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

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        <div xml:id="echoid-div367" type="section" level="1" n="161">
          <p>
            <s xml:id="echoid-s4424" xml:space="preserve">
              <pb o="116" file="128" n="128" rhead=""/>
            ſegmento AGB; </s>
            <s xml:id="echoid-s4425" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s4426" xml:space="preserve">AEB, tertia pars duorum rectorum. </s>
            <s xml:id="echoid-s4427" xml:space="preserve">Deinde, quo-
              <lb/>
            niam latera DB, BA, trianguli DBA, lateribus FB, BA, trianguli FBA,
              <lb/>
            æqualia ſunt, angulosq́ue continent æquales; </s>
            <s xml:id="echoid-s4428" xml:space="preserve">erunt baſes AD, AF, inter ſe
              <lb/>
              <note position="left" xlink:label="note-128-01" xlink:href="note-128-01a" xml:space="preserve">27. tertij.</note>
            æquales. </s>
            <s xml:id="echoid-s4429" xml:space="preserve">Cum ergo AD, ipſi AE, æqualis ſit, propter æquales arcus AD,
              <lb/>
              <note position="left" xlink:label="note-128-02" xlink:href="note-128-02a" xml:space="preserve">29.tertij.</note>
            AE; </s>
            <s xml:id="echoid-s4430" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s4431" xml:space="preserve">AF, eidem AE, æqualis; </s>
            <s xml:id="echoid-s4432" xml:space="preserve">ac propterea anguli AEF, AFE, æqua
              <lb/>
              <note position="left" xlink:label="note-128-03" xlink:href="note-128-03a" xml:space="preserve">5.primi.</note>
            les inter ſe erunt: </s>
            <s xml:id="echoid-s4433" xml:space="preserve">Eſt autem AEF, vt oſtendimus, tertia pars duorum recto-
              <lb/>
            rum. </s>
            <s xml:id="echoid-s4434" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s4435" xml:space="preserve">AFE, tertia pars erit duorum rectorum; </s>
            <s xml:id="echoid-s4436" xml:space="preserve">atque adeo & </s>
            <s xml:id="echoid-s4437" xml:space="preserve">reliquus
              <lb/>
            EAF, tertia pars erit duorum rectorum. </s>
            <s xml:id="echoid-s4438" xml:space="preserve">Quare triangulum AEF, æquilate-
              <lb/>
              <note position="left" xlink:label="note-128-04" xlink:href="note-128-04a" xml:space="preserve">32. primi.</note>
            rum erit, ex coroll. </s>
            <s xml:id="echoid-s4439" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s4440" xml:space="preserve">6. </s>
            <s xml:id="echoid-s4441" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4442" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4443" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4444" xml:space="preserve">ideoque recta EF, differentia chorda-
              <lb/>
            rum BD, BE, chordæ AE, vel AD, æqualis erit. </s>
            <s xml:id="echoid-s4445" xml:space="preserve">Differentia ergo chorda-
              <lb/>
            rum duorum arcuum ſemicirculi, &</s>
            <s xml:id="echoid-s4446" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4447" xml:space="preserve">quod erat demonſtrandum.</s>
            <s xml:id="echoid-s4448" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div370" type="section" level="1" n="162">
          <head xml:id="echoid-head188" xml:space="preserve">COROLLARIVM.</head>
          <note position="left" xml:space="preserve">Duæ chor-
            <lb/>
          dę duorum
            <lb/>
          arcuũ cõſi-
            <lb/>
          ciẽtiũ gra.
            <lb/>
          120. ſimul
            <lb/>
          ęquales sũt
            <lb/>
          chordęarcꝰ
            <lb/>
          cõpoſiti ex
            <lb/>
          arcu grad.
            <lb/>
          120. & arcu
            <lb/>
          minore il-
            <lb/>
          lorum duo
            <lb/>
          rum.</note>
          <p>
            <s xml:id="echoid-s4449" xml:space="preserve">SEQVITVR hinc, ſi duorum arcuum, qui ſimul grad. </s>
            <s xml:id="echoid-s4450" xml:space="preserve">120. </s>
            <s xml:id="echoid-s4451" xml:space="preserve">conficiant, chordæ ſimul
              <lb/>
            iungantur, effici chordam arcus compoſiti ex arcu grad. </s>
            <s xml:id="echoid-s4452" xml:space="preserve">120, & </s>
            <s xml:id="echoid-s4453" xml:space="preserve">arcu minore illorum duo-
              <lb/>
            rum, ſi in æquales ſint. </s>
            <s xml:id="echoid-s4454" xml:space="preserve">Ita namque vides chordas BD, DA, arcuum BD, DA, conſicientium
              <lb/>
            grad. </s>
            <s xml:id="echoid-s4455" xml:space="preserve">120. </s>
            <s xml:id="echoid-s4456" xml:space="preserve">ſimul ſumptas æquari chordæ BE, arcus BAE, compoſiti ex arcu BA, grad. </s>
            <s xml:id="echoid-s4457" xml:space="preserve">120.
              <lb/>
            </s>
            <s xml:id="echoid-s4458" xml:space="preserve">& </s>
            <s xml:id="echoid-s4459" xml:space="preserve">arcu AE, qui minori AD, æqualis eſt: </s>
            <s xml:id="echoid-s4460" xml:space="preserve">propterea quòd vt demonſtratum eſt, differentia
              <lb/>
            EF, inter choidas BD, BE, æqualis eſt chordæ AD.</s>
            <s xml:id="echoid-s4461" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div371" type="section" level="1" n="163">
          <head xml:id="echoid-head189" xml:space="preserve">THEOR. 5. PROPOS. 7.</head>
          <p>
            <s xml:id="echoid-s4462" xml:space="preserve">SI quantitas quantitatem excedat, ſemiſsis il-
              <lb/>
              <note position="left" xlink:label="note-128-06" xlink:href="note-128-06a" xml:space="preserve">Si quãtitas
                <lb/>
              ſuꝑ & quã-
                <lb/>
              titaté ſemiſ
                <lb/>
              ſis ſemiſs ẽ
                <lb/>
              ſuperabit
                <lb/>
              exceſſus ſe
                <lb/>
              miſſe.</note>
            lius ſemiſsem huius ſuperabit exceſſus ſemiſſe.</s>
            <s xml:id="echoid-s4463" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4464" xml:space="preserve">SVPERET quantitas AB, quantitatẽ C, exceſſu DB, qui bifariã ſecetur in
              <lb/>
              <figure xlink:label="fig-128-01" xlink:href="fig-128-01a" number="125">
                <image file="128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/128-01"/>
              </figure>
            E, & </s>
            <s xml:id="echoid-s4465" xml:space="preserve">ipſi EB, æqualis pona
              <lb/>
            tur AF. </s>
            <s xml:id="echoid-s4466" xml:space="preserve">Quoniã igitur AF,
              <lb/>
            EB, toti exceſſui DB, æ-
              <lb/>
            quales ſunt, erit reliqua
              <lb/>
            FE, ipſi C, æqualis. </s>
            <s xml:id="echoid-s4467" xml:space="preserve">Sece-
              <lb/>
            tur FE, bifaria in G. </s>
            <s xml:id="echoid-s4468" xml:space="preserve">Quia
              <lb/>
            ergo GE, GF, æquales
              <lb/>
            sũt; </s>
            <s xml:id="echoid-s4469" xml:space="preserve">additis æqualibus EB,
              <lb/>
            FA, æquales quoque erũt
              <lb/>
            GB, GA; </s>
            <s xml:id="echoid-s4470" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s4471" xml:space="preserve">AB,
              <lb/>
            in G, ſecta erit bifariã. </s>
            <s xml:id="echoid-s4472" xml:space="preserve">Se-
              <lb/>
            miſsis igitur BG, ipſius
              <lb/>
            AB, ſuperat GE, ſemiſſem
              <lb/>
            ipſius FE, hoc eſt, ipſius C, exceſſu EB, qui ſem iſsis eſt exceſſus DB. </s>
            <s xml:id="echoid-s4473" xml:space="preserve">Si quan-
              <lb/>
            titas ergo quantitatem excedat, &</s>
            <s xml:id="echoid-s4474" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4475" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s4476" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div374" type="section" level="1" n="164">
          <head xml:id="echoid-head190" xml:space="preserve">THEOR. 6. PROPOS. 8.</head>
          <p>
            <s xml:id="echoid-s4477" xml:space="preserve">DIFFERENTIA ſinuum duorum </s>
          </p>
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