Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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rimus, vt vtraque in omnibus caſibus demonſtraretur: </
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">tico rectan
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gulo, cuius
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omnes ar-
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cus ſint qua
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drante mi-
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nores, locũ
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etiã habet i
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omni trian
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gulo ſphæ-
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rico rectan
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gulo.</
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in primo caſu, exiſtentibus nimirum omnibus arcubus quadrante minoribus, demon
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ſtratione fuiſſet confirmata. </
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<
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nibus alijs caſibus accommodabimus. </
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<
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que triangulum ſphæricum quodcunq; </
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gulum ACD, habens angulum C, rectum. </
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<
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igitur duo arcus AC, CD, circa angulum re-
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ctum quadrãte ſunt minores, ac proinde & </
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<
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tius arcus AD, quadrante quoque minor; </
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<
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vnus quadrante maior, & </
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denique ambo quadrante maiores: </
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ſolo ſphærico triangulo rectangulo agimus, in
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quo nullus arcus eſt quadrãs. </
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<
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arcus AC, CD, circa angulum rectum quadrante minores: </
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<
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">quo poſito, erit vterque
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">34. huius.</
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angulus D, A, acutus, proptereaque triangulo ACD, demonſtratio vtriuſque pro-
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poſitionis conueniet, quo ad primum caſum.</
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DA, donec coeant in B; </
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">erunt
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, DCB, ſemicirculi; </
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<
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drante minor. </
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<
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">Sunt ergo in triangulo ACB, duo arcus
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, circa angulum re-
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ctum C, quadrante minores. </
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<
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">Quare, vt proxime oſtendimus, ei vtriuſque propoſi-
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tionis demonſtratio, quo ad primum caſum, conueniet. </
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<
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cti, quam complementorum, ſint arcuum, & </
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<
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B, qui arcuum,
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& </
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D; </
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<
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">(Nam, vt in ſinubus diximus, arcus CD, CB, eun-
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dem ſinum habent tam rectum, quam complementi, necnon & </
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D, AB. </
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tam recti anguli ad C, eundem ſinum habent, nempe totum, quam anguli obliqui ad
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A, cum duobus rectis ſint æquales. </
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cum ſint inter ſe æquales: </
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, vtrique triangulo communis eſt.) </
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quido conſtat, quicquid de ſinubus arcuum, angulorumq́; </
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<
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oſtenſum, idem in ſinubus arcuum, & </
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D, locum habere.</
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<
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, quadrante maiores: </
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<
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arcus CB, minor quadrante. </
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<
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CB, arcum
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, circa
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angulum rectum
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, quadrante maiorem, & </
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<
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, minorem. </
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<
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">Quare ei, vt proxime
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eſt demonſtratum, vtraque propoſitio conueniet. </
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<
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">Cum ergo ijdem ſinus tam recti,
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quam complementorum, ſint arcuum, & </
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<
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& </
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<
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">angulorum trianguli ACD, vt paulo ante diximus, liquet eaſdem propoſitiones
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triangulo quoque ACD, conuenire. </
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<
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angulorumq́; </
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<
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quadrante ſint minores, demonſtratum fuerit, locum etiam habere in quocunq; </
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<
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triangulo ſphærico rectangulo.</
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demonſtraſſe in triangulo rectangulo, cuius omnes arcus ſunt quadrante minores,
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quale est triangulum ſecundæ figuræ propoſ. </
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<
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ſtratio omnibus alijs conueniat, vt ex demonſtratis in hoc ſcholio eſt manifeſtum.</
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<
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">EX his, quæ proximis tribus propoſitionibus demonſtrauimus, abſolutus iam per
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ſinus eſt calculus triangulorum ſphæricorum rectangulorum: </
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<
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gulorum calculus ſequi deberet. </
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<
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plerunque triangulorum rectangulorum calculus, quam per ſinus, expeditur, </
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