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

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        <div xml:id="echoid-div320" type="section" level="1" n="149">
          <p style="it">
            <s xml:id="echoid-s3864" xml:space="preserve">
              <pb o="105" file="117" n="117" rhead=""/>
            quadrata æqualia. </s>
            <s xml:id="echoid-s3865" xml:space="preserve">Cum ergo quadrata rectarum _EF, EG,_ æqualia quoque ſint, & </s>
            <s xml:id="echoid-s3866" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-117-01" xlink:href="note-117-01a" xml:space="preserve">47. primi.</note>
            illi quidem æqualia ſint quadrata ex _FK, KE,_ huic vero quadrata ex _GL, LE;_ </s>
            <s xml:id="echoid-s3867" xml:space="preserve">ac
              <lb/>
            proinde duo quadrata ex _FK, KE,_ duobus quadratis ex _GL, LE,_ æqualia: </s>
            <s xml:id="echoid-s3868" xml:space="preserve">ſiau-
              <lb/>
            ferantur duo æqualia quadrata rectarum _FK, GL,_ æqualia remanebunt quadra-
              <lb/>
            ta ex _EK, EL;_ </s>
            <s xml:id="echoid-s3869" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s3870" xml:space="preserve">rectæ _EK, EL,_ æquales erunt Quare cum latera _EF,_
              <lb/>
            _EK,_ lateribus _EG, EL,_ æqualia ſint, & </s>
            <s xml:id="echoid-s3871" xml:space="preserve">baſis _FK,_ baſi _GL,_ æqualis; </s>
            <s xml:id="echoid-s3872" xml:space="preserve">erunt anguli
              <lb/>
              <note position="right" xlink:label="note-117-02" xlink:href="note-117-02a" xml:space="preserve">8. primi.</note>
            _FEB, GED,_ æquales; </s>
            <s xml:id="echoid-s3873" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s3874" xml:space="preserve">arcus _BF, DG,_ æquales erunt. </s>
            <s xml:id="echoid-s3875" xml:space="preserve">Quod ſi ſinus
              <lb/>
              <note position="right" xlink:label="note-117-03" xlink:href="note-117-03a" xml:space="preserve">26. tertij.</note>
            complementorum _EK, EL,_ ſint æquales, oſtendemus eodem modo, rectas _FK, GL,_
              <lb/>
            æquales eſſe. </s>
            <s xml:id="echoid-s3876" xml:space="preserve">Quare vt prius, erunt arcus _BF, DG,_ æquales. </s>
            <s xml:id="echoid-s3877" xml:space="preserve">Si tandem ſinus verſi _KB,_
              <lb/>
            _LD,_ ponantur æquales; </s>
            <s xml:id="echoid-s3878" xml:space="preserve">ijs ablatis ex ſemidiametris _EB, ED,_ relinquentur ſinus
              <lb/>
            complementorum _EK, EL,_ æquales. </s>
            <s xml:id="echoid-s3879" xml:space="preserve">Quare rurſus oſtendemus, vt prius, arcus _BF,_
              <lb/>
            _DG,_ æquales eſſe. </s>
            <s xml:id="echoid-s3880" xml:space="preserve">quoderat oſtendendum. </s>
            <s xml:id="echoid-s3881" xml:space="preserve">Iam vero ſit arcus _BF,_ maior arcu _DM,_
              <lb/>
            & </s>
            <s xml:id="echoid-s3882" xml:space="preserve">ducatur ſinus _MO._ </s>
            <s xml:id="echoid-s3883" xml:space="preserve">Dico ſinum rectum _FK,_ maiorẽ eſſe ſinu recto _MO:_ </s>
            <s xml:id="echoid-s3884" xml:space="preserve">Item ſinũ
              <lb/>
            verſum _KB,_ maiorẽ ſinu verſo _OD:_ </s>
            <s xml:id="echoid-s3885" xml:space="preserve">ſinũ vero cõplementi _EK,_ minorẽ ſinu cõplementi
              <lb/>
            _EO._ </s>
            <s xml:id="echoid-s3886" xml:space="preserve">Poſito enim arcu _DG,_ æquali arcui _BF,_ erunt, vt demonſtr auimus, tam ſinus recti
              <lb/>
            _FK,
              <emph style="sc">G</emph>
            L,_ quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </s>
            <s xml:id="echoid-s3887" xml:space="preserve">Item cũ _EL,_
              <lb/>
            maior ſit, quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </s>
            <s xml:id="echoid-s3888" xml:space="preserve">Item cũ _EL,_
              <lb/>
            minor ſit, quàm _EO,_ erit quoque ſinus cõplementi _EK,_ minor ſinu cõplementi _EO._
              <lb/>
            </s>
            <s xml:id="echoid-s3889" xml:space="preserve">Ducatur _MN,_ ad _GL,_ perpendicularis, eritque _NL,_ ipſi _MO,_ æqualis. </s>
            <s xml:id="echoid-s3890" xml:space="preserve">Cum ergo _GL,_
              <lb/>
              <note position="right" xlink:label="note-117-04" xlink:href="note-117-04a" xml:space="preserve">34. primi.</note>
            maior ſit, quàm _NL,_ hoc eſt, quàm _MO,_ erit quoq; </s>
            <s xml:id="echoid-s3891" xml:space="preserve">ſinus rectus _FK,_ maior ſinu recto
              <lb/>
            _MO._ </s>
            <s xml:id="echoid-s3892" xml:space="preserve">quod demonſt randum er at. </s>
            <s xml:id="echoid-s3893" xml:space="preserve">Sit denique tam ſinus rectus _FK,_ maior ſinu recto
              <lb/>
            _MO,_ quàm ſinus verſus _KB,_ ſinu verſo _OD;_ </s>
            <s xml:id="echoid-s3894" xml:space="preserve">& </s>
            <s xml:id="echoid-s3895" xml:space="preserve">ſinus complementi _EO,_ maior ſinu
              <lb/>
            complementi _EK._ </s>
            <s xml:id="echoid-s3896" xml:space="preserve">Dico ſinui maiori tam recto, quàm verſo reſpondentem arcum _BF,_
              <lb/>
            maiorem eſſe arcu _DM,_ qui minori ſinui tam recto, quàm verſo reſpondet. </s>
            <s xml:id="echoid-s3897" xml:space="preserve">At maio-
              <lb/>
            ri ſinui complementi arcum reſpondentem _DM,_ minorem eſſe arcu _BF,_ qui minori
              <lb/>
            ſinui complementi reſpondet. </s>
            <s xml:id="echoid-s3898" xml:space="preserve">Nam ſi _FK,_ maior ſit, quàm _MO,_ auferatur _KP,_ ipſi
              <lb/>
            _MO,_ æqualis, & </s>
            <s xml:id="echoid-s3899" xml:space="preserve">ducatur _PQ,_ ad _FK,_ perpendicularis, ducaturque _QR,_ ad _BE,_
              <lb/>
            perpendicularis, quæipſi _PK,_ hoc eſt, ipſi _MO,_ æqualis erit; </s>
            <s xml:id="echoid-s3900" xml:space="preserve">ac proinde, vt paulò ante
              <lb/>
              <note position="right" xlink:label="note-117-05" xlink:href="note-117-05a" xml:space="preserve">34. ptimi.</note>
            oſtenſum eſt, erunt arcus _BQ, DM,_ æquales, propter æqualitatem ſinuum rectorum
              <lb/>
            _QR, MO._ </s>
            <s xml:id="echoid-s3901" xml:space="preserve">Cum ergoarcus _BF,_ arcu _BQ,_ maior ſit, erit idem arcus _BF,_ arcu _DM,_
              <lb/>
            maior. </s>
            <s xml:id="echoid-s3902" xml:space="preserve">Quòd ſi _KB,_ maior ſit, quàm _OD,_ abſcindatur _BR,_ ipſi _DO,_ æqualis, duca-
              <lb/>
            turque _RQ,_ ad _BE,_ perpendicularis: </s>
            <s xml:id="echoid-s3903" xml:space="preserve">eruntq́ arcus _BQ, DM,_ vt paulo antemon-
              <lb/>
            ſtrauimus, æquales, ob æqualitatem ſinuum verſorum _RB, OD._ </s>
            <s xml:id="echoid-s3904" xml:space="preserve">Quare cũ arcus _BF,_
              <lb/>
            maior ſit arcu _
              <emph style="sc">B</emph>
            Q,_ eritidem arcus _
              <emph style="sc">B</emph>
            F,_ arcu _DM,_ maior. </s>
            <s xml:id="echoid-s3905" xml:space="preserve">Si tandem maior ſit _EO,_
              <lb/>
            quàm _EK,_ detrahatur _EL,_ ipsi _
              <emph style="sc">E</emph>
            K,_ æqualis, ducaturque ad _
              <emph style="sc">E</emph>
            D,_ perpendicularis
              <lb/>
              <note position="right" xlink:label="note-117-06" xlink:href="note-117-06a" xml:space="preserve">Anguli æ-
                <lb/>
              quales ha-
                <lb/>
              bent ſinus
                <lb/>
              ęquales, &c.</note>
            _LG:_ </s>
            <s xml:id="echoid-s3906" xml:space="preserve">Eruntq́; </s>
            <s xml:id="echoid-s3907" xml:space="preserve">arcus _BF,
              <emph style="sc">Dg</emph>
            ,_ ob æqualitatem sinuum complementorum _
              <emph style="sc">E</emph>
            K,
              <emph style="sc">E</emph>
            L,_
              <lb/>
            æquales, vt paulo ante fuit oſtenſum. </s>
            <s xml:id="echoid-s3908" xml:space="preserve">Quam ob rem cum arcus _DM,_ arcu _DG,_ sit
              <lb/>
            minor, erit idem arcus _DM,_ arcn _BF,_ minor. </s>
            <s xml:id="echoid-s3909" xml:space="preserve">Quod eſt propositum.
              <lb/>
            </s>
            <s xml:id="echoid-s3910" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3911" xml:space="preserve">_
              <emph style="sc">IDe</emph>
            M_ prorſus dicendum eſt de sinubus angulorum. </s>
            <s xml:id="echoid-s3912" xml:space="preserve">Nam & </s>
            <s xml:id="echoid-s3913" xml:space="preserve">anguli æquales ha-
              <lb/>
              <note position="right" xlink:label="note-117-07" xlink:href="note-117-07a" xml:space="preserve">Si in trian
                <lb/>
              gulo rectã
                <lb/>
              gulo latus
                <lb/>
              recto angu
                <lb/>
              lo oppoſitũ
                <lb/>
              ſit ſinus to
                <lb/>
              tus, erit
                <lb/>
              vtrumuis
                <lb/>
              laterum re
                <lb/>
              liquorum
                <lb/>
              ſinus rectꝰ
                <lb/>
              anguli acu
                <lb/>
              ti oppoſiti.</note>
            bent sinus æquales tam rectos, quam complemẽtorum, & </s>
            <s xml:id="echoid-s3914" xml:space="preserve">verſos, &</s>
            <s xml:id="echoid-s3915" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3916" xml:space="preserve">propterea quod
              <lb/>
            æquales anguli insiſtunt in centro æqualibus arcubus, &</s>
            <s xml:id="echoid-s3917" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3918" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3919" xml:space="preserve">_
              <emph style="sc">POSTRe</emph>
            MO_ in omni triangulo rectangulo, si latus recto angulo oppositum
              <lb/>
            ponatur sinus totus, reliqua duo latera ſunt sinus recti reliquorum angulorum acu-
              <lb/>
            torum, quibus opponuntur. </s>
            <s xml:id="echoid-s3920" xml:space="preserve">Vt in triangulo rectangulo _EKF,_ in quo _EF,_ eſt sinus to-
              <lb/>
            tus, vtpote ſemidiameter circuli ex F, deſcripti, latus _
              <emph style="sc">F</emph>
            K,_ eſtsinus rectus anguli
              <lb/>
            _
              <emph style="sc">FE</emph>
            K,_ ex deſin. </s>
            <s xml:id="echoid-s3921" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3922" xml:space="preserve">Sic quoque si idem circulus ex _
              <emph style="sc">F</emph>
            ,_ deſcriberetur, eſſet latus _
              <emph style="sc">E</emph>
            K,_ si-
              <lb/>
            nus reclus anguli _
              <emph style="sc">EF</emph>
            K,_ ex eadem deſin. </s>
            <s xml:id="echoid-s3923" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3924" xml:space="preserve">Quod etiam hinc patet, quèd angulus _
              <emph style="sc">EF</emph>
            </s>
          </p>
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      </text>
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