Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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complementi dati arcus BD, auferatur ex ſinu toto AB, notus relinquetur ſinus verſus EB,
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dati arcus BD. </
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<
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">Pari ratione, ſi ſinus rectus DE, hoc eſt, AF, dati arcus BD, dematur ex ſinu
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toto AC, notus relinquetur ſinus verſus CF, complementi dati arcus BD.</
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cꝰ quadrá-
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te minoris
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finꝰ rectus
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medioloco
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{pro} portiona
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lis eſt inter
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ſemiſsé ſi-
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nus totius,
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& ſinũ ver-
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ſum alteriꝰ
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arcꝰ, ꝗ prio
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ris duplus
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eſt, & qua-
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drante quo
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queminor.</
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noris medio loco proportionalis eſt inter ſemiſ-
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ſem ſemidiametri, ſeu ſinus totius, & </
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ſum arcus alterius, qui prioris arcus duplus eſt, & </
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quadrante quoque minor.</
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<
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">SIT arcus quicunque CE, quadrante minor, cuius dimidium ſit CD. </
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<
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uiſa autem ſemidiametro AC, bifariã in G, ducatur ex E, ad AC, perpendi-
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cularis EF, iungaturque recta AD, quæ ductã chordam CE, ſecabit in H, bifa-
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riam, ex lemmate à nobis ad definitiones ſupra demonſtrato, atque adeo & </
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angulos rectos. </
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arcus CE, qui duplus eſt arcus CD, cum EF, ſit
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eiuſdem arcus CE, ſinus rectus: </
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conſtat. </
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loco eſſe proportionalẽ inter CG, dimidiũ ſinus to-
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tius, & </
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<
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">CF, ſinum verſum arcus CE, qui arcus CD,
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duplus eſt. </
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<
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xml:space
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">Quoniã enim duo anguli ACH, AHC,
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trianguli ACH, æquales ſunt duobus angulis ECF,
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EFC, trianguli ECF, quod angulus C, vtrique
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triangulo ſit communis, & </
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<
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<
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gula erunt triangula ACH, ECF. </
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<
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AC, ad CH, ita EC, ad CF: </
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">4.ſexti.</
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CF. </
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<
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">Vt autem AC, ad CE, ita eſt CG, dimidium ipſius AC, ad CH, dimi-
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<
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dium ipſius CE. </
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<
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<
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">ac pro-
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pterea CH, ſinus rectus arcus CD, medio loco proportionalis eſt inter CG,
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ſemiſſem ſinus totius, & </
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<
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">CF, ſinum verſum arcus CE, qui arcus CD, duplus
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<
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">Ex ſinu re-
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cto cuiuſ-
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uis arcus
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cognito no
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tus fit ſinus
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rectus alte-
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rius arcus,
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qui illius
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dimidiũ ſit
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33. primi.
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3. huius,
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27. ſexti.</
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eſt. </
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<
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<
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monſtrandum erat.</
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<
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<
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">COLLIGITVR hinc, ſi ſinus rectus alicuius arcus cognitus ſit, notum etiam ſieri
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ſinum rectum alterius arcus, qui illius di midium ſit: </
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<
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">ita vt ex EF, ſinu recto arcus CE,
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cognito cognoſcatur etiam CH, ſinus rectus arcus CD, qui dimidium eſt arcus CE. </
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<
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ex noto ſinu recto EF, notus fiet ſinus EI, complementi: </
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<
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">quo ablato ex ſinu toto AC,
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(æqualis enim eſt ſinus EI, rectæ AF.) </
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<
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coroll. </
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<
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<
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<
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inter medietatem ſinus totius, & </
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<
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<
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">erit rectangulum ſub di
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midio ſinus totius, & </
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<
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<
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multiplicetur medietas ſinus totius in ſinum verſum CF, producetur quadratus </
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