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

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        <div xml:id="echoid-div342" type="section" level="1" n="151">
          <head xml:id="echoid-head177" xml:space="preserve">THEOR. 1. PROPOS. 1.</head>
          <p>
            <s xml:id="echoid-s4064" xml:space="preserve">IN Quadrante circuli ſumptis arcubus æqua-
              <lb/>
              <note position="right" xlink:label="note-121-01" xlink:href="note-121-01a" xml:space="preserve">Perpĕdicu-
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              lares ex ar-
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              cubus qua-
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              drãtis ęqua
                <lb/>
              libus ad al-
                <lb/>
              terutrá ſe-
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              midiame -
                <lb/>
              trorum, vel
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              ad rectam
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              ſemidiame
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              tro paral -
                <lb/>
              lelam du-
                <lb/>
              ctę auferũt
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              fegm enta
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              inæqualia,
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              maiufq́ eſt
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              illud, qd al
                <lb/>
              teri femi-
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              diametro
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              {pro}pinquius
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              eſt.</note>
            libus, ſi ab eorum terminis ad alterutram ſemidia-
              <lb/>
            metrorum, vel ad rectam ſemidiametro paralle-
              <lb/>
            lam, perpendiculares ducantur; </s>
            <s xml:id="echoid-s4065" xml:space="preserve">erunt ſegmenta
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            ſemidiametri, velillius parallelæ interillas perpen-
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            diculares intercepta, inæqualia, maiusq́ erit illud,
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            quod alteri ſemidiametro propinquius elt.</s>
            <s xml:id="echoid-s4066" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4067" xml:space="preserve">SIT Quadrans ABC, in quo arcus æquales ſint DE, EF, à quorum ter-
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            minis ad ſemidiametrum AC, vel ad rectam RS, ipſi AC, parallelam per-
              <lb/>
            pendiculares ducantur DKG, ELH, FMI. </s>
            <s xml:id="echoid-s4068" xml:space="preserve">Dico ſegmenta GH, HI, vel
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            KL, LM, inæqualia eſſe, maiusq́ue eſſe GH,
              <lb/>
              <figure xlink:label="fig-121-01" xlink:href="fig-121-01a" number="118">
                <image file="121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/121-01"/>
              </figure>
            quàm HI, vel KL, maius, quàm LM. </s>
            <s xml:id="echoid-s4069" xml:space="preserve">Com-
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            pleto enim ſemicirculo BCN, producantur
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            rectæ DG, EH, FI, vſque ad O, P, Q. </s>
            <s xml:id="echoid-s4070" xml:space="preserve">Du-
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            ctis quoque rectis ET, FV, ad DO, EP, per-
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            pendicularibus, iungantur rectæ EO, FP.
              <lb/>
            </s>
            <s xml:id="echoid-s4071" xml:space="preserve">Et quoniam arcus DE, EF, æquales ſunt,
              <lb/>
              <note position="right" xlink:label="note-121-02" xlink:href="note-121-02a" xml:space="preserve">27. tertij.</note>
            erunt anguli quoque DOE, EPF, illis inſi-
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            ſtentes, æquales: </s>
            <s xml:id="echoid-s4072" xml:space="preserve">Sunt autem & </s>
            <s xml:id="echoid-s4073" xml:space="preserve">recti anguli
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            T, V, æquales. </s>
            <s xml:id="echoid-s4074" xml:space="preserve">Igitur cum tres anguli trian-
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            guli EOT, tribus angulis trianguli FPV,
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            ſint æquales; </s>
            <s xml:id="echoid-s4075" xml:space="preserve">quòd tam illi, quàm hiduobus
              <lb/>
              <note position="right" xlink:label="note-121-03" xlink:href="note-121-03a" xml:space="preserve">32. primi.</note>
            rectis ſint æquales; </s>
            <s xml:id="echoid-s4076" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s4077" xml:space="preserve">reliquus angulus
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            TEO, reliquo angulo VFP, æqualis: </s>
            <s xml:id="echoid-s4078" xml:space="preserve">ac
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            propterea æquiangula erũt triãgula EOT,
              <lb/>
              <note position="right" xlink:label="note-121-04" xlink:href="note-121-04a" xml:space="preserve">4. @exti.</note>
            FPV. </s>
            <s xml:id="echoid-s4079" xml:space="preserve">Quare erit vt OE, ad ET, ita PF, ad
              <lb/>
              <note position="right" xlink:label="note-121-05" xlink:href="note-121-05a" xml:space="preserve">15.tertij.</note>
            EV: </s>
            <s xml:id="echoid-s4080" xml:space="preserve">Eſt auté recta OE, maior, quàm recta
              <lb/>
            PF; </s>
            <s xml:id="echoid-s4081" xml:space="preserve">quod illa centro propinquior ſit, quàm
              <lb/>
            hęc. </s>
            <s xml:id="echoid-s4082" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s4083" xml:space="preserve">recta ET, maior eſt, quàm re-
              <lb/>
            cta FV. </s>
            <s xml:id="echoid-s4084" xml:space="preserve">Cum ergo recta ET, æqualis ſit
              <lb/>
              <note position="right" xlink:label="note-121-06" xlink:href="note-121-06a" xml:space="preserve">34. primi.</note>
            ſegmentis GH, KL, ob parallelogramma
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            TH, TL; </s>
            <s xml:id="echoid-s4085" xml:space="preserve">& </s>
            <s xml:id="echoid-s4086" xml:space="preserve">recta FV, ſegmentis HI, LM,
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            ob parallelogramma VI, VM; </s>
            <s xml:id="echoid-s4087" xml:space="preserve">erit quoque ſegmentum GH, maius ſegmen-
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            to HI, & </s>
            <s xml:id="echoid-s4088" xml:space="preserve">ſegmentum KL, ſegmento LM. </s>
            <s xml:id="echoid-s4089" xml:space="preserve">In quadrante ergo circuli ſumptis
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            arcubus æqualibus, &</s>
            <s xml:id="echoid-s4090" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4091" xml:space="preserve">Quod erat den. </s>
            <s xml:id="echoid-s4092" xml:space="preserve">onſtrandum.</s>
            <s xml:id="echoid-s4093" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s4094" xml:space="preserve">BREVIVS. </s>
            <s xml:id="echoid-s4095" xml:space="preserve">Ducatur recta DF, ſecans ſemidiametrum ductam AE, in
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            Z, & </s>
            <s xml:id="echoid-s4096" xml:space="preserve">rectam EH, in a, producaturq́ue recta FV, vſque ad b. </s>
            <s xml:id="echoid-s4097" xml:space="preserve">Quoniam igi-
              <lb/>
            tur arcus DF, ſectus eſt biſariam in E, ſecta quoque erit recta DF, biſariam
              <lb/>
            in Z, ex lemmate in definitionibus poſito, ac proinde Da, maior erit </s>
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