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

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          <p>
            <s xml:id="echoid-s8825" xml:space="preserve">
              <pb o="301" file="313" n="313" rhead=""/>
            rint, vel alter eorũ obtuſus: </s>
            <s xml:id="echoid-s8826" xml:space="preserve">erit in triangulo rectangulo ABD, vt latus AB,
              <lb/>
            ad latus AD, ita ſinus anguli recti D, ad ſinum anguli B, vt ſupra eſt demon-
              <lb/>
            ſtratum: </s>
            <s xml:id="echoid-s8827" xml:space="preserve">Item in triangulo rectangulo ADC, vt latus AD, ad latus AC, ita
              <lb/>
            ſinus anguli C, ad ſinum anguli recti D. </s>
            <s xml:id="echoid-s8828" xml:space="preserve">Ex æqualitate ergo, & </s>
            <s xml:id="echoid-s8829" xml:space="preserve">perturbata
              <lb/>
            proportione erit, vt latus AB, ad latus AC, ita ſinus anguli C, (Habẽt enim
              <lb/>
            duo anguli ad C, in obtuſangulo triangulo eundem ſinum, vt in tractatione
              <lb/>
            ſinuum oſtendimus. </s>
            <s xml:id="echoid-s8830" xml:space="preserve">ad ſinum anguli B, vt in formula ſuprapoſita apparet: </s>
            <s xml:id="echoid-s8831" xml:space="preserve">Et
              <lb/>
            conuertendo quoq;</s>
            <s xml:id="echoid-s8832" xml:space="preserve">, vt latus AC, ad latus AB, ita ſinus anguli B, ad ſinum
              <lb/>
            anguli C. </s>
            <s xml:id="echoid-s8833" xml:space="preserve">Eodem modo concludemus eſſe, vt latus AB, ad latus BC, ita ſi-
              <lb/>
            num anguli C, ad ſinum anguli BAC: </s>
            <s xml:id="echoid-s8834" xml:space="preserve">Vel vt latus BC, ad latus AB, ita ſi-
              <lb/>
            num anguli BAC, ad ſinum anguli C, &</s>
            <s xml:id="echoid-s8835" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8836" xml:space="preserve">Quocirca in omni triangulo re-
              <lb/>
            ctilineo latera quæuis duo eandem proportionem habent, quam ſinus angu-
              <lb/>
            lorum illis oppoſitorum. </s>
            <s xml:id="echoid-s8837" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s8838" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div784" type="section" level="1" n="454">
          <head xml:id="echoid-head486" xml:space="preserve">SCHOLIVM.</head>
          <note position="right" xml:space="preserve">Qua rŏne
            <lb/>
          ex @@ibꝰ vel
            <lb/>
          duobusan-
            <lb/>
          gulis notis
            <lb/>
          cuiuſuis
            <lb/>
          triãguli co-
            <lb/>
          gnoſcantur
            <lb/>
          propottio-
            <lb/>
          nes laterũ.</note>
          <p style="it">
            <s xml:id="echoid-s8839" xml:space="preserve">_EX_ hac propoſ facile colligemus proportiones laterum cuiuſuis trianguli recti-
              <lb/>
            linei, cuius omnes anguli cogniti ſint, vel duo tantum Sint entm omnes anguli in
              <lb/>
            triangulo _ABC,_ noti. </s>
            <s xml:id="echoid-s8840" xml:space="preserve">Dico proportiones laterum notas eſſe. </s>
            <s xml:id="echoid-s8841" xml:space="preserve">Cum enim eadem
              <lb/>
            ſit proportio lateris _AB,_ ad latus
              <lb/>
              <figure xlink:label="fig-313-01" xlink:href="fig-313-01a" number="163">
                <image file="313-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/313-01"/>
              </figure>
            _AC,_ quæ ſinus anguli _C,_ ad ſinum
              <lb/>
            anguli _B;_ </s>
            <s xml:id="echoid-s8842" xml:space="preserve">ſint autem ſinus angulorũ
              <lb/>
              <note position="right" xlink:label="note-313-02" xlink:href="note-313-02a" xml:space="preserve">@.huius.</note>
            _C, B,_ notorum cogniti ex tabula ſi-
              <lb/>
            nuum; </s>
            <s xml:id="echoid-s8843" xml:space="preserve">nota erit proportio lateris _AB,_
              <lb/>
            ad latus _AC,_ &</s>
            <s xml:id="echoid-s8844" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8845" xml:space="preserve">Exempli cauſa,
              <lb/>
            ponatur in primo triangulo angulus
              <lb/>
            _C,_ grad. </s>
            <s xml:id="echoid-s8846" xml:space="preserve">_60. </s>
            <s xml:id="echoid-s8847" xml:space="preserve">B,_ grad. </s>
            <s xml:id="echoid-s8848" xml:space="preserve">_50. </s>
            <s xml:id="echoid-s8849" xml:space="preserve">& </s>
            <s xml:id="echoid-s8850" xml:space="preserve">A,_
              <lb/>
            grad. </s>
            <s xml:id="echoid-s8851" xml:space="preserve">_70._ </s>
            <s xml:id="echoid-s8852" xml:space="preserve">Horum ſinus ſunt _86602._
              <lb/>
            </s>
            <s xml:id="echoid-s8853" xml:space="preserve">_76604. </s>
            <s xml:id="echoid-s8854" xml:space="preserve">93969._ </s>
            <s xml:id="echoid-s8855" xml:space="preserve">Eſt ergo proportio
              <lb/>
            _AB,_ ad _AC,_ eadem, quæ _86602._ </s>
            <s xml:id="echoid-s8856" xml:space="preserve">ad _76604._ </s>
            <s xml:id="echoid-s8857" xml:space="preserve">& </s>
            <s xml:id="echoid-s8858" xml:space="preserve">_AB,_ ad _BC,_ eadem, quæ _86602._ </s>
            <s xml:id="echoid-s8859" xml:space="preserve">
              <lb/>
            ad _93969._ </s>
            <s xml:id="echoid-s8860" xml:space="preserve">& </s>
            <s xml:id="echoid-s8861" xml:space="preserve">_AC,_ ad _BC,_ eadem, quæ _76604._ </s>
            <s xml:id="echoid-s8862" xml:space="preserve">ad _93969._ </s>
            <s xml:id="echoid-s8863" xml:space="preserve">In triangulo vero ſecun
              <lb/>
            do ponatur angulus _B,_ rectus, ac proinde grad. </s>
            <s xml:id="echoid-s8864" xml:space="preserve">_90. </s>
            <s xml:id="echoid-s8865" xml:space="preserve">C,_ grad. </s>
            <s xml:id="echoid-s8866" xml:space="preserve">_50._ </s>
            <s xml:id="echoid-s8867" xml:space="preserve">& </s>
            <s xml:id="echoid-s8868" xml:space="preserve">_A,_ grad _40._ </s>
            <s xml:id="echoid-s8869" xml:space="preserve">
              <lb/>
            Horum ſinus ſunt _100000. </s>
            <s xml:id="echoid-s8870" xml:space="preserve">76604. </s>
            <s xml:id="echoid-s8871" xml:space="preserve">64278._ </s>
            <s xml:id="echoid-s8872" xml:space="preserve">Eritigitur _AB,_ ad _AC,_ vt _76604._ </s>
            <s xml:id="echoid-s8873" xml:space="preserve">ad
              <lb/>
            _100000._ </s>
            <s xml:id="echoid-s8874" xml:space="preserve">& </s>
            <s xml:id="echoid-s8875" xml:space="preserve">_AB,_ ad _BC,_ vt _76604._ </s>
            <s xml:id="echoid-s8876" xml:space="preserve">ad _64278._ </s>
            <s xml:id="echoid-s8877" xml:space="preserve">& </s>
            <s xml:id="echoid-s8878" xml:space="preserve">_AC,_ ad _BC,_ vt _100000._ </s>
            <s xml:id="echoid-s8879" xml:space="preserve">ad
              <lb/>
            _64278._ </s>
            <s xml:id="echoid-s8880" xml:space="preserve">In triangulo deniq; </s>
            <s xml:id="echoid-s8881" xml:space="preserve">tertio ſtatuatur angulus _B,_ obtuſus & </s>
            <s xml:id="echoid-s8882" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s8883" xml:space="preserve">_124 C,_
              <lb/>
            grad. </s>
            <s xml:id="echoid-s8884" xml:space="preserve">_30_ & </s>
            <s xml:id="echoid-s8885" xml:space="preserve">_A,_ grad _26_ Horum ſinus ſunt (ſi pro ſinu anguli obtuſi accipiatur ſi-
              <lb/>
            nus complementi ipſius vſq; </s>
            <s xml:id="echoid-s8886" xml:space="preserve">ad grad. </s>
            <s xml:id="echoid-s8887" xml:space="preserve">_180_ nempe ſinus grad _56._) </s>
            <s xml:id="echoid-s8888" xml:space="preserve">_82903. </s>
            <s xml:id="echoid-s8889" xml:space="preserve">50000._ </s>
            <s xml:id="echoid-s8890" xml:space="preserve">
              <lb/>
            _43837_ Quare erit _AB,_ ad _AC,_ vt _50000._ </s>
            <s xml:id="echoid-s8891" xml:space="preserve">ad _82903._ </s>
            <s xml:id="echoid-s8892" xml:space="preserve">& </s>
            <s xml:id="echoid-s8893" xml:space="preserve">_AB,_ ad _BC,_ vt _50000._ </s>
            <s xml:id="echoid-s8894" xml:space="preserve">
              <lb/>
            ad _43837._ </s>
            <s xml:id="echoid-s8895" xml:space="preserve">& </s>
            <s xml:id="echoid-s8896" xml:space="preserve">_AC,_ ad _BC,_ vt _82903._ </s>
            <s xml:id="echoid-s8897" xml:space="preserve">ad _43837._</s>
            <s xml:id="echoid-s8898" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8899" xml:space="preserve">IT AQVE vt facile pro@ortiones laterum habeantur, ſatis eſt, ſila-
              <lb/>
              <note position="right" xlink:label="note-313-03" xlink:href="note-313-03a" xml:space="preserve">Praxis.</note>
            teribus ſinus angulorum oppoſitorum aſcribantur: </s>
            <s xml:id="echoid-s8900" xml:space="preserve">propterea quòd late-
              <lb/>
            ra eandem proportionem habent, quam oppoſitorum angulorum ſinus, vt
              <lb/>
            demonſtratum eſt.</s>
            <s xml:id="echoid-s8901" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8902" xml:space="preserve">_QVOD_ ſi duo tantum anguli cogniti ſint, erit reliquus tertius quoq; </s>
            <s xml:id="echoid-s8903" xml:space="preserve">notus.
              <lb/>
            </s>
            <s xml:id="echoid-s8904" xml:space="preserve">Quare, vt prius, laterum proportiones cognoſcentur.</s>
            <s xml:id="echoid-s8905" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8906" xml:space="preserve">_POSSVMVS_ eaſdem proportiones laterum cognoſcere ex angulis datis, ſine
              <lb/>
            auxilio antecedentis propoſ. </s>
            <s xml:id="echoid-s8907" xml:space="preserve">hoc modo. </s>
            <s xml:id="echoid-s8908" xml:space="preserve">Circa datum triangulum _ABC,_ deſcriba-
              <lb/>
              <note position="right" xlink:label="note-313-04" xlink:href="note-313-04a" xml:space="preserve">5. quarti.</note>
            </s>
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