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

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            <s xml:id="echoid-s17232" xml:space="preserve">
              <pb o="478" file="490" n="490" rhead=""/>
            arcus AD, intra, vel extra triangulum ceciderit, cognitus fiet arcus BC.</s>
            <s xml:id="echoid-s17233" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s17234" xml:space="preserve">AD extremum, per praxim problematis 1. </s>
            <s xml:id="echoid-s17235" xml:space="preserve">ſcholij propoſ. </s>
            <s xml:id="echoid-s17236" xml:space="preserve">41. </s>
            <s xml:id="echoid-s17237" xml:space="preserve">erue-
              <lb/>
            mus angulum CAD; </s>
            <s xml:id="echoid-s17238" xml:space="preserve">qui additus angulo inuento BAD, vel ab eo ſubtra-
              <lb/>
            ctus, prout arcus perpendicularis AD, intra triangulum ceciderit, vel
              <lb/>
            extra, notum faciet angulum BAC.</s>
            <s xml:id="echoid-s17239" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17240" xml:space="preserve">QVOD ſi quando arcus AC, alteriangulo B, dato oppoſitus, ſit qua-
              <lb/>
            drans, quod euenire poteſt, non exiſtente quadrante AB; </s>
            <s xml:id="echoid-s17241" xml:space="preserve">erit alter ſaltem
              <lb/>
            reliquorum quoque arcuum AD, CD, in triangulo ACD, quadrans. </s>
            <s xml:id="echoid-s17242" xml:space="preserve">Cum
              <lb/>
              <note position="left" xlink:label="note-490-01" xlink:href="note-490-01a" xml:space="preserve">36. huius.</note>
            ergo AD, eſſe non poſsit quadrans, erit CD, quadrans; </s>
            <s xml:id="echoid-s17243" xml:space="preserve">ac proinde angulus
              <lb/>
              <note position="left" xlink:label="note-490-02" xlink:href="note-490-02a" xml:space="preserve">46. huius.</note>
            ei oppoſitus CAD, rectus. </s>
            <s xml:id="echoid-s17244" xml:space="preserve">Itaque tunc inuentus erit & </s>
            <s xml:id="echoid-s17245" xml:space="preserve">arcus CD, & </s>
            <s xml:id="echoid-s17246" xml:space="preserve">angu-
              <lb/>
              <note position="left" xlink:label="note-490-03" xlink:href="note-490-03a" xml:space="preserve">Quãdo duo
                <lb/>
              anguli dati
                <lb/>
              inæquales
                <lb/>
              sũt, & datus
                <lb/>
              arcꝰ vni eo-
                <lb/>
              rũ oppoſitꝰ,
                <lb/>
              quadrans.</note>
            lus CAD, ſine vllo alio labore: </s>
            <s xml:id="echoid-s17247" xml:space="preserve">ex quibus & </s>
            <s xml:id="echoid-s17248" xml:space="preserve">arcus BC, & </s>
            <s xml:id="echoid-s17249" xml:space="preserve">angulus BAC, de-
              <lb/>
            prehendentur, vt dictum eſt.</s>
            <s xml:id="echoid-s17250" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17251" xml:space="preserve">SIT iam arcus datus AB, quadrans, & </s>
            <s xml:id="echoid-s17252" xml:space="preserve">adhuc duo anguli dati B, C, inæ-
              <lb/>
            quales. </s>
            <s xml:id="echoid-s17253" xml:space="preserve">Erit arcus BD, quadrans etiam, & </s>
            <s xml:id="echoid-s17254" xml:space="preserve">angulus BAD, rectus. </s>
            <s xml:id="echoid-s17255" xml:space="preserve">Cum enim
              <lb/>
            in triangulo ABD, arcus AB, angulo recto oppoſitus quadrans ponatur; </s>
            <s xml:id="echoid-s17256" xml:space="preserve">erit
              <lb/>
              <note position="left" xlink:label="note-490-04" xlink:href="note-490-04a" xml:space="preserve">36. huius.</note>
            ſaltem & </s>
            <s xml:id="echoid-s17257" xml:space="preserve">alter reliquorum arcuum AD, BD, quadrans. </s>
            <s xml:id="echoid-s17258" xml:space="preserve">Non poteſt autẽ AD,
              <lb/>
              <note position="left" xlink:label="note-490-05" xlink:href="note-490-05a" xml:space="preserve">25. huius.</note>
            eſſe quadrans: </s>
            <s xml:id="echoid-s17259" xml:space="preserve">quia duo anguli B, D, ob quadrantes AB, AD, recti eſſent,
              <lb/>
            ideoq́; </s>
            <s xml:id="echoid-s17260" xml:space="preserve">triangulum ABC, rectangulum, quod non ponitur. </s>
            <s xml:id="echoid-s17261" xml:space="preserve">Erit ergo BD,
              <lb/>
            quadrans, ac proinde angulus oppoſitus BAD, rectus. </s>
            <s xml:id="echoid-s17262" xml:space="preserve">Erit quoque B, po-
              <lb/>
              <note position="left" xlink:label="note-490-06" xlink:href="note-490-06a" xml:space="preserve">34. huius.</note>
            lus arcus AD, ob quadrantes AB, BD; </s>
            <s xml:id="echoid-s17263" xml:space="preserve">proptereaq́; </s>
            <s xml:id="echoid-s17264" xml:space="preserve">datus angulus B, arcum
              <lb/>
              <note position="left" xlink:label="note-490-07" xlink:href="note-490-07a" xml:space="preserve">26. huius.</note>
            BD, notum efficiet. </s>
            <s xml:id="echoid-s17265" xml:space="preserve">Inuẽtis autem arcubus AD, BD, cum angulo recto BAD,
              <lb/>
            ſine vlla multiplicationis moleſtia, in uenientur reliqua, vt prius. </s>
            <s xml:id="echoid-s17266" xml:space="preserve">In hoc ta-
              <lb/>
            men caſu arcus AC, nullo pacto quadrans erit, ne duo quadrantes ſint AB,
              <lb/>
              <note position="left" xlink:label="note-490-08" xlink:href="note-490-08a" xml:space="preserve">25. huius.</note>
            AC, in triangulo ABC, ac proinde duo anguli B, C, recti. </s>
            <s xml:id="echoid-s17267" xml:space="preserve">Quod eſtet contra
              <lb/>
            hypotheſim, cum triangulum ponatur non rectangulum.</s>
            <s xml:id="echoid-s17268" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17269" xml:space="preserve">VERVM ſint iam in triangulo ABC, dati duo anguli B, C, æquales.
              <lb/>
            </s>
            <s xml:id="echoid-s17270" xml:space="preserve">
              <note position="left" xlink:label="note-490-09" xlink:href="note-490-09a" xml:space="preserve">Quãdo duc
                <lb/>
              dati anguli
                <lb/>
              squales sũt.</note>
            Erunt duo arcus AB, AC, a quales; </s>
            <s xml:id="echoid-s17271" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s17272" xml:space="preserve">adeo neuter
              <lb/>
            eorum quadrans, ne duo anguli B, C, recti exiſtant.
              <lb/>
            </s>
            <s xml:id="echoid-s17273" xml:space="preserve">
              <figure xlink:label="fig-490-01" xlink:href="fig-490-01a" number="352">
                <image file="490-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/490-01"/>
              </figure>
            Demiſſus igitur arcus perpendicularis AD, ex tertio
              <lb/>
            angulo A, intra triangulum cadet, diuidetq; </s>
            <s xml:id="echoid-s17274" xml:space="preserve">tam ar-
              <lb/>
            cum BC, quam angulum BAC, bifariam, vt ſupra in
              <lb/>
            ſecundo caſu propoſ. </s>
            <s xml:id="echoid-s17275" xml:space="preserve">62. </s>
            <s xml:id="echoid-s17276" xml:space="preserve">oſtendimus. </s>
            <s xml:id="echoid-s17277" xml:space="preserve">Igitur quia in
              <lb/>
            triangulo ABD, rectum habente angulum D, datus
              <lb/>
            eſt arcus AB, angulo recto oppoſitus, cum angulo
              <lb/>
              <note position="left" xlink:label="note-490-10" xlink:href="note-490-10a" xml:space="preserve">Schol. 45.
                <lb/>
              huius.</note>
            B; </s>
            <s xml:id="echoid-s17278" xml:space="preserve">cognitus erit & </s>
            <s xml:id="echoid-s17279" xml:space="preserve">arcus BD: </s>
            <s xml:id="echoid-s17280" xml:space="preserve">qui duplicatus totum
              <lb/>
            arcum BC, notum efficiet: </s>
            <s xml:id="echoid-s17281" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s17282" xml:space="preserve">AC, notus eſt,
              <lb/>
            cum dato arcui AB, æqualis ſit. </s>
            <s xml:id="echoid-s17283" xml:space="preserve">Deinde quoniam in eodem triangulo ABD,
              <lb/>
              <note position="left" xlink:label="note-490-11" xlink:href="note-490-11a" xml:space="preserve">Schol. 41.
                <lb/>
              vel 55. huiꝰ.</note>
            datus eſt arcus AB, angulo recto oppoſitus, cum arcu BD, circa angulum re-
              <lb/>
            ctum proximè inuento;
              <lb/>
            </s>
            <s xml:id="echoid-s17284" xml:space="preserve">VEL, quia datus eſt arcus BD, circa angulum re-
              <lb/>
              <note position="left" xlink:label="note-490-12" xlink:href="note-490-12a" xml:space="preserve">Schol. 42.
                <lb/>
              huius.</note>
            ctum, cum angulo non recto adiacente B:
              <lb/>
            </s>
            <s xml:id="echoid-s17285" xml:space="preserve">VEL denique, quoniam datus eſt arcus AB, recto
              <lb/>
              <note position="left" xlink:label="note-490-13" xlink:href="note-490-13a" xml:space="preserve">Schol. 47.
                <lb/>
              huius.</note>
            angulo oppoſitus, cum angulo non recto B;
              <lb/>
            </s>
            <s xml:id="echoid-s17286" xml:space="preserve">dabitur quoque, per ſcholia in margine adducta, angulus BAD: </s>
            <s xml:id="echoid-s17287" xml:space="preserve">qui duplica-
              <lb/>
            tus totum BAC, quæſitum præbebit.</s>
            <s xml:id="echoid-s17288" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s17289" xml:space="preserve">ITA autem ſolis ſinubus in hoc caſu vtemur. </s>
            <s xml:id="echoid-s17290" xml:space="preserve">Per praxim </s>
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