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

Table of contents

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[241.] THEOR. 7. PROPOS. 10.
Page: 187 (175)
[242.] SCHOLIVM.
Page: 189 (177)
[243.] COROLLARIVM.
Page: 189 (177)
[244.] THEOR. 8. PROPOS. II.
Page: 189 (177)
[245.] SCHOLIVM.
Page: 190 (178)
[246.] PROBL. 4. PROP. 12.
Page: 190 (178)
[247.] PROBL. 5. PROP. 13.
Page: 191 (179)
[248.] COROLLARIVM.
Page: 192 (180)
[249.] PROBL. 6. PROPOS. 14.
Page: 192 (180)
[250.] COROLLARIVM.
Page: 192 (180)
[251.] PROBL. 7. PROPOS. 15.
Page: 193 (181)
[252.] COROLLARIVM.
Page: 194 (182)
[253.] PROBL. 8. PROPOS. 16.
Page: 194 (182)
[254.] SCHOLIVM.
Page: 197 (185)
[255.] LINAE TANGENTES, atque Secantes.
Page: 200 (188)
[256.] THEOR. .9. PROPOS. 17.
Page: 201 (189)
[257.] SCHOLIVM.
Page: 202 (190)
[258.] THEOR. 10. PROPOS. 18.
Page: 203 (191)
[259.] SCHOLIVM.
Page: 204 (192)
[260.] THEOR 11. PROPOS. 19.
Page: 205 (193)
[261.] SCHOLIVM.
Page: 205 (193)
[262.] THEOR. 12. PROPOS. 20.
Page: 206 (194)
[263.] SCHOLIVM.
Page: 207 (195)
[264.] THEOR. 13. PROPOS. 21.
Page: 210 (198)
[265.] THEOR. 14. PROPOS. 22.
Page: 211 (199)
[266.] THEOR. 15. PROPOS. 23.
Page: 211 (199)
[267.] COROLLARIVM.
Page: 212 (200)
[268.] THEOR. 16. PROPOS. 24.
Page: 213 (201)
[269.] SEQVVNTVR TABVLAE TANGEN-tium atque ſecantium.
Page: 213 (201)
[270.] Gradus Quadrantis pro tangentibus
Page: 214 (202)
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              <pb o="402" file="414" n="414" rhead=""/>
            larem ex B, in planum circuli
              <emph style="sc">AECf</emph>
            , demiſſam, ita ſinus arcus AG, ad perpendi-
              <lb/>
            cularem ex G, in planum circuli AECF, demiſſam: </s>
            <s xml:id="echoid-s13870" xml:space="preserve">propterea quòd ſinus arcuum
              <lb/>
            AB, AG, ſuntipſæmet perpendiculares ex B, G, in planum circuli AECF, demiſſæ
              <lb/>
            cadentes in rectam AC, communem circulorum ſectionem, vt patet.</s>
            <s xml:id="echoid-s13871" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">38. vndee.</note>
          <p style="it">
            <s xml:id="echoid-s13872" xml:space="preserve">HINC facile demonſtrari poterunt ſequentia theoremata, quorum nonnulla pl@
              <lb/>
            rimum ad ſphæricorum triangulerum calculum conducunt. </s>
            <s xml:id="echoid-s13873" xml:space="preserve">Primum autem ac ſecun-
              <lb/>
            dum ſunt duo Theoremata Ptolemæi Cyclica in primo lib. </s>
            <s xml:id="echoid-s13874" xml:space="preserve">Almageſti, ſed multo bre-
              <lb/>
            uius, ac facilius demonſtrata ex ijs, quæ in hoc ſcholio oſtenſa ſunt. </s>
            <s xml:id="echoid-s13875" xml:space="preserve">Vnde omittend@
              <lb/>
            @on videbantur, licet eorum vſus in hiſce triangulis non appareat.</s>
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        <div xml:id="echoid-div1101" type="section" level="1" n="541">
          <head xml:id="echoid-head576" xml:space="preserve">I.</head>
          <p>
            <s xml:id="echoid-s13877" xml:space="preserve">SI in ſphæræ ſuperficie ab vno puncto duo arcus maximorum
              <lb/>
            circulorum educantur, quorum vterque ſemicitculo ſit minor, & </s>
            <s xml:id="echoid-s13878" xml:space="preserve">ab
              <lb/>
            eorum terminis in ipſos reflectantur alij duo arcus maximorum cir-
              <lb/>
            culorum ſe inter duos illos priores arcus interſecantes: </s>
            <s xml:id="echoid-s13879" xml:space="preserve">proportio,
              <lb/>
            quam ſinus ſegmẽti vnius eductorum arcuum inter terminum eius,
              <lb/>
            & </s>
            <s xml:id="echoid-s13880" xml:space="preserve">arcum reflexum habet ad ſinum alterius ſegmenti eiuſdem arcus
              <lb/>
            educti, componitur ex proportione, quam ſinus ſegmenti arcus re-
              <lb/>
            flexi inter eundem terminum, & </s>
            <s xml:id="echoid-s13881" xml:space="preserve">alterum arcum reflexum habet ad
              <lb/>
            ſinum alterius ſegmenti eiuſdem arcus reflexi, & </s>
            <s xml:id="echoid-s13882" xml:space="preserve">ex proportione,
              <lb/>
            quam ſinus ſegmenti alterius eductorum arcuum inter eius termi-
              <lb/>
            num, & </s>
            <s xml:id="echoid-s13883" xml:space="preserve">arcum reflexum habet ad ſinũ totius eiuſdem arcus educti.</s>
            <s xml:id="echoid-s13884" xml:space="preserve"/>
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          <p style="it">
            <s xml:id="echoid-s13885" xml:space="preserve">E X puncto A, in ſuperficie ſpharæ educantur duo arcus AB, AC, ſemicirculis
              <lb/>
            minores, & </s>
            <s xml:id="echoid-s13886" xml:space="preserve">à terminis B, C, reflectantur adipſos duo arcus BD, CE, ſe interſe-
              <lb/>
            cantes in F.</s>
            <s xml:id="echoid-s13887" xml:space="preserve"> Dico proportionem ſinus arcus BE, ad ſinum arcus EA, componi ex pro-
              <lb/>
            portione ſinus arcus
              <emph style="sc">Bf</emph>
            , ad ſinum arcus
              <emph style="sc">F</emph>
            D, & </s>
            <s xml:id="echoid-s13888" xml:space="preserve">ex proportione ſinus arcus CD, ad
              <lb/>
            ſinum arcus CA. </s>
            <s xml:id="echoid-s13889" xml:space="preserve">Ductis enim ex punctis B, A, D, ad planum
              <lb/>
              <figure xlink:label="fig-414-01" xlink:href="fig-414-01a" number="262">
                <image file="414-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/414-01"/>
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            circuli CE, tribus perpendicularibus BG, AH, DI; </s>
            <s xml:id="echoid-s13890" xml:space="preserve">quoniam
              <lb/>
            duo circuli AB, CE, ſe mutuo ſecant in E, & </s>
            <s xml:id="echoid-s13891" xml:space="preserve">ex punctis B, A,
              <lb/>
            in planum circuli
              <emph style="sc">Ce</emph>
            , demiſſæ ſunt perpendiculares
              <emph style="sc">B</emph>
            G, AH;
              <lb/>
            </s>
            <s xml:id="echoid-s13892" xml:space="preserve">erit vt ſinus arcus EB, ad ſinum arcus EA, ita recta BG, ad
              <lb/>
              <note position="left" xlink:label="note-414-02" xlink:href="note-414-02a" xml:space="preserve">Schol. 40.
                <lb/>
              huius. &
                <lb/>
              permutan.
                <lb/>
              do.</note>
            rectam AH: </s>
            <s xml:id="echoid-s13893" xml:space="preserve">Item quoniam duo circuli BD,
              <emph style="sc">C</emph>
            E, ſe mutuo
              <lb/>
            ſecant in
              <emph style="sc">F</emph>
            , & </s>
            <s xml:id="echoid-s13894" xml:space="preserve">ex punctis B,D, in planum circuli CE, deductæ
              <lb/>
            ſunt perpendiculares BG, DI; </s>
            <s xml:id="echoid-s13895" xml:space="preserve">erit eadem ratione, vt ſinus ar-
              <lb/>
            cus
              <emph style="sc">FB</emph>
            , ad ſinum arcus FD, ita recta BG, ad rectam DI: </s>
            <s xml:id="echoid-s13896" xml:space="preserve">De-
              <lb/>
            nique quia duo circuli AC,
              <emph style="sc">CE</emph>
            , ſe interſecant in C, & </s>
            <s xml:id="echoid-s13897" xml:space="preserve">ex
              <lb/>
            punctis D,
              <emph style="sc">A,</emph>
            in planum circuli
              <emph style="sc">CE</emph>
            , demiſſæ ſunt perpendi-
              <lb/>
            culares rectæ lineæ DI, AH; </s>
            <s xml:id="echoid-s13898" xml:space="preserve">erit ſimiliter, vt ſinus arcus
              <lb/>
            CD, ad ſinum arcus CA, ita recta DI, ad rectam AH. </s>
            <s xml:id="echoid-s13899" xml:space="preserve">Pro-
              <lb/>
            p@rtio autem recta BG, ad rectam AH, (poſita media linea DI.) </s>
            <s xml:id="echoid-s13900" xml:space="preserve">componitur ex pro-
              <lb/>
            portione rectæ BG, ad rectam DI, & </s>
            <s xml:id="echoid-s13901" xml:space="preserve">ex proportione rectæ DI, ad rectam AH. </s>
            <s xml:id="echoid-s13902" xml:space="preserve">Igi-
              <lb/>
            tur & </s>
            <s xml:id="echoid-s13903" xml:space="preserve">proportio ſinus arcus
              <emph style="sc">BE</emph>
            , ad ſinum arcus EA, (quæ eadem eſt, quæ propor-
              <lb/>
            tio BG, ad AH.) </s>
            <s xml:id="echoid-s13904" xml:space="preserve">componetur ex proportione ſinus arcus BF, ad ſinum arcus FD,
              <lb/>
            (quæ eadem eſt, quæ proportio BG, ad DI.) </s>
            <s xml:id="echoid-s13905" xml:space="preserve">& </s>
            <s xml:id="echoid-s13906" xml:space="preserve">ex proportione ſinus arcus CD, ad
              <lb/>
            @inum arcus AC, (quæ eadem eſt, quæ DI, ad AH.) </s>
            <s xml:id="echoid-s13907" xml:space="preserve">quod eſt propoſitum.</s>
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