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

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          <p>
            <s xml:id="echoid-s15780" xml:space="preserve">
              <pb o="446" file="458" n="458" rhead=""/>
            uallis autem AC, BC, circuli non maximi delineentur KCN, OCP, qui il-
              <lb/>
            lis maximis paralleli erunt: </s>
            <s xml:id="echoid-s15781" xml:space="preserve">& </s>
            <s xml:id="echoid-s15782" xml:space="preserve">tam hi, quam illi ad circulum ABDGH, re-
              <lb/>
              <note position="left" xlink:label="note-458-01" xlink:href="note-458-01a" xml:space="preserve">2. 2. Theod.</note>
            cti erunt, cum ille per horũ polos trãſiens ad ipſos ſit rectus. </s>
            <s xml:id="echoid-s15783" xml:space="preserve">Poſt hæc, vt con-
              <lb/>
              <note position="left" xlink:label="note-458-02" xlink:href="note-458-02a" xml:space="preserve">15. 1. Theo.</note>
            fuſio vitetur, in circulo ABDGH, ſeorſum deſcripto ſint communes ſectio-
              <lb/>
            nes ipſius, & </s>
            <s xml:id="echoid-s15784" xml:space="preserve">circulorum ex polis A, B, deſcriptorum, nempe DF, GH, com-
              <lb/>
            munes ſectiones ipſius, & </s>
            <s xml:id="echoid-s15785" xml:space="preserve">maximorum circulorum DLEF, GMEH, quæ
              <lb/>
            ipſorum diametri erunt ſeſe in centro ſphæræ X, interſecantes: </s>
            <s xml:id="echoid-s15786" xml:space="preserve">At KN, OP,
              <lb/>
            communes ſectiones eiuſdem, & </s>
            <s xml:id="echoid-s15787" xml:space="preserve">circulorum KCN, OCP, ſe interſecantes
              <lb/>
            in S; </s>
            <s xml:id="echoid-s15788" xml:space="preserve">quæ ipſis DF, GH, parallelæ erunt; </s>
            <s xml:id="echoid-s15789" xml:space="preserve">& </s>
            <s xml:id="echoid-s15790" xml:space="preserve">diametri circulorum KCN,
              <lb/>
              <note position="left" xlink:label="note-458-03" xlink:href="note-458-03a" xml:space="preserve">16. vndec.</note>
            OCP; </s>
            <s xml:id="echoid-s15791" xml:space="preserve">quòd maximus circulus ABDGH, per eorum polos tranſiens eos
              <lb/>
            bifariam ſecet, nimirum per eorum diametros. </s>
            <s xml:id="echoid-s15792" xml:space="preserve">Ducantur quoque ſemidiame-
              <lb/>
              <note position="left" xlink:label="note-458-04" xlink:href="note-458-04a" xml:space="preserve">15.1 Theod.</note>
            tri AX, ſecans KN, in Y; </s>
            <s xml:id="echoid-s15793" xml:space="preserve">& </s>
            <s xml:id="echoid-s15794" xml:space="preserve">BX, ſecans KN, OP, in I, R. </s>
            <s xml:id="echoid-s15795" xml:space="preserve">Eruntque ſemi-
              <lb/>
            diametri AX, BX, perpendiculares ad circulos per DF, KN, GH, OP,
              <lb/>
              <note position="left" xlink:label="note-458-05" xlink:href="note-458-05a" xml:space="preserve">Schol. 10.</note>
            ductos; </s>
            <s xml:id="echoid-s15796" xml:space="preserve">cum ab eorum polis A,
              <lb/>
              <note position="left" xlink:label="note-458-06" xlink:href="note-458-06a" xml:space="preserve">1. Theod.</note>
            B, ducantur per X, ſphæræcen-
              <lb/>
              <figure xlink:label="fig-458-01" xlink:href="fig-458-01a" number="318">
                <image file="458-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/458-01"/>
              </figure>
            trum: </s>
            <s xml:id="echoid-s15797" xml:space="preserve">ac proinde anguli ad Y, & </s>
            <s xml:id="echoid-s15798" xml:space="preserve">
              <lb/>
            R, recti erunt, ex defin. </s>
            <s xml:id="echoid-s15799" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15800" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s15801" xml:space="preserve">11.
              <lb/>
            </s>
            <s xml:id="echoid-s15802" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s15803" xml:space="preserve">Ducantur denique ad BX,
              <lb/>
            OP, perpendiculares AV, KQ,
              <lb/>
            KT. </s>
            <s xml:id="echoid-s15804" xml:space="preserve">Erit igitur, per ea, quæ in
              <lb/>
            tractatione ſinuum ſcripſimus,
              <lb/>
            AV, ſinus rectus arcus AB; </s>
            <s xml:id="echoid-s15805" xml:space="preserve">& </s>
            <s xml:id="echoid-s15806" xml:space="preserve">
              <lb/>
            KY, ſinus rectus arcus AK, hoc
              <lb/>
            eſt, arcus AC, cum arcus AK,
              <lb/>
            AC, ex defin. </s>
            <s xml:id="echoid-s15807" xml:space="preserve">poli, æquales ſint, vt in primo circulo apparet. </s>
            <s xml:id="echoid-s15808" xml:space="preserve">BR, erit ſinus
              <lb/>
            verſus arcus BO, id eſt, arcus BC, cum arcus BO, BC, æquales ſint, ex defin. </s>
            <s xml:id="echoid-s15809" xml:space="preserve">
              <lb/>
            poli. </s>
            <s xml:id="echoid-s15810" xml:space="preserve">BQ, ſinus verſus erit arcus BK, qui differentia eſt arcuum inæqualium
              <lb/>
            AB, AC, propterea quod, ex defin. </s>
            <s xml:id="echoid-s15811" xml:space="preserve">poli, arcus AK, arcum AB, arcu BQ, ſupe-
              <lb/>
            rans, æqualis eſt arcui AC: </s>
            <s xml:id="echoid-s15812" xml:space="preserve">ac proinde QR, vel KT, differentia erit inter BR,
              <lb/>
            ſinum verſum tertij arcus BC, & </s>
            <s xml:id="echoid-s15813" xml:space="preserve">BQ, ſinum verſum differentiæ arcuum inæ-
              <lb/>
            qualium AB, AC, hoc eſt, ſinum verſum arcus BK. </s>
            <s xml:id="echoid-s15814" xml:space="preserve">Poſtremo erit KS, ſinus
              <lb/>
            verſus arcus KC, in circulo non maximo KCN, cum recta ex C, in commu-
              <lb/>
            nes ſectiones circulorum KCN, OCP, cum circulo ABDGH, hoc eſt, in
              <lb/>
            punctum S, cadens, (quæ quidem ad circulũ ABDGH, recta eſt, vtpote com-
              <lb/>
            munis ſectio circulorum KCN, OCP, ad eundem circulum ABDGH, re-
              <lb/>
              <note position="left" xlink:label="note-458-07" xlink:href="note-458-07a" xml:space="preserve">19. vndec.</note>
            ctorum) ſinus rectus ſit eiuſdem arcus KC. </s>
            <s xml:id="echoid-s15815" xml:space="preserve">Sumatur quoque DZ, ſinus ver-
              <lb/>
            ſus arcus DL, hoc eſt, anguli A, qui quidem arcus arcui KC, ſimilis eſt. </s>
            <s xml:id="echoid-s15816" xml:space="preserve">De-
              <lb/>
              <note position="left" xlink:label="note-458-08" xlink:href="note-458-08a" xml:space="preserve">10.2. Theo.</note>
            monſtrandum igitur eſt, ita eſſe quadratum ſinus totius, hoc eſt, rectangulum
              <lb/>
            ſub DX, XA, contentum, ad rectangulum ſub ſinubus rectis AV, KY, ar-
              <lb/>
            cuum AB, AC, contentum, vt eſt ſinus verſus DZ, anguli A, ad KT, diſ-
              <lb/>
            fer entiam inter BR, ſinum verſum arcus BC, & </s>
            <s xml:id="echoid-s15817" xml:space="preserve">BQ, ſinum verſum arcus BK,
              <lb/>
            differentiæ arcuum inæqualium AB, AC. </s>
            <s xml:id="echoid-s15818" xml:space="preserve">quod ita fiet.</s>
            <s xml:id="echoid-s15819" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15820" xml:space="preserve">QVONIAM angulus XIY, angulo RIS, æqualis eſt, & </s>
            <s xml:id="echoid-s15821" xml:space="preserve">angulus re-
              <lb/>
              <note position="left" xlink:label="note-458-09" xlink:href="note-458-09a" xml:space="preserve">15. primi.</note>
            ctus Y, angulo recto R; </s>
            <s xml:id="echoid-s15822" xml:space="preserve">erit reliquus angulus IXY, trianguli IXY, reliquo
              <lb/>
              <note position="left" xlink:label="note-458-10" xlink:href="note-458-10a" xml:space="preserve">32. primi.</note>
            angulo ISR, trianguli ISR, æqualis, hoc eſt, angulo ad verticem KST.
              <lb/>
            </s>
            <s xml:id="echoid-s15823" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s15824" xml:space="preserve">angulus rectus V, recto angulo T, æqualis ſit, erit & </s>
            <s xml:id="echoid-s15825" xml:space="preserve">reliquus
              <lb/>
            angulus XAV, trianguli XAV, reliquo angulo SKT, trianguli SKT,
              <lb/>
            æqualis Quam ob rem erit, vt XA, ad AV, ita SK, ad KT. </s>
            <s xml:id="echoid-s15826" xml:space="preserve">Rurſus quia DZ,
              <lb/>
              <note position="left" xlink:label="note-458-11" xlink:href="note-458-11a" xml:space="preserve">4. ſexti.</note>
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