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

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          <p>
            <s xml:id="echoid-s16012" xml:space="preserve">
              <pb o="452" file="464" n="464" rhead=""/>
            ſinus verſus anguli A, ſiue arcus BL, ad KT, differentiam ſinuum verſorum
              <lb/>
            BR, BQ, arcuum BC, BK; </s>
            <s xml:id="echoid-s16013" xml:space="preserve">quamuis nulla hic appareant triangula æquian-
              <lb/>
            gula, ſed XA, AV, inter ſe non differant, quemadmodum neque KS, KT.</s>
            <s xml:id="echoid-s16014" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16015" xml:space="preserve">13. </s>
            <s xml:id="echoid-s16016" xml:space="preserve">SINT arcus AC, AB, quadrante maiores, at BC, minor quadran-
              <lb/>
              <note position="left" xlink:label="note-464-01" xlink:href="note-464-01a" xml:space="preserve">13. caſus.</note>
            te. </s>
            <s xml:id="echoid-s16017" xml:space="preserve">Completo circulo ABGF, & </s>
            <s xml:id="echoid-s16018" xml:space="preserve">reſecto quadrante AL, ex AC, producto
              <lb/>
            item arcu BC, vt fiat quadrans BM, reliqua fiant, vt in ſuperioribus. </s>
            <s xml:id="echoid-s16019" xml:space="preserve">Demon
              <lb/>
            ſtrabimus iam, vt in primo caſu,
              <lb/>
            ita eſſe quadratum ſinus totius,
              <lb/>
              <figure xlink:label="fig-464-01" xlink:href="fig-464-01a" number="330">
                <image file="464-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/464-01"/>
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            nimirum rectangulum ſub DX,
              <lb/>
            XA, ad rectangulum ſub AV,
              <lb/>
            KY, ſinubus rectis arcuum AB,
              <lb/>
            AC, vt eſt DZ, ſinus verſus an-
              <lb/>
            guli A, ſeuarcus DL, ad KT,
              <lb/>
            differentiam ſinuum verſorum
              <lb/>
            BR, BQ, arcuum BC, Bk; </s>
            <s xml:id="echoid-s16020" xml:space="preserve">ni-
              <lb/>
            ſi quòd triagulum XAV, trian
              <lb/>
            gulo
              <emph style="sc">Sk</emph>
            T, demonſtrandum eſt
              <lb/>
            eſſe æquiangulum hac ratione. </s>
            <s xml:id="echoid-s16021" xml:space="preserve">Quoniam angulus
              <emph style="sc">K</emph>
            ST, angulo oppoſito, & </s>
            <s xml:id="echoid-s16022" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-464-02" xlink:href="note-464-02a" xml:space="preserve">29 primi.</note>
            interno YIX, æqualis eſt, erit in triangulis rectangulis
              <emph style="sc">Sk</emph>
            T, IXY, & </s>
            <s xml:id="echoid-s16023" xml:space="preserve">re-
              <lb/>
            liquus angulus
              <emph style="sc">Sk</emph>
            T, reliquo angulo
              <emph style="sc">IXy</emph>
            , hoc eſt, angulo oppoſito, & </s>
            <s xml:id="echoid-s16024" xml:space="preserve">inter-
              <lb/>
            no VAX, (cum parallelæ ſint AV, GH.) </s>
            <s xml:id="echoid-s16025" xml:space="preserve">æqualis. </s>
            <s xml:id="echoid-s16026" xml:space="preserve">Igitur in triangulis rectan
              <lb/>
            gulis
              <emph style="sc">Sk</emph>
            T, XAV, anguli quoque reliqui
              <emph style="sc">K</emph>
            ST, AXV, æquales erunt, ac
              <lb/>
            proinde æquiangula erunt triangula
              <emph style="sc">Sk</emph>
            T, XAV.</s>
            <s xml:id="echoid-s16027" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16028" xml:space="preserve">14. </s>
            <s xml:id="echoid-s16029" xml:space="preserve">SINT rurſum AC, AB, maiores quadrante, at BC, quadrans. </s>
            <s xml:id="echoid-s16030" xml:space="preserve">Com-
              <lb/>
              <note position="left" xlink:label="note-464-03" xlink:href="note-464-03a" xml:space="preserve">14. caſus.</note>
            pleto circulo ABGF, & </s>
            <s xml:id="echoid-s16031" xml:space="preserve">abſciſſo quadrante AL, ex AC, necnon deſcriptis
              <lb/>
            circulis DEF,
              <emph style="sc">K</emph>
            CN, ex polo A, ad interualla AL, AC, deſcribatur quoq;
              <lb/>
            </s>
            <s xml:id="echoid-s16032" xml:space="preserve">ex polo B, ad interuallum quadrantis BC, circulus maximus GEH, atq; </s>
            <s xml:id="echoid-s16033" xml:space="preserve">alia
              <lb/>
            frant, vt ſupra. </s>
            <s xml:id="echoid-s16034" xml:space="preserve">Demonſtrandum
              <lb/>
            ergo eſt, ita eſſe quadratum ſinus
              <lb/>
              <figure xlink:label="fig-464-02" xlink:href="fig-464-02a" number="331">
                <image file="464-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/464-02"/>
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            totius, id eſt, rectágulum ſub DX,
              <lb/>
            XA, ad rectangulum ſub AV,
              <emph style="sc">KY</emph>
            ,
              <lb/>
            ſinubus arcuum AB, AC, vt eſt
              <lb/>
            DZ, ſinus verſus anguli A, arcuſ-
              <lb/>
            ve DL, ad
              <emph style="sc">K</emph>
            T, differentiam in-
              <lb/>
            ter ſinus verſos BR, BQ, ar-
              <lb/>
            cuum BC,
              <emph style="sc">Bk</emph>
            . </s>
            <s xml:id="echoid-s16035" xml:space="preserve">Quod quidem o-
              <lb/>
            ſtendemus, vtin primo caſu. </s>
            <s xml:id="echoid-s16036" xml:space="preserve">So-
              <lb/>
            lum triangula
              <emph style="sc">Sk</emph>
            T, XAV, pro-
              <lb/>
            babũtur æquiangula eſſe, hoc mo
              <lb/>
            do. </s>
            <s xml:id="echoid-s16037" xml:space="preserve">Angulus S, communis eſt vtrique triangulo rectangulo
              <emph style="sc">Sk</emph>
            T,
              <emph style="sc">SRy</emph>
            . </s>
            <s xml:id="echoid-s16038" xml:space="preserve">Igi-
              <lb/>
            tur angulus reliquus
              <emph style="sc">Sk</emph>
            T, reliquo angulo
              <emph style="sc">SRy</emph>
            , hoceſt, angulo oppoſito, & </s>
            <s xml:id="echoid-s16039" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-464-04" xlink:href="note-464-04a" xml:space="preserve">29. primi.</note>
            interno VAX, (cum parallelę ſint AV, GH.) </s>
            <s xml:id="echoid-s16040" xml:space="preserve">æqualis erit. </s>
            <s xml:id="echoid-s16041" xml:space="preserve">Quare rectangu
              <lb/>
            la triangula
              <emph style="sc">Sk</emph>
            T, XAV, æquiangula erunt.</s>
            <s xml:id="echoid-s16042" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16043" xml:space="preserve">15. </s>
            <s xml:id="echoid-s16044" xml:space="preserve">SINT poſtremo omnes tres arcus trianguli ABC, quadrante ma-
              <lb/>
              <note position="left" xlink:label="note-464-05" xlink:href="note-464-05a" xml:space="preserve">25. caſus.</note>
            iores. </s>
            <s xml:id="echoid-s16045" xml:space="preserve">Completo circulo ABGF, & </s>
            <s xml:id="echoid-s16046" xml:space="preserve">reſectis quadrantibus AL, BM, ex ar-
              <lb/>
            cubus AC, BC, fiant omnia alia, vt prius. </s>
            <s xml:id="echoid-s16047" xml:space="preserve">Oſtendemus non ſecus, ac in pri-
              <lb/>
            mo caſu, ita eſſe quadratum ſinus totius, nempe rectangulum ſub DX,
              <emph style="sc">Xa</emph>
            ,
              <lb/>
            ad rectangulum ſub AV,
              <emph style="sc">KY</emph>
            , ſinubus rectis arcuum AB, AC, vt eſt DZ, </s>
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