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

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            crũt ſinus arcuum circa reliquum angulum vnius
              <lb/>
            ſinubus arcuum circa reliquum angulum alterius
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            proportionales, homologiq́; </s>
            <s xml:id="echoid-s16119" xml:space="preserve">erunt ſinus arcuum
              <lb/>
            æquales angulos ſubtendentium. </s>
            <s xml:id="echoid-s16120" xml:space="preserve">Et ſi vnus angu-
              <lb/>
            lus vnius vniangulo alterius ſit æqualis, ſinusq́; </s>
            <s xml:id="echoid-s16121" xml:space="preserve">ar-
              <lb/>
            cuum circa alium angulum vnius ſinubus arcuum
              <lb/>
            circa alium angulum alterius proportionales, ita
              <lb/>
            vt ſinus arcuum æquales angulos ſubtendentium
              <lb/>
            ſint homologi: </s>
            <s xml:id="echoid-s16122" xml:space="preserve">erunt & </s>
            <s xml:id="echoid-s16123" xml:space="preserve">anguli arcubus reliquo-
              <lb/>
            rum ſinuum homologorum oppoſiti inter ſe æ-
              <lb/>
            quales, vel æquales duobus rectis.</s>
            <s xml:id="echoid-s16124" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16125" xml:space="preserve">SINT in duobus triangulis ſphæricis ABC, DEF, duo anguli inter ſe
              <lb/>
            æquales B, E, necnon duo C, F. </s>
            <s xml:id="echoid-s16126" xml:space="preserve">Dico ita eſſe ſinum arcus AB, ad ſinum arcus
              <lb/>
            AC, vt eſt ſinus arcus DE, ad ſinum arcus DF. </s>
            <s xml:id="echoid-s16127" xml:space="preserve">Quia enim eſt, vt ſinus arcus
              <lb/>
            AB, ad ſinum anguli C, ita ſinus arcus AC, ad ſinum
              <lb/>
              <note position="right" xlink:label="note-467-01" xlink:href="note-467-01a" xml:space="preserve">41. huius.</note>
            anguli B; </s>
            <s xml:id="echoid-s16128" xml:space="preserve">erit permutando, vt ſinus arcus AB, ad ſi-
              <lb/>
              <figure xlink:label="fig-467-01" xlink:href="fig-467-01a" number="335">
                <image file="467-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/467-01"/>
              </figure>
            num arcus AC, ita ſinus anguli C, ad ſinum anguli B,
              <lb/>
            hoc eſt, ita ſinus anguli F, ad ſinum anguli E, cum hi
              <lb/>
            anguli illis ponantur æquales. </s>
            <s xml:id="echoid-s16129" xml:space="preserve">Item quia eſt, vt ſi-
              <lb/>
            nus arcus DE, ad ſinum anguli F, ita ſinus arcus DF,
              <lb/>
              <note position="right" xlink:label="note-467-02" xlink:href="note-467-02a" xml:space="preserve">41. huius.</note>
            ad ſinum anguli E; </s>
            <s xml:id="echoid-s16130" xml:space="preserve">erit permutando, vt ſinus arcus
              <lb/>
            DE, ad ſinum arcus DF, ita ſinus anguli F, ad ſinum
              <lb/>
            anguli E. </s>
            <s xml:id="echoid-s16131" xml:space="preserve">Oſtenſum autem eſt, ita etiam eſſe ſinum
              <lb/>
            arcus AB, ad ſinum arcus AC, vt eſt ſinus anguli F,
              <lb/>
            ad ſinum anguli E. </s>
            <s xml:id="echoid-s16132" xml:space="preserve">Igitur erit, vt ſinus arcus AB, ad
              <lb/>
            ſinum arcus AC, ita ſinus arcus DE, ad ſinum arcus DF. </s>
            <s xml:id="echoid-s16133" xml:space="preserve">Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s16134" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16135" xml:space="preserve">SED ſint iam anguli B, E, æquales, & </s>
            <s xml:id="echoid-s16136" xml:space="preserve">ita ſit ſinus arcus AB, ad ſinus ar-
              <lb/>
            cus AC, vt eſt ſinus arcus DE, ad ſinum arcus DF. </s>
            <s xml:id="echoid-s16137" xml:space="preserve">Dico angulos quoq; </s>
            <s xml:id="echoid-s16138" xml:space="preserve">C,
              <lb/>
            F, æquales eſſe, vel certe duobus rectis æquales. </s>
            <s xml:id="echoid-s16139" xml:space="preserve">Oſtendemus enim, vt prius,
              <lb/>
            ita eſſe ſinum arcus AB, ad ſinum arcus AC, vt eſt ſinus anguli C, ad ſinum
              <lb/>
              <note position="right" xlink:label="note-467-03" xlink:href="note-467-03a" xml:space="preserve">41. huius.
                <lb/>
              Et permu-
                <lb/>
              tando.</note>
            anguli B. </s>
            <s xml:id="echoid-s16140" xml:space="preserve">Item ita eſſe ſinum arcus DE, ad ſinum arcus DF, vt eſt ſinus angu-
              <lb/>
            li F, ad ſinum anguli E. </s>
            <s xml:id="echoid-s16141" xml:space="preserve">Quare cum ponatur, vt ſinus arcus AB, ad ſinum ar-
              <lb/>
            cus AC, ita ſinus arcus DE, ad ſinum arcus DF; </s>
            <s xml:id="echoid-s16142" xml:space="preserve">erit, vt ſinus anguli C, ad ſi-
              <lb/>
            num anguli B, ita ſinus anguli F, ad ſinum anguli E: </s>
            <s xml:id="echoid-s16143" xml:space="preserve">Et conuertendo, vt ſinus
              <lb/>
            anguli B, ad ſinum anguli C, ita ſinus anguli E, ad ſinum anguli F. </s>
            <s xml:id="echoid-s16144" xml:space="preserve">Cum ergo
              <lb/>
            ſinus æqualium angulorum B, E, æquales ſint, erunt & </s>
            <s xml:id="echoid-s16145" xml:space="preserve">ſinus angulorum C,
              <lb/>
              <note position="right" xlink:label="note-467-04" xlink:href="note-467-04a" xml:space="preserve">14. quinti.</note>
            F, æquales; </s>
            <s xml:id="echoid-s16146" xml:space="preserve">ac proinde vel anguli C, F, æquales erunt, vel duobus rectis æqua-
              <lb/>
            les. </s>
            <s xml:id="echoid-s16147" xml:space="preserve">Quod eſt propoſitum. </s>
            <s xml:id="echoid-s16148" xml:space="preserve">Itaque ſi duo triangula ſphærica duos angulos,
              <lb/>
            &</s>
            <s xml:id="echoid-s16149" xml:space="preserve">c. </s>
            <s xml:id="echoid-s16150" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s16151" xml:space="preserve"/>
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