb/>
            ergo CF, dimidium ſit ipſius CB, vt oſtendimus,
              <lb/>
            erit & </s>
            <s xml:id="echoid-s6810" xml:space="preserve">EF, dimidium ipſius AB: </s>
            <s xml:id="echoid-s6811" xml:space="preserve">ac propterea cum
              <lb/>
            AB, data ſit ex data BC, data quoq; </s>
            <s xml:id="echoid-s6812" xml:space="preserve">erit EF; </s>
            <s xml:id="echoid-s6813" xml:space="preserve">qua
              <lb/>
              <note position="right" xlink:label="note-193-05" xlink:href="note-193-05a" xml:space="preserve">3. huius.</note>
            dempta ex ſemidiametro ED, nota, data erit quoq;
              <lb/>
            </s>
            <s xml:id="echoid-s6814" xml:space="preserve">reliqua FD. </s>
            <s xml:id="echoid-s6815" xml:space="preserve">Quoniam vero in triangulo GBD, an-
              <lb/>
            gulus B, rectus eſt, à quo demiſſa eſt BF, ad baſim
              <lb/>
              <note position="right" xlink:label="note-193-06" xlink:href="note-193-06a" xml:space="preserve">31. tertij.</note>
            GD, perpendicularis; </s>
            <s xml:id="echoid-s6816" xml:space="preserve">erit recta DB, media propor
              <lb/>
              <note position="right" xlink:label="note-193-07" xlink:href="note-193-07a" xml:space="preserve">Coroll.8.6.</note>
            tionalis inter GD, & </s>
            <s xml:id="echoid-s6817" xml:space="preserve">FD: </s>
            <s xml:id="echoid-s6818" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s6819" xml:space="preserve">adeo rectangulum
              <lb/>
            ſub GD, FD, notis quadrato rectæ DB, æquale.
              <lb/>
            </s>
            <s xml:id="echoid-s6820" xml:space="preserve">
              <note position="right" xlink:label="note-193-08" xlink:href="note-193-08a" xml:space="preserve">17. ſexti.</note>
            Notum ergo erit quadratum rectæ DB; </s>
            <s xml:id="echoid-s6821" xml:space="preserve">proptereaq́; </s>
            <s xml:id="echoid-s6822" xml:space="preserve">radix eius quadrata re-
              <lb/>
            ctam DB, notam exhibebit. </s>
            <s xml:id="echoid-s6823" xml:space="preserve">Quam etiam ita cognoſcemus. </s>
            <s xml:id="echoid-s6824" xml:space="preserve">Quoniam FD,
              <lb/>
            nota facta eſt, erunt quadrata rectarum FD, FB, nota: </s>
            <s xml:id="echoid-s6825" xml:space="preserve">quæ cum æqualia ſint
              <lb/>
              <note position="right" xlink:label="note-193-09" xlink:href="note-193-09a" xml:space="preserve">47. primi.</note>
            quadrato rectæ BD; </s>
            <s xml:id="echoid-s6826" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s6827" xml:space="preserve">hoc quadratum notum, cuius radix quadrata ite-
              <lb/>
            rum rectam BD, efficiet notam. </s>
            <s xml:id="echoid-s6828" xml:space="preserve">quod eſt propoſitum.</s>
            <s xml:id="echoid-s6829" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6830" xml:space="preserve">ALITER. </s>
            <s xml:id="echoid-s6831" xml:space="preserve">SIT rurſus in ſemicirculo ABC, data chorda BC, arcus
              <lb/>
            BDC, cuius ſemiſsis ſit arcus DC, ciusq́; </s>
            <s xml:id="echoid-s6832" xml:space="preserve">chorda DC, quam oporteat dari.
              <lb/>
            </s>
            <s xml:id="echoid-s6833" xml:space="preserve">Ducta chorda AB, abſcindatur ei æqualis AE, iunganturq; </s>
            <s xml:id="echoid-s6834" xml:space="preserve">rectæ BD, DE; </s>
            <s xml:id="echoid-s6835" xml:space="preserve">
              <lb/>
            Diuiſa quoq; </s>
            <s xml:id="echoid-s6836" xml:space="preserve">EC, bifariam in F, demittatur recta
              <lb/>
              <figure xlink:label="fig-193-02" xlink:href="fig-193-02a" number="145">
                <image file="193-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/193-02"/>
              </figure>
            DF. </s>
            <s xml:id="echoid-s6837" xml:space="preserve">Quoniam igitur duo latera BA, AD, equa-
              <lb/>
            lia ſunt duobus lateribus EA, AD, anguloſq;
              <lb/>
            </s>
            <s xml:id="echoid-s6838" xml:space="preserve">comprehendunt æquales, ob arcus æquales BD,
              <lb/>
              <note position="right" xlink:label="note-193-10" xlink:href="note-193-10a" xml:space="preserve">27. tertij.</note>
            DC; </s>
            <s xml:id="echoid-s6839" xml:space="preserve">erunt baſes BD, DE, æquales. </s>
            <s xml:id="echoid-s6840" xml:space="preserve">Eſt autem
              <lb/>
              <note position="right" xlink:label="note-193-11" xlink:href="note-193-11a" xml:space="preserve">4. primi.</note>
            BD, recta rectæ DC, æqualis. </s>
            <s xml:id="echoid-s6841" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s6842" xml:space="preserve">recta DE,
              <lb/>
              <note position="right" xlink:label="note-193-12" xlink:href="note-193-12a" xml:space="preserve">29. tertij.</note>
            eidem DC, æqualis erit. </s>
            <s xml:id="echoid-s6843" xml:space="preserve">Quare cum duo latera
              <lb/>
            EF, FD, duobus lateribus CF, FD, æqualia ſint,
              <lb/>
            baſisq́; </s>
            <s xml:id="echoid-s6844" xml:space="preserve">DE, baſi DC, æqualis; </s>
            <s xml:id="echoid-s6845" xml:space="preserve">erunt anguli ad F,
              <lb/>
              <note position="right" xlink:label="note-193-13" xlink:href="note-193-13a" xml:space="preserve">8.primi.</note>
            æquales, ideoq́; </s>
            <s xml:id="echoid-s6846" xml:space="preserve">recti. </s>
            <s xml:id="echoid-s6847" xml:space="preserve">Quoniam vero chorda AB, nota eſt ex data chorda BC;
              <lb/>
            </s>
            <s xml:id="echoid-s6848" xml:space="preserve">
              <note position="right" xlink:label="note-193-14" xlink:href="note-193-14a" xml:space="preserve">3. huius.</note>
            erit quoq; </s>
            <s xml:id="echoid-s6849" xml:space="preserve">AE, ipſi AB, æqualis, nota: </s>
            <s xml:id="echoid-s6850" xml:space="preserve">qua ablata ex diametro AC, nota re-
              <lb/>
            linquetur EC; </s>
            <s xml:id="echoid-s6851" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s6852" xml:space="preserve">huius medietas FC. </s>
            <s xml:id="echoid-s6853" xml:space="preserve">Iam vero, quia CD, media
              <lb/>
              <note position="right" xlink:label="note-193-15" xlink:href="note-193-15a" xml:space="preserve">Coroll. 8.6.</note>
            </s>
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