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

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          <p>
            <s xml:id="echoid-s13090" xml:space="preserve">SINT iam in eodem triangulo ABC, duo arcus AB, AC, quadrante
              <lb/>
            quidem maiores, at BC, quadrans. </s>
            <s xml:id="echoid-s13091" xml:space="preserve">Autigitur arcus AB, AC, æquales ſunt,
              <lb/>
            aut inæquales. </s>
            <s xml:id="echoid-s13092" xml:space="preserve">Si æquales, erunt duo anguli B, C, obtuſi. </s>
            <s xml:id="echoid-s13093" xml:space="preserve">Sit quadrans BD,
              <lb/>
              <figure xlink:label="fig-396-01" xlink:href="fig-396-01a" number="236">
                <image file="396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/396-01"/>
              </figure>
              <note position="left" xlink:label="note-396-01" xlink:href="note-396-01a" xml:space="preserve">25. huius.</note>
            & </s>
            <s xml:id="echoid-s13094" xml:space="preserve">per puncta C, D, arcus CD, maximi circuli du-
              <lb/>
              <note position="left" xlink:label="note-396-02" xlink:href="note-396-02a" xml:space="preserve">20. 1 Theod.</note>
            catur conueniens cum arcu CA, protracto in E.
              <lb/>
            </s>
            <s xml:id="echoid-s13095" xml:space="preserve">Quia igitur arcus BC, BD, quadrantes ſunt, erũt
              <lb/>
            anguli D, & </s>
            <s xml:id="echoid-s13096" xml:space="preserve">BCD, recti; </s>
            <s xml:id="echoid-s13097" xml:space="preserve">& </s>
            <s xml:id="echoid-s13098" xml:space="preserve">arcus CD, propter
              <lb/>
              <note position="left" xlink:label="note-396-03" xlink:href="note-396-03a" xml:space="preserve">25. huius</note>
            angulum B, quem obtuſum eſſe oſtendimus, qua-
              <lb/>
            drante maior: </s>
            <s xml:id="echoid-s13099" xml:space="preserve">Ponitur autem & </s>
            <s xml:id="echoid-s13100" xml:space="preserve">arcus AC, qua-
              <lb/>
              <note position="left" xlink:label="note-396-04" xlink:href="note-396-04a" xml:space="preserve">26. huius.</note>
            drante maior. </s>
            <s xml:id="echoid-s13101" xml:space="preserve">Igitur arcus CD, CA, ſimul ma-
              <lb/>
            iores ſunt ſemicirculo; </s>
            <s xml:id="echoid-s13102" xml:space="preserve">ac propterea, cum arcus
              <lb/>
            CDE, CAE, circulum conficiant, (quòd vter-
              <lb/>
            que ſemicirculus ſit.) </s>
            <s xml:id="echoid-s13103" xml:space="preserve">erunt arcus ED, EA, ſemi-
              <lb/>
              <note position="left" xlink:label="note-396-05" xlink:href="note-396-05a" xml:space="preserve">11. 1. Theod</note>
            circulo minores. </s>
            <s xml:id="echoid-s13104" xml:space="preserve">Quare angulus EDB, qui rectus
              <lb/>
            eſt, (quòd duo anguli ad D, æquales ſint duobus
              <lb/>
              <note position="left" xlink:label="note-396-06" xlink:href="note-396-06a" xml:space="preserve">5. huius.</note>
            rectis, & </s>
            <s xml:id="echoid-s13105" xml:space="preserve">angulus BDC, oſtenſus ſit rectus.) </s>
            <s xml:id="echoid-s13106" xml:space="preserve">maior
              <lb/>
            erit angulo EAD; </s>
            <s xml:id="echoid-s13107" xml:space="preserve">atque adeo EAD, acutus erit.
              <lb/>
            </s>
            <s xml:id="echoid-s13108" xml:space="preserve">
              <note position="left" xlink:label="note-396-07" xlink:href="note-396-07a" xml:space="preserve">14. huius.</note>
            Cum ergo anguli EAD, DAC, duobus rectis ſint
              <lb/>
              <note position="left" xlink:label="note-396-08" xlink:href="note-396-08a" xml:space="preserve">5. huius.</note>
            æquales, erit BAC, obtuſus. </s>
            <s xml:id="echoid-s13109" xml:space="preserve">Sunt etiam anguli B, C, demonſtrati obtuſi.
              <lb/>
            </s>
            <s xml:id="echoid-s13110" xml:space="preserve">Tres igitur anguli A, B, C, trianguli ABC, obtuſi ſunt.</s>
            <s xml:id="echoid-s13111" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13112" xml:space="preserve">SI verò AB, AC, latera, quæ quadrante maiora ſunt, non ſunt æqualia,
              <lb/>
            ſit maius AC; </s>
            <s xml:id="echoid-s13113" xml:space="preserve">& </s>
            <s xml:id="echoid-s13114" xml:space="preserve">abſcindatur arcus AD, æqualis arcui AB; </s>
            <s xml:id="echoid-s13115" xml:space="preserve">& </s>
            <s xml:id="echoid-s13116" xml:space="preserve">per puncta B,
              <lb/>
              <note position="left" xlink:label="note-396-09" xlink:href="note-396-09a" xml:space="preserve">1. huius.</note>
            D, tranſeat arcus BD, circuli maximi: </s>
            <s xml:id="echoid-s13117" xml:space="preserve">eritq́ue adhuc arcus AD, maior qua-
              <lb/>
              <figure xlink:label="fig-396-02" xlink:href="fig-396-02a" number="237">
                <image file="396-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/396-02"/>
              </figure>
              <note position="left" xlink:label="note-396-10" xlink:href="note-396-10a" xml:space="preserve">20. 1 Theod.</note>
            drante, cum ei æqualis AB, maior etiam ponatur-
              <lb/>
            Anguli igitur ADB, ABD, obtuſi ſunt. </s>
            <s xml:id="echoid-s13118" xml:space="preserve">Multo
              <lb/>
              <note position="left" xlink:label="note-396-11" xlink:href="note-396-11a" xml:space="preserve">25. huius.</note>
            ergo magis obtuſus erit angulus ABC. </s>
            <s xml:id="echoid-s13119" xml:space="preserve">Sit qua-
              <lb/>
            drans BE, & </s>
            <s xml:id="echoid-s13120" xml:space="preserve">per puncta C, E, tranſeat arcus CE,
              <lb/>
              <note position="left" xlink:label="note-396-12" xlink:href="note-396-12a" xml:space="preserve">20. 1 Theod.</note>
            circuli maximi occurrens arcui CA, producto in
              <lb/>
            F. </s>
            <s xml:id="echoid-s13121" xml:space="preserve">Quoniam igitur quadrantes ſunt BE, BC, & </s>
            <s xml:id="echoid-s13122" xml:space="preserve">
              <lb/>
            angulus EBC, oſtẽſus eſt obtuſus, erit arcus EC,
              <lb/>
              <note position="left" xlink:label="note-396-13" xlink:href="note-396-13a" xml:space="preserve">26. huius.</note>
            maior quadrãte, ſed anguli E, & </s>
            <s xml:id="echoid-s13123" xml:space="preserve">BCE, recti erunt.
              <lb/>
            </s>
            <s xml:id="echoid-s13124" xml:space="preserve">
              <note position="left" xlink:label="note-396-14" xlink:href="note-396-14a" xml:space="preserve">25. huius.</note>
            Angulus ergo ACB, obtuſus erit. </s>
            <s xml:id="echoid-s13125" xml:space="preserve">Et quoniam
              <lb/>
            arcus CE, oſtenſus eſt quadrante maior, & </s>
            <s xml:id="echoid-s13126" xml:space="preserve">arcus
              <lb/>
            AC, maior etiam ponitur, quam quadrans; </s>
            <s xml:id="echoid-s13127" xml:space="preserve">erunt
              <lb/>
            arcus CE, CA, ſimul ſemicirculo maiores. </s>
            <s xml:id="echoid-s13128" xml:space="preserve">Cũ ergo
              <lb/>
            arcus CEF, CAF, integro circulo æquales ſint,
              <lb/>
              <note position="left" xlink:label="note-396-15" xlink:href="note-396-15a" xml:space="preserve">21. 1. Theod.</note>
            quòd vterque ſit ſemicirculus, erũt arcus FE, FA,
              <lb/>
            ſimul ſemicirculo minores. </s>
            <s xml:id="echoid-s13129" xml:space="preserve">Quamobrem angulus
              <lb/>
            FEB, quirectus eſt, (ſunt enim duo anguli ad E, duobus rectis æquales, & </s>
            <s xml:id="echoid-s13130" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-396-16" xlink:href="note-396-16a" xml:space="preserve">5. huius.</note>
            angulus BEC, oſtenſus eſt rectus.) </s>
            <s xml:id="echoid-s13131" xml:space="preserve">maior erit angulo FAE. </s>
            <s xml:id="echoid-s13132" xml:space="preserve">Acutus ergo eſt
              <lb/>
              <note position="left" xlink:label="note-396-17" xlink:href="note-396-17a" xml:space="preserve">14. huius.</note>
            angulus FAE; </s>
            <s xml:id="echoid-s13133" xml:space="preserve">ac propterea, cum duo anguli ad A, ſint æquales duobus rectis,
              <lb/>
              <note position="left" xlink:label="note-396-18" xlink:href="note-396-18a" xml:space="preserve">5. huius.</note>
            angulus BAc, obtuſus erit. </s>
            <s xml:id="echoid-s13134" xml:space="preserve">Sunt autem etiam oſtenſi obtuſi anguli ABC,
              <lb/>
            ACB. </s>
            <s xml:id="echoid-s13135" xml:space="preserve">Tres igitur anguli in triangulo ABC, obtuſi ſunt. </s>
            <s xml:id="echoid-s13136" xml:space="preserve">In omni ergo trian
              <lb/>
            gulo ſphærico, cuius omnes arcus, &</s>
            <s xml:id="echoid-s13137" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13138" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s13139" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1038" type="section" level="1" n="522">
          <head xml:id="echoid-head557" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s13140" xml:space="preserve">_HAEC_ propoſitio non conuertitur. </s>
            <s xml:id="echoid-s13141" xml:space="preserve">Non enim omne triangulum ſphæricum, cu-
              <lb/>
            ius omnes anguli ſunt obtuſi, neceſſario habet omnes arcus quadrante maiores, </s>
          </p>
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