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

Table of contents

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[131.] LEMMA.
Page: 102 (90)
[132.] SCHOLIVM.
Page: 103 (91)
[133.] COROLLARIVM.
Page: 104 (92)
[134.] THEOREMA 12. PROPOS 12.
Page: 105 (93)
[135.] SCHOLIVM.
Page: 106 (94)
[136.] THEOR. 13. PROPOS. 13.
Page: 106 (94)
[137.] SCHOLIVM.
Page: 107 (95)
[138.] THEOREMA 14. PROPOS. 14.
Page: 107 (95)
[139.] FINIS LIBRI III. THEODOSII. AD LECTOREM.
Page: 108 (96)
[140.] CHRISTOPHORI CLAVII BAMBERGENSIS E SOCIETATE IESV SINVS, VEL SEMISSES RECTARVM IN CIRCVLO SVBTENSARVM: LINEAE TANGENTES: ATQVE SECANTES.
Page: 109
[141.] CHRISTOPHORI CLAV II BAMBERGENSIS E SOCIETATE IESV SINVS, VEL SEMISSES RECTARVM in circulo ſubtenſarum: LINEÆ TANGENTES, ATQVE SECANTES. PRÆFATIO.
Page: 111 (99)
[142.] DEFINITIONES. I.
Page: 112 (100)
[143.] II.
Page: 113 (101)
[144.] III.
Page: 113 (101)
[145.] Vel aliter.
Page: 113 (101)
[146.] IIII.
Page: 113 (101)
[147.] V.
Page: 113 (101)
[148.] VI.
Page: 113 (101)
[149.] VII.
Page: 113 (101)
[150.] LEMMA.
Page: 118 (106)
[151.] THEOR. 1. PROPOS. 1.
Page: 121 (109)
[152.] COROLLARIVM.
Page: 122 (110)
[153.] PROBL. 1. PROPOS. 2.
Page: 122 (110)
[154.] PROBL. 2. PROPOS. 3.
Page: 123 (111)
[155.] COROLLARIVM.
Page: 123 (111)
[156.] THEOR. 2. PROPOS. 4.
Page: 124 (112)
[157.] COROLLARIVM.
Page: 124 (112)
[158.] SCHOLIVM.
Page: 125 (113)
[159.] THEOR 3. PROPOS. 5.
Page: 125 (113)
[160.] COROLLARIVM.
Page: 127 (115)
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              <pb o="194" file="206" n="206" rhead=""/>
            autem arcu BD, nempe complemente arcus C D, bifariam in F, ducatur ex centre
              <lb/>
            A, per
              <emph style="sc">F</emph>
            , recta A G, ſecans tangentem C
              <emph style="sc">E</emph>
            , productam in
              <emph style="sc">G</emph>
            . </s>
            <s xml:id="echoid-s7876" xml:space="preserve">Eritq́; </s>
            <s xml:id="echoid-s7877" xml:space="preserve">C
              <emph style="sc">G</emph>
            , tangens
              <lb/>
            arcus
              <emph style="sc">Cf</emph>
            , compoſiti ex dato arcu CD, & </s>
            <s xml:id="echoid-s7878" xml:space="preserve">
              <emph style="sc">Df</emph>
            , ſemiſſe complementi DB. </s>
            <s xml:id="echoid-s7879" xml:space="preserve">Dico ſe-
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            cantem
              <emph style="sc">Ae</emph>
            , & </s>
            <s xml:id="echoid-s7880" xml:space="preserve">tangentẽ
              <emph style="sc">Ce</emph>
            . </s>
            <s xml:id="echoid-s7881" xml:space="preserve">ſimul æquales eſſe tangenti
              <emph style="sc">Cg</emph>
            . </s>
            <s xml:id="echoid-s7882" xml:space="preserve">Quia enim anguli BAF,
              <lb/>
              <note position="left" xlink:label="note-206-01" xlink:href="note-206-01a" xml:space="preserve">27. tertij.</note>
            FAE, æquales ſunt, propter æquales arcus
              <emph style="sc">Bf</emph>
            ,
              <emph style="sc">F</emph>
            D; </s>
            <s xml:id="echoid-s7883" xml:space="preserve">& </s>
            <s xml:id="echoid-s7884" xml:space="preserve">angulo BAF, alternus
              <lb/>
              <note position="left" xlink:label="note-206-02" xlink:href="note-206-02a" xml:space="preserve">29. primi.</note>
            angulus G, æqualis eſt; </s>
            <s xml:id="echoid-s7885" xml:space="preserve">erit quoque angulus idem G, angulo
              <emph style="sc">GAe</emph>
            , æqualis Quare
              <lb/>
              <note position="left" xlink:label="note-206-03" xlink:href="note-206-03a" xml:space="preserve">6. primi.</note>
            aquales ſunt rectæ
              <emph style="sc">E</emph>
            A,
              <emph style="sc">E</emph>
            G: </s>
            <s xml:id="echoid-s7886" xml:space="preserve">at que adeo, addita communi
              <emph style="sc">E</emph>
            C, duæ A E,
              <emph style="sc">E</emph>
            C, ſimul
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            toti CG, æquales erunt.</s>
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          <head xml:id="echoid-head289" xml:space="preserve">THEOR. 12. PROPOS. 20.</head>
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            <s xml:id="echoid-s7888" xml:space="preserve">SECANS cuiuſuis arcus æqualis eſt tangen-
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              <note position="left" xlink:label="note-206-04" xlink:href="note-206-04a" xml:space="preserve">Secans cu-
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              iuſuis arcꝰ
                <lb/>
              quorũ duo
                <lb/>
              rú arcuum
                <lb/>
              tangentibꝰ
                <lb/>
              ſit æqualis.</note>
            ti eiuſdem, vna cum tangente ſemiſſis comple-
              <lb/>
            menti arcus eiuſdem.</s>
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          <p>
            <s xml:id="echoid-s7890" xml:space="preserve">IN quadrante ABC, ſit AD, ſecans, & </s>
            <s xml:id="echoid-s7891" xml:space="preserve">CD, tangens arcus CE, cuius
              <lb/>
            complementi EB, ſemiſsis ſit EF, vel FB, & </s>
            <s xml:id="echoid-s7892" xml:space="preserve">huic ſemiſsi æqualis ſit arcus
              <lb/>
            CG. </s>
            <s xml:id="echoid-s7893" xml:space="preserve">Ducta autem recta AG, & </s>
            <s xml:id="echoid-s7894" xml:space="preserve">producta, donec cum DC, protracta coeat in
              <lb/>
            H, erit CH, tangens arcus CG, qui ſemiſsis eſt complementi arcus CE.
              <lb/>
            </s>
            <s xml:id="echoid-s7895" xml:space="preserve">Dico ſecantem AD, æqualem eſſe tangenti
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                <image file="206-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/206-01"/>
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            CD, & </s>
            <s xml:id="echoid-s7896" xml:space="preserve">tangenti CH, ſimul, hoc eſt, toti li-
              <lb/>
            neæ DH. </s>
            <s xml:id="echoid-s7897" xml:space="preserve">Quoniam enim anguli EAF, CAG,
              <lb/>
              <note position="left" xlink:label="note-206-05" xlink:href="note-206-05a" xml:space="preserve">27. tertij.</note>
            æquales ſunt, ob æquales arcus EF, CG;
              <lb/>
            </s>
            <s xml:id="echoid-s7898" xml:space="preserve">addito communi angulo EAC, erunt toti
              <unsure/>
              <lb/>
            anguli FAC, EAH, æquales. </s>
            <s xml:id="echoid-s7899" xml:space="preserve">Rurſus quia
              <lb/>
            in triangulo rectangulo ACH, duo anguli
              <lb/>
              <note position="left" xlink:label="note-206-06" xlink:href="note-206-06a" xml:space="preserve">32. primi.</note>
            A, H, vni recto, nimirũ angulo BAC, æqua-
              <lb/>
            les ſunt; </s>
            <s xml:id="echoid-s7900" xml:space="preserve">ablatis angulis BAF, CAH, qui
              <lb/>
            propter æquales arcus BF, CG, æquales ſunt,
              <lb/>
              <note position="left" xlink:label="note-206-07" xlink:href="note-206-07a" xml:space="preserve">27. tertij.</note>
            erunt reliqui anguli FAC, & </s>
            <s xml:id="echoid-s7901" xml:space="preserve">H, æquales. </s>
            <s xml:id="echoid-s7902" xml:space="preserve">Eſt
              <lb/>
            autem angulus FAC, oſtenſus æqualis angu
              <lb/>
            lo EAH. </s>
            <s xml:id="echoid-s7903" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s7904" xml:space="preserve">angulus H, eidem angulo EAH, æqualis erit: </s>
            <s xml:id="echoid-s7905" xml:space="preserve">ac propte-
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            rea rectæ AD, DH, æquales erunt, hoc eſt, ſecans AD, tangentibus DC, CH,
              <lb/>
              <note position="left" xlink:label="note-206-08" xlink:href="note-206-08a" xml:space="preserve">6. primi.</note>
            æqualis erit. </s>
            <s xml:id="echoid-s7906" xml:space="preserve">quod eſt propoſitum.</s>
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            <s xml:id="echoid-s7908" xml:space="preserve">ALITER. </s>
            <s xml:id="echoid-s7909" xml:space="preserve">Sit rurſus AD, ſecans, & </s>
            <s xml:id="echoid-s7910" xml:space="preserve">CD, tangens arcus CE. </s>
            <s xml:id="echoid-s7911" xml:space="preserve">Dico ſecan-
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            tem AD, æqualem eſſe tangenti CD, vnà cum tangente ſemiſsis complemen
              <lb/>
            ti arcus EC, ſeu anguli DAC, hoc eſt, vna cum tangente ſemiſsis anguli D,
              <lb/>
            qui complementum eſt anguli DAC, cum ambo in triangulo rectágulo ACD,
              <lb/>
            vni recto ſint æquales. </s>
            <s xml:id="echoid-s7912" xml:space="preserve">Centro namque D, & </s>
            <s xml:id="echoid-s7913" xml:space="preserve">interuallo DA, arcus circuli de-
              <lb/>
              <note position="left" xlink:label="note-206-09" xlink:href="note-206-09a" xml:space="preserve">32. primi.</note>
            ſcribatur AHI, ſecans DC, productam in H, & </s>
            <s xml:id="echoid-s7914" xml:space="preserve">AC, productam in I, ducan-
              <lb/>
            turq́; </s>
            <s xml:id="echoid-s7915" xml:space="preserve">rectæ AH, HI. </s>
            <s xml:id="echoid-s7916" xml:space="preserve">Quia igitur recta DC, ex centro D, circuli AHI, edu-
              <lb/>
            cta ſecans rectam AI, ad angulos rectos, ſecat eam bifariam; </s>
            <s xml:id="echoid-s7917" xml:space="preserve">ſecabit eadem
              <lb/>
              <note position="left" xlink:label="note-206-10" xlink:href="note-206-10a" xml:space="preserve">3. tertij.</note>
            DCH, & </s>
            <s xml:id="echoid-s7918" xml:space="preserve">arcum AHI, bifariam, ex lemmate in definitionibus demonſtrato.
              <lb/>
            </s>
            <s xml:id="echoid-s7919" xml:space="preserve">Quare anguli CAH, & </s>
            <s xml:id="echoid-s7920" xml:space="preserve">I, æquales ſunt. </s>
            <s xml:id="echoid-s7921" xml:space="preserve">Quoniam autem, cum anguli D, & </s>
            <s xml:id="echoid-s7922" xml:space="preserve">I,
              <lb/>
              <note position="left" xlink:label="note-206-11" xlink:href="note-206-11a" xml:space="preserve">27. tertij.</note>
            candem habeant baſim arcum AH, & </s>
            <s xml:id="echoid-s7923" xml:space="preserve">ille ſit ad centrum D, hic vero ad </s>
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