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

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        <div xml:id="echoid-div258" type="section" level="1" n="121">
          <p style="it">
            <s xml:id="echoid-s3103" xml:space="preserve">
              <pb o="81" file="093" n="93" rhead=""/>
            & </s>
            <s xml:id="echoid-s3104" xml:space="preserve">arcum N B, vltra arcum D B, ideoque & </s>
            <s xml:id="echoid-s3105" xml:space="preserve">arcum X V, vltra cundem
              <lb/>
            arcum D B: </s>
            <s xml:id="echoid-s3106" xml:space="preserve">propterea quòd maximi circuli Z Y D, M Y, ſe mutuo ſe-
              <lb/>
            cant in Y, polo, & </s>
            <s xml:id="echoid-s3107" xml:space="preserve">punctum M, e§t inter D, & </s>
            <s xml:id="echoid-s3108" xml:space="preserve">Z. </s>
            <s xml:id="echoid-s3109" xml:space="preserve">Quoniam verò circu-
              <lb/>
            lus maximus M Y, ductus per Y, polum paralleli A C, & </s>
            <s xml:id="echoid-s3110" xml:space="preserve">per cont actum
              <lb/>
            M, tranſit etiam per polum circulitangentis N M; </s>
            <s xml:id="echoid-s3111" xml:space="preserve">tranſibit per polos
              <lb/>
              <note position="right" xlink:label="note-093-01" xlink:href="note-093-01a" xml:space="preserve">5. 2. huius</note>
            circulorum X V, & </s>
            <s xml:id="echoid-s3112" xml:space="preserve">N M, ſe mutuo ſecantium in X. </s>
            <s xml:id="echoid-s3113" xml:space="preserve">Quare bifariam ſe-
              <lb/>
            cabit ipſorum ſegmenta. </s>
            <s xml:id="echoid-s3114" xml:space="preserve">Cum ergo vltra punctum V, ſecet ſegmentum ab
              <lb/>
              <note position="right" xlink:label="note-093-02" xlink:href="note-093-02a" xml:space="preserve">9. 2. huius.</note>
            X, per V, vſque ad aliud punctum, vbi ſe mutuo ſecãt circuli X V, N M,
              <lb/>
            vt proxime eſt ostenſum; </s>
            <s xml:id="echoid-s3115" xml:space="preserve">erit X V, arcus minor ſemiſſe ſegmentiab X, per
              <lb/>
            V, vſque ad alteram ſectionem; </s>
            <s xml:id="echoid-s3116" xml:space="preserve">ac proinde multo minor ſemiſſe eiuſdem
              <lb/>
            ſegmenti erit T X. </s>
            <s xml:id="echoid-s3117" xml:space="preserve">quod eſt propoſitum.</s>
            <s xml:id="echoid-s3118" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div260" type="section" level="1" n="122">
          <head xml:id="echoid-head136" xml:space="preserve">LEMMA. I I.</head>
          <p>
            <s xml:id="echoid-s3119" xml:space="preserve">PROPOSITIS duabus magnitudinibus inæqualibus, repe-
              <lb/>
            rire aliam mediam, quæ datæ cuicunque magnitudini commen-
              <lb/>
            ſurabilis ſit.</s>
            <s xml:id="echoid-s3120" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3121" xml:space="preserve">SINT propoſitæ duæ magnitudines inæquales A B, A C, & </s>
            <s xml:id="echoid-s3122" xml:space="preserve">data
              <lb/>
            alia quæcunque D G: </s>
            <s xml:id="echoid-s3123" xml:space="preserve">oporteat{q́ue} inuenire aliam mediam, boc eſt, quæ
              <lb/>
            maior quidem ſit, quàm A C, minor vero, quàm A B, & </s>
            <s xml:id="echoid-s3124" xml:space="preserve">ipſi D G, com-
              <lb/>
            menſurabilis. </s>
            <s xml:id="echoid-s3125" xml:space="preserve">Sit primum D G, minor, quàm B C, exceſſus inter magnitu-
              <lb/>
              <figure xlink:label="fig-093-01" xlink:href="fig-093-01a" number="99">
                <image file="093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/093-01"/>
              </figure>
            dines A B, A C; </s>
            <s xml:id="echoid-s3126" xml:space="preserve">& </s>
            <s xml:id="echoid-s3127" xml:space="preserve">E, multiplex ipſius
              <lb/>
            D G, proxime maior quàm A C. </s>
            <s xml:id="echoid-s3128" xml:space="preserve">Quo po-
              <lb/>
            ſito, erit E, minor, quàm A B. </s>
            <s xml:id="echoid-s3129" xml:space="preserve">Si enim
              <lb/>
            æqualis eſſet, ſi detraheretur ex E, vna
              <lb/>
            magnitudo ipſi D G, æqualis (quę quidem
              <lb/>
            minor ponitur, quàm B C,) maneret adhuc
              <lb/>
            reliqua multiplex ipſius D G, maior quàm
              <lb/>
            A C. </s>
            <s xml:id="echoid-s3130" xml:space="preserve">Non ergo E, eſſet multiplex ipſius D G, proxime maior, quàm A C.
              <lb/>
            </s>
            <s xml:id="echoid-s3131" xml:space="preserve">Quod eſt abſurdum. </s>
            <s xml:id="echoid-s3132" xml:space="preserve">Non ergo æqualis eſt E, ipſi A B; </s>
            <s xml:id="echoid-s3133" xml:space="preserve">atque adeo multo
              <lb/>
            magis neque maior erit. </s>
            <s xml:id="echoid-s3134" xml:space="preserve">Minor igitur eſt, quàm A B; </s>
            <s xml:id="echoid-s3135" xml:space="preserve">atque adeo cum ma-
              <lb/>
            ior quoque ſit quà A C, & </s>
            <s xml:id="echoid-s3136" xml:space="preserve">ipſi D G, commenſurabilis, quòd eius mul-
              <lb/>
            tiplex ſit, conſtat propoſitum.</s>
            <s xml:id="echoid-s3137" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3138" xml:space="preserve">SED iam data magnitudo D G, non minor ſit, quàm B C. </s>
            <s xml:id="echoid-s3139" xml:space="preserve">Diuiſa igi-
              <lb/>
            tur D G, bifariam, & </s>
            <s xml:id="echoid-s3140" xml:space="preserve">dimidia parte rurſus bifariam, & </s>
            <s xml:id="echoid-s3141" xml:space="preserve">ſic deinceps, do-
              <lb/>
            nec relinquatur pars D F, minor quàm B C; </s>
            <s xml:id="echoid-s3142" xml:space="preserve">ſit E, ipſius D F, multiplex
              <lb/>
              <note position="right" xlink:label="note-093-03" xlink:href="note-093-03a" xml:space="preserve">1. decimi.</note>
            proximè maior, quàm A C; </s>
            <s xml:id="echoid-s3143" xml:space="preserve">erit{q́ue} E, ipſi D F, commenſurabilis; </s>
            <s xml:id="echoid-s3144" xml:space="preserve">atque
              <lb/>
            adeo & </s>
            <s xml:id="echoid-s3145" xml:space="preserve">ipſi D G: </s>
            <s xml:id="echoid-s3146" xml:space="preserve">propterea quod vtraque E, & </s>
            <s xml:id="echoid-s3147" xml:space="preserve">D G, ipſi D F, commen
              <lb/>
              <note position="right" xlink:label="note-093-04" xlink:href="note-093-04a" xml:space="preserve">12. decimi.</note>
            ſurabilis eſt. </s>
            <s xml:id="echoid-s3148" xml:space="preserve">Rurſus eodem pacto, vt paulo ante demonſtrauimus, erit E,
              <lb/>
            minor, quam A B. </s>
            <s xml:id="echoid-s3149" xml:space="preserve">Cum ergo maior quoque ſit, quàm A C, & </s>
            <s xml:id="echoid-s3150" xml:space="preserve">ipſi D G,
              <lb/>
            commenſurabilis; </s>
            <s xml:id="echoid-s3151" xml:space="preserve">constat propoſitum.</s>
            <s xml:id="echoid-s3152" xml:space="preserve"/>
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