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

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        <div xml:id="echoid-div1249" type="section" level="1" n="589">
          <head xml:id="echoid-head624" xml:space="preserve">THEOR. 53. PROPOS. 55.</head>
          <p>
            <s xml:id="echoid-s15427" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-
              <lb/>
            ius omnes arcus minores quadrante ſint: </s>
            <s xml:id="echoid-s15428" xml:space="preserve">ſinus to-
              <lb/>
            tus ad ſinum arcus recto angulo oppoſiti eandem
              <lb/>
            proportionem habet, quam ſecans complementi
              <lb/>
            vtriuſlibet arcuum circa angulum rectum ad ſecã-
              <lb/>
            tem complementi anguli huic arcui oppoſiti.</s>
            <s xml:id="echoid-s15429" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15430" xml:space="preserve">IN ſphærico triangulo ABC, cuius omnes arcus ſint minores quadrante,
              <lb/>
            angulus B, rectus ſit. </s>
            <s xml:id="echoid-s15431" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus AC, vt eſt ſe-
              <lb/>
            cans complementi arcus BC, ad ſecantem
              <lb/>
            complementi anguli A, arcui BC, oppoſiti.
              <lb/>
            </s>
            <s xml:id="echoid-s15432" xml:space="preserve">Repetita enim conſtructione figuræ propoſ. </s>
            <s xml:id="echoid-s15433" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-451-01" xlink:href="fig-451-01a" number="312">
                <image file="451-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/451-01"/>
              </figure>
            45. </s>
            <s xml:id="echoid-s15434" xml:space="preserve">erit AE, quadrans; </s>
            <s xml:id="echoid-s15435" xml:space="preserve">DE, arcus anguli A,
              <lb/>
            & </s>
            <s xml:id="echoid-s15436" xml:space="preserve">EF, eius complementum; </s>
            <s xml:id="echoid-s15437" xml:space="preserve">atque CF, com-
              <lb/>
            plementum arcus BC, vt ibi demonſtratum
              <lb/>
            eſt. </s>
            <s xml:id="echoid-s15438" xml:space="preserve">Quia ergo in ſphæra duo maximi circuli
              <lb/>
            AE, AD, ſe interſecant in A, & </s>
            <s xml:id="echoid-s15439" xml:space="preserve">ex punctis
              <lb/>
              <note position="right" xlink:label="note-451-01" xlink:href="note-451-01a" xml:space="preserve">Theor. 8.
                <lb/>
              ſcholij. 40.
                <lb/>
              huius.</note>
            C, E, arcus AE, ad arcum AD, ducti ſunt per-
              <lb/>
            pendiculares arcus CB, ED; </s>
            <s xml:id="echoid-s15440" xml:space="preserve">erit vt ſinus to
              <lb/>
            tus quadrantis AE, ad ſecantem complemen
              <lb/>
            ti arcus CB, hoc eſt, ad ſecantem arcus CF,
              <lb/>
            ita ſinus arcus AC, ad ſecantem complemen-
              <lb/>
            ti arcus DE, ſiue anguli A, id eſt, ad ſecan-
              <lb/>
            tem arcus EF: </s>
            <s xml:id="echoid-s15441" xml:space="preserve">Et permutando, vt ſinus totus ad ſinum arcus AC, ita ſecans
              <lb/>
            complementi arcus BC, ad ſecantem complementi anguli A. </s>
            <s xml:id="echoid-s15442" xml:space="preserve">Pari ratione, ſi
              <lb/>
            aliter figura extruatur, erit, vt ſinus totus ad ſinum arcus AC, ita ſecans com
              <lb/>
            plementi arcus AB, ad ſecantem complementi anguli C. </s>
            <s xml:id="echoid-s15443" xml:space="preserve">Quare in omni trian
              <lb/>
            gulo ſphærico rectangulo, &</s>
            <s xml:id="echoid-s15444" xml:space="preserve">c. </s>
            <s xml:id="echoid-s15445" xml:space="preserve">Quod erat oſtendendum.</s>
            <s xml:id="echoid-s15446" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1251" type="section" level="1" n="590">
          <head xml:id="echoid-head625" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s15447" xml:space="preserve">EX his ſequens problema diſſoluemus, quod alio quoque modo in problemate 1.
              <lb/>
            </s>
            <s xml:id="echoid-s15448" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s15449" xml:space="preserve">41. </s>
            <s xml:id="echoid-s15450" xml:space="preserve">abſolutum fuit.</s>
            <s xml:id="echoid-s15451" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15452" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, qui recto angulo
              <lb/>
            opponitur, cum alterutro arcuum circa rectum angulum, inuenire
              <lb/>
            angulum huic arcui oppoſitum, cum reliquo arcu, & </s>
            <s xml:id="echoid-s15453" xml:space="preserve">angulo.</s>
            <s xml:id="echoid-s15454" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15455" xml:space="preserve">IN triangulo ABC, cuius rectus angulus C, datus ſit tam arcus
              <emph style="sc">AB</emph>
            , quam AC.
              <lb/>
            </s>
            <s xml:id="echoid-s15456" xml:space="preserve">Dico dari quo que angulum
              <emph style="sc">B</emph>
            , vnà cum arcu
              <emph style="sc">B</emph>
            C, & </s>
            <s xml:id="echoid-s15457" xml:space="preserve">angulo
              <emph style="sc">A</emph>
            . </s>
            <s xml:id="echoid-s15458" xml:space="preserve">Quia enim eſt, vt ſi-
              <lb/>
            nus totus ad ſinum arcus
              <emph style="sc">Ab</emph>
            , ita ſecans complementi arcus
              <emph style="sc">A</emph>
            C, ad ſecantem com-
              <lb/>
              <note position="right" xlink:label="note-451-02" xlink:href="note-451-02a" xml:space="preserve">55. huius.</note>
            plementi anguli
              <emph style="sc">B</emph>
            :</s>
            <s xml:id="echoid-s15459" xml:space="preserve"/>
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