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

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            <s xml:id="echoid-s7676" xml:space="preserve">
              <pb o="190" file="202" n="202" rhead=""/>
            CF, minor. </s>
            <s xml:id="echoid-s7677" xml:space="preserve">Ducta autem recta CH, ad AC, perpendiculari, quæ circulum
              <lb/>
            tanget in C, ducantur rectæ AG, AH, AI, per puncta D, E, F. </s>
            <s xml:id="echoid-s7678" xml:space="preserve">Item DK, ad
              <lb/>
              <note position="left" xlink:label="note-202-01" xlink:href="note-202-01a" xml:space="preserve">coroll. 16 3</note>
            AC, perpendicularis. </s>
            <s xml:id="echoid-s7679" xml:space="preserve">Eritq; </s>
            <s xml:id="echoid-s7680" xml:space="preserve">CG, tangens arcus CD; </s>
            <s xml:id="echoid-s7681" xml:space="preserve">& </s>
            <s xml:id="echoid-s7682" xml:space="preserve">CH, tangens arcus
              <lb/>
            CE; </s>
            <s xml:id="echoid-s7683" xml:space="preserve">& </s>
            <s xml:id="echoid-s7684" xml:space="preserve">CI, tangens arcus CF. </s>
            <s xml:id="echoid-s7685" xml:space="preserve">at DK, ſinus rectus
              <lb/>
              <figure xlink:label="fig-202-01" xlink:href="fig-202-01a" number="149">
                <image file="202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/202-01"/>
              </figure>
            arcus CD, & </s>
            <s xml:id="echoid-s7686" xml:space="preserve">AG, eiuſdem ſecans. </s>
            <s xml:id="echoid-s7687" xml:space="preserve">Dico CG, æqua-
              <lb/>
            lem eſſe ſinui toti AC; </s>
            <s xml:id="echoid-s7688" xml:space="preserve">at CH, maiorem, & </s>
            <s xml:id="echoid-s7689" xml:space="preserve">CI, mi-
              <lb/>
            norem. </s>
            <s xml:id="echoid-s7690" xml:space="preserve">Item AG, duplam ſinus DK. </s>
            <s xml:id="echoid-s7691" xml:space="preserve">Quoniam
              <lb/>
              <note position="left" xlink:label="note-202-02" xlink:href="note-202-02a" xml:space="preserve">27. tertij.</note>
            enim anguli CAG, GAB, æquales ſunt, ob arcus
              <lb/>
            æquales CD, DB; </s>
            <s xml:id="echoid-s7692" xml:space="preserve">eſtq́; </s>
            <s xml:id="echoid-s7693" xml:space="preserve">angulus BAC, rectus, erit
              <lb/>
            vterq; </s>
            <s xml:id="echoid-s7694" xml:space="preserve">illorum ſemirectus. </s>
            <s xml:id="echoid-s7695" xml:space="preserve">Quare & </s>
            <s xml:id="echoid-s7696" xml:space="preserve">reliquus angu-
              <lb/>
              <note position="left" xlink:label="note-202-03" xlink:href="note-202-03a" xml:space="preserve">32. primi.</note>
            lus CGA, in triangulo ACG, ſemirectus erit;
              <lb/>
            </s>
            <s xml:id="echoid-s7697" xml:space="preserve">propterea quòd angulus C, rectus eſt. </s>
            <s xml:id="echoid-s7698" xml:space="preserve">Igitur recta
              <lb/>
              <note position="left" xlink:label="note-202-04" xlink:href="note-202-04a" xml:space="preserve">5. primi.</note>
            CG, tangens arcus CD, qui ſemiſsis eſt quadran-
              <lb/>
            tis, ſinui toti AC, æqualis erit. </s>
            <s xml:id="echoid-s7699" xml:space="preserve">Ex quo ſequi-
              <lb/>
            tur, CH, tangentem arcus CE, qui ſemiſſe quadrantis maior eſt, ſinu toto
              <lb/>
            AC, maiorem eſſe; </s>
            <s xml:id="echoid-s7700" xml:space="preserve">& </s>
            <s xml:id="echoid-s7701" xml:space="preserve">CI, tangentem arcus CF, qui ſemiſſe quadrantis mi-
              <lb/>
            nor eſt, minorem: </s>
            <s xml:id="echoid-s7702" xml:space="preserve">cum punctum H, neceſſario cadat ſupra G, & </s>
            <s xml:id="echoid-s7703" xml:space="preserve">punctum I,
              <lb/>
            infra.</s>
            <s xml:id="echoid-s7704" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7705" xml:space="preserve">QVOD tamen ſeorſum ita quoq; </s>
            <s xml:id="echoid-s7706" xml:space="preserve">oſtendi poteſt, nulla habita ratione tan
              <lb/>
            gentis CG, cuius arcus eſt ſemiſsis quadrantis. </s>
            <s xml:id="echoid-s7707" xml:space="preserve">Quoniam arcus CE, ſemiſſe
              <lb/>
            quadrantis maior eſt, erit arcus BE, ſemiſſe quadrantis minor. </s>
            <s xml:id="echoid-s7708" xml:space="preserve">Igitur angulus
              <lb/>
            CAH, angulo HAB, maior eſt, ac proinde maior ſemirecto. </s>
            <s xml:id="echoid-s7709" xml:space="preserve">Cum ergo an-
              <lb/>
              <note position="left" xlink:label="note-202-05" xlink:href="note-202-05a" xml:space="preserve">Schol. 27. 3.</note>
            gulus C, rectus ſit, erit reliquus AHC, in triangulo ACH, ſemirecto minor.
              <lb/>
            </s>
            <s xml:id="echoid-s7710" xml:space="preserve">
              <note position="left" xlink:label="note-202-06" xlink:href="note-202-06a" xml:space="preserve">32. primi.</note>
            Quare recta CH, tangens arcus CE, qui ſemiſſe quadrantis maior eſt, ma-
              <lb/>
              <note position="left" xlink:label="note-202-07" xlink:href="note-202-07a" xml:space="preserve">19. primi.</note>
            ior eſt ſinu toto AC.</s>
            <s xml:id="echoid-s7711" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7712" xml:space="preserve">RVRSVS quia arcus CF, ſemiſſe quadrantis minor eſt, ac proinde BF,
              <lb/>
            maior, erit angulus CAI, angulo IAB, minor; </s>
            <s xml:id="echoid-s7713" xml:space="preserve">atque adeo minor ſemirecto.
              <lb/>
            </s>
            <s xml:id="echoid-s7714" xml:space="preserve">
              <note position="left" xlink:label="note-202-08" xlink:href="note-202-08a" xml:space="preserve">Schol. 27. 3.</note>
            Cum ergo angulus C, ſit rectus, erit reliquus AIC, in triangulo ACI, maior
              <lb/>
              <note position="left" xlink:label="note-202-09" xlink:href="note-202-09a" xml:space="preserve">32. primi.</note>
            ſemirecto: </s>
            <s xml:id="echoid-s7715" xml:space="preserve">ac propterea recta CI, tangens arcus CF, qui quadrantis ſemiſſe
              <lb/>
            minor eſt, minor erit ſinu toto AC.</s>
            <s xml:id="echoid-s7716" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">19. primi.</note>
          <p>
            <s xml:id="echoid-s7717" xml:space="preserve">PRAETEREA quoniam angulus K, rectus eſt, & </s>
            <s xml:id="echoid-s7718" xml:space="preserve">DAK, ſemirectus, vt
              <lb/>
            oſtenſum eſt; </s>
            <s xml:id="echoid-s7719" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s7720" xml:space="preserve">AD k, ſemirectus; </s>
            <s xml:id="echoid-s7721" xml:space="preserve">ac proinde AK, ſinui DK, æqualis
              <lb/>
              <note position="left" xlink:label="note-202-11" xlink:href="note-202-11a" xml:space="preserve">6. primi.</note>
            erit. </s>
            <s xml:id="echoid-s7722" xml:space="preserve">Quia vero triangula GAC, DAK, æquiangula ſunt, erit vt AG, ad AC,
              <lb/>
              <note position="left" xlink:label="note-202-12" xlink:href="note-202-12a" xml:space="preserve">4. ſexti.</note>
            ita AD, hoc eſt, AC, ad AK; </s>
            <s xml:id="echoid-s7723" xml:space="preserve">ac proinde tres rectæ GA, ſecans; </s>
            <s xml:id="echoid-s7724" xml:space="preserve">AC, ſinus
              <lb/>
            totus, & </s>
            <s xml:id="echoid-s7725" xml:space="preserve">AK, ſinus dimidij quadrantis, continue proportionales erunt. </s>
            <s xml:id="echoid-s7726" xml:space="preserve">Ita
              <lb/>
            ergo erit quadratum ex AG, ad quadratum ex AC, vt recta AG, ad rectam
              <lb/>
              <note position="left" xlink:label="note-202-13" xlink:href="note-202-13a" xml:space="preserve">Corol. 20. 6</note>
            AK. </s>
            <s xml:id="echoid-s7727" xml:space="preserve">Eſt autem quadratum ex AG, quadrati ex AC, duplum propterea quòd
              <lb/>
              <note position="left" xlink:label="note-202-14" xlink:href="note-202-14a" xml:space="preserve">47 primi.</note>
            æquale eſt quadratis ex AC, CG, æqualibus. </s>
            <s xml:id="echoid-s7728" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s7729" xml:space="preserve">AG, ſecans dimidij
              <lb/>
            quadrátis dupla eſt ſinus AK, vel DK, eiuſdem dimidij. </s>
            <s xml:id="echoid-s7730" xml:space="preserve">Quocirca tangens di-
              <lb/>
            midij quadrantis ſinui toti æqualis eſt, &</s>
            <s xml:id="echoid-s7731" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7732" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s7733" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Q uæ tan-
            <lb/>
          gentes in ta
            <lb/>
          bula tágen-
            <lb/>
          tiũ mino -
            <lb/>
          res
            <unsure/>
          ſint ſinu
            <lb/>
          toto, & quę
            <lb/>
          maiores.
            <lb/>
          Ité cur ſe-
            <lb/>
          cás gt. 45.
            <lb/>
          dupla ſit ſi-
            <lb/>
          nus gt. 45.</note>
        </div>
        <div xml:id="echoid-div547" type="section" level="1" n="257">
          <head xml:id="echoid-head284" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s7734" xml:space="preserve">E X hac propoſ. </s>
            <s xml:id="echoid-s7735" xml:space="preserve">aperte cauſa colligitur, cur in tabula Tangentium omnes tangen
              <lb/>
            tes arcuum minorum, quàm grad. </s>
            <s xml:id="echoid-s7736" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7737" xml:space="preserve">minores ſint ſinu toto: </s>
            <s xml:id="echoid-s7738" xml:space="preserve">Tangens vero arcus gra.
              <lb/>
            </s>
            <s xml:id="echoid-s7739" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7740" xml:space="preserve">ſinui totiæqualis: </s>
            <s xml:id="echoid-s7741" xml:space="preserve">Tangentes denique omnes arcuum maiorum, quàm grad. </s>
            <s xml:id="echoid-s7742" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7743" xml:space="preserve">
              <lb/>
            ſinu toto maiores. </s>
            <s xml:id="echoid-s7744" xml:space="preserve">Item cur in tabula Secantium ſecans arcus grad. </s>
            <s xml:id="echoid-s7745" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7746" xml:space="preserve">dupla ſit ſinus
              <lb/>
            arcus eiuſdem grad. </s>
            <s xml:id="echoid-s7747" xml:space="preserve">45.</s>
            <s xml:id="echoid-s7748" xml:space="preserve"/>
          </p>
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