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

Table of contents

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[111.] THEOREMA 1. PROPOS. 1.
Page: 77 (65)
[112.] THEOREMA 2. PROPOS. 2.
Page: 79 (67)
[113.] THEOREMA 3. PROPOS. 3.
Page: 80 (68)
[114.] THEOREMA 4. PROPOS. 4.
Page: 81 (69)
[115.] LEMMA.
Page: 83 (71)
[116.] THEOR. 5. PROPOS. 5.
Page: 84 (72)
[117.] THEOREMA 6. PROPOS. 6.
Page: 85 (73)
[118.] LEMMA.
Page: 86 (74)
[119.] THEOR. 7. PROPOS. 7.
Page: 88 (76)
[120.] THEOREMA 8. PROPOS. 8.
Page: 90 (78)
[121.] LEMMA. I.
Page: 92 (80)
[122.] LEMMA. I I.
Page: 93 (81)
[123.] THEOREMA 9. PROPOS. 9.
Page: 94 (82)
[124.] SCHOLIVM.
Page: 95 (83)
[125.] I.
Page: 95 (83)
[126.] II.
Page: 97 (85)
[127.] III.
Page: 97 (85)
[128.] THEOREMA 10. PROPOS. 10.
Page: 98 (86)
[129.] COROLLARIVM.
Page: 100 (88)
[130.] THEOR. 11. PROPOS. 11.
Page: 100 (88)
[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
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          <p style="it">
            <s xml:id="echoid-s3829" xml:space="preserve">
              <pb o="104" file="116" n="116" rhead=""/>
            verſi alterius arcus. </s>
            <s xml:id="echoid-s3830" xml:space="preserve">Vocamus autẽ cõplementum ſinus verſi ſe gmentum diametri, quo
              <lb/>
              <note position="left" xlink:label="note-116-01" xlink:href="note-116-01a" xml:space="preserve">Duotumat
                <lb/>
              euum, quo
                <lb/>
              rum vnus
                <lb/>
              alterius eſt
                <lb/>
              complemẽ
                <lb/>
              tum, ſinus
                <lb/>
              rectꝰ vtriuf
                <lb/>
              libet æqua
                <lb/>
              lis eſt com-
                <lb/>
              plemento
                <lb/>
              ſinus verſi
                <lb/>
              alterius at
                <lb/>
              eus.</note>
            ipſe ſinus verſus à ſemidiametro ſuperatur, ſi eius arcus quadrante minor eſt, vel ſe-
              <lb/>
            midiametrum ſuperat, ſi eius arcus maior eſt quadrante. </s>
            <s xml:id="echoid-s3831" xml:space="preserve">Vt _HE,_ dicimus comple-
              <lb/>
            mentum tam ſinus verſi _AH,_ arcui _FA,_ reſpondentis, quàm ſinus verſi _CH,_ arcui
              <lb/>
            _FC,_ reſpondentis. </s>
            <s xml:id="echoid-s3832" xml:space="preserve">Vides igitur, duorum arcuum _FA, FB,_ quorum vnus alterius
              <lb/>
            eſt complementum, ſinum rectum _FK,_ arcus _FB,_ æqualem eſſe ipſi _HE,_ complemente
              <lb/>
            ſinus verſi _AH,_ alterius arcus _FA:_ </s>
            <s xml:id="echoid-s3833" xml:space="preserve">Et ſinum rectum _FH,_ arcus _FA,_ æqualem eſſe
              <lb/>
            ipſi _EK,_ complemento ſinus verſi _BK,_ alterius arcus _FB._ </s>
            <s xml:id="echoid-s3834" xml:space="preserve">Eadem ratione, quoniam
              <lb/>
            arcus _FB,_ complementum eſt arcus _FC,_ vides ſinum rectum _FH,_ arcus _FC,_ æqualem
              <lb/>
            eſſe ipſi _EK,_ complemente ſinus verſi _BK,_ alterius arcus _FB:_ </s>
            <s xml:id="echoid-s3835" xml:space="preserve">Et ſinum rectum _FK,_
              <lb/>
              <figure xlink:label="fig-116-01" xlink:href="fig-116-01a" number="115">
                <image file="116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/116-01"/>
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            arcus _FB,_ æqualem eſſe ipſi
              <lb/>
              <note position="left" xlink:label="note-116-02" xlink:href="note-116-02a" xml:space="preserve">Comple-
                <lb/>
              mentum ſi
                <lb/>
              nus verſi
                <lb/>
              quid.</note>
            _EH,_ complemento ſinus verſi
              <lb/>
            _CH,_ alterius arcus _FC._ </s>
            <s xml:id="echoid-s3836" xml:space="preserve">Sic
              <lb/>
            etiam ſinus complemẽtiarcus
              <lb/>
            cuiuſuis æqualis eſt complemẽ
              <lb/>
            to ſinus verſi eiuſdem arcus.
              <lb/>
            </s>
            <s xml:id="echoid-s3837" xml:space="preserve">Vt _FK,_ ſinus complementi are
              <lb/>
            cus _AF,_ vel _FC,_ æqualis eſt
              <lb/>
              <note position="left" xlink:label="note-116-03" xlink:href="note-116-03a" xml:space="preserve">14. primi.</note>
            ipſi _HE,_ complemento ſinus
              <lb/>
            verſi _AH,_ vel _CH,_ arcus
              <lb/>
            _AF,_ vel _FC._</s>
            <s xml:id="echoid-s3838" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s3839" xml:space="preserve">_PARI_ ratione in eodem
              <lb/>
            circulo, vel in circulis æqua-
              <lb/>
              <note position="left" xlink:label="note-116-04" xlink:href="note-116-04a" xml:space="preserve">In eodem
                <lb/>
              circulo, aut
                <lb/>
              æqualibus,
                <lb/>
              arcuum æ-
                <lb/>
              qualium
                <lb/>
              finus æqua
                <lb/>
              les ſunt; &
                <lb/>
              contra. At
                <lb/>
              arcuũ in æ-
                <lb/>
              qualium ſi
                <lb/>
              nus inęqua
                <lb/>
              les ſunt; &
                <lb/>
              contra.</note>
            libus, ſinus tam recti, quàm
              <lb/>
            verſi, aut ſinus complemento-
              <lb/>
            rum arcuum æqualium, & </s>
            <s xml:id="echoid-s3840" xml:space="preserve">
              <lb/>
            quadrante minorum, æquales
              <lb/>
            ſunt: </s>
            <s xml:id="echoid-s3841" xml:space="preserve">Et contra, æqualium ſi-
              <lb/>
            nuũ tam rectorum, quàm ver
              <lb/>
            ſorum, aut ſinuum complemen
              <lb/>
            torum, arcus quadrãte mino-
              <lb/>
            res æquales ſunt. </s>
            <s xml:id="echoid-s3842" xml:space="preserve">Arcuũ verò
              <lb/>
            inæqualium, & </s>
            <s xml:id="echoid-s3843" xml:space="preserve">quadrante minorum, ſinus inæquales ſunt, ſinus quidem tam rectus
              <lb/>
            quàm verſus maioris maior, minoris verò minor; </s>
            <s xml:id="echoid-s3844" xml:space="preserve">ſinus autem compleẽti maioris
              <lb/>
            arcus minor, & </s>
            <s xml:id="echoid-s3845" xml:space="preserve">minoris maior; </s>
            <s xml:id="echoid-s3846" xml:space="preserve">Et contra, inæqualium ſinuum tam rectorum,
              <lb/>
            quam verſorum, aut ſinuum complementorum, inæquales arcus ſunt, maioris qui-
              <lb/>
            dem ſinus tam recti quàm verſi maior arcus, & </s>
            <s xml:id="echoid-s3847" xml:space="preserve">minoris minor; </s>
            <s xml:id="echoid-s3848" xml:space="preserve">maioris autem
              <lb/>
            ſinus complementi arcus minor, & </s>
            <s xml:id="echoid-s3849" xml:space="preserve">minoris maior. </s>
            <s xml:id="echoid-s3850" xml:space="preserve">Sint enim arcus æquales _BF,_
              <lb/>
            _DG._ </s>
            <s xml:id="echoid-s3851" xml:space="preserve">Dico eorum ſinus rectos _FK, GL,_ æquales eſſe; </s>
            <s xml:id="echoid-s3852" xml:space="preserve">Item ſinus verſos _KB, LD;_
              <lb/>
            </s>
            <s xml:id="echoid-s3853" xml:space="preserve">nec non ſinus complementorũ _EK, EL._ </s>
            <s xml:id="echoid-s3854" xml:space="preserve">Cum enim arcus _BF, DG,_ æquales ſunt,
              <lb/>
            erunt quoque anguli _BEF, DEG,_ æquales. </s>
            <s xml:id="echoid-s3855" xml:space="preserve">Sunt autẽ & </s>
            <s xml:id="echoid-s3856" xml:space="preserve">recti anguli _K, L,_ æquales,
              <lb/>
              <note position="left" xlink:label="note-116-05" xlink:href="note-116-05a" xml:space="preserve">27. tertij.</note>
            nec non & </s>
            <s xml:id="echoid-s3857" xml:space="preserve">latera _EF, EG,_ æqualia, vtpote ſemidiametri. </s>
            <s xml:id="echoid-s3858" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s3859" xml:space="preserve">tam latera _FK,_
              <lb/>
            _GL,_ quàm latera _EK, EL,_ inter ſe æqualia erunt, nempe ſinus recti inter ſe, & </s>
            <s xml:id="echoid-s3860" xml:space="preserve">ſi-
              <lb/>
              <note position="left" xlink:label="note-116-06" xlink:href="note-116-06a" xml:space="preserve">26. primi.</note>
            nus complementorum inter ſe. </s>
            <s xml:id="echoid-s3861" xml:space="preserve">Detractis autem _EK, EL,_ æqualibus ex ſemidiametru
              <lb/>
            _EB, ED,_ reliqui erunt ſinus verſi _KB, LD,_ æquales. </s>
            <s xml:id="echoid-s3862" xml:space="preserve">quod eſt propoſitum. </s>
            <s xml:id="echoid-s3863" xml:space="preserve">Sint iam
              <lb/>
            ſinus æquales ſiue recti _FK, GL,_ ſiue verſi _KB, LD,_ ſiue ſinus complementorum _EK,_
              <lb/>
            _EL._ </s>
            <s xml:id="echoid-s3864" xml:space="preserve">Dico arcus _BF, DG,_ eſſe æquales, Nam ſi _FK, GL,_ ſint æquales, erunt </s>
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