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

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        <div xml:id="echoid-div1040" type="section" level="1" n="523">
          <p>
            <s xml:id="echoid-s13186" xml:space="preserve">
              <pb o="386" file="398" n="398" rhead=""/>
            quadrantes ſunt, ob angulos rectos B, BAE. </s>
            <s xml:id="echoid-s13187" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s13188" xml:space="preserve">arcum AC, minorem eſ-
              <lb/>
            ſe quadrante, ita oſtendemus. </s>
            <s xml:id="echoid-s13189" xml:space="preserve">Quoniam arcus BE, ducitur per E, polum ar-
              <lb/>
              <figure xlink:label="fig-398-01" xlink:href="fig-398-01a" number="240">
                <image file="398-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/398-01"/>
              </figure>
            cus BD; </s>
            <s xml:id="echoid-s13190" xml:space="preserve">(oſtendemus enim E, eſſe polum arcus AB,
              <lb/>
            vt ſupra, cum BE, quadrans ſit, rectusq́ue ad arcum
              <lb/>
            AB.) </s>
            <s xml:id="echoid-s13191" xml:space="preserve">erit punctum C, intra peripheriam circuli ar-
              <lb/>
            cus BD, in ſuperficie ſphæræ, & </s>
            <s xml:id="echoid-s13192" xml:space="preserve">præter eiuſdem po-
              <lb/>
            lum. </s>
            <s xml:id="echoid-s13193" xml:space="preserve">Quare arcus CA, minor erit arcu CD: </s>
            <s xml:id="echoid-s13194" xml:space="preserve">At CD,
              <lb/>
              <note position="left" xlink:label="note-398-01" xlink:href="note-398-01a" xml:space="preserve">Schol. 21.</note>
            oſtenſus eſt eſſe quadrans. </s>
            <s xml:id="echoid-s13195" xml:space="preserve">Igitur AC, quadrante mi-
              <lb/>
              <note position="left" xlink:label="note-398-02" xlink:href="note-398-02a" xml:space="preserve">2. Theod.</note>
            nor erit. </s>
            <s xml:id="echoid-s13196" xml:space="preserve">Omnes ergo arcus trianguli ABC, qua-
              <lb/>
            drante ſunt minores. </s>
            <s xml:id="echoid-s13197" xml:space="preserve">Quocirca in omni triangulo
              <lb/>
            ſpherico rectangulo, &</s>
            <s xml:id="echoid-s13198" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13199" xml:space="preserve">Quod oſtendendum erat.</s>
            <s xml:id="echoid-s13200" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1043" type="section" level="1" n="524">
          <head xml:id="echoid-head559" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s13201" xml:space="preserve">_PRIMA_ pars buius propoſitionis vera quoque eſt, ſi ſolum vterque arcus circa
              <lb/>
            angulum rectum ponatur quadrante miner, etiamſi ignoretur, reliquum arcum,
              <lb/>
            qui rectum angulum ſubtendit, minorem eſſe quadrante. </s>
            <s xml:id="echoid-s13202" xml:space="preserve">Id quod liquido conſtat ex
              <lb/>
            demonſtratione prioris partis. </s>
            <s xml:id="echoid-s13203" xml:space="preserve">Oſtenſum eſt enim, angulos _BAC, BCA,_ eſſe acu-
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            tos, ex eo ſolum, quòd vterque arcus _BA, BC,_ quadrante minor ponatur, nulla
              <lb/>
            facta mentione arcus _AC._ </s>
            <s xml:id="echoid-s13204" xml:space="preserve">Erit tamen ſemper arcus rectum angulum ſubtendens
              <lb/>
            quadrante minor, ſi duo arcus rectum angulum continentes quadrante minores ſint,
              <lb/>
            vt ex demonſtratione manife ſtum eſt. </s>
            <s xml:id="echoid-s13205" xml:space="preserve">Nam cum ex eo, quòd arcus _BA, BC,_ mino-
              <lb/>
            res ſint quadrante, anguli A, C, acuti ſint, vt in priore parte demonſtratum eſt, ſit,
              <lb/>
            vt & </s>
            <s xml:id="echoid-s13206" xml:space="preserve">arcus AC, minor ſit quadrante, vtin parte poſteriori eſt oſtenſum. </s>
            <s xml:id="echoid-s13207" xml:space="preserve">Itaque
              <lb/>
            proponi poterit etiam buiuſmodi Theorema.</s>
            <s xml:id="echoid-s13208" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13209" xml:space="preserve">IN omni ttiangulo ſphærico rectangulo, cuius duo arcus rectum
              <lb/>
            angulum comprehendentes quadrante ſint minores, erit & </s>
            <s xml:id="echoid-s13210" xml:space="preserve">arcus
              <lb/>
            angulum rectum ſubtendens quadrante minor.</s>
            <s xml:id="echoid-s13211" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1044" type="section" level="1" n="525">
          <head xml:id="echoid-head560" xml:space="preserve">THEOR. 27. PROPOS. 29.</head>
          <p>
            <s xml:id="echoid-s13212" xml:space="preserve">IN omni triangulo ſphærico, cuius omnes an-
              <lb/>
            guli ſint acuti, arcus ſinguli quadrante ſunt mi-
              <lb/>
            nores.</s>
            <s xml:id="echoid-s13213" xml:space="preserve"/>
          </p>
          <figure number="241">
            <image file="398-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/398-02"/>
          </figure>
          <p>
            <s xml:id="echoid-s13214" xml:space="preserve">IN triangulo ſphærico ABC, ſint omnes an-
              <lb/>
            guli acuti. </s>
            <s xml:id="echoid-s13215" xml:space="preserve">Dico ſingulos arcus quadrante mino-
              <lb/>
            res eſſe. </s>
            <s xml:id="echoid-s13216" xml:space="preserve">Sint enim primum omnes anguli acuti
              <lb/>
            æquales. </s>
            <s xml:id="echoid-s13217" xml:space="preserve">Quo poſito, erunt ſinguli arcus qua-
              <lb/>
              <note position="left" xlink:label="note-398-03" xlink:href="note-398-03a" xml:space="preserve">Corollar.
                <lb/>
              25. huius.</note>
            drante minores, vt ſupra demonſtratum eſt.</s>
            <s xml:id="echoid-s13218" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13219" xml:space="preserve">DEINDE ſint duo tantum anguli acuti æ-
              <lb/>
            quales B, C; </s>
            <s xml:id="echoid-s13220" xml:space="preserve">& </s>
            <s xml:id="echoid-s13221" xml:space="preserve">A, minor vtroque illorum. </s>
            <s xml:id="echoid-s13222" xml:space="preserve">Eric
              <lb/>
              <note position="left" xlink:label="note-398-04" xlink:href="note-398-04a" xml:space="preserve">25. huius.</note>
            igitur vterque arcus AB, AC, minor quadrante.</s>
            <s xml:id="echoid-s13223" xml:space="preserve"/>
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