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

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            <s xml:id="echoid-s14689" xml:space="preserve">
              <pb o="421" file="433" n="433" rhead=""/>
            rimus, vt vtraque in omnibus caſibus demonſtraretur: </s>
            <s xml:id="echoid-s14690" xml:space="preserve">ſatis tamen fuißet, ſi vtraq;
              <lb/>
            </s>
            <s xml:id="echoid-s14691" xml:space="preserve">
              <note position="right" xlink:label="note-433-01" xlink:href="note-433-01a" xml:space="preserve">tico rectan
                <lb/>
              gulo, cuius
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              omnes ar-
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              cus ſint qua
                <lb/>
              drante mi-
                <lb/>
              nores, locũ
                <lb/>
              etiã habet i
                <lb/>
              omni trian
                <lb/>
              gulo ſphæ-
                <lb/>
              rico rectan
                <lb/>
              gulo.</note>
            in primo caſu, exiſtentibus nimirum omnibus arcubus quadrante minoribus, demon
              <lb/>
            ſtratione fuiſſet confirmata. </s>
            <s xml:id="echoid-s14692" xml:space="preserve">Eo enim caſu demonſtrato, facile demonſtrationem om-
              <lb/>
            nibus alijs caſibus accommodabimus. </s>
            <s xml:id="echoid-s14693" xml:space="preserve">Sit nam-
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            que triangulum ſphæricum quodcunq; </s>
            <s xml:id="echoid-s14694" xml:space="preserve">rectan
              <lb/>
              <figure xlink:label="fig-433-01" xlink:href="fig-433-01a" number="287">
                <image file="433-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/433-01"/>
              </figure>
            gulum ACD, habens angulum C, rectum. </s>
            <s xml:id="echoid-s14695" xml:space="preserve">Aut
              <lb/>
            igitur duo arcus AC, CD, circa angulum re-
              <lb/>
            ctum quadrãte ſunt minores, ac proinde & </s>
            <s xml:id="echoid-s14696" xml:space="preserve">ter
              <lb/>
            tius arcus AD, quadrante quoque minor; </s>
            <s xml:id="echoid-s14697" xml:space="preserve">aut
              <lb/>
            vnus quadrante maior, & </s>
            <s xml:id="echoid-s14698" xml:space="preserve">alter minor; </s>
            <s xml:id="echoid-s14699" xml:space="preserve">aut
              <lb/>
              <note position="right" xlink:label="note-433-02" xlink:href="note-433-02a" xml:space="preserve">35. huius.,</note>
            denique ambo quadrante maiores: </s>
            <s xml:id="echoid-s14700" xml:space="preserve">Nam de eo
              <lb/>
            ſolo ſphærico triangulo rectangulo agimus, in
              <lb/>
            quo nullus arcus eſt quadrãs. </s>
            <s xml:id="echoid-s14701" xml:space="preserve">Sint primum duo
              <lb/>
            arcus AC, CD, circa angulum rectum quadrante minores: </s>
            <s xml:id="echoid-s14702" xml:space="preserve">quo poſito, erit vterque
              <lb/>
              <note position="right" xlink:label="note-433-03" xlink:href="note-433-03a" xml:space="preserve">34. huius.</note>
            angulus D, A, acutus, proptereaque triangulo ACD, demonſtratio vtriuſque pro-
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            poſitionis conueniet, quo ad primum caſum.</s>
            <s xml:id="echoid-s14703" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14704" xml:space="preserve">SIT deinde arcus DC, quadrante maior, & </s>
            <s xml:id="echoid-s14705" xml:space="preserve">CA, minor. </s>
            <s xml:id="echoid-s14706" xml:space="preserve">Productis arcubus DC,
              <lb/>
            DA, donec coeant in B; </s>
            <s xml:id="echoid-s14707" xml:space="preserve">erunt
              <emph style="sc">DAb</emph>
            , DCB, ſemicirculi; </s>
            <s xml:id="echoid-s14708" xml:space="preserve">atque adeo CB, qua-
              <lb/>
              <note position="right" xlink:label="note-433-04" xlink:href="note-433-04a" xml:space="preserve">11. 1 Theod.</note>
            drante minor. </s>
            <s xml:id="echoid-s14709" xml:space="preserve">Sunt ergo in triangulo ACB, duo arcus
              <emph style="sc">AC, CB</emph>
            , circa angulum re-
              <lb/>
            ctum C, quadrante minores. </s>
            <s xml:id="echoid-s14710" xml:space="preserve">Quare, vt proxime oſtendimus, ei vtriuſque propoſi-
              <lb/>
            tionis demonſtratio, quo ad primum caſum, conueniet. </s>
            <s xml:id="echoid-s14711" xml:space="preserve">Cum ergo ijdem ſinus tam re-
              <lb/>
            cti, quam complementorum, ſint arcuum, & </s>
            <s xml:id="echoid-s14712" xml:space="preserve">angulorum trianguli
              <emph style="sc">AC</emph>
            B, qui arcuum,
              <lb/>
            & </s>
            <s xml:id="echoid-s14713" xml:space="preserve">angulorum trianguli
              <emph style="sc">Ac</emph>
            D; </s>
            <s xml:id="echoid-s14714" xml:space="preserve">(Nam, vt in ſinubus diximus, arcus CD, CB, eun-
              <lb/>
            dem ſinum habent tam rectum, quam complementi, necnon & </s>
            <s xml:id="echoid-s14715" xml:space="preserve">arcus
              <emph style="sc">A</emph>
            D, AB. </s>
            <s xml:id="echoid-s14716" xml:space="preserve">Item
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            tam recti anguli ad C, eundem ſinum habent, nempe totum, quam anguli obliqui ad
              <lb/>
            A, cum duobus rectis ſint æquales. </s>
            <s xml:id="echoid-s14717" xml:space="preserve">Denique & </s>
            <s xml:id="echoid-s14718" xml:space="preserve">anguli D, B, eundem ſinum habent,
              <lb/>
              <note position="right" xlink:label="note-433-05" xlink:href="note-433-05a" xml:space="preserve">5. huius.</note>
            cum ſint inter ſe æquales: </s>
            <s xml:id="echoid-s14719" xml:space="preserve">Arcus autem
              <emph style="sc">Ac</emph>
            , vtrique triangulo communis eſt.) </s>
            <s xml:id="echoid-s14720" xml:space="preserve">li-
              <lb/>
              <note position="right" xlink:label="note-433-06" xlink:href="note-433-06a" xml:space="preserve">13. primi.</note>
            quido conſtat, quicquid de ſinubus arcuum, angulorumq́; </s>
            <s xml:id="echoid-s14721" xml:space="preserve">trianguli ACB, fuerit
              <lb/>
            oſtenſum, idem in ſinubus arcuum, & </s>
            <s xml:id="echoid-s14722" xml:space="preserve">angulorum trianguli
              <emph style="sc">AC</emph>
            D, locum habere.</s>
            <s xml:id="echoid-s14723" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14724" xml:space="preserve">POSTREMO ſint duo arcus DC,
              <emph style="sc">CA</emph>
            , quadrante maiores: </s>
            <s xml:id="echoid-s14725" xml:space="preserve">quo poſito, erit
              <lb/>
            arcus CB, minor quadrante. </s>
            <s xml:id="echoid-s14726" xml:space="preserve">Habet igitur triangulum
              <emph style="sc">A</emph>
            CB, arcum
              <emph style="sc">AC</emph>
            , circa
              <lb/>
            angulum rectum
              <emph style="sc">C</emph>
            , quadrante maiorem, & </s>
            <s xml:id="echoid-s14727" xml:space="preserve">
              <emph style="sc">Cb</emph>
            , minorem. </s>
            <s xml:id="echoid-s14728" xml:space="preserve">Quare ei, vt proxime
              <lb/>
            eſt demonſtratum, vtraque propoſitio conueniet. </s>
            <s xml:id="echoid-s14729" xml:space="preserve">Cum ergo ijdem ſinus tam recti,
              <lb/>
            quam complementorum, ſint arcuum, & </s>
            <s xml:id="echoid-s14730" xml:space="preserve">angulorum trianguli ACB, qui arcuum,
              <lb/>
            & </s>
            <s xml:id="echoid-s14731" xml:space="preserve">angulorum trianguli ACD, vt paulo ante diximus, liquet eaſdem propoſitiones
              <lb/>
            triangulo quoque ACD, conuenire. </s>
            <s xml:id="echoid-s14732" xml:space="preserve">Perſpicuum ergo eſt, quicquid de ſinubus arcuum,
              <lb/>
            angulorumq́; </s>
            <s xml:id="echoid-s14733" xml:space="preserve">trianguli ſphærici rectanguli, cuius duo arcus circa angulum rectum
              <lb/>
            quadrante ſint minores, demonſtratum fuerit, locum etiam habere in quocunq; </s>
            <s xml:id="echoid-s14734" xml:space="preserve">alio
              <lb/>
            triangulo ſphærico rectangulo.</s>
            <s xml:id="echoid-s14735" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14736" xml:space="preserve">IDEM prorſus dicendum eſt de tertio caſu propoſ. </s>
            <s xml:id="echoid-s14737" xml:space="preserve">41. </s>
            <s xml:id="echoid-s14738" xml:space="preserve">Satis enim fuiſſet illum
              <lb/>
            demonſtraſſe in triangulo rectangulo, cuius omnes arcus ſunt quadrante minores,
              <lb/>
            quale est triangulum ſecundæ figuræ propoſ. </s>
            <s xml:id="echoid-s14739" xml:space="preserve">41 dictæ; </s>
            <s xml:id="echoid-s14740" xml:space="preserve">cum eius trianguli demon-
              <lb/>
            ſtratio omnibus alijs conueniat, vt ex demonſtratis in hoc ſcholio eſt manifeſtum.</s>
            <s xml:id="echoid-s14741" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14742" xml:space="preserve">EX his, quæ proximis tribus propoſitionibus demonſtrauimus, abſolutus iam per
              <lb/>
            ſinus eſt calculus triangulorum ſphæricorum rectangulorum: </s>
            <s xml:id="echoid-s14743" xml:space="preserve">quareiam non rectan
              <lb/>
            gulorum calculus ſequi deberet. </s>
            <s xml:id="echoid-s14744" xml:space="preserve">Sed quia per lineas tangentes, ac ſecantes breuius
              <lb/>
            plerunque triangulorum rectangulorum calculus, quam per ſinus, expeditur, </s>
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