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

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            <s xml:id="echoid-s2758" xml:space="preserve">
              <pb o="71" file="083" n="83" rhead=""/>
            ſunt autem & </s>
            <s xml:id="echoid-s2759" xml:space="preserve">anguli A K M, C K N, ad verticem æquales, & </s>
            <s xml:id="echoid-s2760" xml:space="preserve">latera K A, K C,
              <lb/>
              <note position="right" xlink:label="note-083-01" xlink:href="note-083-01a" xml:space="preserve">15. primi.</note>
            æqualia, cum ſint ſemidiametri circuli A D C E. </s>
            <s xml:id="echoid-s2761" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s2762" xml:space="preserve">latera K M, K N,
              <lb/>
              <note position="right" xlink:label="note-083-02" xlink:href="note-083-02a" xml:space="preserve">26. primi.</note>
            æqualia erunt:</s>
            <s xml:id="echoid-s2763" xml:space="preserve">ſunt autem & </s>
            <s xml:id="echoid-s2764" xml:space="preserve">ſemidiametri K D, K E, æquales. </s>
            <s xml:id="echoid-s2765" xml:space="preserve">Reliquæ ergo
              <lb/>
            rectæ D M, E N, æquales erunt. </s>
            <s xml:id="echoid-s2766" xml:space="preserve">Rurſus quoniam recta B K, ex B, polo circuli
              <lb/>
            A D C E, ad eiuſdem centrum K, ducta, recta eſt ad planum circuli, erit an-
              <lb/>
              <note position="right" xlink:label="note-083-03" xlink:href="note-083-03a" xml:space="preserve">Schol. 8. 1.
                <lb/>
              huius.</note>
            gulus M K L, in triangulo K L M, rectus, ex defin. </s>
            <s xml:id="echoid-s2767" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2768" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2769" xml:space="preserve">11. </s>
            <s xml:id="echoid-s2770" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s2771" xml:space="preserve">Angulus igi-
              <lb/>
              <note position="right" xlink:label="note-083-04" xlink:href="note-083-04a" xml:space="preserve">17. primi.</note>
            tur K M L, acutus erit. </s>
            <s xml:id="echoid-s2772" xml:space="preserve">Cum ergo duo anguli F M N, H N M, duobus ſint
              <lb/>
              <note position="right" xlink:label="note-083-05" xlink:href="note-083-05a" xml:space="preserve">29. primi.</note>
            rectis æquales; </s>
            <s xml:id="echoid-s2773" xml:space="preserve">erit angulus H N M, obtuſus. </s>
            <s xml:id="echoid-s2774" xml:space="preserve">Quare, vt mox, lemmate ſequen
              <lb/>
            ti oſtendemus, arcus E H, minor erit, arcu D F; </s>
            <s xml:id="echoid-s2775" xml:space="preserve">atque adeo, cum æquales
              <lb/>
            ſint arcus B D, B E, quòd rectæ ſubtenſæ B D, B E, ex defin. </s>
            <s xml:id="echoid-s2776" xml:space="preserve">poli, ſint æqua-
              <lb/>
              <note position="right" xlink:label="note-083-06" xlink:href="note-083-06a" xml:space="preserve">28. tertij.</note>
            les, maior erit arcus B H, arcu B F. </s>
            <s xml:id="echoid-s2777" xml:space="preserve">Si igitur in ſphæra duo maximi circuli ſe
              <lb/>
            mutuo ſecent, &</s>
            <s xml:id="echoid-s2778" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2779" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s2780" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div244" type="section" level="1" n="115">
          <head xml:id="echoid-head129" xml:space="preserve">LEMMA.</head>
          <p style="it">
            <s xml:id="echoid-s2781" xml:space="preserve">_QVOD_ autem arcus _E H,_ arcw _D F,_ minor ſit, facile demonſtrabimus, hoc propoſi-
              <lb/>
            to theoremate prius demonſtrato.</s>
            <s xml:id="echoid-s2782" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2783" xml:space="preserve">SI arcui circuli recta ſubtendatur, ad quam ex arcu duæ perpen-
              <lb/>
            diculares demittantur auferentes verſus terminos arcus duos arcus
              <lb/>
            æquales; </s>
            <s xml:id="echoid-s2784" xml:space="preserve">auferent eædem duas rectas ex recta ſubtenſa æquales. </s>
            <s xml:id="echoid-s2785" xml:space="preserve">Et ſi
              <lb/>
            duæ perpendiculares ad rectam ſubtenſam ducantur auferẽtes duas
              <lb/>
            rectas æquales; </s>
            <s xml:id="echoid-s2786" xml:space="preserve">auferent eædem duos arcus æquales.</s>
            <s xml:id="echoid-s2787" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2788" xml:space="preserve">ARCVI circuli A B C D, ſubtendatur recta A D, ad quam ex
              <lb/>
            arcu demittantur duæ perpendiculares B E, C F, auferentes duos arcus
              <lb/>
              <figure xlink:label="fig-083-01" xlink:href="fig-083-01a" number="91">
                <image file="083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/083-01"/>
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            A B, D C, æquales. </s>
            <s xml:id="echoid-s2789" xml:space="preserve">Dico eaſdem auferre
              <lb/>
            æquales rectas A E, D F. </s>
            <s xml:id="echoid-s2790" xml:space="preserve">Ducta enim
              <lb/>
              <note position="right" xlink:label="note-083-07" xlink:href="note-083-07a" xml:space="preserve">Schol. 27.
                <lb/>
              tertij.</note>
            recta B C, erunt A D, B C, parallelæ,
              <lb/>
            ob æqualitatem arcuum A B, D C: </s>
            <s xml:id="echoid-s2791" xml:space="preserve">ſunt
              <lb/>
            autem & </s>
            <s xml:id="echoid-s2792" xml:space="preserve">B E, C F, parallelæ. </s>
            <s xml:id="echoid-s2793" xml:space="preserve">Parallelo-
              <lb/>
              <note position="right" xlink:label="note-083-08" xlink:href="note-083-08a" xml:space="preserve">28. primi.</note>
            grammum igitur eſt B E F C, atque adeò
              <lb/>
            & </s>
            <s xml:id="echoid-s2794" xml:space="preserve">rectæ B E, C F, æquales. </s>
            <s xml:id="echoid-s2795" xml:space="preserve">Et quoniam æqualibus arcubus A B, D C, re-
              <lb/>
              <note position="right" xlink:label="note-083-09" xlink:href="note-083-09a" xml:space="preserve">34. primi.</note>
            ctæ ſubtenſæ A B, D C, æquales ſunt; </s>
            <s xml:id="echoid-s2796" xml:space="preserve">erunt quadrata ex A B, D C, æqua-
              <lb/>
              <note position="right" xlink:label="note-083-10" xlink:href="note-083-10a" xml:space="preserve">29. tertij.</note>
            lia. </s>
            <s xml:id="echoid-s2797" xml:space="preserve">Cum ergo tam illud æquale ſit quadratis ex A E, B E, quàm boc qua-
              <lb/>
              <note position="right" xlink:label="note-083-11" xlink:href="note-083-11a" xml:space="preserve">47. primi</note>
            dratis ex D F, C F; </s>
            <s xml:id="echoid-s2798" xml:space="preserve">ſi auferantur æqualia quadrata rectarum B E, C F,
              <lb/>
            æqualia erunt quadrata rectarum A E, D F; </s>
            <s xml:id="echoid-s2799" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s2800" xml:space="preserve">rectæ A E,
              <lb/>
            D F, æquales erunt. </s>
            <s xml:id="echoid-s2801" xml:space="preserve">quod primo loco proponebatur.</s>
            <s xml:id="echoid-s2802" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2803" xml:space="preserve">SED iam perpendiculares B E, C F, auferant æquales rectas A E,
              <lb/>
            D F. </s>
            <s xml:id="echoid-s2804" xml:space="preserve">Dico eaſdem auferre æquales arcus A B, D C. </s>
            <s xml:id="echoid-s2805" xml:space="preserve">Si enim non ſunt
              <lb/>
            æquales, ſit, ſi fieri potest, maior arcus A B, à quo æqualis abſcindatur
              <lb/>
            A G, & </s>
            <s xml:id="echoid-s2806" xml:space="preserve">ex G, ad A D, perpendicularis ducatur G H. </s>
            <s xml:id="echoid-s2807" xml:space="preserve">Erit igitur, vt
              <lb/>
            proxime demonſtr atum eſt, recta A H, rectæ D F, æqualis, atque adeò
              <lb/>
            & </s>
            <s xml:id="echoid-s2808" xml:space="preserve">rectæ A E, pars toti: </s>
            <s xml:id="echoid-s2809" xml:space="preserve">Quod eſt abſurdum. </s>
            <s xml:id="echoid-s2810" xml:space="preserve">Non eſt ergo arcus A B,
              <lb/>
            maior arcu D C: </s>
            <s xml:id="echoid-s2811" xml:space="preserve">eademque ratione neque minor erit. </s>
            <s xml:id="echoid-s2812" xml:space="preserve">Aequalis ergo eſt.</s>
            <s xml:id="echoid-s2813" xml:space="preserve"/>
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