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

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          <p>
            <s xml:id="echoid-s8504" xml:space="preserve">
              <pb o="200" file="212" n="212" rhead=""/>
            AG, AH, AI.</s>
            <s xml:id="echoid-s8505" xml:space="preserve">Et quia in triangulo ACH, angulus C, rectus eſt; </s>
            <s xml:id="echoid-s8506" xml:space="preserve">erit AHC,
              <lb/>
            recto minor, cum ambo ſint duobus rectis minores. </s>
            <s xml:id="echoid-s8507" xml:space="preserve">Cum ergo duo anguli
              <lb/>
              <note position="left" xlink:label="note-212-01" xlink:href="note-212-01a" xml:space="preserve">17. primi.</note>
            ad H, ſint duobus rectis æquales, erit AHI, maior recto, ac proinde angu-
              <lb/>
              <note position="left" xlink:label="note-212-02" xlink:href="note-212-02a" xml:space="preserve">13. primi.</note>
            lus I, in triangulo AHI, recto minor. </s>
            <s xml:id="echoid-s8508" xml:space="preserve">Quare maior erit ſecans AI, ſecante
              <lb/>
              <note position="left" xlink:label="note-212-03" xlink:href="note-212-03a" xml:space="preserve">17. primi.</note>
              <figure xlink:label="fig-212-01" xlink:href="fig-212-01a" number="156">
                <image file="212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/212-01"/>
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            AH. </s>
            <s xml:id="echoid-s8509" xml:space="preserve">Eadẽ ratione maior erit quam AG: </s>
            <s xml:id="echoid-s8510" xml:space="preserve">Item
              <lb/>
              <note position="left" xlink:label="note-212-04" xlink:href="note-212-04a" xml:space="preserve">19. primi.</note>
            AH, maior, quã AG. </s>
            <s xml:id="echoid-s8511" xml:space="preserve">Abſcindatur ergo. </s>
            <s xml:id="echoid-s8512" xml:space="preserve">AK, ipſi
              <lb/>
            AH, & </s>
            <s xml:id="echoid-s8513" xml:space="preserve">AL, ipſi AG, æqualis. </s>
            <s xml:id="echoid-s8514" xml:space="preserve">Dico IH, differẽ
              <lb/>
            tiã tangentiũ CI, CH, arcuũ maiorũ CF, CE,
              <lb/>
            maiorem eſſe differentia HG, tãgentium CH,
              <lb/>
            CG, minorum arcuum CE, CD: </s>
            <s xml:id="echoid-s8515" xml:space="preserve">Item KI,
              <lb/>
            differentiam ſecantium AI, AH, arcuum ma-
              <lb/>
            iorum CF, CE, maiorem eſſe differentia LH,
              <lb/>
            ſecantium AH, AG, minorum arcuum CE,
              <lb/>
            CD. </s>
            <s xml:id="echoid-s8516" xml:space="preserve">Cum enim arcus DE, EF, æquales ſint,
              <lb/>
              <note position="left" xlink:label="note-212-05" xlink:href="note-212-05a" xml:space="preserve">27. tertij.</note>
            erunt & </s>
            <s xml:id="echoid-s8517" xml:space="preserve">anguli DAE, EAF, æquales: </s>
            <s xml:id="echoid-s8518" xml:space="preserve">ac pro-
              <lb/>
            inde angulus IAG, ſectus erit bifariam per re-
              <lb/>
              <note position="left" xlink:label="note-212-06" xlink:href="note-212-06a" xml:space="preserve">3. fexti.</note>
            ctam AH. </s>
            <s xml:id="echoid-s8519" xml:space="preserve">Igitur erit, vt IA, ad AG, ita IH,
              <lb/>
            ad HG: </s>
            <s xml:id="echoid-s8520" xml:space="preserve">Eſt autem AI, maior, quàm AG, vt oſtenſum eſt. </s>
            <s xml:id="echoid-s8521" xml:space="preserve">Recta ergo IH,
              <lb/>
            maior quoque erit, quàm HG. </s>
            <s xml:id="echoid-s8522" xml:space="preserve">quod eſt primum.</s>
            <s xml:id="echoid-s8523" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8524" xml:space="preserve">DVCTIS iam FM, EN, DO, ad AC, perpendicularibus, nempe fi-
              <lb/>
            nubus rectis arcuum CF, CE, CD; </s>
            <s xml:id="echoid-s8525" xml:space="preserve">erit AM, ſinus complementi arcus CF;
              <lb/>
            </s>
            <s xml:id="echoid-s8526" xml:space="preserve">& </s>
            <s xml:id="echoid-s8527" xml:space="preserve">AN, ſinus complementi arcus CE; </s>
            <s xml:id="echoid-s8528" xml:space="preserve">& </s>
            <s xml:id="echoid-s8529" xml:space="preserve">AO, ſinus complementi arcus CD,
              <lb/>
            vt in expoſitione definitionum dictum eſt. </s>
            <s xml:id="echoid-s8530" xml:space="preserve">Quoniam vero recta MN, maior
              <lb/>
              <note position="left" xlink:label="note-212-07" xlink:href="note-212-07a" xml:space="preserve">1. huius.</note>
            eſt, quam NO; </s>
            <s xml:id="echoid-s8531" xml:space="preserve">maior erit proportio AN, ad NO, quàm ad MN: </s>
            <s xml:id="echoid-s8532" xml:space="preserve">Eſt autem
              <lb/>
              <note position="left" xlink:label="note-212-08" xlink:href="note-212-08a" xml:space="preserve">8. quinti.</note>
            adhuc maior proportio AO, ad NO, quàm AN, ad eandem NO. </s>
            <s xml:id="echoid-s8533" xml:space="preserve">Igitur
              <lb/>
            multo maior erit proportio AO, ad NO, quàm AN, ad MN. </s>
            <s xml:id="echoid-s8534" xml:space="preserve">Et per con-
              <lb/>
            uerſionem rationis, minor proportio AO, ad AN, quàm AN, ad AM: </s>
            <s xml:id="echoid-s8535" xml:space="preserve">hoc
              <lb/>
              <note position="left" xlink:label="note-212-09" xlink:href="note-212-09a" xml:space="preserve">30. quinti.</note>
            eſt, maior proportio AN, ad AM, quàm AO, ad AN. </s>
            <s xml:id="echoid-s8536" xml:space="preserve">Cum ergo ſit, vt
              <lb/>
            AN, ad AM, ita AI, ad AH; </s>
            <s xml:id="echoid-s8537" xml:space="preserve">Et vt AO, ad AN, ita AH, ad AG: </s>
            <s xml:id="echoid-s8538" xml:space="preserve">maior
              <lb/>
              <note position="left" xlink:label="note-212-10" xlink:href="note-212-10a" xml:space="preserve">22. huius.</note>
            quoque erit proportio AI, ad AH, hoc eſt, ad AK, quàm AH, ad AG,
              <lb/>
            hoc eſt, ad AL. </s>
            <s xml:id="echoid-s8539" xml:space="preserve">Diuidendo ergo maior etiam proportio erit IK, ad AK, hoc
              <lb/>
              <note position="left" xlink:label="note-212-11" xlink:href="note-212-11a" xml:space="preserve">29. quinti.</note>
            eſt, ad AH, quàm HL, ad AL: </s>
            <s xml:id="echoid-s8540" xml:space="preserve">Et conuertendo minor erit proportio AH,
              <lb/>
              <note position="left" xlink:label="note-212-12" xlink:href="note-212-12a" xml:space="preserve">26. quinti.</note>
            ad KI, quàm AL, ad LH: </s>
            <s xml:id="echoid-s8541" xml:space="preserve">hoc eſt, maior proportio erit AL, ad LH, quàm
              <lb/>
            AH, ad KI. </s>
            <s xml:id="echoid-s8542" xml:space="preserve">Quare cum maior adhuc ſit proportio AH, ad LH, quàm AL,
              <lb/>
              <note position="left" xlink:label="note-212-13" xlink:href="note-212-13a" xml:space="preserve">8. quinti.</note>
            ad eandem LH: </s>
            <s xml:id="echoid-s8543" xml:space="preserve">multo maior proportio erit AH, ad LH, quàm eiuſdem
              <lb/>
            AH, ad KI; </s>
            <s xml:id="echoid-s8544" xml:space="preserve">ac propterea recta LH, minor erit, quàm kI. </s>
            <s xml:id="echoid-s8545" xml:space="preserve">quod eſt ſecun-
              <lb/>
              <note position="left" xlink:label="note-212-14" xlink:href="note-212-14a" xml:space="preserve">10. quinti.</note>
            dum. </s>
            <s xml:id="echoid-s8546" xml:space="preserve">Ex quo fit, differentias tam tangentium, quàm ſecantium in tabula ſem
              <lb/>
            per augeri ad finem vſq; </s>
            <s xml:id="echoid-s8547" xml:space="preserve">quadrantis: </s>
            <s xml:id="echoid-s8548" xml:space="preserve">cuius quidem contrarium in ſinubus ac-
              <lb/>
            cidit, vt ſupra demonſtratum eſt. </s>
            <s xml:id="echoid-s8549" xml:space="preserve">Quamobrem ſi plures ſint arcus æquali
              <lb/>
            exceſſu porgredientes, &</s>
            <s xml:id="echoid-s8550" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8551" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s8552" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div581" type="section" level="1" n="267">
          <head xml:id="echoid-head294" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s8553" xml:space="preserve">SEQVITVR hinc, ſi quotlibet arcuum tangentes æqualiter ſeſe excedant, arcus
              <lb/>
              <note position="left" xlink:label="note-212-15" xlink:href="note-212-15a" xml:space="preserve">Arcus tan-
                <lb/>
              gentium æ
                <lb/>
              quales ex-
                <lb/>
              ceſſus habẽ
                <lb/>
              tium inæ-
                <lb/>
              quales ha</note>
            earum inæqualiter ſeſe excedere, exceſſusq; </s>
            <s xml:id="echoid-s8554" xml:space="preserve">ma orum arcuum eſſe minores: </s>
            <s xml:id="echoid-s8555" xml:space="preserve">quàm ma-
              <lb/>
            iorum Omnium item ſecantium ſegmenta extra quadrantem eſſe inæqualia, minoraq; </s>
            <s xml:id="echoid-s8556" xml:space="preserve">eſſe
              <lb/>
            illa, quæ principio quadrantis ſunt propinquiora. </s>
            <s xml:id="echoid-s8557" xml:space="preserve">Quoniam enim poſi is arcubus DE,
              <lb/>
            EF, æqualibus, oſtenſum fuit, rectam IH, maiorem eſſe quam HG; </s>
            <s xml:id="echoid-s8558" xml:space="preserve">liquido conſtat, ſi ex
              <lb/>
            HI, auferatur recta ipſi HG, æqualis. </s>
            <s xml:id="echoid-s8559" xml:space="preserve">ſecantem inter duas AI, AH, ductam diuidere </s>
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