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

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          <p>
            <s xml:id="echoid-s9906" xml:space="preserve">
              <pb o="316" file="328" n="328" rhead=""/>
            nient hæ lineæ productæ ad partes D, F, vt in puncto G. </s>
            <s xml:id="echoid-s9907" xml:space="preserve">Cum enim ſinuum
              <lb/>
            proportio ſit data maioris inæqualitatis, maior erit ſinus arcus BF, hoc eſt,
              <lb/>
            perpendicularis ex B, ad CF, demiſſa, ſinu arcus DF, hoc eſt, perpendiculari
              <lb/>
            ex D, ad CF, demiſſa. </s>
            <s xml:id="echoid-s9908" xml:space="preserve">Quare minus diſtabit punctum D, à recta CF, quàm pun-
              <lb/>
            ctum B; </s>
            <s xml:id="echoid-s9909" xml:space="preserve">atque adeo tandem coibunt BD, CF, productæ ad partes D, F. </s>
            <s xml:id="echoid-s9910" xml:space="preserve">Quod
              <lb/>
            etiam ita probabitur. </s>
            <s xml:id="echoid-s9911" xml:space="preserve">Si ambo ſinus, hoc eſt, perpendiculares ex B, D, ad CF,
              <lb/>
              <note position="left" xlink:label="note-328-01" xlink:href="note-328-01a" xml:space="preserve">28.primi.</note>
              <figure xlink:label="fig-328-01" xlink:href="fig-328-01a" number="174">
                <image file="328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/328-01"/>
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            demiſſæ eſſent æquales, cum ipſæ ſint paral-
              <lb/>
            lelæ, eſſent quoque BD, CF, parallelæ. </s>
            <s xml:id="echoid-s9912" xml:space="preserve">Cũ
              <lb/>
              <note position="left" xlink:label="note-328-02" xlink:href="note-328-02a" xml:space="preserve">33.primi.</note>
            ergo perpendicularis ex D, demiſſa minor
              <lb/>
            ſit, eſſicitur, vt conueniant, &</s>
            <s xml:id="echoid-s9913" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9914" xml:space="preserve">Diuiſo dein-
              <lb/>
            de arcu BD, bifariam in A, ſecabit ſemidia-
              <lb/>
            meter ducta EA, chordã BD, quoq; </s>
            <s xml:id="echoid-s9915" xml:space="preserve">bifariã
              <lb/>
            in H, ex lemmate ad defin. </s>
            <s xml:id="echoid-s9916" xml:space="preserve">ſinuum demon-
              <lb/>
              <note position="left" xlink:label="note-328-03" xlink:href="note-328-03a" xml:space="preserve">3.tertij.</note>
            ſtrato; </s>
            <s xml:id="echoid-s9917" xml:space="preserve">& </s>
            <s xml:id="echoid-s9918" xml:space="preserve">proinde & </s>
            <s xml:id="echoid-s9919" xml:space="preserve">ad angulos rectos. </s>
            <s xml:id="echoid-s9920" xml:space="preserve">Quo-
              <lb/>
            niam vero proportio ſinus arcus BF, ad ſi-
              <lb/>
            num arcus DF, eſt, ex hypotheſi, vt 11. </s>
            <s xml:id="echoid-s9921" xml:space="preserve">ad
              <lb/>
            5. </s>
            <s xml:id="echoid-s9922" xml:space="preserve">eſtq́ue vt ſinus arcus BF, ad ſinum arcus
              <lb/>
              <note position="left" xlink:label="note-328-04" xlink:href="note-328-04a" xml:space="preserve">5.huius.</note>
            DF, ita BG, ad DG; </s>
            <s xml:id="echoid-s9923" xml:space="preserve">erit quoque BG, ad
              <lb/>
            DG, vt 11. </s>
            <s xml:id="echoid-s9924" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s9925" xml:space="preserve">Poſita igitur recta BG, 11.
              <lb/>
            </s>
            <s xml:id="echoid-s9926" xml:space="preserve">erit DG, 5. </s>
            <s xml:id="echoid-s9927" xml:space="preserve">ac proinde reliqua BD, 6. </s>
            <s xml:id="echoid-s9928" xml:space="preserve">vtraque uero ſemiſsis BH, HD, 3.</s>
            <s xml:id="echoid-s9929" xml:space="preserve">ac
              <lb/>
            denique HG, 8. </s>
            <s xml:id="echoid-s9930" xml:space="preserve">Rurſus quia arcus BD, ponitur grad. </s>
            <s xml:id="echoid-s9931" xml:space="preserve">60. </s>
            <s xml:id="echoid-s9932" xml:space="preserve">erit utraque ſemiſ-
              <lb/>
            ſis BA, AD, grad. </s>
            <s xml:id="echoid-s9933" xml:space="preserve">30. </s>
            <s xml:id="echoid-s9934" xml:space="preserve">proptereaq́ue & </s>
            <s xml:id="echoid-s9935" xml:space="preserve">uterque angulus BEA, AED, gra-
              <lb/>
            duum quoque 30. </s>
            <s xml:id="echoid-s9936" xml:space="preserve">Et quia, poſito ſinu toto EH, recta HD, tangens eſt angu-
              <lb/>
            li DEH, & </s>
            <s xml:id="echoid-s9937" xml:space="preserve">HG, tangens anguli HEG, ut ad initium tangentium, atque
              <lb/>
            ſecãtium monuimus; </s>
            <s xml:id="echoid-s9938" xml:space="preserve">dabitur ex tangentium tabula, tangens grad. </s>
            <s xml:id="echoid-s9939" xml:space="preserve">30. </s>
            <s xml:id="echoid-s9940" xml:space="preserve">hoc eſt,
              <lb/>
            HD, partium 57735. </s>
            <s xml:id="echoid-s9941" xml:space="preserve">Quapropter, ut tangentem HG, anguli HEG, cognoſ-
              <lb/>
            camus, dicemus per auream regulam. </s>
            <s xml:id="echoid-s9942" xml:space="preserve">Si HD, ſemiſsis differentiæ terminorum
              <lb/>
            proportionis datæ, nempe 3. </s>
            <s xml:id="echoid-s9943" xml:space="preserve">dat HD, tangentem ſemiſsis differentiæ datæ
              <lb/>
            arcuum BF, FD, uel angulorum BEF, DEF, partium 57735. </s>
            <s xml:id="echoid-s9944" xml:space="preserve">quid dabit
              <lb/>
            HG, aggregatum ex ſemiſſe differentiæ terminorum datæ proportionis, & </s>
            <s xml:id="echoid-s9945" xml:space="preserve">
              <lb/>
            conſequente eiuſdem proportionis, nimirum 8? </s>
            <s xml:id="echoid-s9946" xml:space="preserve">prouenietq́ue HG, tangens
              <lb/>
            partium 153960. </s>
            <s xml:id="echoid-s9947" xml:space="preserve">ut hic apparet.</s>
            <s xml:id="echoid-s9948" xml:space="preserve"/>
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          <note position="right" xml:space="preserve">
            <lb/>
          HD. # HD. # HG. # # HG.
            <lb/>
          3. # 57735? # 8? # ſit. # 153960.
            <lb/>
          </note>
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            <s xml:id="echoid-s9949" xml:space="preserve">In tangentium autem tabula hæc tangens inuenta offert angulum AEF, ſiue
              <lb/>
            arcum AF, grad. </s>
            <s xml:id="echoid-s9950" xml:space="preserve">57. </s>
            <s xml:id="echoid-s9951" xml:space="preserve">cui ſi addatur ſemiſsis AB, grad. </s>
            <s xml:id="echoid-s9952" xml:space="preserve">30. </s>
            <s xml:id="echoid-s9953" xml:space="preserve">dabitur maior arcus
              <lb/>
            BF, ſiue angulus BEF, grad. </s>
            <s xml:id="echoid-s9954" xml:space="preserve">87. </s>
            <s xml:id="echoid-s9955" xml:space="preserve">ſi uero ab eodem ſubtrahatur ſemiſsis AD,
              <lb/>
            grad.</s>
            <s xml:id="echoid-s9956" xml:space="preserve">30. </s>
            <s xml:id="echoid-s9957" xml:space="preserve">remanebit minor areus DF, uel angulus DEF, grad.</s>
            <s xml:id="echoid-s9958" xml:space="preserve">27.</s>
            <s xml:id="echoid-s9959" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9960" xml:space="preserve">IGITVR quando proportio ſinus maioris arcus, vel anguli, ad ſi-
              <lb/>
              <note position="left" xlink:label="note-328-06" xlink:href="note-328-06a" xml:space="preserve">Praxis.</note>
            nũ minoris eſt maioris inæqualitatis: </s>
            <s xml:id="echoid-s9961" xml:space="preserve">Si ſiat, vt ſemiſſis differentiæ termi-
              <lb/>
            norum proportionis datæ ad tangentem ſemiſſis differẽtiæ arcuum, vel an-
              <lb/>
            gulorum datæ, ita aggregatum ex ſemiſſe differentiæ terminorum propor-
              <lb/>
            tionis, & </s>
            <s xml:id="echoid-s9962" xml:space="preserve">conſequente proportionis ad aliud, producetur tangens arcus,
              <lb/>
            vel anguli, qui ſemiſſi differentiæ arcuum, vel angulorum datæ additus
              <lb/>
            componit maiorem arcum, ſeu angulum; </s>
            <s xml:id="echoid-s9963" xml:space="preserve">& </s>
            <s xml:id="echoid-s9964" xml:space="preserve">ſi ab eodem ſemiſſis dicta
              <lb/>
            ſubducatur, remanet arcus, vel angulus minor.</s>
            <s xml:id="echoid-s9965" xml:space="preserve"/>
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