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

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          <p>
            <s xml:id="echoid-s12427" xml:space="preserve">
              <pb o="371" file="383" n="383" rhead=""/>
            bunt & </s>
            <s xml:id="echoid-s12428" xml:space="preserve">tria latera tribus lateribus æqualia, ſingu-
              <lb/>
            laſingulis, quæ æquales angulos ſubrendunt.</s>
            <s xml:id="echoid-s12429" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12430" xml:space="preserve">HABEANT duo triangula ſphærica ABC, DEF, tres angulos A,
              <lb/>
            B, C, tribus angulis D, E, F, ſingulos ſingulis, æquales. </s>
            <s xml:id="echoid-s12431" xml:space="preserve">Dico & </s>
            <s xml:id="echoid-s12432" xml:space="preserve">tria latera
              <lb/>
            AB, AC, BC, tribus lateribus DE, DF, EF, eſſe æqualia, ſingula ſingulis,
              <lb/>
            quæ angulos æquales ſubtendunt. </s>
            <s xml:id="echoid-s12433" xml:space="preserve">Sienim la-
              <lb/>
              <figure xlink:label="fig-383-01" xlink:href="fig-383-01a" number="219">
                <image file="383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/383-01"/>
              </figure>
            tera BC, EF, (vt ab his lateribus exordiamur.)
              <lb/>
            </s>
            <s xml:id="echoid-s12434" xml:space="preserve">non ſunt æqualia, ſit BC, ſi fieri poteſt, maius; </s>
            <s xml:id="echoid-s12435" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-383-01" xlink:href="note-383-01a" xml:space="preserve">1. huius.</note>
            & </s>
            <s xml:id="echoid-s12436" xml:space="preserve">abſcindatur arcus BG, arcui EF, æqua-
              <lb/>
            lis. </s>
            <s xml:id="echoid-s12437" xml:space="preserve">Aut ergo arcus BA, æqualis eſt arcui ED,
              <lb/>
            aut maior, aut minor. </s>
            <s xml:id="echoid-s12438" xml:space="preserve">Quodcunque horũ di-
              <lb/>
            catur, ſequetur abſurdũ ex eo, quòd inæqua-
              <lb/>
            lia dicuntur eſſe latera BC, EF, nempe BC,
              <lb/>
            maius, quàm E F. </s>
            <s xml:id="echoid-s12439" xml:space="preserve">Sit enim primum arcus BA,
              <lb/>
              <note position="right" xlink:label="note-383-02" xlink:href="note-383-02a" xml:space="preserve">20. 1. Theo.</note>
            arcui ED, æqualis; </s>
            <s xml:id="echoid-s12440" xml:space="preserve">ducaturq́ue per puncta
              <lb/>
            A, G, arcus maximi circuli AG. </s>
            <s xml:id="echoid-s12441" xml:space="preserve">Igitur cum latera BA, BG, æqualia ſint la-
              <lb/>
            teribus ED, EF, angulosq́ue contineãt æquales B,E, ex hypotheſi; </s>
            <s xml:id="echoid-s12442" xml:space="preserve">erunt an-
              <lb/>
            guli BAG, & </s>
            <s xml:id="echoid-s12443" xml:space="preserve">D, æquales: </s>
            <s xml:id="echoid-s12444" xml:space="preserve">Eſt autem angulus D, poſitus æqualis angulo BAC.
              <lb/>
            </s>
            <s xml:id="echoid-s12445" xml:space="preserve">
              <note position="right" xlink:label="note-383-03" xlink:href="note-383-03a" xml:space="preserve">7. huius.</note>
            Angulus igitur BAG, æqualis erit quoque angulo BAC, pars toti. </s>
            <s xml:id="echoid-s12446" xml:space="preserve">Quod
              <lb/>
            eſt abſurdum.</s>
            <s xml:id="echoid-s12447" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">1. huius.</note>
          <p>
            <s xml:id="echoid-s12448" xml:space="preserve">SIT deinde arcus BA, maior arcu ED, & </s>
            <s xml:id="echoid-s12449" xml:space="preserve">abſcindatur arcus BI, æqua-
              <lb/>
            lis ipſi ED; </s>
            <s xml:id="echoid-s12450" xml:space="preserve">ac per puncta G,I, arcus circuli maximi ducatur GI, conueniens
              <lb/>
              <note position="right" xlink:label="note-383-05" xlink:href="note-383-05a" xml:space="preserve">20. 1. Theo.</note>
            cum arcu CA, protracto in H. </s>
            <s xml:id="echoid-s12451" xml:space="preserve">Quoniam igitur latera BI, BG, æqualia ſunt
              <lb/>
              <note position="right" xlink:label="note-383-06" xlink:href="note-383-06a" xml:space="preserve">7. huius.</note>
            lateribus ED, EF, angulosq́ue continent æquales B, E; </s>
            <s xml:id="echoid-s12452" xml:space="preserve">erunt anguli BIG,
              <lb/>
            BGI, angulis D, F, hoc eſt, angulis BAC, BCA, æquales; </s>
            <s xml:id="echoid-s12453" xml:space="preserve">quod his duobus
              <lb/>
              <note position="right" xlink:label="note-383-07" xlink:href="note-383-07a" xml:space="preserve">6. huius.</note>
            æquales ſint poſiti D, & </s>
            <s xml:id="echoid-s12454" xml:space="preserve">F; </s>
            <s xml:id="echoid-s12455" xml:space="preserve">ſunt autem anguli BIG, BAC, angulis HIA,
              <lb/>
            HAK, ad verticem æquales. </s>
            <s xml:id="echoid-s12456" xml:space="preserve">Aequales ergo ſunt & </s>
            <s xml:id="echoid-s12457" xml:space="preserve">anguli HAK, HIA. </s>
            <s xml:id="echoid-s12458" xml:space="preserve">Igi-
              <lb/>
            tur cum & </s>
            <s xml:id="echoid-s12459" xml:space="preserve">angulus BGH, externus æqualis ſit interno BCH, & </s>
            <s xml:id="echoid-s12460" xml:space="preserve">externus
              <lb/>
            HAK, interno HIK, vt oſtendimus: </s>
            <s xml:id="echoid-s12461" xml:space="preserve">erunt tam arcus AH, HI, quam arcus
              <lb/>
              <note position="right" xlink:label="note-383-08" xlink:href="note-383-08a" xml:space="preserve">15. huius.</note>
            CH, HG, ſemicirculo æquales; </s>
            <s xml:id="echoid-s12462" xml:space="preserve">atque adeo arcus AH, HI, arcubus CH,
              <lb/>
            HG, æquales erunt, pars toti. </s>
            <s xml:id="echoid-s12463" xml:space="preserve">Quod eſt abſurdum.</s>
            <s xml:id="echoid-s12464" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s12465" xml:space="preserve">SIT tandem arcus BA, minor arcu ED, producaturq́ue vltra A, & </s>
            <s xml:id="echoid-s12466" xml:space="preserve">ex eo
              <lb/>
            abſcindatur arcus BK, æqualis arcui ED; </s>
            <s xml:id="echoid-s12467" xml:space="preserve">atque per puncta G, K, arcus cir-
              <lb/>
              <note position="right" xlink:label="note-383-09" xlink:href="note-383-09a" xml:space="preserve">1. huius.</note>
            culi maximi ducatur GK, ſecans arcum AC, in L. </s>
            <s xml:id="echoid-s12468" xml:space="preserve">Quoniam ergo latera BK,
              <lb/>
              <note position="right" xlink:label="note-383-10" xlink:href="note-383-10a" xml:space="preserve">20. 1. Theo.</note>
            BG, lateribus ED, EF, æqualia ſunt, anguloſque continent æquales B, E,
              <lb/>
            erunt & </s>
            <s xml:id="echoid-s12469" xml:space="preserve">anguli BKG, BGK, angulis D, F, hoc eſt, angulis BAC, BCA,
              <lb/>
              <note position="right" xlink:label="note-383-11" xlink:href="note-383-11a" xml:space="preserve">7. huius.</note>
            (quòd his duobus æquales ſint poſiti anguli D, F.) </s>
            <s xml:id="echoid-s12470" xml:space="preserve">æquales. </s>
            <s xml:id="echoid-s12471" xml:space="preserve">Itaque cum & </s>
            <s xml:id="echoid-s12472" xml:space="preserve">
              <lb/>
            angulus BAL, externus æqualis ſit interno BKL, & </s>
            <s xml:id="echoid-s12473" xml:space="preserve">externus BGL, inter-
              <lb/>
            no BCL, vt oſtendimus, erunt tam arcus AL, LK, quàm arcus CL, LG,
              <lb/>
              <note position="right" xlink:label="note-383-12" xlink:href="note-383-12a" xml:space="preserve">15, huius.</note>
            ſemicirculo æquales; </s>
            <s xml:id="echoid-s12474" xml:space="preserve">ac proinde duo arcus AC, GK, integro circulo æqua-
              <lb/>
            les erunt. </s>
            <s xml:id="echoid-s12475" xml:space="preserve">Quod eſt abſurdum, cum vterque arcus AC, GK, ſemicirculo ſit
              <lb/>
              <note position="right" xlink:label="note-383-13" xlink:href="note-383-13a" xml:space="preserve">2. huius.</note>
            minor. </s>
            <s xml:id="echoid-s12476" xml:space="preserve">Non ergo inæqualia ſunt latera BC, EF, ſed æqualia. </s>
            <s xml:id="echoid-s12477" xml:space="preserve">Eodemq́ue mo-
              <lb/>
            do oſtendemus, latera AC, DF, nec non AB, DE, æqualia eſſe. </s>
            <s xml:id="echoid-s12478" xml:space="preserve">Tria ergo
              <lb/>
            latera trianguli ABC, tribus lateribus trianguli DEF, æqualia ſunt. </s>
            <s xml:id="echoid-s12479" xml:space="preserve">Quare
              <lb/>
            ſi duo triangula ſphærica, &</s>
            <s xml:id="echoid-s12480" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12481" xml:space="preserve">Quod oſtendendum erat.</s>
            <s xml:id="echoid-s12482" xml:space="preserve"/>
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