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

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          <p>
            <s xml:id="echoid-s4226" xml:space="preserve">
              <pb o="113" file="125" n="125" rhead=""/>
            ſinus CH, cuius radix quadrata notum dabit ſinum rectum CH. </s>
            <s xml:id="echoid-s4227" xml:space="preserve">Eademq́ue ratio eſt
              <lb/>
            de cæteris.</s>
            <s xml:id="echoid-s4228" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4229" xml:space="preserve">IDEM hac etiam ratione oſtendi poteſt. </s>
            <s xml:id="echoid-s4230" xml:space="preserve">Quoniam enim EF, ſinus rectus arcus
              <lb/>
            CE, notus ponitur, cognoſcetur & </s>
            <s xml:id="echoid-s4231" xml:space="preserve">EI, ſinus complementi ei uſd em arcus, hoc eſt, re@
              <lb/>
              <note position="right" xlink:label="note-125-01" xlink:href="note-125-01a" xml:space="preserve">3. huius.</note>
            cta AF, illi æqualis. </s>
            <s xml:id="echoid-s4232" xml:space="preserve">Detracta igitur recta AF, hoc eſt, ſinu complementi arcus CE,
              <lb/>
              <note position="right" xlink:label="note-125-02" xlink:href="note-125-02a" xml:space="preserve">34. primi.</note>
            ex ſinu toto AC, cognitus erit ſinus verſus FC, arcus eiuſdem CE, vt etiam in coroll.
              <lb/>
            </s>
            <s xml:id="echoid-s4233" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4234" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4235" xml:space="preserve">oſtendinaus. </s>
            <s xml:id="echoid-s4236" xml:space="preserve">Quia vero quadratum rectæ CE, æquale eſt quadratis rectarum EF,
              <lb/>
            FC; </s>
            <s xml:id="echoid-s4237" xml:space="preserve">fit, vt quadrata rectarum EF, FC, nota rum in vnam ſummam collecta efficiant
              <lb/>
            quadratum rectæ CE: </s>
            <s xml:id="echoid-s4238" xml:space="preserve">cuius radix quadrata ipſam rectam CE, reddet notam; </s>
            <s xml:id="echoid-s4239" xml:space="preserve">ac proinde
              <lb/>
            huius radicis dimidium dabit CH, ſinum rectum arcus CD, qui dimidium eſt dati arcus
              <lb/>
            CE, notum.</s>
            <s xml:id="echoid-s4240" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4241" xml:space="preserve">VICISSIM ex hac eadem propof. </s>
            <s xml:id="echoid-s4242" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4243" xml:space="preserve">colligitur, ſi ſinus rectus alicuius arcus cognitus
              <lb/>
              <note position="right" xlink:label="note-125-03" xlink:href="note-125-03a" xml:space="preserve">Ex ſinu re-
                <lb/>
              cto cuiuſ-
                <lb/>
              uis arcꝰ co-
                <lb/>
              gnito notꝰ
                <lb/>
              ſit ſinus re
                <lb/>
              ctus alteriꝰ
                <lb/>
              arcꝰ, qui il-
                <lb/>
              liꝰ ſit duplꝰ
                <lb/>
              dummodo
                <lb/>
              quadrante
                <lb/>
              minor ſit.</note>
            ſit, notum etiam fieri ſinum rectum alrerius arcus, qui illius duplus ſit, dummodo quadran-
              <lb/>
            te ſit minor: </s>
            <s xml:id="echoid-s4244" xml:space="preserve">ita vt ex CH, ſinu recto arcus CD, cognito cognoſcatur etiam EF, ſinus rectus
              <lb/>
            arcus CE, qui arcus CD, eſt duplus. </s>
            <s xml:id="echoid-s4245" xml:space="preserve">Cum enim ſin us CH, ſit medio loco proportionalis in-
              <lb/>
            ter medictatem ſinus totius, & </s>
            <s xml:id="echoid-s4246" xml:space="preserve">ſinum verſum FC, vt oſtendimus; </s>
            <s xml:id="echoid-s4247" xml:space="preserve">erit rectangulum ſub di-
              <lb/>
            midio ſinus totius, & </s>
            <s xml:id="echoid-s4248" xml:space="preserve">ſinu verſo FC, contentum æquale quadrato ſinus recti CH. </s>
            <s xml:id="echoid-s4249" xml:space="preserve">Quare
              <lb/>
            quadratum ſinus CH, noti erit illud rectangulum; </s>
            <s xml:id="echoid-s4250" xml:space="preserve">quo diuiſo per dimidium ſinus totius,
              <lb/>
            notus euadet ſinus verſus FC. </s>
            <s xml:id="echoid-s4251" xml:space="preserve">Quia vero recta CE, cum ſit dupla ſinus CH, noti nota eſt,
              <lb/>
            erit & </s>
            <s xml:id="echoid-s4252" xml:space="preserve">eius quadratum notum: </s>
            <s xml:id="echoid-s4253" xml:space="preserve">à quo ſi auferatur quadratum ſinus verſi FC, noti, relinque-
              <lb/>
            tur etiam quadratum rectæ EF, notum; </s>
            <s xml:id="echoid-s4254" xml:space="preserve">(cum quadratũ rectæ CE, quadratis rectarum CF,
              <lb/>
            FE, ſit æquale.) </s>
            <s xml:id="echoid-s4255" xml:space="preserve">ac proinde radix quadrata illius notum dabit ſinum rectum EF.</s>
            <s xml:id="echoid-s4256" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div361" type="section" level="1" n="158">
          <head xml:id="echoid-head184" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s4257" xml:space="preserve">QVOD _ſi quando perpendicularis_
              <emph style="sc">Ef</emph>
            , _ſemidiametrum_ AC, _ſecet bifariam, vt_
              <lb/>
            _in hac figura contingit, erit adhuc_ CH, _ſinus arcus_ CD,
              <lb/>
              <figure xlink:label="fig-125-01" xlink:href="fig-125-01a" number="122">
                <image file="125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/125-01"/>
              </figure>
            _medio loco proporlionalis inter_
              <emph style="sc">Cf</emph>
            , _ſemiſſem ſinus totius,_
              <lb/>
            & </s>
            <s xml:id="echoid-s4258" xml:space="preserve">
              <emph style="sc">Cf</emph>
            , _ſinũ verſum arcus_ CE, _qui arcus_ CD, _duplus eſt. </s>
            <s xml:id="echoid-s4259" xml:space="preserve">
              <emph style="sc">E</emph>
            rũt_
              <lb/>
            _enim rurſum triangula_ ACH,
              <emph style="sc">E</emph>
              <emph style="sc">Cf</emph>
            , _æquiangula; </s>
            <s xml:id="echoid-s4260" xml:space="preserve">ac_
              <lb/>
            _proinde, vt_ AC, _ad_ CH, _ita_ EC, _ad_
              <emph style="sc">Cf</emph>
            : </s>
            <s xml:id="echoid-s4261" xml:space="preserve">_Et permutan-_
              <lb/>
              <note position="right" xlink:label="note-125-04" xlink:href="note-125-04a" xml:space="preserve">4. fexti.</note>
            _do, vt_ AC, _ad_ CE, _ita_ CH, _ad_
              <emph style="sc">Cf</emph>
            . </s>
            <s xml:id="echoid-s4262" xml:space="preserve">_Cum ergo ſit, vt_
              <lb/>
            AC, _ad_
              <emph style="sc">Cf</emph>
            , _dimidium ipſius_ AC, _ad_ CH, _dimi-_
              <lb/>
              <note position="right" xlink:label="note-125-05" xlink:href="note-125-05a" xml:space="preserve">15. quinti.</note>
            _dium ipſius_
              <emph style="sc">Ce</emph>
            ; </s>
            <s xml:id="echoid-s4263" xml:space="preserve">_erit quoq; </s>
            <s xml:id="echoid-s4264" xml:space="preserve">vt_
              <emph style="sc">Cf</emph>
            , _ad_ CH, _ita_ CH, _ad_
              <emph style="sc">Cf</emph>
            :
              <lb/>
            </s>
            <s xml:id="echoid-s4265" xml:space="preserve">_proptereaq́;_ </s>
            <s xml:id="echoid-s4266" xml:space="preserve">CH, _ſinus rectus arcus_ CD, _medio loco propor-_
              <lb/>
            _tionalis eſt inter_
              <emph style="sc">Cf</emph>
            , _ſemiſſem ſinus totius,_ & </s>
            <s xml:id="echoid-s4267" xml:space="preserve">
              <emph style="sc">Cf</emph>
            , _ſinum ver_
              <lb/>
            _ſum arcus_ CG, _qui duplus eſt arcus_ CD.</s>
            <s xml:id="echoid-s4268" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4269" xml:space="preserve">HINC _fit, ſiperpendicularis_ EF, _ſemidiametrum_ AC, _ſecet bifariam, rectam_
              <lb/>
            CH, _æqualem eſſe rectæ_
              <emph style="sc">Cf</emph>
            . </s>
            <s xml:id="echoid-s4270" xml:space="preserve">_Si enim maior eſſet, aut minor, non poßet eſſe, vt_
              <emph style="sc">Cf</emph>
            ,
              <lb/>
            _ad_ CH, _ita_ CH, _ad_
              <emph style="sc">Cf</emph>
            : </s>
            <s xml:id="echoid-s4271" xml:space="preserve">_cum vna proportio eſſet maioris inæqualitatis, & </s>
            <s xml:id="echoid-s4272" xml:space="preserve">altera_
              <lb/>
            _minoris inæqualitatis._</s>
            <s xml:id="echoid-s4273" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Sinus rectꝰ
            <lb/>
          grad. 54. æ-
            <lb/>
          qualis eſt
            <lb/>
          ſemiſsi ſinꝰ
            <lb/>
          totiꝰ, & ſi-
            <lb/>
          nui gra. 18.
            <lb/>
          ſimul. Sinꝰ
            <lb/>
          aũt verſus
            <lb/>
          grad. 72 æ-
            <lb/>
          qualiseſt ſe
            <lb/>
          miſſi ſinus
            <lb/>
          totius, & ſi-
            <lb/>
          nui verſo
            <lb/>
          grad. 36. ſi-
            <lb/>
          mul.</note>
        </div>
        <div xml:id="echoid-div363" type="section" level="1" n="159">
          <head xml:id="echoid-head185" xml:space="preserve">THEOR 3. PROPOS. 5.</head>
          <p>
            <s xml:id="echoid-s4274" xml:space="preserve">SINVS rectus arcus graduum 54. </s>
            <s xml:id="echoid-s4275" xml:space="preserve">componi-
              <lb/>
            tur ex ſemiſſe ſinus totius, & </s>
            <s xml:id="echoid-s4276" xml:space="preserve">ſinu recto arcus grad
              <lb/>
            18. </s>
            <s xml:id="echoid-s4277" xml:space="preserve">Sinus autem verſus arcus grad. </s>
            <s xml:id="echoid-s4278" xml:space="preserve">72. </s>
            <s xml:id="echoid-s4279" xml:space="preserve">componitur
              <lb/>
            ex ſemiſſe ſinus totius, & </s>
            <s xml:id="echoid-s4280" xml:space="preserve">ſinu verſo arcus grad. </s>
            <s xml:id="echoid-s4281" xml:space="preserve">36.</s>
            <s xml:id="echoid-s4282" xml:space="preserve"/>
          </p>
        </div>
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