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

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          <p>
            <s xml:id="echoid-s10524" xml:space="preserve">
              <pb o="327" file="339" n="339" rhead=""/>
            B, grad. </s>
            <s xml:id="echoid-s10525" xml:space="preserve">36. </s>
            <s xml:id="echoid-s10526" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s10527" xml:space="preserve">52. </s>
            <s xml:id="echoid-s10528" xml:space="preserve">C, grad. </s>
            <s xml:id="echoid-s10529" xml:space="preserve">30. </s>
            <s xml:id="echoid-s10530" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s10531" xml:space="preserve">31. </s>
            <s xml:id="echoid-s10532" xml:space="preserve">& </s>
            <s xml:id="echoid-s10533" xml:space="preserve">BAC, grad. </s>
            <s xml:id="echoid-s10534" xml:space="preserve">112. </s>
            <s xml:id="echoid-s10535" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s10536" xml:space="preserve">37. </s>
            <s xml:id="echoid-s10537" xml:space="preserve">vt
              <lb/>
            hic vides.</s>
            <s xml:id="echoid-s10538" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          AB. # AB. # # BD. # # BD.
            <lb/>
          11. # 100000. # # 8 {4/5}? # fit # 80000.
            <lb/>
          # # Item.
            <lb/>
          AC. # AC. # # CD. # # CD.
            <lb/>
          13. # 100000. # # 11 {1/5}? # fit # 86154.
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s10539" xml:space="preserve">Complemétum arcus, quem prior ſinus inuentus offert, dat angulum B, grad.
              <lb/>
            </s>
            <s xml:id="echoid-s10540" xml:space="preserve">36. </s>
            <s xml:id="echoid-s10541" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s10542" xml:space="preserve">52. </s>
            <s xml:id="echoid-s10543" xml:space="preserve">At complementum arcus poſterioris ſinus inuenti dat angulum
              <lb/>
            C, grad. </s>
            <s xml:id="echoid-s10544" xml:space="preserve">30. </s>
            <s xml:id="echoid-s10545" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s10546" xml:space="preserve">31. </s>
            <s xml:id="echoid-s10547" xml:space="preserve">&</s>
            <s xml:id="echoid-s10548" xml:space="preserve">c. </s>
            <s xml:id="echoid-s10549" xml:space="preserve">Eſt ergo doctrina huius propoſitionis generalis, ſi-
              <lb/>
              <note position="right" xlink:label="note-339-02" xlink:href="note-339-02a" xml:space="preserve">Generalitas
                <lb/>
              huius pro-
                <lb/>
              poſ.</note>
            ue angulus maximus A, acutus ſit, vt in priori triangulo, ſiue obtuſus, vt in
              <lb/>
            poſteriori, ſiue deniq; </s>
            <s xml:id="echoid-s10550" xml:space="preserve">rectus ſit; </s>
            <s xml:id="echoid-s10551" xml:space="preserve">quamuis in rectangulo triangulo iam ſupra
              <lb/>
            traditum ſit propoſ. </s>
            <s xml:id="echoid-s10552" xml:space="preserve">3. </s>
            <s xml:id="echoid-s10553" xml:space="preserve">quo pacto ex duobus lateribus cognitis facilius an-
              <lb/>
            guli duo acuti inueniantur.</s>
            <s xml:id="echoid-s10554" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Quando la
            <lb/>
          terum pro-
            <lb/>
          portiones
            <lb/>
          datæ ſunt.</note>
          <p>
            <s xml:id="echoid-s10555" xml:space="preserve">IAM ſi dentur laterum proportiones, ſaltem duæ, continuabimus eas in
              <lb/>
            tribus minimis numeris, ſi proportionum numeri minimi non ſint, vt Eucl.
              <lb/>
            </s>
            <s xml:id="echoid-s10556" xml:space="preserve">docuit propoſ. </s>
            <s xml:id="echoid-s10557" xml:space="preserve">4. </s>
            <s xml:id="echoid-s10558" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10559" xml:space="preserve">8. </s>
            <s xml:id="echoid-s10560" xml:space="preserve">eosq́; </s>
            <s xml:id="echoid-s10561" xml:space="preserve">numeros lateribus aſcribemus, perinde ac ſi in
              <lb/>
              <note position="right" xlink:label="note-339-04" xlink:href="note-339-04a" xml:space="preserve">35. ſeptimi.</note>
            illis numeris darentur. </s>
            <s xml:id="echoid-s10562" xml:space="preserve">Vt ſi in priori triangulo proportio AB, ad BC, ſit,
              <lb/>
            quæ 26. </s>
            <s xml:id="echoid-s10563" xml:space="preserve">ad 42. </s>
            <s xml:id="echoid-s10564" xml:space="preserve">At AB, ad AC, quæ 39. </s>
            <s xml:id="echoid-s10565" xml:space="preserve">ad 60. </s>
            <s xml:id="echoid-s10566" xml:space="preserve">reuocabuntur hæ proportiones
              <lb/>
            ad minimos hoſce numeros 13. </s>
            <s xml:id="echoid-s10567" xml:space="preserve">21. </s>
            <s xml:id="echoid-s10568" xml:space="preserve">& </s>
            <s xml:id="echoid-s10569" xml:space="preserve">13. </s>
            <s xml:id="echoid-s10570" xml:space="preserve">20. </s>
            <s xml:id="echoid-s10571" xml:space="preserve">Dabitur ergo AB, 13. </s>
            <s xml:id="echoid-s10572" xml:space="preserve">AC, 20.
              <lb/>
            </s>
            <s xml:id="echoid-s10573" xml:space="preserve">
              <note position="right" xlink:label="note-339-05" xlink:href="note-339-05a" xml:space="preserve">Quãdo triã
                <lb/>
              gulũ eſt Iſo
                <lb/>
              ſceles.
                <lb/>
              coroll. 8.
                <lb/>
              huius.</note>
            & </s>
            <s xml:id="echoid-s10574" xml:space="preserve">BC, 21. </s>
            <s xml:id="echoid-s10575" xml:space="preserve">Ex quibus angulos eruemus, vt prius.</s>
            <s xml:id="echoid-s10576" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10577" xml:space="preserve">PORRO in Iſoſcele datorum laterum ducenda eſt perpendicularis ad
              <lb/>
            baſim, ſiue ea ſit maximum latus, ſiue minimum: </s>
            <s xml:id="echoid-s10578" xml:space="preserve">quæ diuidet baſim bifariam.
              <lb/>
            </s>
            <s xml:id="echoid-s10579" xml:space="preserve">Quare ſi fiat, vt vnum æqualium laterum ad ſinũ totum, ita dimidium baſis
              <lb/>
            ad aliud, inuenietur ſinus cuiuſdam arcus, cuius complementum dabit vnum
              <lb/>
            æ qualium angulorum ſupra baſim, vt ex demonſtratis liquet. </s>
            <s xml:id="echoid-s10580" xml:space="preserve">Ergo & </s>
            <s xml:id="echoid-s10581" xml:space="preserve">alter da-
              <lb/>
            bitur: </s>
            <s xml:id="echoid-s10582" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s10583" xml:space="preserve">tertius baſi oppoſitus, vtpote reliquus duorũ rectorum.</s>
            <s xml:id="echoid-s10584" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10585" xml:space="preserve">IN æquilatero deniq; </s>
            <s xml:id="echoid-s10586" xml:space="preserve">triangulo dabuntur anguli, etiamſi latera non den-
              <lb/>
              <note position="right" xlink:label="note-339-06" xlink:href="note-339-06a" xml:space="preserve">Quãdo triã
                <lb/>
              gulũ eſt æ-
                <lb/>
              quilaterũ.</note>
            tur, cum quilibet ſit tertia pars duorum rectorum, hoc eſt, contineat grad.
              <lb/>
            </s>
            <s xml:id="echoid-s10587" xml:space="preserve">60. </s>
            <s xml:id="echoid-s10588" xml:space="preserve">Datis igitur omnibus trianguli non rectanguli lateribus, &</s>
            <s xml:id="echoid-s10589" xml:space="preserve">c. </s>
            <s xml:id="echoid-s10590" xml:space="preserve">Quod fa-
              <lb/>
            ciendum erat.</s>
            <s xml:id="echoid-s10591" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div876" type="section" level="1" n="469">
          <head xml:id="echoid-head501" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s10592" xml:space="preserve">_ETSI_ in hac propoſ. </s>
            <s xml:id="echoid-s10593" xml:space="preserve">præcepimus, perpendicularem ad maximum latus eſſe du-
              <lb/>
            cendam ex angulo oppoſito, vt intra triangulum cadat, fiatq; </s>
            <s xml:id="echoid-s10594" xml:space="preserve">calculus facilior: </s>
            <s xml:id="echoid-s10595" xml:space="preserve">ta-
              <lb/>
            men eadem fere via problema abſoluemus, ſi in triangulo obtuſangulo perpendicula-
              <lb/>
            ris non ducatur ab obtſo angulo in maximum latus, ſed ab alterutro acutorum an-
              <lb/>
              <note position="right" xlink:label="note-339-07" xlink:href="note-339-07a" xml:space="preserve">Quádo per
                <lb/>
              pendicũla-
                <lb/>
              ris in obtu-
                <lb/>
              sãgulo triã
                <lb/>
              gulo cadit
                <lb/>
              extta trian
                <lb/>
              gulum.</note>
            gulorum in latus oppoſitum protractum, ita vt cadat extra
              <lb/>
              <figure xlink:label="fig-339-01" xlink:href="fig-339-01a" number="184">
                <image file="339-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/339-01"/>
              </figure>
            triangulum, vt in hoc triangulo _ABC,_ manifeſtum eſt, in
              <lb/>
            quo latus _AB,_ datur _22. </s>
            <s xml:id="echoid-s10596" xml:space="preserve">AC, 31._ </s>
            <s xml:id="echoid-s10597" xml:space="preserve">& </s>
            <s xml:id="echoid-s10598" xml:space="preserve">_BC, 14._ </s>
            <s xml:id="echoid-s10599" xml:space="preserve">_N_am ſi fiat,
              <lb/>
            vt _BC, 14._ </s>
            <s xml:id="echoid-s10600" xml:space="preserve">(in quod latus perpendicularis eſt ducta) ad
              <lb/>
            _53._ </s>
            <s xml:id="echoid-s10601" xml:space="preserve">ſummam aliorum laterum _AB, AC,_ ita _9._ </s>
            <s xml:id="echoid-s10602" xml:space="preserve">differentia
              <lb/>
            eorundem laterum ad aliud, reperietur numerus _34 {1/14}._ </s>
            <s xml:id="echoid-s10603" xml:space="preserve">à
              <lb/>
            quo ſi ſubducatur latus _
              <emph style="sc">B</emph>
            C,_ remanebit numerus _20 {1/14}._
              <lb/>
            </s>
            <s xml:id="echoid-s10604" xml:space="preserve">cuius ſemiſsis _10 {1/28}._ </s>
            <s xml:id="echoid-s10605" xml:space="preserve">erit recta _BD,_ ac proinde _CD,_
              <lb/>
            _24 {1/28}._ </s>
            <s xml:id="echoid-s10606" xml:space="preserve">Quam obrem ſi iam fiat, vt _
              <emph style="sc">Ab</emph>
            ,_ _22._ </s>
            <s xml:id="echoid-s10607" xml:space="preserve">ad _
              <emph style="sc">Ab</emph>
            ,_ ſi-
              <lb/>
            num totum 100000. </s>
            <s xml:id="echoid-s10608" xml:space="preserve">ita _BD, 10 {1/28}._ </s>
            <s xml:id="echoid-s10609" xml:space="preserve">ad aliud, innenietur _
              <emph style="sc">B</emph>
            D,_ ſinus _45617._</s>
            <s xml:id="echoid-s10610" xml:space="preserve"/>
          </p>
        </div>
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    </echo>
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