Full text: Bithynius, Theodosius: Theodosii Tripolitae Sphaericorum libri tres

IN quadrante ABC, ſit BD, arcus grad. 54. ac proinde eius cõplementum
CD, grad. 36. quod diuidatur bifariam in H, vt vterq; arcuũ CH, HD, habeat
grad. 18. Ducatur DM, ad AB, perpendicularis pro ſinu arcus grad. 54. & DE,
ad AC, perpédicularis pro ſinu arcus grad. 36. Iunga
tur quoq; recta AH, quæ per lẽma in definitionibus
demonſtratũ ſecabit rectã CD, in I, bifariam, ac pro
inde & ad angulos rectos: eritq́ propterea CI, ſinus
rectus arcus CH, grad. 18. Sũpta tandẽ recta EF, ipſi
EC, æquali, diuidantur AC, AF, bifariã in G, K, & ex K, ad AC, perpendicularis ducatur KL. Dico ſi-
num rectũ DM, arcus grad. 54. hoc eſt, rectam AE,
illi ęqualẽ, componi ex AG, dimidio ſinus totius, & ex CI, ſinu recto arcus grad. 18. hoc eſt, rectam GE,
(quæ cũ AG, conſtituit totam rectã AE,) ęqualẽ eſ
ſe ſinui recto CI. Item ſinũ verſum arcus grad. 72. componi ex dimidio ſinus to
tius, & ex CE, ſinu verſo arcus CD, grad. 36. hoc eſt, rectam EK, (quæ cum
ſinu verſo CE, rectam CK, componit) æqualem eſſe dimidio ſinus totius, ip-
ſam vero CK, eſſe ſinum verſum arcus grad. 72. hoc eſt, arcum CL, (cuius ſi-
nus verſus eſt CK,) eſſe grad. 72. Ducta enim recta LN, ad AB, perpendicu-
lari, pro ſinu arcus BL, iungantur rectæ AD, DF. Quoniam igitur arcus
CH, grad. 18. continet {1/5}. quadrantis BC, (quòd quinquies 18. faciant 90.) continebit arcus CD, {2/5}. eiuſdem quadrantis, ac proinde proportio arcus
CD, ad arcum BC, erit vt 2. ad 5. Eſt autem, vt arcus CD, ad arcum BC, ita
angulus CAD, ad rectum angulum BAC. Igitur proportio anguli CAD,
ad angulum rectum BAC, erit quoque, vt 2. ad 5. ac proinde angulus CAD,
continebit {2/5}. vnius anguli recti. Cum ergo tres anguli trianguli CAD, con-
tineant {10/5}. vnius recti, hoc eſt, æquales ſint duobus rectis, ſintq́ue inter ſe
æquales duo anguli ACD, ADC; continebit vterque eorum {4/5}. vnius recti. Et quoniam angulus DFC, angulo DCF, eſt æqualis, quòd & rectę DF, DC,
æquales ſint; (cum enim DE, EF, latera trianguli DEF, æqualia ſint lateri-
bus DE, EC, trianguli DEC, angulosq́ue ad E, contineant æquales, vtpo-
te rectos; æquales erunt baſes DF, DC,) continebit quoque angulus DFC,
{4/5}. vnius recti; ac proinde reliquus angulus DFA, ex duobus rectis, hoc eſt, ex
{10/5}. vnius recti, continebit {6/5}. vnius recti. Cum ergo angulus DAF, oſtenſus
ſit continere {2/5}. vnius recti, & omnes tres anguli in triangulo AFD, conti-
neant {10/5}. vnius recti, continebit angulus ADF, {2/5}. vnius recti, propte-
reaq́ue angulo DAF, æqualis erit. Quare æqualia erunt latera DF, AF. Cum ergo recta DF, rectæ DC, oſtenſa ſit æqualis, erit & recta AF, rectæ DC,
æqualis: ideoque & k F, medietas ipſius AF, ipſi CI, medietati ipſius DC,
æqualis erit.

159.1.

126-01
3. tertij.
34. primi.
33. fexti.
32. primi.
5.primi.
5.primi.
4. primi.
32. primi.
6. primi.

RVRSVS quoniam AK, KF, æquales ſunt; additis æqualibus EC, FE,
erit recta compoſita ex Ak, EC, æqualis rectæ KE: ac proinde KE, medie-
tas erit ſemidiametri AC; quandoquidem AC, diuiſa eſt in duas partes æqua
les, quarum vna eſt KE, altera vero, recta ex AK, EC, compoſita. Eſt igi-
tur KE, æqualis ipſi CG. Ablata ergo communi recta GE, remanebunt
æquales GK, EC. Eſt autem EC, ſumpta ipſi EF, æqualis. Igitur & GK, ipſi EF, æqualis erit; additaque communi recta FG, erit EG, ipſi FK,
æqualis, hoc eſt, ipſi CI, cui oſtendimus ſupra rectam k F, eſſe æqualem. Com-

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